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\centerline{\titlefont The Langlands Classification and Irreducible}
\centerline{\titlefont Characters for Real Reductive Groups}
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\define\cal{\Cal }
\centerline{\titlefont {JEFFREY ADAMS}}
\footnote{Supported in part by NSF grant DMS-8802586 and a grant from the Sloan
Foundation.}

\centerline{\titlefont {DAN BARBASCH}}
\footnote{Supported in part by an NSF grant.}

\centerline{\titlefont {DAVID A. VOGAN, JR.}}
\footnote{Supported in part by NSF grant DMS-8805665.}
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\centerline{\bf 1. Introduction.}
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In [Arthur1] and [Arthur2], Arthur has formulated a number of conjectures
about automorphic forms.  These conjectures would have profound
consequences for the unitary representation theory of the group $G({\Bbb
R})$ of real points of a connected reductive algebraic group $G$ defined over
${\Bbb R}$.  Our purpose in this paper is to establish a few of these local
consequences.  In order to do that, we have been led to combine the ideas of
Langlands and Shelstad (concerning dual groups and endoscopy) with those of
Kazhdan and Lusztig (concerning the fine structure of irreducible
representations).

We will recall Arthur's conjectures in detail in sections 22 and 26, but for
the moment it is enough to understand their general shape. We begin by
recalling the form of the Langlands classification.  Define
$$\Pi(G({\Bbb R})) \supset \Pi_{\rom unit}(G({\Bbb R}))\supset \Pi_{\rom
temp}(G({\Bbb R})) \eqno(1.1)$$
to be the set of equivalence classes of irreducible admissible (respectively
unitary or tempered) representations of $G({\Bbb R})$.  Now define
$$\Phi(G({\Bbb R})) \supset \Phi_{\rom temp}(G({\Bbb R})) \eqno(1.2)$$
to be the set of Langlands parameters for irreducible admissible
(respectively tempered) representations of $G({\Bbb R})$ (see [Langlands],
[Borel],  [AV2], section 5, and Definition 22.3).  To each $\phi\in
\Phi(G({\Bbb R}))$, Langlands attaches a finite set $\Pi_\phi\subset
\Pi(G({\Bbb R}))$, called an {\it L-packet of representations}.  The L-packets
$\Pi_\phi$ partition $\Pi(G({\Bbb R}))$.  If $\phi \in \Phi_{\rom
temp}(G({\Bbb R}))$, then the representations in $\Pi_\phi$ are all
tempered, and in this way one gets also a partition of $\Pi_{\rom
temp}(G({\Bbb R}))$.

Now the classification of the unitary representations of $G({\Bbb R})$ is one
of the most interesting unsolved problems in harmonic analysis.  Langlands'
results immediately suggest that one should look for a set between
$\Phi(G({\Bbb R}))$ and $\Phi_{\rom temp}(G({\Bbb R}))$ parametrizing
exactly the unitary representations.  Unfortunately, nothing quite so
complete is possible: Knapp has found examples in which some members of
the set $\Pi_\phi$ are unitary and some are not.

The next most interesting possibility is to describe a set of parameters
giving rise to a large (but incomplete) family of unitary representations. 
This is the local aim of Arthur's conjectures.  A little more precisely,
Arthur defines a new set
$$\Psi(G/{\Bbb R})\eqno(1.3)(a)$$
of parameters (Definition 22.4).  (We write $G/{\Bbb R}$ rather than
$G({\Bbb R})$ because Arthur's parameters depend only on an inner class of
real forms, and not on one particular real form.)  Now assume that $G({\Bbb
R})$ is quasisplit.  Then Arthur defines an inclusion
$$\Psi(G/{\Bbb R})\hookrightarrow\Phi(G({\Bbb R})),\quad \psi\mapsto
\phi_\psi.\eqno(1.3)(b)$$
Write $\Phi_{\rom Arthur}(G({\Bbb R}))$ for the image of this inclusion. 
Then
$$\Phi(G({\Bbb R}))\supset \Phi_{\rom Arthur}(G({\Bbb R}))\supset
\Phi_{\rom temp}(G({\Bbb R})).\eqno(1.3)(c)$$
Roughly speaking, Arthur proposes that $\Psi(G/{\Bbb R})$ should
parametrize all the unitary representations of $G({\Bbb R})$ that are of
interest for global applications.  More specifically, he proposed the
following problems (still for $G({\Bbb R})$ quasisplit at first).
\smallskip
\item{} {\bf Problem A.}  Associate to each parameter $\psi\in \Psi(G/{\Bbb
R})$ a finite set $\Pi_\psi \subset \Pi(G({\Bbb R}))$.  This set (which we
might call an {\it Arthur packet}) should contain the L-packet
$\Pi_{\phi_\psi}$ (cf. (1.3)(b)) and should have other nice properties, some
of which are specified below.
\smallskip
\noindent The Arthur packet will not in general turn out to be a union of
L-packets; so we cannot hope to define it simply by attaching some
additional Langlands parameters to $\psi$.

Associated to each Arthur parameter is a certain finite group $A_\psi$
(Definition 21.4).
\smallskip
\item{}{\bf Problem B.}  Associate to each $\pi\in \Pi_\psi$ a non-zero
finite-dimensional representation $\tau_\pi$ of $A_\psi$.
\smallskip
\item{}{\bf Problem C.} Show that the distribution on $G({\Bbb R})$
$$\sum_{\pi\in \Pi_\psi}\left(\epsilon_\pi{\rom dim}(\tau_\pi)\right)
\Theta(\pi)$$
is a stable distribution in the sense of Langlands and Shelstad ([Langlands2],
[Shelstad]).  Here $\epsilon_\pi = \pm 1$ is also to be defined.
\smallskip
\item{}{\bf Problem D.}  Prove analogues of Shelstad's theorems on lifting
tempered characters (cf. [Shelstad]) in this setting.
\smallskip
\item{}{\bf Problem E.}  Extend the definition of $\Pi_\psi$ to
non-quasisplit $G$, in a manner consistent with appropriate generalizations
of Problems B, C, and D.
\smallskip
\item{}{\bf Problem F.}  Show that every representation $\pi\in\Pi_\psi$ is
unitary.
\smallskip

We give here complete solutions of problems A, B, C, D, and E.  Our methods
offer no information about Problem F.  (In that direction the best results are
those of [Barbasch], where Problem F is solved for complex classical
groups.)

The central idea of the proofs is by now a familiar one in the representation
theory of reductive groups.  It is to describe the representations of $G({\Bbb
R})$ in terms of an appropriate geometry on an L-group.  So let ${^\vee G}$
be the (complex reductive) dual group of $G$, and ${^\vee G^\Gamma}$ the
(Galois form of the) L-group attached to the real form $G({\Bbb R})$.  The
L-group is a complex Lie group, and we have a short exact sequence
$$1\rightarrow {^\vee G} \rightarrow {^\vee G^\Gamma} \rightarrow {\rom
Gal}({\Bbb C}/{\Bbb R})\rightarrow 1\eqno(1.4)(a)$$
We also need the Weil group $W_{\Bbb R}$ of ${\Bbb C}/{\Bbb R}$; this is a
real Lie group, and there is a short exact sequence
$$1\rightarrow {\Bbb C}^\times \rightarrow W_{\Bbb R} \rightarrow {\rom
Gal}({\Bbb C}/{\Bbb R})\rightarrow 1.\eqno(1.4)(b)$$
(The Weil group is not a complex Lie group because the action of the Galois
group on ${\Bbb C}^\times$ is the non-trivial one, which does not preserve
the complex structure.)
\smallskip
{\bf Definition 1.5} ([Langlands], [Borel]).  A {\it quasiadmissible
homomorphism} $\phi$ from $W_{\Bbb R}$ to ${^\vee G^\Gamma}$ is a
continuous group homomorphism satisfying
\smallskip
\item{(a)}  $\phi$ respects the homomorphisms to ${\rom Gal}({\Bbb C}/{\Bbb
R})$ defined by (1.4); and
\item{(b)}  $\phi({\Bbb C}^\times)$ consists of semisimple elements of
${^\vee G}$.
\smallskip

\noindent (Langlands' notion of ``admissible homomorphism'' includes an
additional ``relevance'' hypothesis on $\phi$, which will not concern us. 
This additional hypothesis is empty if $G({\Bbb R})$ is quasisplit.)  Define 
$$P({^\vee G^\Gamma})=\{\,\phi:W_{\Bbb R}\rightarrow {^\vee
G^\Gamma}\mid \phi \hbox{ is quasiadmissible}\,\}.$$ 
Clearly ${^\vee G}$ acts on $P({^\vee G^\Gamma})$ by conjugation on the
range of a homomorphism, and we define
$$\Phi(G/{\Bbb R}) = \{\,{^\vee G}\hbox{ orbits on } P({^\vee
G^\Gamma})\,\}.$$
(If $G({\Bbb R})$ is quasisplit, this is precisely the set of parameters in
(1.2).  In general Langlands omits the ``irrelevant'' orbits.)
\smallskip

Now a homomorphism $\phi$ is determined by the value of its differential
on a basis of the real Lie algebra of ${\Bbb C}^\times$, together with its
value at a single specified element of the non-identity component of
$W_{\Bbb R}$; that is, by an element of the complex manifold ${^\vee{\frak
g}} \times {^\vee{\frak g}}\times{^\vee G^\Gamma}$.  The conditions (a) and
(b) of Definition 1.5 amount to requiring the first two factors to be
semisimple, and the third to lie in the non-identity component.  Requiring
that these elements define a group homomorphism imposes a finite number
of complex-analytic relations, such as commutativity of the first two
factors.  Pursuing this analysis, we will prove in section 5 
\medskip
{\bf Proposition 1.6.} {\it Suppose ${^\vee G^\Gamma}$ is an L-group.  The
set $P({^\vee G^\Gamma})$ of quasiadmissible homomorphisms from
$W_{\Bbb R}$ into ${^\vee G^\Gamma}$ may be identified with the set of
pairs $(y,\lambda)$ satisfying the following conditions:
\item{a)} $y \in {^\vee G^\Gamma} - {^\vee G}$, and $\lambda\in {^\vee{\frak
g}}$ is a semisimple element;
\item{b)} $y^2 = {\rom exp}(2\pi i \lambda)$; and
\item{c)} $[\lambda, {\rom Ad}(y)\lambda] = 0$.}
\medskip
The Langlands classification described after (1.2) is thus already geometric:
L-packets are parametrized by the orbits of a reductive group acting on a
topological space.  Subsequent work of Langlands and Shelstad supports the
importance of this geometry.  For example, one can interpret some of the
results of [Shelstad] as saying that the L-packet $\Pi_\phi$ may be
parametrized using ${^\vee G}$-equivariant local systems on the ${^\vee G}$
orbit of $\phi$.

By analogy with the theory created by Kazhdan-Lusztig and
Beilinson-Bernstein in [KL] and [BB], one might hope that information about
irreducible characters is encoded by perverse sheaves on the closures of
${^\vee G}$-orbits on $P({^\vee G^\Gamma})$.  Unfortunately, it turns out
that the orbits are already closed, so these perverse sheaves are nothing but
the local systems mentioned above.  On the other hand, one can often
parametrize the orbits of several rather different group actions using the
same parameters; so we sought a different space with a ${^\vee G}$ action,
having the same set of orbits as $P({^\vee G^\Gamma})$, but with a more
interesting geometry.  

In order to define our new space, we need some simple structure theory for
reductive groups.  (This will be applied in a moment to ${^\vee G}$.)
\medskip
{\bf Definition 1.7.}  Suppose $H$ is a complex reductive group, with Lie
algebra ${\frak h}$, and $\lambda\in {\frak h}$ is a semisimple element. 
Set
$${\frak h}(\lambda)_n = \{\,\mu\in {\frak h}\mid [\lambda,\mu] =
n\mu\,\}\quad(n \in {\Bbb Z})\eqno(1.7)(a)$$
$${\frak n}(\lambda) = \sum_{n=1,2,\ldots} {\frak h}(\lambda)_n
\eqno(1.7)(b)$$
$$e(\lambda) = {\rom exp}(2\pi i\lambda)\in H.\eqno(1.7)(c)$$
The {\it canonical flat through $\lambda$} is the affine subspace
$${\cal F}(\lambda) = \lambda + {\frak n}(\lambda)\subset {\frak
h}.\eqno(1.7)(d)$$
We will see in section 6 that the canonical flats partition the semisimple
elements of ${\frak h}$ --- in fact they partition each conjugacy class ---
and that the map $e$ is constant on each canonical flat.  If $\Lambda$ is a
canonical flat, we may therefore write
$$e(\Lambda) = {\rom exp}(2\pi i\lambda)\qquad \hbox{(any $\lambda \in
\Lambda$)}. \eqno(1.7)(e)$$
Finally, write ${\cal F}({\frak h})$ for the set of all canonical flats in
${\frak h}$.  
\medskip
{\bf Definition 1.8}  Suppose ${^\vee G^\Gamma}$ is the L-group of a real
reductive group (cf. (1.4)).  The {\it geometric parameter space for ${^\vee
G^\Gamma}$} is the set
$$X = X({^\vee G^\Gamma}) = \{\,(y,\Lambda)\mid y\in {^\vee G^\Gamma} -
{^\vee G},\,\Lambda\in {\cal F}({^\vee {\frak g}}),\,y^2 = e(\Lambda)\,\}.$$ 
\medskip

This is our proposed substitute for Langlands' space $P({^\vee G^\Gamma})$. 
The set ${\cal F}({^\vee {\frak g}})$ is difficult to topologize nicely, as one
can see already for $SL(2)$; this difficulty is inherited by $X$.  To make use
of geometric methods we will always restrict to the subspaces appearing in
the following lemma.
\medskip
{\bf Lemma 1.9} (cf. Proposition 6.16 below).  {\it In the setting of
Definition 1.8, fix a single orbit ${\cal O}$ of ${^\vee G}$ on the semisimple
elements of ${^\vee {\frak g}}$, and set
$$X({\cal O},{^\vee G^\Gamma}) = \{\,(y,\Lambda)\in X\mid \Lambda \subset
{\cal O}\,\}.$$
Then $X({\cal O},{^\vee G^\Gamma})$ has in a natural way the structure of a
smooth complex algebraic variety, on which ${^\vee G}$ acts with a finite
number of orbits.}
\medskip
\noindent  The variety $X({\cal O},{^\vee G^\Gamma})$ need not be connected
or equidimensional, but this will cause no difficulties.  We topologize $X$ by
making the subsets $X({\cal O},{^\vee G^\Gamma})$ open and closed.  (It
seems likely that a more subtle topology will be important for harmonic
analysis, as soon as continuous families of representations are involved.)

Of course the first problem is to check that the original Langlands
classification still holds.
\medskip
{\bf Proposition 1.10.} (cf. Proposition 6.17 below).  {\it Suppose ${^\vee
G^\Gamma}$ is an L-group.  Then there is a natural ${^\vee G}$-equivariant
map
$$p:P({^\vee G^\Gamma})\rightarrow X({^\vee G^\Gamma}), \quad
p(y,\lambda) = (y,{\cal F}(\lambda))$$
inducing a bijection on the level of ${^\vee G}$-orbits.  The fibers of $p$ are
principal homogeneous spaces for unipotent algebraic groups.   More
precisely, suppose $x = p(\phi)$.  Then the isotropy group ${^\vee G}_{\phi}$
is a Levi subgroup of ${^\vee G}_x$.}
\medskip

This proposition shows that (always locally over ${\Bbb R}$!) the geometric
parameter space $X$ shares all the formal properties of $P({^\vee
G^\Gamma})$ needed for the Langlands classification.  In particular, if
$G({\Bbb R})$ is quasisplit,  L-packets in $\Pi(G({\Bbb R}))$ are
parametrized precisely by ${^\vee G}$-orbits on $X$.  What has changed is
that the orbits on the new space $X$ are not closed; so the first new
question to consider is the meaning of the closure relation.
\medskip
{\bf Proposition 1.11.}  {\it Suppose $G({\Bbb R})$ is quasisplit.  Let $\phi,
\,\phi'\in \Phi(G/{\Bbb R})$ be two Langlands parameters, and
$S,\,S'\subset X$ the corresponding ${^\vee G}$-orbits.  Then the following
conditions are equivalent:
\item{i)}   $S$ is contained in the closure of $S'$.
\item{ii)}  there are irreducible representations $\pi\in \Pi_\phi$ and
$\pi'\in \Pi_{\phi'}$ with the property that $\pi'$ is a composition factor of
the standard representation of which $\pi$ is the unique quotient.

\noindent If $\phi$ is a tempered parameter, then the orbit $S$ is open in
the variety $X({\cal O},{^\vee G^\Gamma})$ containing it (cf. Lemma 1.9).}
\medskip

\noindent (In the interest of mathematical honesty, we should admit that
this result is included only for expository purposes; we will not give a
complete proof.  That (ii) implies (i) (even for $G({\Bbb R})$ not quasisplit)
follows from Corollary 1.25(b) and (7.11).  The other implication in the
quasisplit case can be established by a subtle and not very interesting trick. 
The last assertion follows from Proposition 22.9(b) (applied to an Arthur
parameter with trivial $SL(2)$ part).)

Proposition 1.11 suggests the possibility of a deeper relationship between
irreducible representations and the geometry of orbit closures on $X$.  To
make the cleanest statements, we need to introduce some auxiliary ideas. 
(These have not been emphasized in the existing literature on the Langlands
classification, because they reflect phenomena over ${\Bbb R}$ that are
non-existent or uninteresting globally.)   The reader should assume at first
that $G$ is adjoint.  In that case the notion of ``strong real form'' introduced
below amounts to the usual notion of real form, and the ``algebraic universal
covering'' of ${^\vee G}$ is trivial.
\medskip
{\bf Definition 1.12.}  Suppose $G$ is a complex connected reductive
algebraic group.  An {\it extended group for $G/{\Bbb R}$} is a pair $(
G^\Gamma,{\cal W})$, subject to the following conditions.
\item{(a)} $G^\Gamma$ is a real Lie group containing $G$ as a subgroup of
index two, and every element of $G^\Gamma - G$ acts on $G$ (by
conjugation) by antiholomorphic automorphisms.
\item{(b)} ${\cal W}$ is a $G$-conjugacy class of triples $(\delta,N,\chi)$,
with
\itemitem{(1)} The element $\delta$ belongs to $G^\Gamma - G$, and
$\delta^2\in Z(G)$ has finite order.  (Write $\sigma=\sigma(\delta)$ for the
conjugation action of $\delta$ on $G$, and $G({\Bbb R})$ or $G({\Bbb
R},\delta)$ for the fixed points of $\sigma$; this is a real form of $G$.)
\itemitem{(2)} $N\subset G$ is a maximal unipotent subgroup, and $\delta$
normalizes $N$.  (Then $N$ is defined over ${\Bbb R}$; write $N({\Bbb
R})=N({\Bbb R},\delta)$ for the subgroup of real points.)
\itemitem{(3)}  The element $\chi$ is a one-dimensional non-degenerate
unitary character of $N({\Bbb R})$.  (Here ``non-degenerate'' means
non-trivial on each simple restricted root subgroup of $N$.)

\medskip
We will discuss this definition in more detail in sections 2 and 3.  For now
it suffices to know that each inner class of real forms of $G$ gives rise to
an extended group.  The groups $G({\Bbb R})$ appearing in the definition are
quasisplit (because of (b)(2)) and the pair $(N({\Bbb R}),\chi)$ is the set of
data needed to define a Whittaker model for $G({\Bbb R})$.
\medskip
{\bf Definition 1.13.}  Suppose $(G^\Gamma,{\cal W})$ is an extended group. 
A {\it strong real form} of $(G^\Gamma,{\cal W})$ (briefly, of $G$) is an
element $\delta\in G^\Gamma-G$ such that $\delta^2 \in Z(G)$ has finite
order.  Given such a $\delta$, we write $\sigma=\sigma(\delta)$ for its
conjugation action on $G$, and
$$G({\Bbb R})=G({\Bbb R},\delta)$$
for the fixed points of $\sigma$.  Two strong real forms $\delta$ and
$\delta'$ are called {\it equivalent} if they are conjugate by $G$; we write
$\delta\sim\delta'$.  (The elements $\delta$ of Definition 1.12 constitute a
single equivalence class of strong real forms, but in general there will be
many others.)
\medskip
The usual notion of a real form can be described as an antiholomorphic
involution $\sigma$ of $G$.  Two such are equivalent if they differ by the
conjugation action of $G$.   This is exactly the same as our definition if $G$
is adjoint.  The various groups $G({\Bbb R},\delta)$ (for $\delta$ a strong
real form of $(G^\Gamma,{\cal W})$) constitute exactly one inner class of
real forms of $G$.
\medskip
{\bf Definition 1.14.}  Suppose $(G^\Gamma,{\cal W})$ is an extended group.
A {\it representation of a strong real form} of $(G^\Gamma,{\cal W})$
(briefly, of $G$) is a pair $(\pi,\delta)$, subject to
\item{(a)} $\delta$ is a strong real form of $(G^\Gamma,{\cal W})$
(Definition 1.13); and
\item{(b)} $\pi$ is an admissible representation of $G({\Bbb R},\delta)$.

Two such representations $(\pi,\delta)$ and $(\pi',\delta')$ are said to be
{\it (infinitesimally) equivalent} if there is an element $g\in G$ such that
$g\delta g^{-1} = \delta'$, and $\pi\circ {\rom Ad}(g^{-1})$ is
(infinitesimally) equivalent to $\pi'$.  (In particular, this is possible only if
the strong real forms are equivalent.)  Finally, define
$$\Pi(G^\Gamma,{\cal W}) = \Pi(G/{\Bbb R})$$
to be the set of (infinitesimal) equivalence classes of irreducible
representations of strong real forms of $G$.
\medskip
{\bf Lemma 1.15.}  {\it Suppose $(G^\Gamma,{\cal W})$ is an extended group
for $G$ (Definition 1.12).  Choose representatives $\{\,\delta_s \mid s\in
\Sigma\,\}$ for the equivalence classes of strong real forms of $G$
(Definition 1.13).  Then the natural map from left to right induces a
bijection
$$\coprod_{s\in \Sigma} \Pi(G({\Bbb R},\delta_s)) \simeq \Pi(G/{\Bbb R})$$
(Definition 1.14; the set on the left is a disjoint union).}
\medskip
\noindent This lemma is an immediate consequence of the definitions; we
will give the argument in section 2.  

The set $\Pi(G/{\Bbb R})$ is the set of representations we wish to
parametrize.  To do so requires one more definition on the geometric side.
\medskip
{\bf Definition 1.16.}  Suppose ${^\vee G^\Gamma}$ is the L-group of the
inner class of real forms represented by the extended group $G^\Gamma$ (cf.
(1.4) and Definition 1.13).  The {\it algebraic universal covering} ${^\vee
G}^{alg}$ is the projective limit of all the finite coverings of ${^\vee G}$. 
This is a pro-algebraic group, of which each finite-dimensional
representation factors to some finite cover of ${^\vee G}$.
\medskip

With the algebraic universal covering in hand, we can define a complete set
of geometric parameters for representations.
\medskip
{\bf Definition 1.17.} Suppose $G$ is a connected reductive algebraic group
endowed with an inner class of real forms, and ${^\vee G^\Gamma}$ is a
corresponding L-group for $G$.  A {\it complete geometric parameter} for
$G$ is a pair
$$\xi = (S,{\cal V}),$$
where
\smallskip
\item{(a)}  $S$ is an orbit of ${^\vee G}$ on $X({^\vee G^\Gamma})$
(Definition 1.8); and
\item{(b)}  ${\cal V}$ is an irreducible ${^\vee {\widetilde G}}$-equivariant
local system on $S$, for some finite covering ${^\vee {\widetilde G}}$ of
${^\vee G}$.
\smallskip

\noindent  We may write $(S_\xi,{\cal V}_\xi)$ to emphasize the
dependence on $\xi$.  In (b), it is equivalent to require ${\cal V}$ to be
${^\vee G}^{alg}$-equivariant.  Write $\Xi(G/{\Bbb R})$ for the set of all
complete geometric parameters.

A slightly different formulation of this definition is sometimes helpful.  Fix
a ${^\vee G}$-orbit $S$ on $X$, and a point $x\in S$.  Write ${^\vee
G}^{alg}_x$ for the stabilizer of $x$ in ${^\vee G}^{alg}$, and define
$$A_S^{loc,alg} = {^\vee G}^{alg}_x/\left({^\vee G}^{alg}_x\right)_0$$
for its (pro-finite) component group.  We call $A_S^{loc,alg}$ the {\it
equivariant fundamental group of $S$}; like a fundamental group, it is
defined only up to inner automorphism (because of its dependence on $x$). 
Representations of $A_S^{loc,alg}$ classify equivariant local systems on
$S$, so we may also define a complete geometric parameter for $G$ as a
pair
$$\xi = (S,\tau),$$
where
\smallskip
\item{(a)}  $S$ is an orbit of ${^\vee G}$ on $X({^\vee G^\Gamma})$; and
\item{(b)}  $\tau$ is an irreducible representation of $A_S^{loc,alg}$.

\noindent Again we may write $(S_\xi,\tau_\xi)$.
\medskip
{\bf Theorem 1.18.}  {\it Suppose $(G^\Gamma,{\cal W})$ is an extended
group for $G$ (Definition 1.12), and ${^\vee G^\Gamma}$ is an L-group for the
corresponding inner class of real forms.   Then there is a natural bijection
between the set $\Pi(G/{\Bbb R})$ of equivalence classes of  irreducible
representations of strong real forms of $G$ (Definition 1.14), and the set
$\Xi(G/{\Bbb R})$ of complete geometric parameters for $G$ (Definition
1.17).  In this parametrization, the set of representations of a fixed real
form $G({\Bbb R})$ corresponding to complete geometric parameters
supported on a single orbit is precisely the L-packet for $G({\Bbb R})$
attached to that orbit (Proposition 1.10).}
\medskip
As we remarked after Proposition 1.6, one can find results of this nature in
[Shelstad].

For each complete geometric parameter $\xi$, we define (using Theorem
1.18 and Definition 1.14)
$$(\pi(\xi),\delta(\xi)) = \hbox{ some irreducible representation
parametrized by $\xi$}\eqno(1.19)(a)$$
$$M(\xi) = \hbox{ standard representation with Langlands quotient
$\pi(\xi)$.}\eqno(1.19)(b)$$
As a natural setting in which to study character theory, we will also use
$$K\Pi(G/{\Bbb R}) = \hbox{ free ${\Bbb Z}$-module with basis }\Pi(G/{\Bbb
R}).\eqno(1.19)(c)$$
We will sometimes call this the {\it lattice of virtual characters}.  One can
think of it as a Grothendieck group of an appropriate category of
representations of strong real forms.  In particular, any such representation
$\rho$ has a well-defined image
$$[\rho] \in K\Pi(G/{\Bbb R}).$$
By abuse of notation, we will usually drop the brackets, writing for example
$M(\xi) \in K\Pi(G/{\Bbb R})$.  (All of these definitions are discussed in
somewhat more depth in sections 11 and 15.)

In order to write character formulas, we will also need a slight variant on
the notation of (1.19)(a).  Fix a strong real form $\delta$ of $G$, and a
complete geometric parameter $\xi$.  By the proof of Lemma 1.15, there is
at most one irreducible representation $\pi$ of $G({\Bbb R},\delta)$ so that
$(\pi,\delta)$ is equivalent to $(\pi(\xi),\delta(\xi))$.  We define
$$\pi(\xi,\delta) = \pi.\eqno(1.19(d)$$
If no such representation $\pi$ exists, then we define
$$\pi(\xi,\delta) = 0.\eqno(1.19)(e)$$
Similarly we can define $M(\xi,\delta)$.
\medskip
{\bf Lemma 1.20} (Langlands --- see [Green], [IC4]).  {\it The (image in
$K\Pi(G/{\Bbb R})$ of the) set
$$\{\,M(\xi)\mid \xi \in \Xi(G/{\Bbb R})\,\}$$
is a basis for $K\Pi(G/{\Bbb R})$.}
\medskip
Because the standard representations of real groups are fairly well
understood, it is natural to try to describe the irreducible representations
in terms of the standard ones.  On the level of character theory, this means
relating the two bases $\{\pi(\xi)\}$ and $\{M(\xi)\}$ of $K\Pi(G/{\Bbb R})$:
$$M(\xi) = \sum_{\gamma \in \Xi} m_r(\gamma,\xi)\pi(\gamma).
\eqno(1.21)$$ 
(The subscript $r$ stands for ``representation-theoretic,'' and is included to
distinguish this matrix from an analogous one to be introduced in Definition
1.22.)  Here the {\it multiplicity matrix} $m_r(\gamma,\xi)$ is what we
want.  The Kazhdan-Lusztig conjectures (now proved) provide a way to
compute the multiplicity matrix, and a geometric interpretation of it ---
the ``Beilinson-Bernstein picture'' of [BB].  Unfortunately, this geometric
interpretation is more complicated than one would like in the case of
non-integral infinitesimal character, and it has some fairly serious
technical shortcomings in the case of singular infinitesimal character. 
(What one has to do is compute first at nonsingular infinitesimal character,
then apply the ``translation principle.''  The translation principle can
introduce substantial cancellations, which are not easy to understand in the
Beilinson-Bernstein picture.)  We have therefore sought a somewhat
different geometric interpretation of the multiplicity matrix.  Here are the
ingredients.  (A more detailed discussion appears in section 7.)

\medskip
{\bf Definition 1.22.}  Suppose $Y$ is a complex algebraic variety on which
the pro-algebraic group $H$ acts with finitely many orbits.  Define
$${\cal C}(Y,H) = \hbox{ category of $H$-equivariant constructible sheaves
on $Y$.}\eqno(1.22)(a)$$
$${\cal P}(Y,H) = \hbox{ category of $H$-equivariant perverse sheaves on
$Y$.}\eqno(1.22)(b)$$
(For the definition of perverse sheaves we refer to [BBD].  The definition of
$H$-equivariant requires some care in the perverse case; see [LInt], section
0, or [LChar], (1.9.1) for the case of connected $H$.)  Each of these
categories is abelian, and every object has finite length.  (One does not
ordinarily expect the latter property in a category of constructible sheaves;
it is a consequence of the strong assumption about the group action.)  The
simple objects in the two categories may be parametrized in exactly the
same way: by the set of pairs
$$\xi = (S_\xi,{\cal V}_\xi) = (S,{\cal V})\eqno(1.22)(c)$$
with $S$ an orbit of $H$ on $Y$, and ${\cal V}$ an irreducible
$H$-equivariant local system on $S$.  The set of all such pairs will be
written $\Xi(Y,H)$, the set of {\it complete geometric parameters for $H$
acting on $Y$}.  Just as in Definition 1.17, we may formulate this definition
in terms of the equivariant fundamental group
$$A_S^{loc} = H_y/(H_y)_0\qquad(y\in S)$$
and its representations.  We write $\mu(\xi)$ for the irreducible
constructible sheaf corresponding to $\xi$ (the extension of $\xi$ by zero),
and $P(\xi)$ for the irreducible perverse sheaf (the ``intermediate
extension'' of $\xi$ --- cf. [BBD], Definition 1.4.22).

The Grothendieck groups of the two categories ${\cal P}(Y,H)$ and ${\cal
C}(Y,H)$ are naturally isomorphic (by the map sending a perverse sheaf to
the alternating sum of its cohomology sheaves, which are constructible). 
Write $K(Y,H)$ for this free abelian group.  The two sets $\{P(\xi)\mid
\xi\in \Xi\}$ and $\{\mu(\xi)\mid \xi\in \Xi\}$ are obviously bases of their
respective Grothendieck groups, but they are {\it not} identified by the
isomorphism.  Write $d(\xi)$ for the dimension of the underlying orbit
$S_\xi$.  We can write in $K(Y,H)$
$$\mu(\xi) = (-1)^{d(\xi)}\sum_{\gamma \in \Xi(Y,H)}
m_g(\gamma,\xi)P(\gamma)\eqno(1.22)(d)$$
with $m_g(\gamma,\xi)$ an integer.  (The subscript $g$ stands for
``geometric.'') In this formula, it follows easily from the definitions that
$$m_g(\xi,\xi)=1,\qquad m_g(\gamma,\xi)\ne 0\hbox{ only if } S_\gamma
\subset \left( \overline{S_\xi} - S_\xi\right)\quad(\gamma\ne
\xi).\eqno(1.22)(e)$$
\medskip

The matrix $m_g(\gamma,\xi)$ is essentially the matrix relating our two
bases of $K(Y,H)$.  It is clearly analogous to (1.21).  In each case, we have a
relationship between something uncomplicated (the standard
representations, or the extensions by zero) and something interesting (the
irreducible representations, or the simple perverse sheaves).  One can
expect the matrix $m$ to contain interesting information, and to be
difficult to compute explicitly.
\medskip
{\bf Definition 1.23.} In the setting of Definition 1.8, define
$${\cal C}(X({^\vee G}^\Gamma),{^\vee G}^{alg})$$
to be the direct sum over semisimple orbits ${\cal O} \subset {^\vee {\frak
g}}$ of the categories ${\cal C}(X({\cal O},{^\vee G^\Gamma}),{^\vee G}^{alg})$
of Definition 1.22.  The objects of this category are called (by a slight abuse
of terminology) {\it ${^\vee G}^{alg}$-equivariant constructible sheaves on
$X$}.  Similarly we define
$${\cal P}(X({^\vee G}^\Gamma),{^\vee G}^{alg}),$$
the {\it ${^\vee G}^{alg}$-equivariant perverse sheaves on $X$}.  The
irreducible objects in either category are parametrized by $\Xi(G/{\Bbb R})$
(cf. Definition 1.22), and we write
$$KX({^\vee G}^{alg})$$
for their common Grothendieck group.  We write $\mu(\xi)$ and $P(\xi)$ for
the irreducible objects constructed in Definition 1.22, or their images in
$KX({^\vee G}^{alg})$.  These satisfy (1.22)(d) and (e).
\medskip

Since Theorem 1.18 tells us that the two Grothendieck groups $KX({^\vee
G}^{alg})$ and $K\Pi(G/{\Bbb R})$ have bases in natural one-to-one
correspondence, it is natural to look for a functorial relationship between a
category of representations of strong real forms of $G$, and one of the
geometric categories of Definition 1.23.  We do not know what form such a
relationship should take, or how one might hope to establish it directly.
What we are able to establish is a formal relationship on the level of
Grothendieck groups.  This will be sufficient for studying character theory.
\medskip
{\bf Theorem 1.24}  {\it Suppose $(G^\Gamma,{\cal W})$ is an extended group
for $G$ (Definition 1.12), and ${^\vee G^\Gamma}$ is an L-group for the
corresponding inner class of real forms.   Then there is a natural perfect
pairing
$$<,>:K\Pi(G/{\Bbb R})\times KX({^\vee G}^{alg})\rightarrow {\Bbb Z}$$
between the Grothendieck group of the category of finite length
representations of strong real forms of $G$, and that of ${^\vee
G}^{alg}$-equivariant (constructible or perverse) sheaves on $X$ (cf. (1.19)
and Definition 1.23).  This pairing is defined on the level of basis vectors by 
$$<M(\xi),\mu(\xi')> = e(G({\Bbb R},\delta(\xi)))\delta_{\xi,\xi'}.$$
Here we use the notation of (1.19) and Definition 1.22.  The group $G({\Bbb
R},\delta(\xi))$ is the real form represented by $M(\xi)$; the constant
$e(G({\Bbb R})) = \pm 1$ is the one defined in [Kott1] (see also Definition
15.8), and the last $\delta$ is a Kronecker delta.  In terms of the other
bases of (1.19) and Definition 1.23, we have
$$<\pi(\xi),P(\xi')> = e(G({\Bbb R}, \delta(\xi)))(-1)^{d(\xi)}
\delta_{\xi,\xi'}.$$}
\medskip
\noindent The content of this theorem is in the equivalence of the two
possible definitions of the pairing.  We will deduce it from the main result
of [IC4].  As an indication of what the theorem says, here are three simple
reformulations.
\medskip
{\bf Corollary 1.25.}  {\it 
\item{a)} The matrices $m_r$ and $m_g$ of (1.21) and Definition 1.22(d) are
essentially inverse transposes of each other:
$$\sum_\gamma (-1)^{d(\gamma)} m_r(\gamma,\xi)m_g(\gamma,\xi') =
(-1)^{d(\xi)}\delta_{\xi,\xi'}.$$
\item{b)} The multiplicity of the irreducible representation $\pi(\gamma)$
in the standard representation $M(\xi)$ is up to a sign the multiplicity of
the local system ${\cal V}_\xi$ in the restriction to $S_\xi$ of the Euler
characteristic of the perverse sheaf $P(\gamma)$:
$$m_r(\gamma,\xi) = (-1)^{d(\gamma)-d(\xi)}\sum_i (-1)^i(\hbox
{multiplicity of ${\cal V}_\xi$ in } H^iP(\gamma) \mid_{S_\xi}).$$
\item{c)} The coefficient of the standard representation $M(\gamma)$ in the
expression of the irreducible representation $\pi(\xi)$ is equal to
$(-1)^{d(\gamma) - d(\xi)}$ times the multiplicity of the perverse sheaf
$P(\xi)$ in the expression of $\mu(\gamma)[-d(\gamma)]$.
\medskip}
\noindent Here part (c) refers to the expansion of $\pi(\xi)$ in the
Grothendieck group as a linear combination of standard representations (cf.
Lemma 1.20); and similarly for $\mu(\gamma)$. 

Another way to think of Theorem 1.24 is this.
\medskip
{\bf Corollary 1.26.}  {\it In the setting of Theorem 1.24, write
$$\overline{K}= \overline{K}\Pi(G/{\Bbb R})$$
for the set of (possibly infinite) integer combinations of irreducible
representations of strong real forms of $G$.  Then  $\overline{K}$ may be
identified with the space of ${\Bbb Z}$-linear functionals on the
Grothendieck group $KX({^\vee G}^{alg})$:
$$\overline{K}\Pi(G/{\Bbb R})\simeq {\rom Hom}_{\Bbb Z}(KX({^\vee
G}^{alg}),{\Bbb Z}).$$  In this identification,
\item{a)} the standard representation $M(\xi)$ of $G({\Bbb R},\delta(\xi))$
corresponds to $e(G({\Bbb R}, \delta(\xi)))$ times the linear functional
``multiplicity of ${\cal V}_\xi$ in the restriction to $S_\xi$ of the
constructible sheaf $C$;'' and 
\item{b)} the irreducible representation $\pi(\xi)$ of $G({\Bbb
R},\delta(\xi))$ corresponds to $e(G({\Bbb R}, \delta(\xi)))(-1)^{d(\xi)}$
times the linear functional ``multiplicity of $P(\xi)$ as a composition
factor of the perverse sheaf $Q$.''
\smallskip
\noindent Here in (a) we are interpreting $KX({^\vee G}^{alg})$ as the
Grothendieck group of constructible sheaves, and in (b) as the Grothendieck
group of perverse sheaves.}
\medskip
\noindent  We call elements of $\overline{K}\Pi(G/{\Bbb R})$ {\it formal
virtual characters of strong real forms of $G$}.  

In order to bring Langlands' notion of stability into this picture, we must
first reformulate it slightly.
\medskip
{\bf Definition 1.27.}  In the setting of Definition 1.14 and (1.19), suppose
$$\eta = \sum_{\xi\in \Xi} n(\xi)(\pi(\xi),\delta(\xi))$$
is a formal virtual character.  We say that $\eta$ is {\it locally finite} if
for each strong real form $\delta$ there are only finitely many $\xi$ with
$n(\xi)\ne 0$ and $\delta(\xi)$ equivalent to $\delta$.  Suppose that $\eta$
is locally finite, and that $\delta$ is a strong real form of $G$.  There is a
finite set $\pi_1,\ldots,\pi_r$ of inequivalent irreducible representations
of $G({\Bbb R},\delta)$ so that each $(\pi_j,\delta)$ is equivalent to some
$(\pi(\xi_j),\delta(\xi_j))$ with $n(\xi_j)\ne 0$.  Each of these
representations has a character $\Theta(\pi_j)$,  a generalized function on
$G({\Bbb R},\delta)$; and we define
$$\Theta(\eta,\delta) = \sum_j n(\xi_j)\Theta(\pi_j),$$
a generalized function on $G({\Bbb R},\delta)$.  This generalized function
has well-defined values at the regular semisimple elements of $G({\Bbb
R},\delta)$, and these values determine $\Theta(\eta,\delta)$.  In the
notation of (1.19)(d,e), we can write
$$\Theta(\eta,\delta) = \sum_\xi n(\xi)\Theta(\pi(\xi,\delta)).$$

We say that $\eta$ is {\it strongly stable} if it is locally finite, and the
following condition is satisfied.  Suppose $\delta$ and $\delta'$ are strong
real forms of $G$, and $g \in G({\Bbb R},\delta)\cap G({\Bbb R},\delta')$ is a
strongly regular semsimple element.  Then
$$\Theta(\eta,\delta)(g) = \Theta(\eta,\delta')(g).$$
\medskip
A necessary condition for $\eta$ to be strongly stable is that each
$\Theta(\eta,\delta)$ should be stable in Langlands' sense.  Conversely,
Shelstad's results in [Shelstad] imply that if $\Theta$ is a stable finite
integer combination of characters on a real form $G({\Bbb R},\delta)$, then
there is a strongly stable $\eta$ with $\Theta=\Theta(\eta,\delta)$.

Corollary 1.26 gives a geometric interpretation of formal virtual
characters.  We can now give a geometric interpretation of the notion of
stability.
\medskip
{\bf Definition 1.28.}  In the setting of Definition 1.22, fix an $H$-orbit
$S\subset Y$, and a point $y\in S$.  For a constructible sheaf $C$ on $Y$,
write $C_y$ for the stalk of $C$ at $y$, a finite-dimensional vector space. 
The map
$$\chi^{loc}_S:{\rom Ob}\,{\cal C}(Y,H) \rightarrow {\Bbb N}, \quad
\chi^{loc}_S(C) = {\rom dim}(C_y)$$
is independent of the choice of $y$ in $S$.  It is additive for short exact
sequences, and so defines a ${\Bbb Z}$-linear map
$$\chi^{loc}_S:K(Y,H)\rightarrow {\Bbb Z},$$
the {\it local multiplicity along $S$.}   If we regard $K(Y,H)$ as a
Grothendieck group of perverse sheaves, then the formula for $\chi^{loc}_S$
on a perverse sheaf $P$ is
$$\chi^{loc}_S(P) = \sum (-1)^i {\rom dim}(H^iP)_y.$$
Any ${\Bbb Z}$-linear functional $\eta$ on $K(Y,H)$ is called {\it
geometrically stable} if it is in the ${\Bbb Z}$-span of the various
$\chi^{loc}_S$.

In the setting of Definition 1.23, a ${\Bbb Z}$-linear functional $\eta$ on
$K(Y,H)$ is called {\it geometrically  stable} if its restriction to each
summand $K(X({\cal O}),{^\vee G}^{alg})$ is geometrically stable, and
vanishes for all but finitely many ${\cal O}$.
\medskip
{\bf Theorem 1.29.}  {\it In the identification of Corollary 1.26, the strongly
stable formal virtual characters correspond precisely to the geometrically
stable linear functionals.}
\medskip
\noindent This is an immediate consequence of Corollary 1.26 and Shelstad's
description of stable characters in [Shelstad1].  (It is less easy to give a
geometric description of the stable characters on a single real form of $G$,
even a quasisplit one.)

In a sense Arthur's conjectures concern the search for interesting new
stable characters.  We have now formulated that problem geometrically, but
the formulation alone offers little help.  The only obvious geometrically
stable linear functionals are the $\chi^{loc}_S$.  For $S$ corresponding to an
L-packet by Proposition 1.10, the corresponding strongly stable formal
virtual character is essentially the sum of all the standard representations
attached to the L-packet.  This sum is stable and interesting, but not new,
and not what is needed for Arthur's conjectures.  To continue, we need a
different construction of geometrically stable linear functionals on
$K(Y,H)$.
\medskip
{\bf Definition 1.30.}  Suppose $Y$ is a smooth complex algebraic variety on
which the connected pro-algebraic group $H$ acts with finitely many orbits. 
To each orbit $S$ we associate its conormal bundle
$$T^*_S(Y)\subset T^*(Y);$$
this is an $H$-invariant smooth Lagrangian subvariety of the cotangent
bundle.  Attached to every $H$-equivariant perverse sheaf $P$ on $Y$ is a
{\it characteristic cycle}
$${\rom Ch}(P) = \sum_S \chi^{mic}_S(P) \overline{T^*_S(Y)}.$$
Here the coefficients $\chi^{mic}_S(P)$ are non-negative integers, equal to
zero unless $S$ is contained in the support of $P$.  One way to construct
${\rom Ch}(P)$ is through the Riemann-Hilbert correspondence ([BorelD]): the
category of $H$-equivariant perverse sheaves on $Y$ is equivalent to the
category of $H$-equivariant regular holonomic $D$-modules on $Y$, and the
characteristic cycle of a $D$-module is fairly easy to define (see for
example [Ginsburg] or [Kashiwara]).  The functions $\chi^{mic}_S$ are
additive for short exact sequences, and so define ${\Bbb Z}$-linear
functionals
$$\chi^{mic}_S:K(Y,H) \rightarrow {\Bbb Z},$$
the {\it microlocal multiplicity along $S$.}
\medskip

{\bf Theorem 1.31} (Kashiwara --- see [Kold], [Kashiwara], Theorem 6.3.1, or
[Ginsburg], Theorem 8.2.)  {\it The linear functionals $\chi^{mic}_S$ of
Definition 1.30 are geometrically stable.  More precisely, for every
$H$-orbit $S'$ such that $\overline{S'}\supset S$ there is an integer
$c(S,S')$ so that for every $H$-equivariant perverse sheaf on $Y$,
$$\chi^{mic}_S(P) = \sum_{S'} c(S,S') \chi^{loc}_{S'}(P).$$
Here $\chi^{loc}_{S'}$ is defined in Definition 1.28.}
\medskip
\noindent In fact Kashiwara's interest was in an inverted form of this
relationship, expressing $\chi^{loc}_S$ in terms of the various
$\chi^{mic}_{S'}$. (The invertibility of the matrix $c(S,S')$ is an immediate
consequence of the facts that $c(S,S)=(-1)^{{\rom dim}\,S}$, and that
$c(S,S')\ne 0$ only if $\overline{S'}\supset S$.)  We could therefore have
defined geometrically stable in terms of the linear functionals
$\chi^{mic}_S$.

Perhaps the main difficulty in Theorem 1.31 is the definition of the matrix
$c(S,S')$.  That definition is due independently to Macpherson in [Ch]. 
Although the $D$-module approach to characteristic cycles is intuitively
very simple, it entails some great technical problems (notably that of
lifting [SKK]).  We will therefore find it convenient to use a geometric
definition of $\chi^{mic}_S$ due to MacPherson (see (24.10) and Definition
24.11 below).  With this definition, Theorem 1.31 has a very simple proof
due to MacPherson; we reproduce it at the end of section 24.

The matrix $c(S,S')$ and its inverse have been extensively studied from
several points of view (see for example the references in [Ginsburg]).  If
$S\ne S'$ is contained in the smooth part of $\overline{S'}$, then $c(S,S') =
0$.  Nevertheless (and in contrast with the multiplicity matrices of
(1.22)(d)) there is no algorithm known for computing it in all the cases of
interest to us.
\medskip

{\bf Corollary 1.32.} {\it Suppose $({^\vee G},{\cal W})$ is an extended group
for $G$ (Definition 1.12), and ${^\vee G^\Gamma}$ is an L-group for the
corresponding inner class of real forms.  Fix an orbit $S$ of ${^\vee G}$ on
$X({^\vee G^\Gamma})$ (Definition 1.8) (or, equivalently, an L-packet for the
quasisplit form of $G$ (Proposition 1.10)).  Then the linear functional
$\chi^{mic}_S$ on $KX({^\vee G}^{alg})$ (Definition 1.30) corresponds via
Corollary 1.26 to  a strongly stable formal virtual representation
$\eta^{mic}_S$.  The irreducible representations of strong real forms
occurring in $\eta^{mic}_S$ are those for which the corresponding perverse
sheaf $P$ has $\chi^{mic}_S(P)\ne 0$.  This includes all perverse sheaves
attached to the orbit $S$ itself, and certain sheaves attached to orbits $S'$
containing $S$ in their closures.  With notation as in (1.19) and Definition
1.27, the corresponding stable distribution on $G({\Bbb R},\delta)$ is
$$\Theta(\eta^{mic}_S,\delta) = e(G({\Bbb R},\delta))\sum_{\xi'\in
\Xi}(-1)^{d(\xi')-\dim S}\chi_S^{mic}(P(\xi'))\Theta(\pi(\xi',\delta)).$$
In terms of standard representations, this distribution may be expressed as
$$\Theta(\eta^{mic}_S,\delta) = e(G({\Bbb R},\delta))(-1)^{\dim
S}\sum_{\xi'\in \Xi}c(S,S_{\xi'})\Theta(M(\xi',\delta)).$$

The set $\{\,\eta^{mic}_S\,\}$ (as $S$ varies) is a basis of the lattice of
strongly stable formal virtual representations.}
\medskip
\noindent (Recall that the tempered representations correspond to open
orbits; in that case $\chi^{mic}_S = (-1)^{{\rom dim}\,S}\chi^{loc}_S$, and we
get nothing new.)

As the second formula of Corollary 1.32 shows, obtaining explicit character
formulas for $\eta^{mic}_S$ amounts to computing the matrix $c(S,S')$ in
Theorem 1.31.

Our approach to Arthur's conjectures is now fairly straightforward.  Arthur
attaches to his parameter $\psi$ a Langlands parameter $\phi_\psi$, and
thus (by Proposition 1.10) an orbit $S_\psi$.  We define $\Pi_\psi$ to
consist of all those representations appearing in $\eta^{mic}_{S_\psi}$;
that is, representations for which the corresponding perverse sheaf has the
conormal bundle of $S_\psi$ in its characteristic cycle.  We will show
(Corollary 27.13) that this agrees with the previous definition of Barbasch
and Vogan (in the case of ``unipotent'' parameters) in terms of primitive
ideals.  We will show in Proposition 22.9 that Arthur's group $A_\psi$ turns
out (for $S=S_\psi$) to be a quotient of a certain geometrically defined
group $A^{mic,alg}_S$; the difference arises only from our use of the
algebraic universal covering of ${^\vee G}$, and for these local purposes our
choice seems preferable.  In section 24, we extend the analysis of Definition
1.30 to define a representation $\tau_S^{mic}(P)$ of $A^{mic,alg}_S$, of
dimension equal to $\chi^{mic}_S(P)$.  Now Problems A, B, C, and E are
resolved as special cases of Corollary 1.32 and the preceding definitions.

This approach to character theory lends itself very well to the local study
of endoscopy and the functoriality principle:  the space $X({^\vee
G^\Gamma})$ obviously depends covariantly on ${^\vee G^\Gamma}$.  There is
no difficulty in obtaining a formulation of Shelstad's definition of
endoscopic lifting (Definition 26.18).  To complete the solution of Problem
D, we use an appropriate ``unstable'' version of Theorem 1.31 (Theorem 25.8). 
This we deduce from the version of the Lefschetz fixed point formula in
[GMFP].  Since this introduction is already much too long, we postpone a
detailed discussion to sections 25 and 26.

It is a great pleasure to thank a few of the many mathematicians who helped
us to write a paper we barely feel competent to read.  Mike Artin and Steve
Kleiman led us patiently through the algebraic geometry, answering
hundreds of foolish questions as well as several serious ones.  Bob
MacPherson, Kari Vilonen, Victor Ginsburg, and Jean-Luc Brylinski taught us
about perverse sheaves, ${\cal D}$-modules, and microlocal geometry. 
George Lusztig knows everything about reductive groups, and so does Bert
Kostant; they have helped us through many treacherous swamps.

The specific form of this paper we owe to Martin Andler.  He suggested that
something like Proposition 1.11 had to be true; this led to the definition of
the geometric parameter space, and so made everything else possible.  He
has also offered many corrections and improvements to the manuscript.  

Most of all, we are grateful to Jim Arthur.  In papers, in lectures, and in
conversation, he has opened up for us some of the mysteries of automorphic
representation theory.  His gentle and encouraging manner allowed us to
persist through long periods when we felt (no doubt correctly) that we were
in far over our heads.  Working with his ideas has been a privilege
mathematically and personally.
%\end
%\input mssymb.tex
%\font\Bbb=cmr10
%\font\frak=cmr10
%\hsize=27pc
%\vsize=42pc
%\hoffset=3.5pc
%\voffset=5.5pc
%\pageno=178
\vskip 3pc
\centerline{\bf Index of notation.}
\vskip 1pc
We list here some of our principal notation and terminology, with a very
brief summary of definitions and references to more complete discussion in
the body of the paper.
\medskip
$A_\phi^{loc,can}$: group of connected components of the centralizer in
${^\vee G}^{can}$ of a Langlands parameter $\phi$ (Definition 5.11).  A
Langlands parameter $\phi$ gives a point $x$ of the geoemetric parameter
space $X({^\vee G^\Gamma})$ (Proposition 6.17), which generates an orbit
$S$ under ${^\vee G}$.  The ${^\vee G}^{can}$-equivariant fundamental group
of $S$ based at $x$ (Definition 7.1) is written $A_S^{loc,can}$ or
$A_x^{loc,can}$; it is naturally isomorphic to $A_\phi^{loc,can}$ (Lemma
7.5).
\smallskip
$A^{mic,can}_\phi$: micro-component group for a Langlands parameter
$\phi$ (Definition 23.13).  Write $S$ for the orbit on $X({^\vee G^\Gamma})$
corresponding to $\phi$, and $(x,\nu)$ for a sufficiently generic conormal
vector to $S$ at $x\in S$.  Then $A^{mic,can}_\phi$ is the group of connected
components of the stablizer of $(x,\nu)$ in ${^\vee G}^{can}$.
\smallskip
$A^{can}_\psi$: group of connected components of the centralizer in ${^\vee
G}^{can}$ of an Arthur parameter $\psi$ (Definition 21.4).  Naturally
isomorphic to the micro-component group $A^{mic,can}_{\phi_\psi}$ for the
corresponding Langlands parameter (Proposition 21.9 and Definition 23.7).
\smallskip
$d(\xi)$: dimension of the underlying orbit for a geometric parameter $\xi$
((1.22),(7.10)).
\smallskip 
$e(G_{\Bbb R})$, $e(\xi)$: Kottwitz' sign attached to a complete geometric
parameter or to the corresponding real form of $G$ (Definition 15.8, Lemma
15.9).
\smallskip
$e(\cdot)$: normalized exponential map $\exp(2\pi i\cdot)$; usually from
semisimple elements or canonical flats in ${^\vee {\frak g}}$ ((6.2), Lemma
6.11).
\smallskip
$G$: connected complex reductive algebraic group.
\smallskip
$G^\Gamma$: weak extended group for $G$ (Definition 2.13).
\smallskip
$(G^\Gamma,{\cal W})$: extended group for $G$ (Definition 1.12, Proposition
3.6).
\smallskip
$G({\Bbb R},\delta)$: real points of $G$ with real form defined by $\delta
\in G^\Gamma - G$ (Definition 2.13).
\smallskip
${^\vee G}$: dual group for $G$ (Definition 4.2).
\smallskip
${^\vee G^\Gamma}$: weak L-group or E-group for $G$; an algebraic extension
of the Galois group of ${\Bbb C}/{\Bbb R}$ by a dual group (Definition 4.3).
\smallskip
$({^\vee G^\Gamma},{\cal D})$: L-group or E-group for $G$, essentially as
defined by Langlands (in Galois form); ${\cal D}$ is a class of distinguished
splittings.
\smallskip
$({^\vee G^\Gamma},{\cal S})$: L-group or E-group for $G$, as defined in
[AV2] (Definition 4.14).  Equivalent to giving an L-group or E-group as in the
preceding definition, but the set ${\cal S}$ is different from ${\cal D}$.
\smallskip
${\cal H}({\cal O})$: Hecke algebra attached to a regular semisimple
conjugacy class ${\cal O}$ in ${^\vee {\frak g}}$ ((16.10)).
\smallskip
$KX({\cal O},{^\vee G^\Gamma})$: Grothendieck group of ${^\vee
G}^{can}$-equivariant perverse sheaves (or constructible sheaves, or
holonomic ${\cal D}$-modules) on $X({\cal O},{^\vee G^\Gamma})$ (Definition
7.13, (7.10)).
\smallskip
$K\Pi^z({\cal O},G_{\Bbb R})$: Grothendieck group of finite-length canonical
projective representations of $G_{\Bbb R}$ of type $z$ and infinitesimal
character ${\cal O}$ ((15.5)).  Without the ${\cal O}$, no assumption on
infinitesimal character.  With $G/{\Bbb R}$, the sum of these over all strong
real forms ((15.7)). With $\overline{K}$, infinite sums (still with finite
multiplicities) are allowed (Corollary 1.26, (16.12)).  With $\overline{K}_f$,
these infinite sums are required to include only finitely many
representations of each strong real form (Definition 18.5).  With
$\overline{K}_{\Bbb C}$, complex-linear combinations of irreducible
representations are allowed (Proposition 25.1).
\smallskip
${\cal K}\Pi^z({\cal O},G/{\Bbb R})$: mixed Grothendieck group of
finite-length canonical projective representations of strong real forms of
$G$ of type $z$ and infinitesimal character ${\cal O}$ ((16.12)).  (Formally,
${\cal K}$ denotes extension of scalars to ${\Bbb Z}[u^{1/2},u^{-1/2}]$.) 
\smallskip
$l^I(\Lambda)$, $l^I(\xi)$: integral length for a standard limit character of
regular infinitesimal character, or for the corresponding complete
geometric parameter (Definition 16.16).
\smallskip
$L^z(G_{\Bbb R})$:  equivalence classes of final standard limit characters of
Cartan subgroups of $G_{\Bbb R}$ (Definition 11.13).
\smallskip
$L^z(G/{\Bbb R})$:  equivalence classes of final standard limit characters of
Cartan subgroups of $G^\Gamma$ (Definition 12.1).
\smallskip
${\cal M}^z(G_{\Bbb R})$, ${\cal M}^z(G/{\Bbb R})$: category of (canonical
projective of type $z$) representations of a real reductive group $G_{\Bbb
R}$, or of strong real forms of $G^\Gamma$ ((15.2) and (15.7)).
\smallskip
$M(\Lambda)$, $M(\xi)$: standard limit representation attached to a limit
character $\Lambda$, or to the corresponding complete geometric parameter
$\xi$ (Definition 11.2, (15.7)).
\smallskip
${\cal O}$: semisimple ${^\vee G}$-conjugacy class in ${^\vee {\frak g}}$;
parametrizes an infinitesimal character for ${\frak g}$.  Inserted in other
notation, it restricts attention to objects living on that orbit, or having that
infinitesimal character ((6.10), Proposition 6.16, Lemma 15.4).
\smallskip
$T^\Gamma$: Cartan subgroup of $G^\Gamma$ (Definition 12.1).
\smallskip
$\Lambda$: limit character of a Cartan subgroup of a real reductive group;
parameter for a standard representation (Definitions 11.2, 11.6, 11.10,
11.13; Definition 12.1). 
\smallskip
$\chi_\phi^{mic}(\pi)(\sigma)$: trace at $\sigma\in A_\phi^{mic,can}$ of
the representation $\tau_\phi^{mic,can}$ (Definition 23.13).  Without the
argument $\sigma$, the dimension of this representation, which is equal to
the multiplicity of the conormal bundle to the orbit $S_\phi$ in the
characteristic cycle of the perverse sheaf attached to $\pi$ (Definition
19.13, Theorem 23.8).
\smallskip
$\eta^{loc}_\phi$: strongly stable standard formal virtual representation of
strong real forms of $G$ attached to a Langlands parameter $\phi$
(Definition 18.13).  We may replace $\phi$ by the corresponding ${^\vee
G}$-orbit $S$ on the geometric parameter space $X({^\vee G^\Gamma})$.
\smallskip
$\eta^{mic}_\phi$, $\eta^{mic}_S$: strongly stable formal virtual
representation of strong real forms of $G$ attached to a Langlands
parameter $\phi$ or to the corresponding orbit $S$ on the geometric
parameter space $X({^\vee G^\Gamma})$ (Corollary 19.16, Corollary 1.32).
\smallskip
$\xi$: complete geometric parameter $(S_\xi,{\cal V}_\xi)$.  Here $S_\xi$
is an orbit of ${^\vee G}$ on the geometric parameter space $X({^\vee
G^\Gamma})$, and ${\cal V}_\xi$ is an irreducible equivariant local system
on $S_\xi$ (Definition 1.17, Definition 7.6).
\smallskip
$\pi(\xi)$: irreducible representation corresponding to the complete
geometric parameter $\xi$ (Theorem 10.4, Definition 10.8, and (1.19)).  This
is a representation of $G({\Bbb R},\delta(\xi))^{can}$.
\smallskip
$\pi(\Lambda)$: Langlands quotient of the standard representation
$M(\Lambda)$ (Definition 11.2(e), Theorem 11.14).
\smallskip
$\Pi^z(G/{\Bbb R})$: equivalence classes of pairs $(\pi,\delta)$, with
$\delta$ a strong real form (for a fixed extended group $G^\Gamma$) and
$\pi$ an irreducible representation of $G({\Bbb R},\delta)^{can}$ of type $z$
(Definition 10.3, Definition 1.14).  This set is in one-to-one correspondence
with $\Xi^z(G/{\Bbb R})$.
\smallskip
$\Pi^z(G/{\Bbb R})_\phi$: the L-packet attached to the Langlands parameter
$\phi$; that is, the subset of $\Pi^z(G/{\Bbb R})$ parametrized by  complete
Langlands parameters of the form $(\phi,\tau)$, or by complete geometric
parameters supported on the orbit attached to $\phi$ (Definition 10.8). 
Includes at most finitely many representations of each strong real form.
\smallskip
$\Pi^z(G/{\Bbb R})^{mic}_\phi$: the micro-packet attached to the Langlands
parameter $\phi$ (Definition 19.15); the set of irreducible constituents of
the stable virtual representation $\eta_\phi^{mic}$.  Contains the L-packet.
\smallskip 
$\Xi({^\vee G^\Gamma})$: complete geometric parameters; that is, pairs $\xi
= (S,{\cal V})$ with $S$ an orbit of ${^\vee G}$ on the geometric parameter
space $X({^\vee G^\Gamma})$, and ${\cal V}$ an irreducible ${^\vee
G}^{can}$-equivariant local system on $S$ (Definition 7.6).  Equivalently,
complete Langlands parameters; that is, equivalence classes of pairs
$(\phi,\tau)$, with $\phi$ a Langlands parameter and $\tau$ an irreducible
representation of $A_\phi^{loc,can}$ (Definition 5.11).  When ${^\vee
G^\Gamma}$ has the structure of an E-group with second invariant $z$, we
may write instead $\Xi^z(G/{\Bbb R})$.
\smallskip
$\Phi({^\vee G^\Gamma})$: equivalence classes of Langlands parameters;
that is, ${^\vee G}$-conjugacy classes of quasiadmissible homomorphisms
$\phi$ from $W_{\Bbb R}$ to ${^\vee G^\Gamma}$ (Definition 5.2). 
Equivalently, equivalence classes of geometric parameters; that is, ${^\vee
G}$-orbits on $X({^\vee G^\Gamma})$ (Definition 7.6).  When ${^\vee
G^\Gamma}$ has the structure of an E-group with second invariant $z$, we
write instead $\Phi^z(G/{\Bbb R})$.
\smallskip
$\Psi({^\vee G^\Gamma})$: equivalence classes of Arthur parameters; that
is, ${^\vee G}$-conjugacy classes of homomorphisms from $W_{\Bbb
R}\times SL(2,{\Bbb C})$ satisfying some additional conditions (Definition
21.4).  We sometimes write $\Psi^z(G/{\Bbb R})$ as for Langlands
parameters.
\smallskip
$\tau_\phi^{mic}(\pi)$: representation (possibly reducible) of
$A^{mic,can}_\phi$ attached to an irreducible representation $\pi$ of a
strong real form of $G$ (Definition 23.13).  Non-zero if and only if $\pi \in
\Pi^z(G/{\Bbb R})^{mic}_\phi$.
\smallskip
$\Theta(\eta)$: distribution character of the virtual representation $\eta$
((18.1)).  When the argument includes a strong real form $\delta$, $\eta$ is
a (locally finite) virtual representation of all strong real forms, and
$\Theta(\eta,\delta)$ is the character of those defined on the group
$G({\Bbb R},\delta)^{can}$.
\smallskip
$\Theta_{SR}(\eta)$: smooth function on the strongly regular elements
obtained by restricting the character of $\eta$.
\smallskip
%\end
%\input mssymb.tex
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\vskip  3pc
\centerline{\bf 2. Structure theory: real forms.}
\vskip 1pc
In this section we review the basic facts about real forms of reductive
groups.  Since our concern is entirely with local problems over ${\Bbb R}$,
we have included proofs of several well-known results that are perhaps less
familiar to experts on real groups.  (The definitions, statements and some of
the proofs have nevertheless been constructed with more general fields in
mind.)
 
Suppose $G$ is a connected reductive complex algebraic group.  A {\it real
form} of $G$ is an antiholomorphic involutive automorphism
$$\sigma:G\rightarrow G.\eqno(2.1)(a)$$
Here ``antiholomorphic'' means that if $f$ is any algebraic function on $G$,
then the function
$$g\mapsto \overline{f(\sigma g)}\eqno(2.1)(b)$$
is also algebraic.  Equivalently, the differential of $\sigma$ (an
automorphism of ${\frak g} = {\rom Lie}(G)$) satisfies
$$ d\sigma(ix) = -id\sigma(x).\eqno(2.1)(b')$$
``Involutive'' means that $\sigma^2$ is the identity.  The {\it group of real
points of $\sigma$} is
$$G({\Bbb R},\sigma) = G({\Bbb R}) = \{\,g\in G\mid \sigma g =
g\,\}.\eqno(2.1)(c)$$

Condition $(2.1)(b')$ shows that the Lie algebra ${\frak g}({\Bbb R})$
determines $d\sigma$, and hence determines $\sigma$.   We may therefore
speak of $G({\Bbb R})$ as the real form of $G$ without danger of confusion. 
However, not every real form of the Lie algebra ${\frak g}$ exponentiates to
a real form of $G$.  The multiplicative group $G={\Bbb C}^\times$ has
exactly two real forms, given by the automorphisms
$$\sigma_s(z) = \overline{z},\qquad \sigma_c(z) =
\overline{z}^{-1}.\eqno(2.2)(a)$$
The corresponding groups are
$$G({\Bbb R},\sigma_s) = {\Bbb R}^\times, \qquad G({\Bbb R},\sigma_c) =
S^1.\eqno(2.2)(b)$$
The real forms of the Lie algebra ${\Bbb C}$, on the other hand, are
parametrized by real lines in ${\Bbb C}$.  The corresponding subgroups
$\{e^{tz}\mid t\in {\Bbb R}\}$ are not (identity components of) real forms
unless $z$ is real or imaginary.

It is worth remarking that this definition is quite restrictive in some
slightly surprising ways.  For example, the identity component of a
disconnected group of real points is {\it not} usually a group of real points. 
Thus familiar linear groups like $SO(p,q)_0$ are excluded.  This is necessary
to get the very clean character formulas discussed in the introduction.  (Of
course it is a routine matter --- though not a trivial one --- to relate the
representations of $SO(p,q)_0$ to those of $SO(p,q)$, which {\it is} a group
of real points.)

Two real forms $\sigma$ and $\sigma'$ are called {\it equivalent} if there
is an element $g\in G$ such that
$$\sigma' = {\rom Ad}(g)\circ \sigma\circ {Ad}(g^{-1}).\eqno(2.3)(a)$$
We can write this as
$$\sigma' = {\rom Ad}(g\sigma(g^{-1}))\circ \sigma.\eqno(2.3)(a')$$
In terms of the groups of real points, it is equivalent to
$$G({\Bbb R},\sigma') = gG({\Bbb R},\sigma)g^{-1}.\eqno(2.3)(a'')$$
The set of real forms equivalent to a fixed real form $\sigma$ is therefore
a homogeneous space $G/H$; the isotropy group is
$$H=\{\,g\in G\mid g\sigma(g^{-1})\in Z(G)\,\}.\eqno(2.3)(b)$$
This isotropy group contains $G({\Bbb R},\sigma)Z(G)$, but may be larger. 
(For example, if $G({\Bbb R})= SL(2,{\Bbb R})$, then the group $H$ contains
the diagonal matrix with entries $(i,-i)$.)  The point of the theory of strong
real forms is to find a notion of equivalence for which the corresponding
isotropy group is just $G({\Bbb R})$.  This in turn will allow us to formulate
results like Lemma 1.15.

This notion of equivalence is already a little more subtle than the one
sometimes encountered in the classification of real forms of simple Lie
algebras.  There one is interested in the question of when the groups of real
points of two real forms are isomorphic.  The group $SO(4n,{\Bbb C})$ has
two isomorphic but inequivalent real forms (the isomorphism class is
represented by $SO^*(4n)$) if $n\ge 1$.  The involutions are conjugate by the
non-identity component of $O(4n,{\Bbb C})$.  (There is a unique equivalence
class of real forms of $SO(4n+2,{\Bbb C})$ with $G({\Bbb R})$ isomorphic to
$SO^*(4n+2).$)  A less subtle example is provided by the inequivalent real
forms ${\Bbb R}^\times\times S^1$ and $S^1\times {\Bbb R}^\times$ of
${\Bbb C}^\times\times{\Bbb C}^\times$.

Two real forms $\sigma$ and $\sigma'$ are said to be {\it inner} to each
other if there is an element $g\in G$ such that
$$\sigma' = Ad(g)\circ\sigma.\eqno(2.4)$$
This is an equivalence relation.  (If $\sigma$ is a real form, the
automorphism $\sigma'$ defined by (2.4) will certainly {\it not} be a real
form for arbitrary $g$.)  Because of $(2.3)(a')$, equivalent real forms are
inner to each other.  We therefore get an equivalence relation on equivalence
classes of real forms.  Obviously the relation is trivial if $G$ is abelian. 
The two isomorphic inequivalent real forms of $SO(4n,{\Bbb C})$ mentioned
earlier are inner to each other, and to the real forms $SO(2p,2q)$.  The real
forms $SO(2p+1,2q-1)$ constitute a separate inner class.

\medskip
{\bf Example 2.5.}  Suppose $G= GL(n,{\Bbb C})$, and $\sigma_c$ is the
compact real form
$$\sigma_c(g) = ^t{\overline {g}}^{-1}, \qquad G({\Bbb R},\sigma_c) = U(n).$$
For $0\le p\le n$, let $g_p$ be the diagonal matrix with $p$ entries equal to
$1$ and $n-p$ equal to $-1$.  Set
$$\sigma_p=Ad(g_p)\circ\sigma_c.$$
Then
$$G({\Bbb R},\sigma_p) = U(p,q),$$
so all these forms are inner to each other.  On the other hand, the split real
form
$$\sigma_s(g) = \overline{g},\qquad G({\Bbb R},\sigma_s) = GL(n,{\Bbb R})$$
is not inner to $\sigma_c$ (as one can see --- rather unfairly --- by
examining their restrictions to the center of $G$).
\medskip

We recall that a real form $\sigma$ is called {\it quasisplit} if there is a
Borel subgroup $B\subset G$ such that
$$\sigma(B) = B.\eqno(2.6)$$

\medskip
{\bf Proposition 2.7.}  {\it Suppose $\sigma'$ is a real form of the connected
reductive complex algebraic group $G$.  Then there is a unique equivalence
class of quasisplit real forms of $G$ inner to $\sigma'$.  Specifically, there
is a quasisplit real form $\sigma$ of $G$ and an element $g$ of $G$ so that
$$\sigma' = {\rom Ad}(g)\circ\sigma.$$}
\medskip
Before giving the proof, we recall the basic structural fact on which it is
based.
\medskip
{\bf Proposition 2.8.} {\it Suppose $G$ is a connected reductive complex
algebraic group.  Fix a Borel subgroup  $B$ of $G$, a maximal torus
$T\subset B$, and  a set of basis vectors $\{X_\alpha\}$ for the simple root
spaces of $T$ in the Lie algebra ${\frak b}$.  Let $B', T', \{{X_\alpha}'\}$ be
another set of choices of these objects.  Then there is an element $g\in G$
such that
$$gBg^{-1} = B',\qquad gTg^{-1} = T',\qquad {\rom Ad}(g)(\{X_\alpha\}) = 
\{{X_\alpha}'\}.$$
The inner automorphism ${\rom Ad}(g)$ is uniquely determined by these
requirements; that is, any two choices of $g$ differ by $Z(G)$.  If we require
only the first two conditions, then the coset $gT$ is uniquely determined;
and if we require only the first condition, then the coset $gB$ is uniquely
determined.}
\medskip

To get a formulation valid over any algebraically closed field $k$, one need
only replace the Lie algebra elements $X_\alpha$ by one-parameter
subgroups $x_\alpha:k\rightarrow B$.

{\it Proof of Proposition 2.7.}  Fix $B$, $T\subset B$, and $\{X_\alpha\}$ as
in Proposition 2.8.  Then the automorphism $\sigma'$ carries these objects
to others of the same kind:
$$\sigma'(B) = B',\qquad \sigma'(T) = T'\subset B',\qquad d\sigma(X_\alpha)
= {X_\alpha}'.\eqno(2.9)(a)$$
By Proposition 2.8, we can therefore find an element $g\in G$ such that
$${\rom Ad}(g^{-1})(B') = B,\qquad {\rom Ad}(g^{-1})(T') = T, \qquad {\rom
Ad}(g^{-1})(\{{X_\alpha}'\}) =  \{X_\alpha\}.\eqno(2.9)(b)$$
Define $\sigma = {\rom Ad}(g^{-1})\circ\sigma'$; then $\sigma$ is an
antiholomorphic automorphism of $G$, and
$$\sigma(B) = B, \qquad \sigma(T) = T, \qquad \sigma(\{X_\alpha\}) =
\{X_\alpha\}.\eqno(2.9)(c)$$
Furthermore $\sigma^2$ is an inner automorphism of $G$ (namely ${\rom
Ad}(g^{-1})\circ{\rom Ad}(\sigma'(g^{-1}))$) with the properties in $(2.9)(c)$. 
By Proposition 2.8, such an inner automorphism is necessarily trivial, so
$\sigma$ is an involution.  It is therefore a real form.  Since $\sigma$
preserves $B$, it is quasisplit.  By construction it is inner to $\sigma'$.

It remains to prove the uniqueness of the equivalence class of $\sigma$.  So
suppose $\sigma''$ is another quasisplit real form in the inner class of
$\sigma$.  By definition of quasisplit, this means that there is a Borel
subgroup $B''$ fixed by $\sigma''$.  The first problem is to find a
$\sigma''$-stable maximal torus in $B''$.  Write $N''$ for the unipotent
radical of $B''$, and $\overline{T''} = B''/N''$; this torus inherits a quotient
real form $\overline{\sigma''}$, and is isomorphic by the quotient map to
any maximal torus of $B''$.  Using any of these isomorphisms, we get a
well-defined set of positive roots (characters of $\overline{T''}$).   These
roots are permuted by the action of $\overline{\sigma''}$ on the character
lattice $X^*(\overline{T''})$.  Let $\overline{Z}$ be any element of the Lie
algebra $\overline{{\frak t}''}$ on which all the positive roots take positive
values; then $\overline{Z}+\overline{\sigma''}(\overline{Z})$  has the same
property.  It follows that any preimage of $\overline{Z}+ \overline{\sigma''}
(\overline{Z})$ in ${\frak b''}$ is a regular semisimple element.  Let $Z\in
{\frak b}$ be any preimage of $\overline{Z}$; then $Z+\sigma''(Z)$ is a
preimage of $\overline{Z}+\overline{\sigma}(\overline{Z})$, so it is a
$\sigma''$-fixed regular semisimple element of ${\frak b''}$.   Its
centralizer in $B''$ is therefore a $\sigma''$-stable maximal torus.

Finally, we want to find a $\sigma''$-stable set of basis vectors for the
simple root spaces of $T''$ in $B''$.  We know that $\sigma''$ permutes these
root spaces by a permutation of order 2.  We consider first a pair
$\{\alpha,\beta\}$ of distinct simple roots interchanged by $\sigma''$. 
Choose any basis vector $X_\alpha$ for the $\alpha$ root space, and define
$X_\beta = \sigma''(X_\alpha)$.  Then the relation $(\sigma'')^2=1$ forces
$X_\alpha = \sigma''(X_\beta)$.  Next, suppose the root $\alpha$ is fixed by
$\sigma''$.  Let $Y_\alpha$ be any basis vector for the root space.  Then
there is a complex number $c_\alpha$ with $\sigma''(Y_\alpha) = c_\alpha
Y_\alpha$.  Applying $\sigma''$ to this relation, and using the fact that
$\sigma''$ is an antiholomorphic involution, we get
$c_\alpha\overline{c_\alpha} = 1$.  Let $z_\alpha$ be a square root of
$c_\alpha$; then it follows easily that $X_\alpha = z_\alpha Y_\alpha$ is
fixed by $\sigma''$.

We have shown how to construct $B''$, $T''\subset B''$, and $\{{X_\alpha}''\}$
as in Proposition 2.8 preserved by $\sigma''$.  By Proposition 2.8, we can
find a $g\in G$ carrying $B$ to $B''$, etc.  It follows that the
antiholomorphic involution
$$\widetilde{\sigma''} = {\rom Ad}(g)\circ\sigma\circ{\rom Ad}(g^{-1})$$
preserves $B''$, etc.  Therefore $(\sigma'')^{-1}\widetilde{\sigma''}$
preserves them as well.  But the assumption that $\sigma$ and $\sigma''$
are in the same inner class means that this last automorphism is inner.  By
Proposition 2.8, it is trivial.  So $\sigma'' = \widetilde{\sigma''}$.  The right
side here is equivalent to $\sigma$, as we wished to show.  Q.E.D.
\medskip
{\bf Definition 2.10} (see [Springer]). Suppose $G$ is a complex connected
reductive algebraic group, $B\subset G$ is a Borel subgroup, and $T\subset
G$ is a maximal torus.  The {\it based root datum for $G$ defined by $B$ and
$T$} is the quadruple
$$ \Psi_0(G,B,T) = (X^*(T),\Delta(B,T),X_*(T),\Delta^\vee(B,T)).$$
Here $X^*(T)$ is the lattice of rational characters of $T$,
$\Delta(B,T)\subset X^*(T)$ is the set of simple roots of $T$ in ${\frak b}$,
$X_*(T)$ is the lattice of rational one-parameter subgroups of $T$, and
$\Delta^\vee(B,T)\subset X_*(T)$ is the set of simple coroots.

It is a consequence of Proposition 2.8 that any two based root data are
canonically isomorphic.  Following [Kott2], we use these canonical
isomorphisms to define the {\it based root datum for $G$}, $\Psi_0(G)$, as
the projective limit over all such $T\subset B$:
$$\Psi_0(G)= (X^*,\Delta,X_*,\Delta^\vee) = \lim_{B,T} \Psi_0(G,B,T).$$
The structure of the based root datum consists of the lattice structures on
$X^*$ and $X_*$, the containments $\Delta\subset X^*$ and
$\Delta^\vee\subset X_*$, and the perfect pairing
$$\langle,\rangle:X^*\times X_* \rightarrow{\Bbb Z}.$$
By an {\it isomorphism} or {\it automorphism} of based root data we will
understand a map preserving these structures.

\medskip
{\bf Proposition 2.11} (see [Springer], Corollary 2.14.)  {\it Suppose $G$ is a
complex connected reductive algebraic group.  Write ${\rom Aut}(G)$ for the
(complex) group of rational (equivalently, holomorphic) automorphisms of
$G$, and ${\rom Aut}(\Psi_0(G))$ for the (discrete) group of automorphisms
of the based root datum of $G$.  Then there is a natural short exact sequence
$$1\rightarrow {\rom Int}(G)\rightarrow {\rom Aut}(G)\buildrel
\Psi_0\over\rightarrow {\rom Aut}(\Psi_0(G))\rightarrow 1.$$

This sequence splits (but not canonically), as follows. Choose a Borel
subgroup  $B$ of $G$, a maximal torus $T\subset B$, and  a set of basis
vectors $\{X_\alpha\}$ for the simple root spaces of $T$ in the Lie algebra
${\frak b}$; and define ${\rom Aut}(G,B,T,\{X_\alpha\})$ to be the set of
holomorphic automorphisms of $G$ preserving $B$, $T$, and $\{X_\alpha\}$
as sets.   Then the restriction of $\Psi_0$ to ${\rom
Aut}(G,B,T,\{X_\alpha\})$ is an isomorphism.}
\medskip
For the reader's convenience we recall how the map $\Psi_0$ is defined.  Fix
$\tau \in {\rom Aut}(G)$, and choose any pair $T \subset B$ as in Definition
2.10.  (We assume no relationship between $\tau$ and $B$ or $T$.)  If
$\lambda$ is any rational character of $T$, then $\lambda\circ \tau^{-1}$ is
a rational character of $\tau(T)$.  This defines an isomorphism
$$X^*(T) \rightarrow X^*(\tau(T))$$
carrying $\Delta(B,T)$ to $\Delta(\tau(B),\tau(T))$.
Similarly, composition with $\tau$ carries one-parameter subgroups of $T$
to one-parameter subgroups of $\tau(T)$.  Assembling these maps, we get an
isomorphism
$$\Psi_0(\tau,B,T): \Psi_0(G,B,T) \rightarrow \Psi_0(G,\tau(B),\tau(T)).$$
Since both range and domain are {\it canonically} isomorphic to $\Psi_0(G)$,
this map provides the automorphism $\Psi_0(\tau)$ that we want.

An automorphism belonging to one of the sets ${\rom Aut}(G,B,T,
\{X_\alpha\})$ is called {\it distinguished}.

We have seen that the inner classes of real forms of $G$ are in one-to-one
correspondence with the equivalence classes of quasisplit real forms. The
quasisplit real forms are easy to classify.
\medskip
{\bf Proposition 2.12.}  {\it Suppose $G$ is a complex connected reductive
algebraic group.  Then the equivalence classes of quasisplit real forms of
$G$ (and therefore also the inner classes of all real forms) are in
one-to-one correspondence with the involutive automorphisms of
$\Psi_0(G)$.}
\medskip
{\it Proof.}  Suppose $\sigma$ is any antiholomorphic automorphism of $G$;
we will show how to define an automorphism $\Psi_0(\sigma)$ of
$\Psi_0(G)$.  (This is not immediately handled by Proposition 2.11, since
$\sigma$ is not a holomorphic automorphism of $G$.)  Fix a Borel subgroup
$B$ of $G$, and $T\subset B$ a maximal torus.  If $\lambda$ is a
holomorphic character of $T$, then $\lambda\circ\sigma^{-1}$ is an
antiholomorphic character of $\sigma(T)$; so $\overline{\lambda
\circ\sigma^{-1}}$ is a holomorphic character of $\sigma(T)$.  The map
sending $\lambda$ to $\overline{\lambda \circ\sigma^{-1}}$ is an
isomorphism
$$X^*(T) \rightarrow X^*(\sigma(T)),$$
and it carries $\Delta(B,T)$ to $\Delta(\sigma(B),\sigma(T))$.  Continuing in
this way, we find that $\sigma$ induces an isomorphism
$$\Psi_0(\sigma,B,T):\Psi_0(G,B,T)\rightarrow
\Psi_0(G,\sigma(B),\sigma(T)).$$
Since both of these objects are canonically isomorphic to $\Psi_0(G)$, we
have the automorphism we want.  If $\sigma'$ is another antiholomorphic
automorphism of $G$, then it is clear from the definitions that
$$\Psi_0(\sigma\circ(\sigma')^{-1}) =
\Psi_0(\sigma)\circ\Psi_0(\sigma')^{-1}.$$
Here the map on the left is given by Proposition 2.11, and those on the right
by the preceding construction.  It follows from Proposition 2.11 that
$\Psi_0(\sigma) = \Psi_0(\sigma')$ if and only if $\sigma$ and $\sigma'$
differ by an inner automorphism.   This proves the injectivity of our map
from inner classes of real forms to involutive automorphisms of the based
root datum.

For the surjectivity, fix an automorphism $a$ of $\Psi_0(G)$ of order 2, and
data $B$, $T$, and $\{X_\alpha\}$ as in Proposition 2.8.  We must find a real
form $\sigma$ of $G$ with $\Psi_0(\sigma) = a$.  The theory of Chevalley
forms (forms of a reductive group defined and split over any field) shows
that there is a real form $\sigma_s$ of $G$ such that
$$\sigma_s(B) = B, \qquad \sigma_s(T) = T,\qquad\sigma_s(X_\alpha) =
X_\alpha,$$
and $\sigma_s$ induces the identity automorphism of $X^*(T)$ and $X_*(T)$.
On the other hand, the splitting of the exact sequence of Proposition 2.11
produces a holomorphic automorphism $t_a$ of $G$ such that
$$t_a(B) = B, \qquad t_a(T) = T,\qquad t_a(X_\alpha) = X_{a(\alpha)},$$
and $t_a$ induces the automorphism $a$ on $X^*(T)$ and $X_*(T)$. The
composition $\sigma = t_a\circ\sigma_s$ is an antiholomorphic
automorphism of $G$ satisfying
$$\sigma(B) = B, \qquad \sigma(T) = T,\qquad\sigma(X_\alpha) =
X_{t(\alpha)},$$
and $\sigma$ induces the automorphism $a$ on $X^*(T)$ and $X_*(T)$.  This
implies first of all that $\Psi_0(\sigma) = a$.  Since $a$ has order 2, it
follows that $\sigma^2$ is a holomorphic automorphism of $G$ acting
trivially on $T$ and fixing the various $X_\alpha$.  Such an automorphism is
trivial by Proposition 2.8; so $\sigma^2 = 1$, and $\sigma$ is the real form
we want.  Q.E.D.

We can now describe approximately the context in which we will do
representation theory.  We fix the connected reductive complex algebraic
group $G$, and an inner class of real forms of $G$.  This inner class is
specified by an automorphism of order 2 of the based root datum
$\Psi_0(G)$.  We need to consider at the same time representations of
various real forms of $G$.  It is natural therefore to consider pairs
$(\pi,\sigma)$, with $\sigma$ a real form of $G$ (say in the specified inner
class) and $\pi$ a representation of $G({\Bbb R},\sigma)$.  It is natural to
define two such pairs $(\pi,\sigma)$ and $(\pi',\sigma')$ to be equivalent if
there is an element $g\in G$ such that $\sigma' = {\rom Ad}(g)\circ
\sigma\circ {Ad}(g^{-1})$, and $\pi\circ {Ad}(g^{-1})$ (which is a
representation of $G({\Bbb R},\sigma')$ on the space of $\pi$) is equivalent
to $\pi'$.  The difficulty with this definition first appears in the example of
$SL(2,{\Bbb R})$, discussed after (2.3).  If we let $\pi$ be a holomorphic
discrete series representation of $G({\Bbb R},\sigma)=SL(2,{\Bbb R})$, and
$\pi'$ the corresponding antiholomorphic discrete series representation,
then this definition makes $(\pi,\sigma)$ equivalent to $(\pi',\sigma)$. 
Clearly this will lead to inconvenience at least when we try to use the
theory to write precise character formulas.  The next definition, taken from
Definitions 1.12, 1.13, and 1.14 of section 1,  provides a way around the
problem.
\medskip
{\bf Definition 2.13.}  Suppose $G$ is a connected reductive complex
algebraic group.  A {\it (weak) extended group containing $G$} is a real Lie
group $G^\Gamma$ subject to the following conditions.
\smallskip
\item{(a1)}  $G^\Gamma$ contains $G$ as a subgroup of index two.
\item{(a2)}  Every element of $G^\Gamma - G$ acts on $G$ (by conjugation)
as an antiholomorphic automorphism.
\smallskip
Condition (a1) may be rephrased as follows. Write $\Gamma = {\rom
Gal}({\Bbb C}/{\Bbb R})$ for the Galois group.
\smallskip
\item{(a1$'$)}  There is a short exact sequence
$$1\rightarrow G \rightarrow G^\Gamma \rightarrow \Gamma\rightarrow
1.$$
\smallskip
A {\it strong real form} of $G^\Gamma$ is an element $\delta\in
G^\Gamma-G$ such that $\delta^2\in Z(G)$ has finite order.  The {\it
associated real form} for $\delta$ is the (antiholomorphic involutive)
automorphism $\sigma(\delta)$ of $G$ defined by conjugation by $\delta$:
$$\sigma(\delta)(g) = \delta g\delta^{-1}.$$
The {\it group of real points of $\delta$} is defined to be the group of real
points of $\sigma(\delta)$:
$$G({\Bbb R},\delta) = \{\, g\in G\mid \delta g\delta^{-1} = g\,\}.$$
Two strong real forms $\delta$ and $\delta'$ of $G^\Gamma$ are called {\it
equivalent} if they are conjugate by $G$:
$$\delta\sim\delta'\hbox{  if and only if  } \delta' = g\delta g^{-1}\hbox{ 
for some $g\in G$.}$$
Thus the set of strong real forms equivalent to a fixed strong real form
$\delta$ is a homogeneous space $G/H$; the isotropy group is
$$H = \{\,g\in G\mid g\delta g^{-1} = \delta\,\} = G({\Bbb R},\delta)$$
(cf. (2.3)).

A {\it representation of a strong real form} of $G^\Gamma$ is a pair
$(\pi,\delta)$, subject to
\smallskip
\item{(a)} $\delta$ is a strong real form of $G^\Gamma$ ; and
\item{(b)} $\pi$ is an admissible representation of $G({\Bbb R},\delta)$.
\smallskip
\noindent Two such representations $(\pi,\delta)$ and $(\pi',\delta')$ are
said to be {\it (infinitesimally) equivalent} if there is an element $g\in G$
such that $g\delta g^{-1} = \delta'$, and $\pi \circ{\rom Ad}(g^{-1})$ is
(infinitesimally) equivalent to $\pi'$.  (In particular, this is possible only if
the strong real forms are equivalent.)  Finally, define
$$\Pi(G^\Gamma)=\Pi(G/{\Bbb R})$$
to be the set of (infinitesimal) equivalence classes of irreducible
representations of strong real forms of $G^\Gamma$.  (Here when we use the
notation on the right, we must have in mind a particular weak extended
group $G^\Gamma$.)
\medskip
Clearly the equivalence of strong real forms implies the equivalence of the
associated real forms. We have not formulated the obvious definition of
``inner'' for strong real forms, since any two strong real forms for the same
$G^\Gamma$ are automatically inner to each other.  In order for this to be a
reasonable definition, we need to know that every real form is represented
by a strong real form of some extended group.  This is a consequence of
Proposition 2.14 and Corollary 2.16 below.

We pause now to give the proof of Lemma 1.15.  (The extra datum ${\cal W}$
needed to complete the definition of an extended group plays no r\^ole in
this lemma.)  Surjectivity of the map is clear; what must be established is
injectivity.  So suppose that $\pi$ and $\pi'$ are irreducible representations
of $G({\Bbb R},\delta_s)$ and $G({\Bbb R},\delta_{s'})$, respectively, and
that the pairs $(\pi,\delta_s)$ and $(\pi',\delta_{s'})$ are equivalent.  We
must show that $\delta_s = \delta_{s'}$, and that $\pi$ is equivalent to
$\pi'$.  The hypothesis means that there is an element $g\in G$ such that
$$g\delta_s g^{-1} =\delta_{s'},\qquad \pi\circ {Ad}(g^{-1})\sim \pi'$$
The first condition says that the strong real forms $\delta_s$ and
$\delta_{s'}$ are equivalent. Since they belong to a set of representatives
for the equivalence classes, they must in fact coincide.  Consequently $g$
commutes with $\delta_s$, and so belongs to $G({\Bbb R},\delta_s)$.  But
this means that $\pi\circ {Ad}(g^{-1})$ is equivalent to $\pi$, the
equivalence being implemented by the operator $\pi(g)$ on the space of
$\pi$.  The second condition therefore implies that $\pi'$ is equivalent to
$\pi$ as representations of $G({\Bbb R},\delta_s)$, as we wished to show. 
Q.E.D.

We conclude this section by investigating the possible structures of weak
extended groups.

\medskip
{\bf Proposition 2.14.}  {\it Suppose $G^\Gamma$ is a weak extended group. 
Then the set of real forms of $G$ associated to strong real forms of
$G^\Gamma$ constitutes exactly one inner class of real forms.}
\medskip
{\it Proof.} The conjugation action of any element $\delta$ of
$G^\Gamma-G$ defines an antiholomorphic automorphism $\sigma(\delta)$
of $G$.  This automorphism preserves $Z(G)$, and in fact its restriction
$\sigma_Z$ to $Z(G)$ is independent of the choice of $\delta$.  

The proof of Proposition 2.12 attaches to each element $\delta$ of
$G^\Gamma-G$ an automorphism $a = \Psi_0(\delta)$ of the based root
datum of $G$.   The proof also shows that $a$ is independent of the choice of
$\delta$, and that the various conjugation actions $\sigma(\delta)$ give all
antiholomorphic automorphisms $\sigma$ of $G$ such that $\Psi_0(\sigma)
= a$.  In particular, Proposition 2.12 implies that for any real form $\sigma$
in the inner class defined by $a$, there is a $\delta_1\in G^\Gamma-G$ with
$\sigma(\delta_1) = \sigma$.  Since $\sigma$ is an involution, this implies
that $\delta_1^2 = z_1\in Z(G)$.  To complete the proof, we must show that
our choice of  $\delta_1$ can be modified to make $z_1$ have finite order. 
We first compute
$$\sigma_Z(z_1) = \delta_1z\delta_1^{-1} =
\delta_1\delta_1^2\delta_1^{-1} = \delta_1^2 = z_1.$$
That is, $z_1\in Z^{\sigma_Z}$.  We can now apply the following elementary
lemma.
\medskip
{\bf Lemma 2.15.}  {\it Suppose $Z$ is a (possibly disconnected) complex
reductive abelian algebraic group, and $\sigma$ is an antiholomorphic
involutive automorphism of $Z$.  Put
$$(1+\sigma)Z = \{\,z\sigma(z)\mid z\in Z\,\}.$$  
Then the quotient group $Z^\sigma/(1+\sigma)Z$ is finite, and each coset
has a representative of finite order in $Z$.}
\medskip
\noindent Of course the quotient in the lemma is a Galois cohomology group.

We can now complete the proof of Proposition 2.14.  By Lemma 2.15, there is
an element $z_2\in Z(G)$ such that $z=z_1z_2\sigma_Z(z_2)$  has finite
order.  Set
$$\delta = z_2\delta_1.$$
Then $\sigma(\delta) = \sigma$, and $\delta^2 = z$ has finite order; so
$\delta$ is the required strong real form.  Q.E.D.

Here is the classification of weak extended groups.
\medskip
{\bf Corollary 2.16.}  {\it Suppose $G$ is a connected reductive complex
algebraic group. 
\item{a)} Fix a weak extended group $G^\Gamma$ for $G$.  Let $\sigma_Z$
be the antiholomorphic involution of $Z(G)$ defined by the conjugation
action of any element of $G^\Gamma-G$.  We can attach to $G^\Gamma$ two
invariants.  The first of these is an involutive automorphism
$$a\in {\rom Aut}(\Psi_0(G))$$
of the based root datum of $G$.  The second is a class 
$$\overline{z}\in Z(G)^{\sigma_Z}/(1+\sigma_Z)Z(G).$$
\item{b)} Suppose $G^\Gamma$ and $\left(G^\Gamma\right)'$ are weak
extended groups for $G$ with the same invariants $(a,\overline{z})$.  Then
the identity map on $G$ extends to an isomorphism from $G^\Gamma$ to
$\left(G^\Gamma\right)'$.
\item{c)} Suppose $a\in {\rom Aut}(\Psi_0(G))$ is an involutive
automorphism.  Write $\sigma_Z$ for the antiholomorphic involution of
$Z(G)$ defined by the action of any real form $\sigma$ in the inner class
\hang corresponding to $a$ (Proposition 2.12); and suppose
$$\overline{z}\in Z(G)^{\sigma_Z}/(1+\sigma_Z)Z(G).$$
Then there is a weak extended group $G^\Gamma$ with invariants
$(a,\overline{z})$.
\medskip}
{\it Proof.}  For (a), any element $\delta$ of $G^\Gamma-G$  defines by
conjugation an antiholomorphic automorphism $\sigma(\delta)$ of $G$.  By
Proposition 2.12 and its proof, the corresponding automorphism
$a=\Psi_0(\sigma(\delta))$ is independent of the choice of $\delta$.  To
define $\overline{z}$, fix a quasisplit real form $\sigma_q$ in the inner
class defined by $a$, and choose $\delta_q\in G^\Gamma-G$ so that
$$\sigma(\delta_q) = \sigma_q$$
(as is possible by Proposition 2.14.)  Since $\sigma_q^2=1$, 
$$\delta_q^2 = z\in Z(G);$$
and using $\delta_q$ to compute $\sigma_Z$, we see that
$$\sigma_Z(z) = z.$$
By Proposition 2.7, any other choice of $\sigma'_q$ differs from $\sigma_q$
by conjugating with an element ${\rom Ad}(g)$; so any other $\delta'_q$ is of
the form
$$\delta'_q = z_1g\delta_qg^{-1}.$$
We compute immediately that
$$(\delta'_q)^2 = z_1\sigma_Z(z_1)\delta_q^2,$$
so the class $\overline{z}$ is independent of all choices.

For (b), fix a quasisplit real form $\sigma_q$ of $G$ as in (a), and choose
elements $\delta_q\in G^\Gamma-G$, ${\delta_q}''\in
\left(G^\Gamma\right)'$ so that
$$\sigma(\delta_q) =\sigma_q,\qquad \sigma({\delta_q}'') = \sigma_q.$$
Write $z = \delta_q^2$, $z' = ({\delta_q}'')^2$.   By  the hypothesis on the
invariants of the two weak extended groups, there is an element $z_1\in
Z(G)$ so that $z = z'\left(z_1\sigma_Z(z_1)\right)$.  Set $\delta'_q =
z_1{\delta_q}''$; then 
$$\sigma(\delta'_q) = \sigma(\delta_q)=\sigma_q,\qquad \delta_q^2 =
(\delta'_q)^2=z.\eqno(2.17)(a)$$
Now the group $G^\Gamma$ is the disjoint union of $G$ and the coset
$G\delta_q$; these are multiplied according to the rules
$$(g_1\delta_q)(g_2\delta_q) = g_1\sigma_q(g_2)z,\qquad
(g_1\delta_q)(g_2) = g_1\sigma_q(g_2)\delta_q\eqno(2.17)(b)$$
and the obvious rules for the other two kinds of product.  We can define a
bijection from $G^\Gamma$ to $\left(G^\Gamma\right)'$ by using the
identity on $G$, and sending $g\delta_q$ to $g\delta'_q$.  By (2.17), this
bijection is a group homomorphism, proving (b).

For (c), choose a quasisplit real form $\sigma_q$ of $G$ in the inner class
corresponding to $a$ (Proposition 2.12), and a representative $z\in Z(G)$ of
the class $\overline{z}$. Define $G^\Gamma$ to consist of the union of $G$
and the set of formal symbols $g\delta_q$ (topologized as the union of two
copies of $G$).  Introduce a multiplication on $G^\Gamma$ by the rules in
$(2.17)(b)$; then it is a simple matter to check that $G^\Gamma$ is a weak
extended group with the desired invariants.  Q.E.D.  
%\end
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\centerline{\bf 3.  Structure theory: extended groups and Whittaker models.}
\vskip 1pc
Part of the goal of the Langlands classification is a parametrization of the
representations of real forms of $G$ in terms of L-groups.  A difficulty with
this goal is that several different pairs (real form, representation) may be
isomorphic.  The basic example is $(SL(2,{\Bbb R}),\hbox{discrete series})$,
where we may have a holomorphic or an antiholomorphic discrete series
representation with the same infinitesimal character.  Even though the
notion of strong real form allows us to separate these pairs (Lemma 1.15),
it still gives no reason to prefer one over another.  The L-group parameters
we find for these representations (typically local systems of some kind)
{\it do} include a distinguished parameter (a trivial local system).  In order
to establish a parametrization like Theorem 1.18, we therefore need
(roughly speaking) a way to specify a preferred representation in each
L-packet.

Langlands' program suggests a way to approach this problem.  One can
specify a ``Whittaker model'' for a quasisplit real form $G({\Bbb R})$ of $G$
(or rather an equivalence class of such models for an equivalence class of
quasisplit strong real forms).  Two such models must differ by an
automorphism of $G({\Bbb R})$, but not necessarily by an inner
automorphism.  It is essentially known from [Kostant] that each tempered
L-packet for $G({\Bbb R})$ contains exactly one representation admitting a
Whittaker model.  (Such a result is expected over any local field, but of
course it is unlikely to be established in the absence of a definition of
L-packets.)  In the case of non-tempered L-packets, often no representation
admits a Whittaker model.  Nevertheless, there will be exactly one
irreducible representation in the L-packet for which the corresponding
standard representation admits a Whittaker model.  This representation will
be taken as the ``base point'' corresponding to the trivial local system in
Theorem 1.18.

\medskip
{\bf Definition 3.1.}  Suppose $G({\Bbb R})$ is a quasisplit real form of a
complex connected reductive algebraic group.  Fix $T\subset B$ Cartan and
Borel subgroups of $G$ defined over ${\Bbb R}$, and write $A\subset T$ for
the maximal split torus in $T$.   (Thus the identity component of $A({\Bbb
R})$ is an ``Iwasawa $A$'' for $G({\Bbb R})$.)  Write $N$ for the unipotent
radical of $B$.  List the simple (restricted) roots of $A$ in $N$ as
$\{\alpha_1,\ldots,\alpha_l\}$, and write ${\frak g}_{\alpha_j}\subset
{\frak n}$ for the corresponding restricted root spaces.  Then
$${\frak n}/[{\frak n},{\frak n}] \simeq \sum_{j=1}^l  {\frak
g}_{\alpha_j}.\eqno(3.1)(a)$$
All the spaces here are defined over ${\Bbb R}$, so we get a corresponding
decomposition of ${\frak n}({\Bbb R})/[{\frak n}({\Bbb R}),{\frak n}({\Bbb
R})]$. In particular, the space of (one-dimensional) unitary characters of
$N({\Bbb R})$ is isomorphic (by taking differentials) to
$$\sum_{j=1}^l  i{\frak g}_{\alpha_j}({\Bbb R})^*,\eqno(3.1)(b)$$
the space of imaginary-valued linear functionals on the simple restricted
real root spaces.  A unitary character $\chi$ of $N({\Bbb R})$ is called {\it
non-degenerate} if its restriction to each simple restricted root subgroup is
non-trivial.

Suppose  $\chi$ is a non-degenerate unitary character of $N({\Bbb R})$. 
Write ${\Bbb C}_\chi$ for the one-dimensional space on which $\chi$ acts,
and
$${\cal L}_\chi = G({\Bbb R}) \times_{N({\Bbb R})} {\Bbb
C}_\chi\eqno(3.1)(c)$$
for the corresponding line bundle on $G({\Bbb R})/N({\Bbb R})$.  The {\it
Whittaker model ${\rom Wh}(\chi)$ for $G({\Bbb R})$ defined by $\chi$} is the
space of smooth sections of ${\cal L}_\chi$:
$${\rom Wh}(\chi)\simeq \{\,f\in C^\infty(G({\Bbb R}))\mid f(gn) =
\chi(n)^{-1}f(g)\,\}\eqno(3.1)(d).$$
We make ${\rom Wh}(\chi)$ into a smooth representation of $G({\Bbb R})$ by
left translation.

A Hilbert space representation $(\pi,{\cal H}_\pi)$ of $G({\Bbb R})$ is said
to {\it admit a Whittaker model of type $\chi$} if there is a non-zero
continuous map
$${\cal H}_\pi^\infty \rightarrow {\rom Wh}(\chi)\eqno(3.1)(e)$$
respecting the action of $G({\Bbb R})$.  (Equivalently, $\pi$ should admit a
non-zero distribution vector transforming according to the character $\chi$
under $N({\Bbb R})$.)
\medskip
 
We consider first the uniqueness of non-degenerate characters.  The
following lemma is well illustrated by the examples of $GL(2,{\Bbb R})$
(which has up to conjugacy only one kind of Whittaker model) and
$SL(2,{\Bbb R})$ (which has two).  The element $t$ of (c) below can always
be chosen in $T({\Bbb R})$ in the first case, but not in the second.

\medskip
{\bf Lemma 3.2}([Kostant], Lemma 6.2.1). {\it Suppose $G({\Bbb R})$ is a
quasisplit real form of a complex connected reductive algebraic group, and
$T\subset B$ are Cartan and Borel subgroups defined over ${\Bbb R}$.
\smallskip
\item{a)} The decomposition (3.1)(a) of ${\frak n}/[{\frak n},{\frak n}]$ is
invariant under ${\rom Ad}(B)$.  Consequently the adjoint action of $B({\Bbb
R})$ on characters of $N({\Bbb R})$ preserves the set of non-degenerate
unitary characters.  In particular, the notion of non-degenerate unitary
character of $N({\Bbb R})$ is independent of the choice of $T$.
\item{b)} Suppose $t\in T$.  Then the automorphism ${\rom Ad}(t)$ of $G$ is
defined over ${\Bbb R}$ if and only if it preserves each of the simple real
restricted root spaces ${\frak g}_{\alpha_j}({\Bbb R})$.
\item{c)}  Suppose $\chi$ and $\chi'$ are non-degenerate unitary characters
of $N({\Bbb R})$.  Then there is an element $t\in T$ such that ${\rom Ad}(t)$
is defined over ${\Bbb R}$ (so that ${\rom Ad}(t)$ defines an automorphism of
$N({\Bbb R})$), and $t\cdot\chi$ is equal to $\chi'$ (the action on characters
being defined by composition of the character with the inverse of the
automorphism of $N({\Bbb R})$).
\item{d)}  In (c), the coset $tZ(G)$is uniquely determined by $\chi$ and
$\chi'$.  In particular, $\chi$ is conjugate to $\chi'$ by $G({\Bbb R})$ if and
only if $t \in T({\Bbb R})Z(G)$.
\medskip}

{\it Proof.}  For (a), ${\rom Ad}(N)$ acts trivially on ${\frak n}/[{\frak
n},{\frak n}]$; so we need only consider ${\rom Ad}(T)$. Because the
decomposition arises from the weights of the action of a subtorus of $T$,
the invariance is clear.  The second claim follows.  For the last, recall that
any two choices of $T({\Bbb R})$ are conjugate by $N({\Bbb R})$.

For (b), the condition is obviously necessary for ${\rom Ad}(t)$ to be defined
over ${\Bbb R}$; so suppose that it holds.  Write $\sigma$ for the complex
conjugation on $G$.  Because $\sigma$ preserves $T$, it must permute the
root spaces in ${\frak g}$; we write $\sigma$ for the corresponding
permutation of the roots.  Then ${\rom Ad}(t)$ is defined over ${\Bbb R}$ if
and only if for every root $\beta$ of $T$ in $G$, we have
$$\beta(\sigma t) = \overline{(\sigma\beta)(t)}.\eqno(3.3)(a)$$
It suffices to verify this condition for simple roots $\beta$.    Write
$\alpha$ for the restriction of $\beta$ to $A$; then the real restricted root
space ${\frak g}_\alpha$ consists of elements of the form 
$$X + \sigma X \qquad (X\in {\frak g}_\beta).$$
Now
$${\rom Ad}(t)(X + \sigma X) = \beta(t)X + (\sigma\beta(t))\sigma
X.\eqno(3.3)(b)$$
It follows that ${\rom Ad}(t)$ preserves the real restricted root space if and
only if
$$(\sigma\beta(t))\sigma X = \sigma(\beta(t)X).$$
Since $\sigma$ is conjugate-linear, this is equivalent to (3.3)(a).

For (c), we regard the complexified differential of $\chi$ as a character of
the Lie algebra ${\frak n}$.  As such, it has a complex value $d\chi(X)\in
{\Bbb C}$ on any $X\in {\frak n}$.  The condition that $\chi$ be unitary is
equivalent to $d\chi$ taking purely imaginary values on each simple
restricted real root space.   By  (3.3)(a), this is equivalent to
$$d\chi(X) = -\overline{d\chi(\sigma X)}\qquad(X\in {\frak
g}_\beta)\eqno(3.4)(a)$$
for every simple root $\beta$.   The non-degeneracy condition is
$$d\chi(X) \ne 0\qquad(X\in {\frak g}_\beta - 0).\eqno(3.4)(b)$$
Suppose now that $\chi$ and $\chi'$ are as in (c) of the lemma.  Since the
simple roots are linearly independent, we can find $t\in T$ with
$$\beta(t) =  d\chi(X)/d\chi'(X)\qquad(X\in {\frak g}_\beta -
0).\eqno(3.4)(c)$$
Combining (3.4)(c) with (3.4)(a) gives exactly the condition in (3.3)(a) for
${\rom Ad}(t)$ to be defined over ${\Bbb R}$.  To compute the action of $t$ on
$\chi$, it is enough to evaluate the differential at an element $X\in {\frak
g}_\beta$.  This is
$$d(t\cdot\chi)(X) = d\chi({\rom Ad}(t^{-1})X) = d\chi(\beta(t^{-1})X) =
\beta(t)^{-1}d\chi(X) = d\chi'(X).$$
It follows that $t\cdot\chi = \chi'$, as we wished to show.

For (d), the proof of (c) shows that the choice (3.4)(c) of $\beta(t)$ (for
every simple $\beta$) is forced by the requirement that $t\cdot\chi = \chi'$. 
This gives the first assertion.  For the second, $\chi$ is conjugate to $\chi'$
by $G({\Bbb R})$ if and only if it is conjugate by the normalizer $B({\Bbb R})
= T({\Bbb R})N({\Bbb R})$ of $N({\Bbb R})$.  Since $N({\Bbb R})$ fixes $\chi$,
(d) follows.  Q.E.D.

Recall now from Definition 1.12 the notion of an {\it extended group}
$(G^\Gamma,{\cal W})$ for $G$.  The {\it invariants} of the extended group
are the automorphism $a$ of the based root datum $\Psi_0(G)$ attached to
the underlying weak extended group $G^\Gamma$ (Corollary 2.16), and the
element
$$z = \delta^2 \in Z(G)^{\sigma_Z}.\eqno(3.5)$$
Here $\delta$ is any element of a triple $(\delta,N,\chi)\in {\cal W}$
(Definition 1.12).  The element $z$ is independent of the choice of $\delta$,
since ${\cal W}$ is a single conjugacy class under $G$.

Here is the classsification of extended groups.
\medskip
{\bf Proposition 3.6.}  {\it Suppose $G$ is a connected reductive complex
algebraic group. 
\item{a)} Suppose $( G^\Gamma,{\cal W})$ and $\left(\left(
G^\Gamma\right)',{\cal W}'\right)$  are extended groups for $G$ with the
same invariants $(a,z)$ (cf. (3.5)).  Then the identity map on $G$ extends to
an isomorphism from $G^\Gamma$ to $\left(G^\Gamma\right)'$ carrying
${\cal W}$ to ${\cal W}'$.  Any two such extensions differ by an inner
automorphism of $G^\Gamma$ from $Z(G)$.
\item{b)} Fix a weak extended group $G^\Gamma$ for $G$ with invariants
$(t,\overline{z})$.  If $(G^\Gamma,{\cal W})$ is an extended group, then its
second invariant is a representative for the class of $\overline{z}$. 
Conversely, if $z \in Z(G)^{\sigma_Z}$ is an element of finite order
representing the class of $\overline{z}$, then there is an extended group
structure on $G^\Gamma$ with second invariant $z$.
\medskip}
{\it Proof.}  For (a), fix $(\delta,N,\chi)\in{\cal W}$.  Since $\delta^2=z\in
Z(G)$, conjugation by $\delta$ defines an antiholomorphic involutive
automorphism $\sigma=\sigma(\delta)$ of $G$, and thus a real form
$G({\Bbb R})$.  By condition (b)(2) in Definition 1.12, $G({\Bbb R})$ is
quasisplit.  The real forms defined analogously using ${\cal W}'$ are also
quasisplit, and inner to $G({\Bbb R})$.  By Proposition 2.12, at least one of
them must coincide with $G({\Bbb R})$.  That is, we can find
$(\delta',N',\chi')\in {\cal W}'$ so that
$$\sigma(\delta') = \sigma.\eqno(3.7)(a)$$
Now $N$ and $N'$ are maximal unipotent subgroups of $G$ defined over
${\Bbb R}$.  They are therefore conjugate by an element $g\in G({\Bbb R})$. 
After replacing $(\delta',N',\chi')$ by their conjugates by $g$ (which does
not change $\delta'$, and therefore preserves (3.7)(a)) we may assume that
$$N' = N.\eqno(3.7)(b)$$
Now $\chi$ and $\chi'$ are non-degenerate unitary characters of $N({\Bbb
R})$.  By Lemma 3.2, there is an element $t\in G$ normalizing $N({\Bbb R})$,
carrying $\chi'$ to $\chi$, and with ${\rom Ad}(t)$ defined over ${\Bbb R}$. 
This last condition means that
$$\sigma(t) = tw$$
for some $w\in Z(G)$.  We finally replace $(\delta',N',\chi')$ by their
conjugates by $t$.  This replaces $\delta'$ by $w^{-1}\delta'$, and therefore
(since $w$ is central) does not affect (3.7)(a).  Condition (3.7)(b) is
preserved since $t$ normalizes $N$.  By the choice of $t$,
$$\chi'=\chi.\eqno(3.7)(c)$$
We can now construct the desired isomorphism between the extended groups
as in the proof of Corollary 2.16(b), by sending $\delta$ to $\delta'$.

For the uniqueness, any other such isomorphism must send $(\delta,N,\chi)$
to a $G$-conjugate $g\cdot (\delta',N,\chi)$ of $(\delta',N,\chi)$ (since it
carries ${\cal W}$ to ${\cal W}'$). We wish to show that $g \in G({\Bbb
R})Z(G)$.  Since the isomorphism is the identity on $G$, we deduce three
facts: the conjugation actions of $\delta$ and $g(\delta')g^{-1}$ must agree
on $G$; $gNg^{-1} = N$; and $g\cdot \chi = \chi$.  The first of these facts
means that ${\rom Ad}(g)$ is defined over ${\Bbb R}$.  The second fact
implies that $g=tn \in B$.  Since ${\rom Ad}(g)$ is defined over ${\Bbb R}$, it
follows that $n\in N({\Bbb R})$ and that ${\rom Ad}(t)$ is defined over ${\Bbb
R}$.  Because of Lemma 3.2(d), the third fact guarantees that  $t\in Z(G)$, as
we wished to show. 

The first part of (b) is clear from the definitions.  For the second part, the
argument before (2.17)(a) shows how to find an element $\delta$ of
$G^\Gamma-G$ with $\sigma(\delta)$ a quasisplit real form of $G$, and
$\delta^2 = z$.  Let $N$ be any maximal unipotent subgroup  of $G$ defined
over ${\Bbb R}$, and let $\chi$ be any non-degenerate unitary character of
$N({\Bbb R})$.  Then the $G$-conjugacy class ${\cal W}$ of the triple 
$(\delta,N,\chi)$ is evidently an extended group structure with second
invariant $z$. Q.E.D.

\medskip
{\bf Corollary 3.8.} {\it Suppose $G$ is a connected complex reductive
algebraic group endowed with an inner class of real forms.  Then the
equivalence classes of extended groups for $G$ are parametrized by
elements of finite order in $Z(G)^{\sigma_Z}$ (cf. Corollary 2.16).}
\medskip

We have found no use for the wide selection of extended groups provided by
Corollary 3.8, and no reason to prefer one choice over another.  This should
probably be taken as evidence that our definitions are imperfect.  The
analogous phenomenon for dual groups (Proposition 4.7) has a much clearer
r\^ole, as we will see.
%\end

 
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\centerline{\bf 4. Structure theory: L-groups.}
\vskip 1pc

The based root datum for a complex connected reductive algebraic group
(Definition 2.10) characterizes that group up to isomorphism.  On the other
hand, based root data of reductive groups may be characterized by some
simple axioms (see [Springer]).  These axioms are symmetric in $X^*$ and
$X_*$; that is, the quadruple $\Psi_0 = (X^*,\Delta,X_*,\Delta^\vee)$ is the
based root datum of a reductive group if and only if ${^\vee\Psi_0} =
(X_*,\Delta^\vee,X^*,\Delta)$ is as well.  We call ${^\vee\Psi_0}$ the {\it
dual based root datum to $\Psi_0$}.  Notice that the automorphism groups of
dual based root data are canonically isomorphic:
$${\rom Aut}(\Psi_0) \simeq {\rom Aut}({^\vee\Psi_0}).\eqno(4.1)$$

\medskip
{\bf Definition 4.2} (see [Langlands] or [Borel]).  Suppose $G$ is a complex
connected reductive algebraic group.  A {\it dual group for G} is a complex
connected reductive algebraic group ${^\vee G}$, together with an
isomorphism from the dual of the based root datum for $G$ to the based root
datum for ${^\vee G}$:
$${^\vee\Psi_0}(G) \simeq \Psi_0({^\vee G}).$$
\medskip

By Proposition 2.8, any two dual groups for $G$ are isomorphic, and the
isomorphism is canonical up to inner automorphism of either group.

\medskip
{\bf Definition 4.3} (cf.~[AV2], Definition 10.5).  Suppose $G$ is a complex
connected reductive algebraic group.  A {\it weak E-group for $G$} is an
algebraic group ${^\vee G^\Gamma}$ containing a dual group ${^\vee G}$ for
$G$ as a subgroup of index 2.  That is, there is a short exact sequence
$$1\rightarrow {^\vee G} \rightarrow {^\vee G^\Gamma} \rightarrow
\Gamma\rightarrow 1.$$ 
Here $\Gamma = {\rom Gal}({\Bbb C}/{\Bbb R})$ as in Definition 2.13.

It will often be convenient for us to use this terminology in a various more
specific ways, which we record here though they require Proposition 4.4
below for their formulation.  Suppose $G^\Gamma$ is a weak extended group
for $G$ (Definition 2.13).   Recall that there is attached to $G^\Gamma$ an
involutive automorphism $a$ of the based root datum of $G$ (Corollary
2.16); of course $a$ really depends only on the inner class of real forms of
$G$ defined by $G^\Gamma$.  A {\it weak E-group for $G^\Gamma$} is a weak
E-group for $G$ with second invariant $a$ (Proposition 4.4(a)).  We may also
call this a {\it weak E-group for $G$ and the specified inner class of real
forms}, or even a {\it weak E-group for $G({\Bbb R})$} (with $G({\Bbb R})$ a
real form in the specified inner class).

Finally, we will occasionally need to use this terminology in a {\it less}
specific way.  We will say that a {\it weak E-group} is an algebraic group
$H^\Gamma$ containing a complex connected reductive algebraic subgroup
$H$ of index two.  That is, there is a short exact sequence
$$1\rightarrow H \rightarrow H^\Gamma \rightarrow \Gamma\rightarrow
1.$$
Of course every complex connected reductive algebraic group may be
regarded as a dual group, so this definition does not enlarge the class of
weak E-groups.
\medskip
The notion of dual group is symmetric, in the sense that if ${^\vee G}$ is a
dual group for $G$ then $G$ is a dual group for ${^\vee G}$).  We emphasize
that there is no such symmetry between the notions of weak extended
groups and weak E-groups.  An E-group is always a complex algebraic group,
but an extended group never is (because of condition (a2) in Definition 2.13).

Here is the classification of weak E-groups.
\medskip
{\bf Proposition 4.4.}  {\it Suppose $G$ is a connected reductive complex
algebraic group. 
\smallskip
\item{a)} Fix a weak E-group ${^\vee G^\Gamma}$ for $G$.  Let $\theta_Z$ be
the holomorphic involution of $Z({^\vee G})$ defined by the conjugation
action of any element of ${^\vee G^\Gamma}-{^\vee G}$.  We can attach to
${^\vee G^\Gamma}$ two invariants.  The first of these is an involutive
automorphism
$$a\in {\rom Aut}(\Psi_0(G))$$
of the based root datum of $G$.  The second is a class 
$$\overline z\in Z({^\vee G})^{\theta_Z}/(1 + \theta_Z) Z({^\vee G}).$$
\item{b)} Suppose ${^\vee G^\Gamma}$ and $\left({^\vee G^\Gamma}\right)'$ 
are weak E-groups for $G$ with the same invariants $(a,\overline z)$.  Then
any one of the canonical isomorphisms from ${^\vee G}$ to ${^\vee G}'$
described after Definition 4.2 extends to an isomorphism from ${^\vee
G^\Gamma}$ to $\left({^\vee G^\Gamma}\right)'$.
\item{c)} Suppose $a\in {\rom Aut}(\Psi_0(G))$ is an involutive
automorphism, and ${^\vee G}$ is a dual group of $G$.  Write $\theta_Z$ for
the involution of $Z({^\vee G})$ defined by the action of any automorphism
$\sigma$ corresponding to $a$ (Proposition 2.11 and (4.1)); and suppose
$$\overline z\in Z({^\vee G})^{\theta_Z}/(1 + \theta_Z)Z({^\vee G}).$$
Then there is a weak E-group ${^\vee G^\Gamma}$ with invariants
$(a,\overline z)$.}
\medskip

\noindent (In order to make sense of the statement in $(b)$, note that by the
remarks after Definition 4.2, the centers of any two dual groups of $G$ are
canonically isomorphic.) 

The proof is a slightly simpler version of the proof of Corollary 2.16, so we
omit it.  We should record the construction of $\overline z$, however.  Let
$^\vee\delta$ be any element of ${^\vee G^\Gamma} - {^\vee G}$ such that
the conjugation action of $^\vee\delta$ on ${^\vee G}$ is a distinguished
automorphism (as defined after Proposition 2.11).  (Such elements
necessarily exist.)   Then
$${^\vee\delta}^2 = z \in Z({^\vee G})\eqno(4.5)$$
is a representative of $\overline z$.

\medskip
{\bf Definition 4.6.}  Suppose $G$ is a complex connected reductive algebraic
group.  An {\it E-group for $G$} is a pair $({^\vee G^\Gamma},{\cal D})$,
subject to the following conditions.
\smallskip
\item{(a)} ${^\vee G^\Gamma}$ is a weak E-group for $G$ (Definition 4.3).
\item{(b)}  ${\cal D}$ is a ${^\vee G}$-conjugacy class of elements of finite
order in ${^\vee G^\Gamma} - {^\vee G}$.
\item{(c)}  Suppose $^\vee\delta\in {\cal D}$.  Then conjugation by
$^\vee\delta$ is a distinguished involutive automorphism of ${^\vee G}$ (see
the definition after Proposition 2.11).
\smallskip
The {\it invariants} of the E-group are the automorphism $a$ attached to
${^\vee G^\Gamma}$ by Proposition 4.4, and the element
$$z = {^\vee\delta}^2 \in Z({^\vee G})^{\theta_Z};$$
here ${^\vee\delta}$ is any element of ${\cal D}$.

An {\it L-group for G} is an E-group whose second invariant is equal to 1. 
That is, we replace condition (b) above by
\smallskip
\item{(b)$'$} ${\cal D}$ is a ${^\vee G}$-conjugacy class of elements of
order two in ${^\vee G^\Gamma} - {^\vee G}$.

Of course the notion of L-group is due to Langlands (cf.~[Langlands], [Borel]). 
Just as in Definition 4.3, we can speak of E-groups or L-groups attached to
an extended group, to an inner class of real forms, to a single real form, or
to nothing at all.
\medskip
Here is the classification of E-groups.
\medskip
{\bf Proposition 4.7.}  {\it Suppose $G$ is a connected reductive complex
algebraic group. 
\item{a)} Suppose $({^\vee G^\Gamma},{\cal D})$ and $\left(\left({^\vee
G^\Gamma}\right)',{\cal D}'\right)$  are E-groups for $G$ with the same
invariants $(a,z)$.  Then any one of the canonical isomorphisms from ${^\vee
G}$ to ${^\vee G}'$ described after Definition 4.2 extends to an isomorphism
from ${^\vee G^\Gamma}$ to $\left({^\vee G^\Gamma}\right)'$ carrying
${\cal D}$ to ${\cal D}'$.
\item{b)}  In the setting of (a), suppose $\tau_1$ and  $\tau_2$ are
isomorphisms from ${^\vee G^\Gamma}$ to $\left({^\vee G^\Gamma}\right)'$
with the property that $\tau_i\mid_{^\vee G}$ is one of the canonical
isomorphisms described after Definition 4.2, and that $\tau_i({\cal D}) =
{\cal D}'$.  Then there is an element $g\in {^\vee G}$ such that $\tau_1 =
\tau_2\circ{\rom Ad}(g)$.
\item{c)} Fix a weak E-group ${^\vee G^\Gamma}$ for $G$ with invariants
$(a,\overline z)$.  If $({^\vee G^\Gamma},{\cal D})$ is an E-group, then its
second invariant (Definition 4.6) is a representative for the class of
$\overline z$.  Conversely, if $z \in Z({^\vee G})^{\theta_Z}$ is an element of
finite order representing the class of $\overline z$, then there is an E-group
structure on ${^\vee G^\Gamma}$ with second invariant $z$.
\medskip}
\noindent The argument follows that given for Proposition 3.6, and we leave
it to the reader.
\medskip
{\bf Corollary 4.8.}  {\it Suppose $G$ is a connected complex reductive
algebraic group endowed with an inner class of real forms.  Then there is an
L-group for $G$ and this class of real forms (Definition 4.6).  Any two such
L-groups $({^\vee G^\Gamma},{\cal D})$ and $\left(({^\vee G^\Gamma})',{\cal
D}'\right)$ are isomorphic.  We may choose this isomorphism $\tau$ so that
$\tau({\cal D}) = {\cal D}'$, and the restriction of $\tau$ to ${^\vee G}$ is
one of the canonical isomorphisms described after Definition 4.2.  This
choice of $\tau$ is unique up to composition with an inner automorphism
from ${^\vee G}$.}
\medskip
In the course of the proof of Theorem 1.18 (see Definition 13.7 below) it
will be convenient to reformulate the definition of E-group.  We need a
preliminary definition, which will be applied in a moment to a dual group.

\medskip
{\bf Definition 4.9.}  Suppose $G$ is a complex connected reductive algebraic
group, $B$ is a Borel subgroup, and $T\subset B$ is a maximal torus.  Write
$$\chi(2{^\vee \rho}):{\Bbb C}^\times \rightarrow T\eqno(4.9)(a)$$
for the sum of the positive coroots, an element of $X_*(T)$.  Set
$$m({^\vee \rho}) = \chi(2{^\vee \rho})(i)\in T,\quad z({^\vee \rho}) =
m({^\vee \rho})^2 = \chi(2{^\vee \rho})(-1).\eqno(4.9)(b)$$
The familiar fact that the sum of the positive coroots takes the value 2 on
each simple root $\alpha$ means that
$${\rom Ad}(m({^\vee \rho}))(X_\alpha) = i^2X_\alpha = -X_\alpha,\quad {\rom
Ad}(z({^\vee \rho}))(X_\alpha) = X_\alpha\eqno(4.9)(c)$$
for any element $X_\alpha$ of the $\alpha$ root space.  In particular,
$z({^\vee \rho})\in Z(G)$; this element is independent of the choice of $B$
and $T$, so it is preserved by all automorphisms of $G$.

Suppose $\theta$ is an involutive (holomorphic) automorphism of $G$
preserving a Borel subgroup $B$.  Fix a $\theta$-stable maximal torus
$T\subset B$ (as is possible).  We say that the pair $(\theta,B)$ is {\it
large} if $\theta$ acts by $-1$ on each $\theta$-stable simple root space. 
We say that $(\theta,B)$ is {\it distinguished} if $\theta$ acts by $+1$ on
each $\theta$-stable simple root space.  (Since $T$ is unique up to
conjugation by the $\theta$-invariants in $B$, these definitions do not
depend on the choice of $T$.)
\medskip
{\bf Lemma 4.10.}  {\it Suppose $G$ is a complex connected reductive
algebraic group, $B\subset G$ is a Borel subgroup, and $\theta$ is an
involutive holomorphic automorphism of $G$ preserving $B$.  Fix a
$\theta$-stable maximal torus $T\subset B$.
\item{a)} The automorphism $\theta$ fixes the map $\chi(2{^\vee \rho})$ of
Definition 4.9.  In particular, it fixes the element $m({^\vee \rho})$; so
$$\theta' = {\rom Ad}(m({^\vee \rho}))\circ\theta$$
\item{}is another involutive automorphism of $G$ preserving $B$.
\item{b)}  The pair $(\theta,B)$ is large (respectively distinguished) if and
only if $(\theta',B)$ is distinguished (respectively large).
\item{c)}  The pair $(\theta,B)$ is distinguished if and only if $\theta$ is a
distinguished automorphism (as defined after Proposition 2.11).
\medskip}
{\it Proof.}  Parts (a) and (b) are immediate from Definition 4.9 (particularly
(4.9)(c)).  The ``if'' in part (c) is also immediate from Definition 4.9.  For the
``only if,'' suppose $(\theta,B)$ is distinguished in the sense of Definition
4.9, and $T$ is a $\theta$-stable maximal torus in $B$.  To see that $\theta$
is distinguished in the sense of section 2, we must find a set of simple root
vectors $\{X_\alpha\}$ permuted by $\theta$.  Begin with any set
$\{X'_\alpha\}$ of simple root vectors.  If $\alpha$ is fixed by $\theta$,
then the assumption that $(\theta,B)$ is distinguished means that $\theta
X'_\alpha = X'_\alpha$.  For such roots $\alpha$ we define $X_\alpha =
X'_\alpha$.  The remaining simple roots occur in pairs interchanged by
$\theta$; by choosing one representative of each pair, we can list them as
$\{\beta_1,\ldots,\beta_r,\theta\beta_1,\ldots,\theta\beta_r\}$.  We now
define
$$X_{\beta_i} = X'_{\beta_i},\qquad  X_{\theta\beta_i} = \theta
X'_{\beta_i}.$$
Then $\theta \in {\rom Aut}(G,B,T,\{X_\alpha\})$, as we wished to show. 
Q.E.D.
\medskip
{\bf Lemma 4.11.}  {\it Suppose $G$ is a complex connected reductive
algebraic group, and $\theta$ is a distinguished involutive holomorphic
automorphism of $G$.  Then any two $\theta$-stable Borel subgroups of $G$
are conjugate by the invariants of $\theta$ in $G$.}
\medskip
{\it Proof.}  The proof requires a few ideas from the theory of Cartan
involutions, and we only sketch it.  Write $K$ for the group of fixed points
of $\theta$ in $G$, and $K_0$ for its identity component.  Suppose $B$ and
$B'$ are $\theta$-stable.  Choose $\theta$-stable tori $T$ and $T'$ in $B$
and $B'$; then $T\cap K_0$ and $T'\cap K_0$ are maximal tori in  $K_0$,
contained in Borel subgroups $B\cap K_0$ and $B'\cap K_0$.   By conjugacy
of maximal tori and Borel subgroups in the reductive group $K_0$, we may
assume (after replacing $B'$ by a conjugate under $K_0$) that they coincide:
$$T\cap K_0 = T'\cap K_0,\qquad B\cap K_0 = B'\cap K_0.$$
Now $T\cap K_0$ contains regular elements of $G$ (for example in the
image of the map $\chi(2{^\vee \rho})$), so the identity component of its
centralizer is exactly $T$.  It follows that $T=T'$.  

Up until now we have used only the assumption that $\theta$ is an
involutive automorphism.  A root of $T$ in $G$ is $B$-simple if and only if
its restriction to $T\cap K_0$ is simple for $B\cap K_0$; this is a
consequence of the fact that $(\theta,B)$ is distinguished (Definition 4.9
and Lemma 4.10(c)); and similarly for $B'$.  Using this fact, we see that the
simple roots of $T$ in $B$ and in $B'$ coincide, so $B=B'$.  Q.E.D.

Using these two lemmas, we can formulate a first approximation to our new
definition of E-groups.

\medskip
{\bf Definition 4.12.} Suppose $G$ is a complex connected reductive
algebraic group.  An {\it E-group for $G$} is a pair $({^\vee G^\Gamma},{\cal
E})$, subject to the following conditions.
\smallskip
\item{(a)} ${^\vee G^\Gamma}$ is a weak extended group for $G$ (Definition
4.3).
\item{(b)}  ${\cal E}$ is a ${^\vee G}$-conjugacy class of pairs
$({^\vee\delta},{^dB})$, with ${^\vee\delta}$ an element of finite order in
${^\vee G^\Gamma} - {^\vee G}$, and ${^dB}$ a Borel subgroup of ${^\vee G}$.
\item{(c)}  Suppose $({^\vee\delta},{^dB})\in {\cal E}$.  Then conjugation by
$^\vee\delta$ is a distinguished involutive automorphism of ${^\vee G}$ (see
the definition after Proposition 2.11) preserving ${^dB}$.
\medskip
To see that Definition 4.12 is equivalent to Definition 4.6, suppose first
that $({^\vee G^\Gamma},{\cal E})$ is an E-group in the sense of Definition
4.12.  Set
$${\cal D} = \{\,{^\vee\delta}\mid ({^\vee\delta},{^dB})\in {\cal E}\,\}.
\eqno(4.13)(a)$$
Then $({^\vee G^\Gamma},{\cal D})$ is an E-group in the sense of Definition
4.6.  Conversely, suppose $({^\vee G^\Gamma},{\cal D})$ is an E-group in the
sense of Definition 4.6.  Set
$${\cal E} = \{\,({^\vee\delta},{^dB})\mid {^\vee\delta}\in {\cal D}, \hbox{
and ${^dB}$ is a ${^\vee\delta}$-invariant Borel subgroup of ${^\vee G}$}\,\}.
\eqno(4.13)(b)$$
Lemma 4.11 implies that ${\cal E}$ is a single ${^\vee G}$-conjugacy class,
so $({^\vee G^\Gamma},{\cal E})$ is an E-group in the sense of Definition
4.12.

It is now a simple matter to pass to our final definition of E-groups.
\medskip
{\bf Definition 4.14} ([AV2], Definition 10.5).  Suppose $G$ is a complex
connected reductive algebraic group.  An {\it E-group for $G$} is a pair
$({^\vee G^\Gamma},{\cal S})$, subject to the following conditions.
\smallskip
\item{(a)} ${^\vee G^\Gamma}$ is a weak extended group for $G$ (Definition
4.3).
\item{(b)}  ${\cal S}$ is a ${^\vee G}$-conjugacy class of pairs
$({^\vee\delta},{^dB})$, with ${^\vee\delta}$ an element of finite order in
${^\vee G^\Gamma} - {^\vee G}$, and ${^dB}$ a Borel subgroup of ${^\vee G}$.
\item{(c)}  Suppose $({^\vee\delta},{^dB})\in {\cal S}$.  Then conjugation by
$^\vee\delta$ defines an involutive automorphism $\theta$ of ${^\vee G}$ 
preserving ${^dB}$.  The pair $(\theta,{^dB})$ is large (Definition 4.9).
\smallskip
The {\it invariants} of the E-group are the automorphism $a$ attached to
${^\vee G^\Gamma}$ by Proposition 4.5, and the element
$$z = z(\rho){^\vee\delta}^2 \in Z({^\vee G})^{\theta_Z};$$
here $({^\vee\delta},{^dB})$ is any element of ${\cal S}$, and $z(\rho)\in
Z({^\vee G})$ is the distinguished element described in Definition 4.9.
 
An {\it L-group for G} is an E-group whose second invariant is equal to 1. 
That is, the elements $^\vee\delta$ appearing in (b) are required to satisfy
${^\vee\delta}^2 = z(\rho).$
\medskip
We recall from section 9 of [AV2] the proof that Definitions 4.12 and 4.14
are equivalent.  Suppose that $({^\vee G^\Gamma},{\cal E})$ is an E-group in
the sense of Definition 4.12.  Fix $({^\vee\delta},{^dB})\in {\cal E}$, and
write $\theta$ for the automorphism of ${^\vee G}$ defined by conjugation
by ${^\vee\delta}$.  Choose a $\theta$-stable maximal torus ${^dT}\subset
{^dB}$, and define $m(\rho)\in {^dT}$ as in Definition 4.9.   Set
$${^\vee\delta}' = m(\rho)^{-1}{^\vee\delta};\eqno(4.15)(a)$$
the corresponding automorphism of ${^\vee G}$ is
$$\theta' = {\rom Ad}(m(\rho))\theta.\eqno(4.15)(b)$$
Changing the choice of ${^dT}$ changes  ${^\vee\delta}'$ only by conjugation
by ${^dB}$; so the conjugacy class
$${\cal S} = {\rom Ad}({^\vee G})({^\vee\delta}',{^dB})\eqno(4.15)(c)$$
is well-defined.  By hypothesis, the pair $(\theta,{^dB})$ is distinguished, so
by (4.15)(b) and Lemma 4.10 the pair $(\theta',{^dB})$ is large.  Consequently
$({^\vee G^\Gamma},{\cal S})$ is an E-group in the sense of Definition 4.14. 
This argument may be reversed without difficulty, proving the equivalence
of the two definitions.
%\end

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\vskip  3pc
\centerline{\bf 5. Langlands parameters and L-homomorphisms.}
\vskip 1pc

The (local) goal of Langlands' theory of L-groups is the description of
representations of real forms of $G$ in terms of an L-group of $G$.  In this
section we begin our detailed analysis of the parameters that will appear in
this description, without as yet explaining how they are related to
representation theory.  As is explained in [AV2], E-groups play the same
r\^ole with respect to the description of certain projective representations
(corresponding always to linear covering groups) and they can be treated at
the same time without difficulty.  (Indeed it is apparent from Definitions
4.3 and 4.6 that E-groups are in certain respects {\it less} complicated than
L-groups.)  In any case we will need to have E-groups available when we
discuss endoscopy.

\medskip
{\bf Definition 5.1} ([Langlands], section 2).  Suppose ${^\vee H^\Gamma}$
and ${^\vee G^\Gamma}$ are weak E-groups (Definition 4.3).  An {\it
L-homomorphism from ${^\vee H^\Gamma}$ to ${^\vee G^\Gamma}$} is a
morphism $\epsilon:{^\vee H^\Gamma}\rightarrow{^\vee G^\Gamma}$ of
algebraic groups, with the property that the diagram
$$ \matrix {^\vee H^\Gamma}&\buildrel \epsilon\over\longrightarrow
&{^\vee G^\Gamma}\cr\quad \searrow &{}&\swarrow\quad\cr
 {} &\Gamma  &{}\cr\endmatrix$$
(cf. Definition 4.3) commutes.  (Even when ${^\vee H^\Gamma}$ and ${^\vee
G^\Gamma}$ are E-groups, we do {\it not} require that $\epsilon$ should
respect the distinguished conjugacy classes ${\cal D}$, ${\cal E}$, or ${\cal
S}$ of Definitions 4.6, 4.12, and 4.14.)  Two such morphisms $\epsilon$ and
$\epsilon'$ are said to be {\it equivalent} if they are conjugate by the action
of ${^\vee G}$; that is, if there is an element $g\in {^\vee G}$ so that
$\epsilon' = {\rom Ad}(g)\circ\epsilon$.
\medskip

A central feature of Langlands' (local) philosophy is that an equivalence
class of L-homomorphisms should give rise to something like a map
(``transfer'') from representations of (real forms of) $H$ to representations
of $G$.  In particular, any object we construct from an E-group that is
intended to correspond to representations of $G$ ought to be covariant with
respect to L-homomorphisms.  The first object of this kind that we will
consider is the set of ``Langlands parameters'' of Definition 1.5.  To describe
these more carefully, we begin by recalling the definition of the Weil group
for ${\Bbb R}$.

\medskip
{\bf Definition 5.2} (see [Langlands], [Borel], or [Tate]).  The {\it Weil group
of ${\Bbb R}$} is the real Lie group $W_{\Bbb R}$ generated by ${\Bbb
C}^\times$ and a distinguished element $j$, subject to the relations
$$j^2 = -1 \in {\Bbb C}^\times,\quad jzj^{-1} = \overline{z}.\eqno(5.2)(*)$$
We define a homomorphism from $W_{\Bbb R}$ to the Galois group
$\Gamma$ of ${\Bbb C}/{\Bbb R}$ by sending ${\Bbb C}^\times$ to the
identity and $j$ to complex conjugation.  (This gives the exact sequence 
$$1\rightarrow {\Bbb C}^\times \rightarrow W_{\Bbb R} \rightarrow
\Gamma\rightarrow 1$$
of (1.4)(b).)  Again we emphasize that the Weil group is {\it not} a complex
Lie group.

Suppose ${^\vee G^\Gamma}$ is a weak E-group.  A {\it quasiadmissible
homomorphism} (or {\it Langlands parameter}) $\phi$ from $W_{\Bbb R}$ to
${^\vee G^\Gamma}$ is a continuous group homomorphism satisfying
\smallskip
\item{(a)} the diagram
$$ \matrix W_{\Bbb R}&\buildrel \phi\over\longrightarrow &{^\vee
G^\Gamma}\cr\quad \searrow &{}&\swarrow\quad\cr
 {} &\Gamma  &{}\cr\endmatrix$$ 
is commutative; and
\item{(b)} $\phi({\Bbb C}^\times)$ consists of semisimple elements of
${^\vee G}$.
\smallskip
Two such homomorphisms are said to be {\it equivalent} if they are
conjugate by the action of ${^\vee G}$.  The set of Langlands parameters for
${^\vee G^\Gamma}$ is written
$$P({^\vee G^\Gamma})=\{\,\phi:W_{\Bbb R}\rightarrow {^\vee
G^\Gamma}\mid \phi \hbox{ is quasiadmissible}\,\}.$$
We make ${^\vee G}$ act on $P({^\vee G^\Gamma})$ by conjugation on the
range of a homomorphism.  The set of equivalence classes of Langlands
parameters --- that is, the set of orbits of ${^\vee G}$ on $P({^\vee
G^\Gamma})$ --- is written
$$\Phi({^\vee G^\Gamma}) = \{\,{^\vee G} \hbox{ orbits on } P({^\vee
G^\Gamma})\,\}.$$
\medskip
{\bf Definition 5.3} (see [Langlands], [Borel]).  Suppose $G$ is a connected
complex reductive algebraic group endowed with an inner class of real
forms.  If $({^\vee G^\Gamma},{\cal D})$ is any L-group for $G$ (Definition
4.6), then we write
$$\Phi(G/{\Bbb R}) = \Phi({^\vee G^\Gamma}).$$
The omission of ${^\vee G^\Gamma}$ from the notation is justified by
Corollary 4.8: if $\left(({^\vee G^\Gamma})',{\cal D}'\right)$ is any other
L-group for the same inner class of real forms, then the corollary provides a
{\it canonical} bijection from $\Phi({^\vee G^\Gamma})$ to $\Phi(({^\vee
G^\Gamma})')$.  A little more generally, if $({^\vee G^\Gamma},{\cal D})$ is
an E-group with second invariant $z$ (Definition 4.6), then we write
$$\Phi^z(G/{\Bbb R}) = \Phi({^\vee G^\Gamma}).$$
\medskip
The set of Langlands parameters behaves well under L-homomorphisms.
\medskip
{\bf Proposition 5.4.}  {\it Suppose ${^\vee H^\Gamma}$ and ${^\vee
G^\Gamma}$ are weak E-groups, and $\epsilon:{^\vee H^\Gamma}\rightarrow
{^\vee G^\Gamma}$ is an L-ho\-mo\-mor\-phism.  Then composition with
$\epsilon$ defines a map
$$P(\epsilon):P({^\vee H^\Gamma})\rightarrow P({^\vee G^\Gamma})$$
on Langlands parameters, which descends to a map
$$\Phi(\epsilon):\Phi({^\vee H^\Gamma})\rightarrow \Phi({^\vee
G^\Gamma})$$
on equivalence classes.}
\medskip
This is obvious.  We begin now the proof of Proposition 1.6.
\medskip
{\bf Lemma 5.5}  {\it Suppose $H$ is a complex Lie group, with Lie algebra
${\frak h}$.  Then the set of continuous homomorphisms $\phi$ from ${\Bbb
C}^\times$ into $H$ is in one-to-one correspondence with the set of pairs
$(\lambda,\mu)\in {\frak h}\times{\frak h}$, subject to the following
conditions:
\smallskip
\item{a)} $[\lambda,\mu] = 0$; and
\item{b)} ${\rom exp}(2\pi i(\lambda - \mu)) = 1$.
\medskip}
We omit the elementary proof; essentially the same result may be found in
Lemma 2.8 of [Langlands], or Proposition 2.10 of [AV2].  We will need a
formula for $\phi$, however; it is
$$\phi(e^t) = {\rom exp}(t\lambda + \overline{t}\mu)\qquad(t\in {\Bbb
C}).\eqno(5.5)(a)$$
This is usually written more succinctly (if perhaps a little less clearly) as
$$\phi(z) = z^\lambda\overline{z}^\mu\qquad(z\in {\Bbb
C}^\times).\eqno(5.5)(b)$$

We restate Proposition 1.6 in our present slightly more general setting.
\medskip
{\bf Proposition 5.6.} {\it Suppose ${^\vee G^\Gamma}$ is a weak E-group
(Definition 4.3).  The set $P({^\vee G^\Gamma})$ of quasiadmissible
homomorphisms from $W_{\Bbb R}$ into ${^\vee G^\Gamma}$ (Definition 5.2)
may be identified with the set of pairs $(y,\lambda)$ satisfying the
following conditions:
\item{a)} $y \in {^\vee G^\Gamma} - {^\vee G}$, and $\lambda\in {^\vee{\frak
g}}$ is a semisimple element;
\item{b)} $y^2 = {\rom exp}(2\pi i \lambda)$; and
\item{c)} $[\lambda, {\rom Ad}(y)\lambda] = 0$.
\medskip}
{\it Proof.}  Suppose $\phi$ is a quasiadmissible homomorphism.  By Lemma
5.4, the restriction of $\phi$ to ${\Bbb C}^\times$ determines two
commuting elements $\lambda$ and $\mu$ of $^\vee{\frak g}$ such that
$$\phi(z) = z^\lambda\overline{z}^\mu\qquad(z\in {\Bbb
C}^\times).\eqno(5.7)(a)$$
By Definition 5.2(b), the elements $\lambda$ and $\mu$ are semisimple. 
Since $j$ acts on ${\Bbb C}^\times$ by complex conjugation, we have
$${\rom Ad}(\phi(j))(\lambda) = \mu.\eqno(5.7)(b)$$
Define
$$y = {\rom exp}(\pi i\lambda)\phi(j)\in {^\vee G^\Gamma}-{^\vee
G};\eqno(5.7)(c)$$
the last inclusion is a consequence of Definition 5.2(a).  This gives (a) of the
Proposition.  Since ${\rom Ad}({\rom exp}(\pi i\lambda))$ fixes $\lambda$, we
have
$${\rom Ad}(y)(\lambda) = \mu.\eqno(5.7)(d)$$
Since $\lambda$ and $\mu$ commute, this gives (c) of the proposition.  Now
$$\eqalign{y^2 &= {\rom exp}(\pi i\lambda)\phi(j){\rom exp}(\pi
i\lambda)\phi(j)\cr
&={\rom exp}(\pi i\lambda)\phi(j){\rom exp}(\pi
i\lambda)\phi(j)^{-1}\phi(j^2)\cr
&={\rom exp}(\pi i\lambda){\rom exp}(\pi i\mu)\phi(-1)\qquad\qquad\hbox{(by
(5.7)(b) and (5.2)($*$))}\cr
&={\rom exp}(\pi i(\lambda+\mu)){\rom exp}(\pi i(\lambda-\mu))
\qquad\hbox{(by (5.5)(a))}\cr
&={\rom exp}(2\pi i\lambda),\cr}$$
as required by (b) of the proposition.

Conversely, suppose we are given $y$ and $\lambda$ satisfying (a)--(c) of
the proposition.  Define $\mu$ by (5.7)(d); then $\lambda$ and $\mu$ are
commuting semisimple elements of $^\vee{\frak g}$.  By (b) of the
proposition,
$$ {\rom exp}(2\pi i\lambda) = {\rom exp}(2\pi i{\rom Ad}(y)(\lambda)).$$
So $\lambda$ and $\mu$ satisfy the conditions of Lemma 5.5, and we can
define a homomorphism $\phi$ from ${\Bbb C}^\times$ into ${^\vee G}$ by
(5.7)(a).  We extend this to $W_{\Bbb R}$ by setting
$$\phi(j) = {\rom exp}(-\pi i\lambda)y.\eqno(5.7)(e)$$
That the relations (5.2)($*$) are preserved, and that $\phi$ is a
quasiadmissible homomorphism, can be proved by reversing the arguments
given for the first half of the proposition.  We leave the details to the
reader.  Q.E.D.

When $y$, $\lambda$, and $\phi$ are related as in Proposition 5.6 --- that
is, when (5.7)(a)--(e) hold --- we write
$$\phi = \phi(y,\lambda),\quad y=y(\phi),\quad \lambda =
\lambda(\phi)\eqno(5.8)(a)$$
It will also be convenient to write (following (1.7)(c))
$$e(\lambda)={\rom exp}(2\pi i\lambda)\in H\eqno(5.8)(b)$$
whenever $H$ is a complex Lie group and $\lambda\in {\frak h}$.
\medskip
{\bf Corollary 5.9.}  {\it Suppose ${^\vee G^\Gamma}$ is a weak E-group,
$\phi\in P({^\vee G^\Gamma})$ is a Langlands parameter, and $y,\lambda$
are as in Proposition 5.6.  Define
$$\eqalign{{^\vee G}(\lambda) &= \hbox{ centralizer in ${^\vee G}$ of
$e(\lambda)$}\cr
L(\lambda) &= \{\,g\in {^\vee G}\mid {\rom Ad}(g)\lambda = \lambda\,\}\cr
L(\lambda,y\cdot\lambda) &= \{\,g\in {^\vee G}\mid {\rom Ad}(g)\lambda =
\lambda, {\rom Ad}(g)(y\cdot \lambda) = y\cdot\lambda\,\}\cr
{^\vee G}_\phi &= \hbox{ centralizer in ${^\vee G}$ of $\phi(W_{\Bbb R})$}\cr
K(y) &= \hbox{ centralizer in ${^\vee G}$ of $y$.}\cr}$$

\smallskip
\item{a)}  We have
$${^\vee G}_\phi\subset L(\lambda,y\cdot\lambda)\subset
L(\lambda)\subset {^\vee G}(\lambda).$$
All four groups are complex reductive algebraic subgroups of ${^\vee G}$;
the middle two are Levi subgroups of parabolics (and therefore connected).
\item{b)}  Conjugation by $y$ defines an involutive automorphism
${\theta}_y$ of ${^\vee G}(\lambda)$ preserving
$L(\lambda,y\cdot\lambda)$.
\item{c)}  The group of fixed points of ${\theta}_y$ on ${^\vee G}(\lambda)$
is $K(y)$.  We have
$${^\vee G}_\phi=K(y)\cap L(\lambda,y\cdot\lambda) = K(y) \cap
L(\lambda).$$
In particular, 
\item{d)} ${^\vee G}_\phi$ may be described as the set of fixed points of an
involutive automorphism of the connected reductive (Levi) subgroup
$L(\lambda,y\cdot\lambda)$ of ${^\vee G}$.
\medskip}
This is immediate from Proposition 5.6.

Recall now from Definition 1.16 the algebraic universal cover ${^\vee
G}^{alg}$ of ${^\vee G}$.  We write
$$1\rightarrow \pi_1({^\vee G})^{alg}\rightarrow {^\vee G}^{alg}\rightarrow
{^\vee G} \rightarrow 1.\eqno(5.10)(a)$$
(Recall that $\pi_1({^\vee G})^{alg}$ is the projective limit of the finite
quotient groups of $\pi_1({^\vee G})$.)  If $H$ is any subgroup of ${^\vee G}$,
then we write $H^{alg}$ for its inverse image in ${^\vee G}^{alg}$.  (When we
consider $H$ as a subgroup of several different weak E-groups, this notation
is ambiguous; we will write instead $H^{alg,{^\vee G}}$.)  This gives an exact
sequence
$$1\rightarrow \pi_1({^\vee G})^{alg}\rightarrow H^{alg}\rightarrow H
\rightarrow 1.\eqno(5.10)(b)$$

We should offer here some justification for the introduction of these
covering groups.  One way to eliminate them is to change the definition of
strong real form (Definition 2.13) to require $\delta^2 = 1$.  Then many of
our principal results (beginning with Theorem 1.18) hold with no covering
groups at all.  The problem with this approach is that not every real form is
represented by a strong real form with $\delta^2 =1$.  (For example, the real
form $SU(2)$ of $SL(2)$ is eliminated.)   For this reason we believe that the
use of some covering groups is unavoidable.  It is still of interest to
understand exactly which real forms require which coverings of ${^\vee G}$,
and we will investigate this carefully in section 10 (Theorem 10.11).

We can now introduce the parameters for the complete Langlands
classification (Theorem 1.18).
\medskip
{\bf Definition 5.11.}  Suppose ${^\vee G^\Gamma}$ is a weak E-group, and
$\phi\in P({^\vee G^\Gamma})$.  Use the notation of Corollary 5.9.  Define
$$A^{loc}_\phi = {^\vee G}_\phi/\left({^\vee G}_\phi\right)_0,$$
the {\it Langlands component group for $\phi$}.  Similarly, set
$$A^{loc,alg}_\phi = {^\vee G}^{alg}_\phi/\left({^\vee
G}^{alg}_\phi\right)_0,$$
the {\it universal component group for $\phi$}.  (We will occasionally need
to write $A^{loc,alg}_{\phi,{^\vee G}}$ to avoid ambiguity.  When ${^\vee
G^\Gamma}$ is an E-group for $G$, we simplify this to
$A^{loc,alg}_{\phi,G}$.)  It is a consequence of Corollary  5.9(d) that
$A^{loc}_\phi$ is a (finite) product of copies of ${\Bbb Z}/2{\Bbb Z}$, and
that $A^{loc,alg}_\phi$ is abelian.  This is not entirely obvious; the
argument is given at (12.11)(e) below. By (5.10)(b), there is a right exact
sequence
$$\pi_1({^\vee G})^{alg}\rightarrow A^{loc,alg}_\phi\rightarrow A^{loc}_\phi
\rightarrow 1.$$
(The kernel of the first map is the intersection of $\pi_1({^\vee G})^{alg}$
with the identity component of ${^\vee G}^{alg}_\phi$.)  A {\it complete
Langlands parameter for ${^\vee G^\Gamma}$} is a pair $(\phi,\tau)$, with
$\tau$ an irreducible representation of $A^{loc,alg}_\phi$.  Two such
parameters are called equivalent if they are conjugate under the obvious
action of ${^\vee G}^{alg}$.  We write $\Xi({^\vee G^\Gamma})$ for the set of
equivalence classes of complete Langlands parameters.  As in Definition 5.3,
the case of L-groups merits special attention: if $G$ is a connected complex
reductive group endowed with an inner class of real forms, we write
$$\Xi(G/{\Bbb R}) = \{\hbox{ equivalence classes of complete Langlands
parameters for the L-group of $G$ }\}.$$
(This is not the definition that was given in the introduction, but we will
see in section 7 that it is equivalent.)  More generally, if $({^\vee
G^\Gamma},{\cal D})$ is an E-group for $G^\Gamma$ with second invariant
$z$ (Definition 4.6) then we write $\Xi^z(G/{\Bbb R})$ for $\Xi({^\vee
G^\Gamma})$.  If $(({^\vee G^\Gamma})',{\cal D}')$ is another E-group with
second invariant $z$, then Proposition 4.7(b) provides a canonical bijection
from $\Xi({^\vee G^\Gamma})$ onto $\Xi(({^\vee G^\Gamma})')$.  This
justifies the omission of the E-group from the notation.
\medskip
Although we will make no use of it, the following proposition may provide a
little motivation for this definition.
\medskip
{\bf Proposition 5.12.} {\it Suppose ${^\vee G}$ is a weak E-group.  Then
there is a natural one-to-one correspondence between equivalence classes
of complete Langlands parameters for ${^\vee G^\Gamma}$ (Definition 5.11),
and irreducible ${^\vee G}^{alg}$-equivariant local systems on ${^\vee
G}$-orbits on $P({^\vee G^\Gamma})$  (Definition 5.2).}
\medskip
This is immediate from the definitions (see also Lemma 7.3 below).

Particularly in connection with endoscopy, we will occasionally need to
consider a slightly different situation.  Suppose that $Q$ is any pro-finite
group, and that we have an extension of pro-algebraic groups
$$1\rightarrow Q \rightarrow {^\vee G}^Q \rightarrow {^\vee G} \rightarrow
1. \eqno(5.13)(a)$$
If $\epsilon:H\rightarrow {^\vee G}$ is any morphism of algebraic groups,
we can pull back the extension (5.13)(a) to 
$$ \matrix 1& \rightarrow &Q&\rightarrow &H^Q & \rightarrow & H &
\rightarrow& 1\cr 
&{}&{}\downarrow &{}&\downarrow{\scriptstyle\epsilon^Q} &{}
&\downarrow {\scriptstyle\epsilon} &{}&{}\cr 
1&\rightarrow &Q &\rightarrow &{^\vee G}^Q &\rightarrow &{^\vee G}
&\rightarrow &1\cr\endmatrix \eqno(5.13)(b)$$
(The case of (5.10) has $Q = \pi_1({^\vee G})^{alg}$, and $\epsilon$ the
inclusion of a subgroup.  In almost all the examples we consider, $Q$ will be
a quotient of $\pi_1({^\vee G})^{alg}$.)  If $\phi \in P({^\vee G^\Gamma})$,
we define the {\it $Q$-component group for $\phi$} by
$$A_\phi^{loc,Q} = {^\vee G}_\phi^Q/({^\vee G}_\phi^Q)_0. \eqno(5.13)(c)$$
If we need to emphasize the group, we write $A_{\phi,{^\vee G}}^{loc,Q}$.
There is a right exact sequence
$$Q \rightarrow A_\phi^{loc,Q} \rightarrow A_\phi^{loc} \rightarrow 1.
\eqno(5.13)(d)$$
A {\it complete Langlands parameter for ${^\vee G^\Gamma}$ of  type $Q$} is
a pair $(\phi,\tau)$, with $\phi$ a Langlands parameter and $\tau$ an
irreducible representation of $A_\phi^{loc,Q}$.  Two such parameters are
called equivalent if they are conjugate by the action of ${^\vee G}^Q$.  We
write $\Xi({^\vee G^\Gamma})^Q$ for the set of equivalence classes of
complete Langlands parameters of type $Q$.

Suppose now that we have an L-homomorphism
$$\epsilon:{^\vee H^\Gamma} \rightarrow {^\vee G^\Gamma},
\eqno(5.14)(a)$$
(Definition 5.1), and that we are given compatible pro-finite extensions
$$ \matrix 1& \rightarrow &Q_H&\rightarrow &{^\vee H}^{Q_H} &
\rightarrow & {^\vee H} & \rightarrow& 1\cr 
&{}&{}\downarrow &{}&\downarrow{\scriptstyle\epsilon^\bullet} &{}
&\downarrow {\scriptstyle\epsilon} &{}&{}\cr 
1&\rightarrow &Q &\rightarrow &{^\vee G}^Q &\rightarrow &{^\vee G}
&\rightarrow &1\cr\endmatrix \eqno(5.14)(b)$$
(In our examples, ${^\vee G}^Q$ will usually be the algebraic universal
covering of ${^\vee G}$, and ${^\vee H^\Gamma}$ will be (roughly) the
centralizer in ${^\vee G^\Gamma}$ of a semsimple element.  We will take
$Q_H=Q$, and ${^\vee H}^Q$ the induced extension (cf.~(5.13)(b)).  In this
case ${^\vee H}^Q$ will be a quotient of the algebraic universal cover of
${^\vee H}$.)  Suppose now that $\phi \in P({^\vee H^\Gamma})$, so that
$\epsilon\circ\phi \in P({^\vee G^\Gamma})$ (Proposition 5.4).  Then the map
$$\epsilon^\bullet:{^\vee H}^{Q_H} \rightarrow {^\vee G}^Q \eqno(5.14)(c)$$
carries the centralizer of the image of $\phi$ to the centralizer of the
image of $\epsilon\circ\phi$; so we get an induced homomorphism of
component groups
$$A^{loc}(\epsilon):A^{loc,Q_H}_{\phi,{^\vee H}} \rightarrow
A^{loc,Q}_{\epsilon\circ\phi,{^\vee G}}. \eqno(5.14)(d)$$ 
Notice that this map appears to go in the wrong direction from the point of
view of Langlands functoriality principle: a complete Langlands parameter
of type $Q_H$ for ${^\vee H^\Gamma}$ induces a Langlands parameter for
${^\vee G^\Gamma}$, but not a representation of the $Q$-component group. 
(The problem is unrelated to our extensions; it occurs even if $Q = Q_H =
\{1\}$.)  We will return to this point in section 26, using the ideas of
Langlands and Shelstad.
%\end
%\input mssymb.tex
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%\font\frak=cmr10
%\hsize=27pc
%\vsize=42pc
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%\nopagenumbers
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\vskip 3pc
\centerline{\bf 6. Geometric parameters.}
\vskip 1pc
We continue the analysis of the preceding section, turning now to the new
geometric parameters defined in the introduction.  A calculation illustrating
several of the technical difficulties is outlined in Example 6.22.  The reader
may wish to refer to it while reading this section.

Definition 1.7 of the introduction was made as succinctly as possible; we
repeat it here, including some useful auxiliary ideas.  Suppose $H$ is a
complex reductive group, with Lie algebra ${\frak h}$, and $\lambda\in
{\frak h}$ is a semisimple element.  Set
$${\frak h}(\lambda)_n = \{\,\mu\in {\frak h}\mid [\lambda,\mu] =
n\mu\,\}\quad(n \in {\Bbb Z})\eqno(6.1)(a)$$
$${\frak h}(\lambda) = \sum_{n\in {\Bbb Z}} {\frak
h}(\lambda)_n\eqno(6.1)(b)$$
$${\frak l}(\lambda) = {\frak h}(\lambda)_0 = \hbox{ centralizer of
$\lambda$ in ${\frak h}$ }\eqno(6.1)(c)$$
$${\frak n}(\lambda) = \sum_{n=1,2,\ldots} {\frak h}(\lambda)_n
\eqno(6.1)(d)$$
$${\frak p}(\lambda) = {\frak l}(\lambda) + {\frak
n}(\lambda)\eqno(6.1)(e)$$
Next, define
$$e(\lambda) = {\rom exp}(2\pi i\lambda)\in H.\eqno(6.2)(a)$$
$$H(\lambda) = \hbox{ centralizer in $H$ of $e(\lambda)$ }\eqno(6.2)(b)$$
$$L(\lambda) = \hbox{ centralizer in $H$ of $\lambda$ }\eqno(6.2)(c)$$  
$$N(\lambda) = \hbox{ connected unipotent subgroup with Lie algebra
${\frak n}(\lambda)$ }\eqno(6.2)(d)$$
$$P(\lambda) = L(\lambda)N(\lambda)\eqno(6.2)(e)$$
\medskip
{\bf Lemma 6.3.} {\it Suppose $H$ is a complex reductive algebraic group,
and $\lambda\in {\frak h}$ is a semisimple element.  Use the notation of
(6.1) and (6.2).
\smallskip
\item{a)} $H(\lambda)$ is a reductive subgroup of $H$ with Lie algebra
${\frak h}(\lambda)$.
\item{b)} $L(\lambda)$ is a reductive subgroup of $H(\lambda)$ with Lie
algebra ${\frak l}(\lambda)$.  It is connected if $H$ is.
\item{c)} $N(\lambda)$ is a connected unipotent subgroup of $H(\lambda)$
with Lie algebra ${\frak n}(\lambda)$.
\item{d)} $P(\lambda)$ is a parabolic subgroup of $H(\lambda)$ with Lie
algebra ${\frak p}(\lambda)$, and Levi decomposition
$P(\lambda)=L(\lambda)N(\lambda)$.  It is connected if $H$ is.
\item{e)} ${\rom Ad}(P(\lambda))\cdot \lambda = {\rom Ad}(N(\lambda))\cdot
\lambda = \lambda + {\frak n}(\lambda).$  More precisely, the adjoint action
of $N(\lambda)$ defines an isomorphism of varieties
$$N(\lambda) \simeq \lambda + {\frak n}(\lambda),\qquad n\mapsto {\rom
Ad}(n)\lambda.$$
\medskip}
This is elementary.  (Perhaps the most subtle part is (e).  This goes back at
least to [Harish-Chandra], Lemma 8; the hypotheses there are a little more
special, but the proof extends without change.  A convenient reference is
[Wallach], equation (4) on page 55.)  The affine space in (e) is just the
canonical flat of Definition 1.7:
$${\cal F}(\lambda) = {\rom Ad}(P(\lambda))\cdot \lambda = {\rom
Ad}(N(\lambda))\cdot \lambda = \lambda + {\frak n}(\lambda).\eqno(6.4)$$
\medskip
{\bf Proposition 6.5.}  {\it Suppose $H$ is a complex reductive algebraic
group, and $\lambda\in {\frak h}$ is a semisimple element.  Use the notation
of (6.1)--(6.4), and suppose $\lambda'\in {\cal F}(\lambda)$.  Then
\smallskip
\item{a)} $\lambda'$ is a semisimple element of ${\frak h}$, conjugate to
$\lambda$ by ${\rom Ad}(N(\lambda))$;
\item{b)}  $e(\lambda) = e(\lambda')$, $H(\lambda) = H(\lambda')$,
$N(\lambda) = N(\lambda')$, $P(\lambda) = P(\lambda')$, ${\cal F}(\lambda)
= {\cal F}(\lambda')$; and
\item{c)} the stabilizer in $H$ of the set ${\cal F}(\lambda)$ is
$P(\lambda)$.   More precisely, if ${\rom Ad}(h)\cdot{\cal F}(\lambda)$ has a
non-trivial intersection with ${\cal F}(\lambda)$, then $h\in P(\lambda)$.
\medskip}
{\it Proof.}  Part (a) is immediate from (6.4).  Write $\lambda' = {\rom
Ad}(n)\cdot \lambda$, for some $n\in N(\lambda)\subset H(\lambda)$. 
Obviously ${\rom Ad}(n)$ carries objects defined in terms of $\lambda$ to
those defined in terms of $\lambda'$.  Since $e(\lambda)$ is central in
$H(\lambda)$, the first claim of (b) follows.  For the next three, use the fact
that $N(\lambda)$ normalizes $H(\lambda)$, $N(\lambda)$, and
$P(\lambda)$.  For the last claim of (b), use the fact that ${\cal
F}(\lambda)$ is already a homogeneous space for $N(\lambda)$.  For (c),
suppose $h\in H$ carries $\lambda$ to another element $\lambda'$ of ${\cal
F}(\lambda)$.  By (a), there is an element $n\in N(\lambda)$ with $\lambda'
= {\rom Ad}(n)\cdot \lambda$.  Since $n$ and $h$ carry $\lambda$ to the
same element, the element $l = n^{-1}h$ belongs to the stabilizer
$L(\lambda)$ of $\lambda$.  Hence $h=nl$ belongs to $P(\lambda)$, as we
wished to show.  Q.E.D.

Suppose now that $\Lambda$ is a canonical flat in ${\frak h}$.  We define
$$e(\Lambda) = e(\lambda),\quad H(\Lambda) = H(\lambda),\quad N(\Lambda)
= N(\lambda),\quad P(\Lambda) = P(\lambda)\eqno(6.6)$$
for any $\lambda\in \Lambda$; this is well-defined by Proposition 6.5(b). 
We write ${\frak h}(\Lambda)$, etc., for the Lie algebras.
\medskip
{\bf Proposition 6.7.} {\it Suppose $H$ is a complex reductive algebraic
group, and $\Lambda\subset {\frak h}$ is a canonical flat.  Use the notation
of (6.6).
\smallskip
\item{a)}  $\Lambda$ is an affine space for the vector space ${\frak
n}(\Lambda)$, and a homogeneous space for $P(\Lambda)$.  These structures
are compatible (via the adjoint action of $P(\Lambda)$ on ${\frak
n}(\Lambda)$).  We can therefore form the induced bundle
$$H\times_{P(\Lambda)} \Lambda\rightarrow H/P(\Lambda),$$
which is an affine bundle for the vector bundle 
$$H\times_{P(\Lambda)} {\frak n}(\Lambda)\rightarrow H/P(\Lambda).$$
\item{b)}  The inclusion of $\Lambda$ in ${\frak h}$ induces a map
$$H\times_{P(\Lambda)} \Lambda\rightarrow{\frak h}, \quad
(h,\lambda)\mapsto {\rom Ad}(h)\cdot \lambda.$$
This map is an isomorphism onto the conjugacy class containing $\Lambda$. 
The fibers of the bundle structure map to the canonical flats in the
conjugacy class.
\medskip}
{\it Proof.}  (A brief discussion of induced bundles can be found in the
appendix of [Dix]; this result has some overlap with those of section 3 in
that paper.  Recall  that a point of the induced bundle is an equivalence class
of pairs $(h,\lambda)$, with $h\in H$ and $\lambda\in \Lambda$; the
equivalence relation is $(hp,\lambda)\sim(h,p\cdot\lambda)$ for $p\in
P(\Lambda)$.)  Part (a) and the existence of the map in (b) are formal (cf.
Proposition A.2(b) in [Dix], for example).  That the map is an isomorphism
follows from Proposition A.2(d) of [Dix], or a direct calculation: the fiber
over $\lambda$ consists of the equivalence classes of pairs $(h,\lambda')$
with ${\rom Ad}(h)\cdot \lambda' = \lambda$.  By (c) of Proposition 6.5, it
follows that $h\in P(\Lambda)$, and thus that $(h,\lambda')$ is equivalent
to $(e,\lambda)$.  This shows that the map is bijective.  Since the domain
and range are homogeneous spaces, it must be an isomorphism.  For the last
claim, obviously the fiber of the bundle over the identity coset
$eP(\Lambda)$ maps to the canonical flat $\Lambda$.  The general
statement then follows from the $H$-equivariance of the map.  Q.E.D.
\medskip
{\bf Example 6.8.}  Most semisimple elements (a dense open set) of a
reductive Lie algebra have no integral eigenvalues except zero; and the zero
eigenspace is the unique Cartan subalgebra to which they belong.  For such
an element $\lambda$, we have $P(\lambda)=L(\lambda)=T$, a Cartan
subgroup of $H$; ${\cal F}(\lambda) = \{\lambda\}$; the unipotent group
$N(\lambda)$ is trivial; the bundles of Proposition 6.7 are trivial on $H/T$;
and Proposition 6.7(b) simply identifies the conjugacy class of $\lambda$
with $H/T$.

At the other extreme, suppose $\lambda$ has all of its eigenvalues integral. 
Then $H(\lambda)$ contains the identity component of $H$, and $P(\lambda)$
is a parabolic subgroup of $H$.  The partition of the ${\rom Ad}(H)$ orbit of
$\lambda$ into canonical flats can be viewed as a Lagrangian foliation of a
symplectic structure on the orbit; but since this is an adjoint rather than a
coadjoint orbit, the symplectic structure is not canonical and this is not a
particularly good point of view.

Suppose for example that $H=SL(2,{\Bbb C})$.  Write $\{h,e,f\}$ for the
standard basis of the Lie algebra, and choose
$$\lambda = {1/2}h=\pmatrix1/2&0\cr 0&-1/2\cr\endpmatrix\eqno(6.8)(a)$$
Then
$${\frak h}(\lambda)_{-1} = {\Bbb C}f,\quad {\frak h}(\lambda)_0 = {\Bbb
C}h,\quad {\frak h}(\lambda)_1 = {\Bbb C}e\eqno(6.8)(b)$$
$${\frak h}(\lambda) = {\frak h},\quad {\frak l}(\lambda) = {\Bbb
C}h,\quad{\frak n}(\lambda) =  {\Bbb C}e,\quad {\frak p}(\lambda) = {\Bbb
C}h + {\Bbb C}e\eqno(6.8)(c)$$
$$e(\lambda) = -I\in SL(2,{\Bbb C}),\quad H(\lambda) = SL(2,{\Bbb
C})\eqno(6.8)(d)$$
$${\cal F}(\lambda) = \left\{ \,\pmatrix1/2&t\cr 0&-1/2\cr\endpmatrix\mid t\in
{\Bbb C}\,\right\}\eqno(6.8)(e)$$  
The orbit of $\lambda$ is the quadric
$${\rom Ad}(H)\cdot\lambda = \left\{ \,\pmatrix a&b\cr c&-a\cr\endpmatrix\mid a^2 +
bc = 1/4\,\right\};\eqno(6.8)(f)$$
its partition into the conjugates of the line (6.8)(e) is just one of the two 
standard rulings of a quadric surface ([Hartshorne], Exercise I.2.15).
\medskip
{\bf Definition 6.9.}  Suppose ${^\vee G^\Gamma}$ is an E-group.  A {\it
geometric parameter} for ${^\vee G^\Gamma}$ is a pair $(y,\Lambda)$
satisfying
\smallskip
\item{(a)} $y\in {^\vee G^\Gamma}-{^\vee G}$;
\item{(b)} $\Lambda\subset {^\vee{\frak g}}$ is a canonical flat (Definition
1.7); and
\item{(c)} $y^2 = e(\Lambda)$ (notation (6.6)).
\smallskip
The set of geometric parameters for ${^\vee G^\Gamma}$ is written
$X({^\vee G^\Gamma})$.  We make ${^\vee G}$ act on $X({^\vee G^\Gamma})$
by conjugation.  Two geometric parameters are called {\it equivalent} if
they are conjugate by this action.  (We do not introduce a separate notation
for the set of equivalence classes, since this will turn out to be naturally
identified with $\Phi({^\vee G^\Gamma})$.)

If ${\cal O}$ is a ${^\vee G}$ orbit of semisimple elements in $^\vee{\frak
g}$, then as in Lemma 1.9 we define
$$X({\cal O},{^\vee G^\Gamma}) = \{\,(y,\Lambda)\in X({^\vee
G^\Gamma})\mid \Lambda\subset {\cal O}\,\}.$$
\medskip 

We begin now our analysis of the spaces $X({\cal O},{^\vee G^\Gamma})$.  Fix
a ${^\vee G}$ orbit of semisimple elements
$${\cal O} \subset {^\vee {\frak g}},\eqno(6.10)(a)$$
and write ${\cal F}({\cal O})$ for the set of canonical flats in ${\cal O}$.  If
we fix such a flat $\Lambda$, then Proposition 6.5(c) provides an
isomorphism
$${\cal F}({\cal O})\simeq {^\vee G}/P(\Lambda).\eqno(6.10)(b)$$
We also need to consider the conjugacy class of $e(\Lambda)$ in ${^\vee G}$. 
This is
$${\cal C}({\cal O}) = \{\,ge(\Lambda)g^{-1}\mid g\in {^\vee
G}\,\}.\eqno(6.10)(c)$$
By (6.2)(b), there is an isomorphism
$${\cal C}({\cal O})\simeq {^\vee G}/{^\vee G}(\Lambda).\eqno(6.10)(d)$$
Because $e(\Lambda)$ is a semisimple element, the conjugacy class ${\cal
C}({\cal O})$ is closed in ${^\vee G}$.  From the form of Definition 1.8, it is
clear that we also need to consider the subvariety
$${\cal I}({\cal O}) = \{\,y\in {^\vee G^\Gamma}-{^\vee G}\mid y^2\in {\cal
C}({\cal O}) \,\}\eqno(6.10)(e)$$
It will be convenient also to consider 
$${\cal I}(\Lambda) = \{\,y\in {^\vee G^\Gamma}-{^\vee G}\mid y^2=
e(\Lambda) \,\}. \eqno(6.10)(f)$$
Finally, for $y\in {\cal I}({\cal O})$, we write
$$K(y) = \hbox{ centralizer of $y$ in ${^\vee G}$.}\eqno(6.10)(g)$$

Now the definition of the map $e$ (cf. (6.2)(a) and (6.6)) on ${\cal F}({\cal
O})$ involves the exponential map, and so is not algebraic in nature. 
Nevertheless, we have
\medskip
{\bf Lemma 6.11.} {\it In the setting of (6.10), the map
$$e:{\cal F}({\cal O})\rightarrow {\cal C}({\cal O}),\quad \Lambda\mapsto
e(\Lambda)$$
is a smooth projective algebraic morphism.}
\medskip
{\it Proof.}  By (6.10)(b) and (d), the map is just the natural quotient map of
algebraic homogeneous spaces
$${^\vee G}/P(\Lambda) \rightarrow {^\vee G}/{^\vee G}(\Lambda).$$
Since $P(\Lambda)$ is parabolic in ${^\vee G}(\Lambda)$, the map is 
projective.  Q.E.D.

The next lemma will be the key to the finiteness claim of Lemma 1.9.
\medskip
{\bf Lemma 6.12.} {\it Suppose $H$ is a complex reductive algebraic group
(possibly disconnected), and $z\in H$ centralizes the identity component
$H_0$.  Consider the set
$$I = \{\,y\in H\mid y^2 =z\,\}.$$
Then $I$ is the union of finitely many orbits under the conjugation action of
$H_0$.}
\medskip
{\it Proof.}  The conjugation action of $y\in I$ on $H_0$ defines an
involutive automorphism $\theta_y$.  Fix a connected component $C$ of $H$,
and consider the subset $I\cap C$ of $I$. Since $H$ has finitely many
components, it suffices to show that $I\cap C$ has finitely many $H_0$
orbits.  The involutions in the set 
$$\Theta(C) = \{\theta_y\mid y\in I\cap C\}$$
all differ by inner automorphisms (since $C$ is a coset of $H_0$).  In
particular, they have a common restriction $\theta_Z$ to the center
$Z(H_0)$, so they are distinguished by their restrictions to the semisimple
commutator subgroup of $H_0$.  A complex semisimple algebraic group
admits only finitely many conjugacy classes of involutive automorphisms;
indeed Cartan showed that these classes correspond bijectively to the
equivalence classes of real forms.  It follows that $\Theta(C)$ is a finite
union of $H_0$ orbits.  To complete the proof, we must study the fibers of
the map
$$I\cap C\rightarrow \Theta(C),\qquad y \mapsto \theta_y.$$
It is enough to show that each fiber is a finite union of orbits under the
conjugation action of $Z(H_0)$.  Now the fiber over $\theta_y$ is clearly
$$\{\,yw\mid w\in Z(H_0), w\theta_Z(w) = e\,\} = yZ_1.$$
On the other hand, the $Z(H_0)$ conjugacy class of $y$ is
$$\{\,yw\mid w = v\theta_Z(v)^{-1},\hbox{ some }v\in Z(H_0)\,\} = yZ_2.$$
The two subgroups of $Z(H_0)$ appearing here are algebraic, and
$Z_2\subset Z_1$.  Since they have the same Lie algebra (namely the $-1$
eigenspace of the differential of $\theta_Z$), it follows that $Z_2$ has
finite index in $Z_1$, as we wished to show.  Q.E.D.
\medskip
{\bf Proposition 6.13.}  {\it In the setting (6.10), the orbits of the
conjugation action of ${^\vee G}$ on ${\cal I}({\cal O})$ are in one-to-one
correspondence (by intersection with ${\cal I}(\Lambda)$) with ${^\vee
G}(\Lambda)$ orbits on ${\cal I}(\Lambda)$.  All of these orbits are closed,
and there are finitely many of them.

Fix a single orbit ${\cal I}_1({\cal O})$ of ${^\vee G}$ on ${\cal I}({\cal O})$. 
Then the natural map
$${\cal I}_1({\cal O})\rightarrow{\cal C}({\cal O}),\qquad y\mapsto y^2$$
is a smooth algebraic morphism.}
\medskip
{\it Proof.}  The bijection is formal.  That the orbits are closed follows
from the fact that they are finite unions of connected components of
semisimple conjugacy classes in the (disconnected) reductive group ${^\vee
G^\Gamma}$.  For the finiteness, we consider the reductive group
$$H = \hbox{ centralizer of $e(\Lambda)$ in ${^\vee G^\Gamma}$.}$$
Obviously any element $y$ of ${\cal I}(\Lambda)$ belongs to  $H$ (since $y$
always commutes with $y^2$).  Furthermore $z=e(\Lambda)$ is central in
$H$.  It follows that the set $I$ of Lemma 6.12 contains ${\cal I}(\Lambda)$. 
Since ${^\vee G}(\Lambda)$ contains the identity component of $H$, the
finiteness we need follows from Lemma 6.12.

For the last claim, fix an element $y_1\in {\cal I}_1({\cal O})$ with $y_1^2
= e(\Lambda)$.  Then
$${\cal I}_1({\cal O})\simeq {^\vee G}/K(y_1),\eqno(6.14)(a)$$
and the morphism in question is just the quotient map
$${^\vee G}/K(y_1) \rightarrow {^\vee G}/{^\vee G}(\Lambda).
\eqno(6.14)(b)$$
Of course this map is smooth.  Q.E.D.

Here is the last general ingredient we need to complete our description of
the geometric parameter space.

\medskip
{\bf Lemma 6.15.}  {\it Suppose $G$ is an algebraic group, $H$ is a closed
subgroup of $G$, and $A$ and $B$ are closed subgroups of $H$.  Then $G/A$
and $G/B$ both carry natural maps to $G/H$, so we can form the fiber
product
$$X = G/A \times_{G/H} G/B\rightarrow G/H.$$
On the other hand, $H$ acts on $H/A$ and on $H/B$, so it acts on the product
$H/A \times H/B$.  We can therefore form the induced bundle
$$Y = G \times_H \left( H/A \times H/B \right)\rightarrow G/H.$$
Finally, $A$ acts on the homogeneous space $H/B$, so we can form the
induced bundle
$$Z = G \times_A H/B\rightarrow G/A \rightarrow G/H$$
(and similarly with $A$ and $B$ exchanged).
\smallskip
\item {a)}  $X$, $Y$, and $Z$ are smooth bundles over $G/H$.
\item {b)} $Y$ is isomorphic to $X$ by the map sending the equivalence class 
of $(g,(h_1A,h_2B))\in Y$ to $(gh_1A,gh_2B)\in X$.  This map is a
$G$-equivariant isomorphism of fiber bundles from $Y$ to $X$.
\item {c)} $Z$ is isomorphic to $X$ by the map sending the equivalence class
of $(g,hB)$ to $(gA,ghB)$.  This map is a $G$-equivariant isomorphism of
fiber bundles from $Y$ to $X$.
\item {d)} The isomorphism in (b) induces a natural bijection from the
orbits of $H$ on $\left( H/A \times H/B \right)$ onto the orbits of $G$ on
$X$.
\item {e)} The isomorphism in (c) induces a natural bijection from the
orbits of $A$ on $H/B$ onto the orbits of $G$ on $X$.
\item{f)}  The following four sets are in natural one-to-one correspondence:
the orbits of $H$ on $\left( H/A \times H/B \right)$; the $(A,B)$ double
cosets in $H$; the orbits of $A$ on $H/B$; and the orbits of $B$ on $H/A$.
\medskip}

{\it Proof.}  An induced bundle is smooth if and only if the inducing space is
smooth; so the smoothness of $Y$ (respectively $Z$) is a consequence of the
smoothness of $(H/A\times H/B)$ (respectively $H/B$).  As for $X$, the
spaces $G/A$ and $G/B$ are both smooth over $G/H$, so their fiber product
is smooth over $G/H$ as well ([Hartshorne], Proposition III.10.1(d)).  For (b),
the $G$-equivariant maps from an induced bundle to a $G$-space are
determined by the $H$-equivariant maps from the inducing space.  The map
in (b) corresponds in this way to the natural inclusion of $(H/A\times H/B)$
in the fiber product $X$.   That this map is a bijection is formal.  That it is
an isomorphism can be deduced by computing its differential (because of the
smoothness established in (a)).  Part (c) is similar; or one can show directly
that $Z$ is isomorphic to $Y$.  For (d) and (e), the $G$-orbits on an induced
bundle are in natural bijection with the orbits on the inducing space.  Part
(f) is elementary group theory.  Q.E.D.

\medskip
{\bf Proposition 6.16.} {\it Suppose ${^\vee G^\Gamma}$ is a weak E-group. 
Fix an orbit ${\cal O}$ of ${^\vee G}$ on the semisimple elements of
$^\vee{\frak g}$, and use the notation of (6.6) and (6.10).  In particular, we
have smooth algebraic maps
$${\cal F}({\cal O})\rightarrow {\cal C}({\cal O}),\qquad {\cal I}({\cal
O})\rightarrow {\cal C}({\cal O}).$$
The variety $X({\cal O},{^\vee G^\Gamma})$ of Definition 6.9 is the fiber
product
$$X({\cal O},{^\vee G^\Gamma}) = {\cal F}({\cal O})\times_{{\cal C}({\cal O})}
{\cal I}({\cal O}).$$

A little more explicitly, it has the following structure.  List the orbits of
${^\vee G}$ on ${\cal I}({\cal O})$ as ${\cal I}_1({\cal O}),\ldots,{\cal
I}_r({\cal O})$ (Proposition 6.13).  Fix a canonical flat $\Lambda\in {\cal
F}({\cal O})$.  This gives a reductive group ${^\vee G}(\Lambda)$ and a
parabolic subgroup $P(\Lambda)$ (cf. (6.6) and Lemma 6.3).  For each $i$,
choose a point
$$y_i\in {\cal I}_i({\cal O}), \quad y_i^2 = e(\Lambda).$$
Then conjugation by $y_i$ defines an involutive automorphism $\theta_i$ of
${^\vee G}(\Lambda)$ with fixed point set $K_i = K(y_i)$ (cf. (6.10)(g)).  The
variety $X({\cal O},{^\vee G^\Gamma})$ is the disjoint union of $r$ closed
subvarieties 
$$X_i({\cal O},{^\vee G^\Gamma}) = {\cal F}({\cal O})\times_{{\cal C}({\cal
O})} {\cal I}_i({\cal O}).$$
The $i$th subvariety looks like
$$\eqalign{X_i({\cal O},{^\vee G^\Gamma}) &\simeq {^\vee G} \times_{{^\vee
G}(\Lambda)} ({^\vee G}(\Lambda)/K_i \times {^\vee
G}(\Lambda)/P(\Lambda))\cr
&\simeq {^\vee G} \times_{K_i} ({^\vee G}(\Lambda)/P(\Lambda)).\cr}$$
In particular, the orbits of ${^\vee G}$ on $X_i({\cal O},{^\vee G^\Gamma})$
are in one-to-one correspondence with the orbits of $K_i$ on the partial
flag variety ${^\vee G}(\Lambda)/P(\Lambda)$.  These orbits are finite in
number.  The isotropy group of the action of ${^\vee G}$ at the point $x =
(y,\Lambda)$ is
$${^\vee G}_x = K(y) \cap P(\Lambda).$$}
\medskip
\noindent We will see later (Proposition 7.14) that the correspondence
between $K_i$ orbits on ${^\vee G}(\Lambda)/P(\Lambda)$ and ${^\vee G}$
orbits on $X_i({\cal O})$ not only is a bijection of sets, but also preserves
the closure relations and the nature of the singularities of closures.

{\it Proof.}  Everything but the last finiteness assertion is immediate from
Lemma 6.15 and Proposition 6.13.  (The description of $X({\cal O},{^\vee
G^\Gamma})$ as a fiber product is just a restatement of Definition 1.8.)  For
the finiteness, we must show that if $H$ is a reductive group, $Z$ is a
partial flag variety for $H$ (that is, a homogeneous space for which an
isotropy group contains a Borel subgroup of $H_0$) and $K\subset H$ is the
group of fixed points of an involution, then $K$ has finitely many orbits on
$Z$.  After straightforward reductions, we may assume that $H$ is
connected and semisimple, and that $Z$ is the variety of Borel subgroups.  In
that case the result is well-known.  (A complete description of the orbits
may be found in [Matsuki]; the finiteness result is even older.)  Q.E.D.

With this description of the geometric parameter space, we can consider
now its relationship to the Langlands parameters of section 5.
\medskip
{\bf Proposition 6.17.}  {\it Suppose ${^\vee G^\Gamma}$ is a weak E-group. 
Then there is a natural ${^\vee G}$-equivariant map $p$ from the set of
Langlands parameters for ${^\vee G^\Gamma}$ (Definition 5.2) onto the set
of geometric parameters for ${^\vee G^\Gamma}$ (Definition 6.9).  In terms
of the description of Langlands parameters in Proposition 5.6, this map is
$$p:P({^\vee G^\Gamma})\rightarrow X({^\vee G^\Gamma}),\quad
p(\phi(y,\lambda)) = (y,{\cal F}(\lambda))$$
(notation (5.8), (6.4)).  The map $p$ induces a bijection from equivalence
classes of Langlands parameters (that is, ${^\vee G}$ orbits on $P({^\vee
G^\Gamma})$) onto equivalence classes of geometric parameters (that is,
${^\vee G}$ orbits on $X({^\vee G^\Gamma})$).  If $x=p(\phi)$, then the
isotropy group ${^\vee G}_\phi$ (Corollary 5.9) is a Levi subgroup of the
isotropy group ${^\vee G}_x$ (Proposition 6.16).  In particular, the fiber
$p^{-1}(x)$ is a principal homogeneous space for the unipotent radical of
${^\vee G}_x$.}
\medskip

Before embarking on the proof, we need some general facts about parabolic
subgroups and involutions.

\medskip
{\bf Lemma 6.18.}  {\it Suppose $G$ is a connected complex reductive
algebraic group, $P\subset G$ is a parabolic subgroup, and $\theta$ is an
involutive automorphism of $G$.  (We do not assume that $\theta$ preserves
$P$.)  Write $K\subset G$ for the group of fixed points of $\theta$.
\smallskip
\item{a)}  There is a maximal torus $T\subset P$ such that $\theta T = T$.
\smallskip
>From now on we fix such a torus.  There is a unique Levi decomposition
$P=LN$ with $T\subset L$.
\smallskip
\item{b)} The group $L_\theta = L\cap \theta L$ is a connected
$\theta$-stable reductive subgroup of $G$ containing $T$.
\item{c)} The group $N_\theta$ generated by $N\cap \theta P$ and $P \cap
\theta N$ is a connected $\theta$-stable unipotent subgroup of $G$,
normalized by $L_\theta$.
\item{d)} The group $P_\theta = P\cap \theta P$ is a connected algebraic
subgroup of $G$, with $\theta$-stable Levi decomposition $P_\theta =
L_\theta N_\theta$.
\item{e)}  We have
$$L\cap K  =  L_\theta \cap K, \quad P\cap K = P_\theta \cap K = (L_\theta
\cap K)(N_\theta\cap K).$$
The last formula is a Levi decomposition of the algebraic group $P\cap K$.
\medskip}

{\it Proof.}  For (a), we may as well replace $P$ by a Borel subgroup
contained in it.  In that case the existence of $T$ is established in [Matsuki]. 
Now write $R=R(G,T)$ for the set of roots.  Because $\theta$ preserves $T$,
$\theta$ acts on $R$ as an automorphism of order 2. Write 
$$R(P,T) = R(L,T) \cup R(N,T)$$
for the corresponding sets of roots.  Fix an element $\lambda\in {\frak t}$
with the property that
$$\alpha(\lambda) = 0\quad (\alpha\in R(L,T)),\qquad \alpha(\lambda) >
0\quad (\alpha \in R(N,T)).$$
Then $\theta\lambda$ also belongs to ${\frak t}$, and has the corresponding
properties for $\theta P$.  For (b), $L$ is the centralizer in $G$ of
$\lambda$, so
$$L\cap \theta L = \{\,g\in G\mid {\rom Ad}(g)\lambda = \lambda, {\rom
Ad}(g)(\theta\lambda) = \theta\lambda\,\}.$$
Since $\lambda$ and $\theta\lambda$ are commuting semisimple elements
of ${\frak g}$, it follows that $L\cap \theta L$ is connected and reductive. 
The rest of (b) is clear.

For (c) and (d), one computes easily that the roots of $T$ in $L_\theta$ and
$P_\theta$ are
$$R(L_\theta,T) = \{\,\alpha\in R\mid \alpha(\lambda) = 0\hbox { and }
\alpha(\theta\lambda) = 0\,\}$$
$$R(P_\theta,T) = \{\,\alpha\in R\mid \alpha(\lambda)\ge 0\hbox { and }
\alpha(\theta\lambda)\ge 0\,\}.$$
The groups $N\cap \theta P$ and $P\cap \theta N$ are obviously unipotent
algebraic, and therefore connected.  The corresponding roots are
$$R(N\cap \theta P,T) = \{\,\alpha\in R\mid \alpha(\lambda) > 0\hbox { and }
\alpha(\theta\lambda) \ge 0\,\}$$
$$R(P\cap \theta N,T) = \{\,\alpha\in R\mid \alpha(\lambda) \ge 0\hbox {
and } \alpha(\theta\lambda) > 0\,\}.$$
Now set
$$R^+_\theta = R(N\cap \theta P,T)\cup R(P\cap \theta N,T).$$
Because this set is easily seen to be closed under addition, it is the set of
roots of $T$ in the group $N_\theta$.  To prove (c), we must show that
$N_\theta$ is unipotent. This is a consequence of the fact that all the roots
in $R^+_\theta$ lie on one side of the hyperplane defined by $\lambda
+\theta \lambda$.

For (d), observe first that
$$R(P_\theta,T) = R(L_\theta,T)\cup R(N_\theta,T),$$
a disjoint union.  It follows from (b) and (c) that $L_\theta N_\theta$ is a
Levi decomposition of the identity component of $P_\theta$; it remains to
show that $P_\theta$ is connected.  So suppose $p\in P_\theta$; we want to
show that $p$ is in the identity component. Now $pTp^{-1}$ is a maximal
torus in $P_\theta$; so by the conjugacy of maximal tori in connected
algebraic groups, we can find an element $p_0$ in the identity component of
$P_\theta$ conjugating $pTp^{-1}$ to $T$.  After replacing $p$ by $p_0p$,
we may therefore assume that $p$ normalizes $T$.  Now the normalizer of
$T$ in $P$ is contained in $L$; so $p\in L$.  Similarly, the normalizer of $T$
in $\theta P$ is contained in $\theta L$, so $p\in \theta L$.  Therefore $p
\in L_\theta$, which we already know is a connected subgroup of
$P_\theta$.

Part (e) is clear: since $\theta$ respects the Levi decomposition in
$P_\theta$, an element of $P_\theta \cap K$ must have both its Levi
components in $K$.  Q.E.D.

{\it Proof of Proposition 6.17.}  That $p$ is a well-defined ${^\vee
G}$-equivariant map is clear from Proposition 5.6 and Definition 6.9.  The
first problem is to show that $p$ is surjective.  So fix $x=(y,\Lambda)\in
X({^\vee G^\Gamma})$.   Obviously
$$p^{-1}(x) = \{\,\phi(y,\lambda)\mid \lambda \in \Lambda,
[\lambda,y\cdot\lambda] = 0\,\}.\eqno(6.19)(a)$$
Use the notation of (6.6); then $P(\Lambda)$ is a parabolic subgroup of
${^\vee G}(\Lambda)$ (Lemma 6.3), and the conjugation action of $y$ defines
an involutive automorphism $\theta$ of ${^\vee G}(\Lambda)$, with fixed
points $K(y)$.  Apply Lemma 6.18 to this situation.  We conclude first of all
that $P(\Lambda)$ contains a $\theta$-stable maximal torus $T$.  The
semisimple $P(\Lambda)$-conjugacy class $\Lambda\subset {\frak
p}(\Lambda)$ must meet the Lie algebra ${\frak t}$ in some element
$\lambda$.  Then $\lambda$ and $y\cdot \lambda$ both belong to ${\frak t}$,
so they commute; so $\phi(y,\lambda)\in p^{-1}(x)$ by (6.19)(a).

To prove that $p$ is a bijection on the level of orbits, we must show that
the isotropy group
$$G_x = P(\Lambda) \cap K(y)\eqno(6.19)(b)$$
(cf. Proposition 6.16) acts transitively on the fiber $p^{-1}(x)$.  Now any
element of $G_x$ is fixed by $\theta$, so it must also belong to $\theta
P(\Lambda)$.  Consequently
$$G_x = P_\theta(\Lambda)\cap K(y) = (L_\theta(\lambda)\cap
K(y))(N_\theta(\Lambda)\cap K(y)).\eqno(6.19)(c)$$

So suppose $\phi(y,\lambda')$ is another point in the fiber.  By Lemma
6.3(e),
$$\lambda' = {\rom Ad}(n)\lambda\eqno(6.20)(a)$$
for a unique $n\in N(\Lambda)$.  By (6.19)(a), we have
$$[n\cdot\lambda,\theta(n\cdot\lambda)] = 0.\eqno(6.20)(b)$$
Now the centralizer of $\lambda$ in $^\vee{\frak g}$ is ${\frak
l}(\lambda)\subset {\frak p}(\Lambda)$.  It follows that the centralizer of
$\theta(n\cdot\lambda)$ is
$$\theta(n\cdot {\frak l}(\lambda))\subset \theta(n\cdot{\frak
p}(\Lambda)) = \theta{\frak p}(\Lambda).$$
>From this and (6.20)(b), we deduce that
$$n\cdot\lambda \in {\frak p}(\Lambda) \cap \theta{\frak p}(\Lambda).$$
The latter Lie algebra is described in Lemma 6.18.  In conjunction with
(6.3)(e), this allows us to conclude (using notation from Lemma 6.18) that
$$n\cdot\lambda \in (\lambda + {\frak n}(\Lambda))\cap ({\frak
l}_\theta(\lambda) + {\frak n}_\theta(\Lambda)) = \lambda + ({\frak
n}(\Lambda) \cap {\frak n}_\theta(\Lambda)).$$
Now the isomorphism in Lemma 6.3(e) respects $T$-stable subgroups of
$N(\Lambda)$; so the conclusion we draw (using the uniqueness in Lemma
6.3(e)) is
$$n\in N(\Lambda)\cap N_\theta(\Lambda).$$

We now rewrite (6.20)(b) by applying $\theta(n^{-1})$ to each term in the
bracket.  The conclusion is
$$[(\theta(n^{-1})n)\cdot\lambda,\theta\lambda] = 0.\eqno(6.20)(c)$$
Since $N_\theta(\Lambda)$ is preserved by $\theta$, the element
$m=\theta(n^{-1})n$ belongs to $N_\theta(\Lambda)$. The discussion of
roots in Lemma 6.18 leads to a factorization
$$N_\theta(\Lambda) = (N(\Lambda)\cap \theta L(\lambda))(N(\Lambda)\cap
\theta N(\Lambda))(L(\lambda)\cap \theta N(\Lambda));$$
write $m = m_1m_2m_3$ accordingly.  Then $m_3$ fixes $\lambda$, and
$m_1$ fixes $\theta\lambda$, so (6.20)(c) leads to
$$[m_2\cdot\lambda,\theta\lambda] = 0.$$
But Lemma 6.3(e) tells us that the first term here is of the form $\lambda +
X_2$, with $X_2\in {\frak n}(\Lambda)\cap \theta{\frak n}(\Lambda)$.  Such
an element commutes with $\theta\lambda$ only if $X_2=0$, which implies
that $m_2 = e$.  On the other hand, the definition of $m$ shows that $\theta
m = m^{-1}$, from which we conclude that $m_1 = \theta(m_3^{-1})$.  Define
$n' = nm_3^{-1}$; then $\theta((n')^{-1})n' = e$.  Assembling all of this, we
find
$$n'\in N_\theta(\Lambda)\cap K(y),\qquad n'\cdot\lambda = \lambda'.
\eqno(6.20)(d)$$ 
Clearly $n'\cdot \phi(y,\lambda) = \phi(y,\lambda')$.  Since $n'$ belongs to
the unipotent radical of $G_x$ (cf. (6.19)(c)), this completes the proof of
Proposition 6.17.  Q.E.D.

\medskip
{\bf Corollary 6.21.}  {\it Suppose ${^\vee H^\Gamma}$ and ${^\vee
G^\Gamma}$ are weak E-groups, and $\epsilon:{^\vee H^\Gamma}\rightarrow
{^\vee G^\Gamma}$ is an L-ho\-mo\-mor\-phism (Definition 5.1).  Then there
is a natural map
$$X(\epsilon):X({^\vee H^\Gamma})\rightarrow X({^\vee G^\Gamma})$$
on geometric parameters, compatible with the maps $P(\epsilon)$,
$\Phi(\epsilon)$ of Proposition 5.4 via the maps $p$ of Proposition 6.17. 
Explicitly, 
$$X(\epsilon)(y,\Lambda) = (\epsilon(y), \epsilon(\Lambda)).$$
(Here $\epsilon(\Lambda)$ denotes the unique canonical flat containing
$d\epsilon(\Lambda)$.)  

Fix a ${^\vee H}$ orbit ${\cal O}\subset {^\vee {\frak h}}$ of semisimple
elements, and define $\epsilon({\cal O})$ to be the unique ${^\vee G}$ orbit
containing $d\epsilon({\cal O})$.  Then $X(\epsilon)$ restricts to a
morphism of algebraic varieties
$$X({\cal O},\epsilon):X({\cal O},{^\vee H^\Gamma}) \rightarrow
X(\epsilon({\cal O}),{^\vee G^\Gamma}).$$
If $\epsilon$ is injective, then $X({\cal O},\epsilon)$ is a closed
immersion.}
\medskip

{\it Proof.}  The first point that is not quite obvious is that the image under
$d\epsilon$ of a canonical flat $\Lambda$ is contained in a single canonical
flat for ${^\vee{\frak g}}$.  To see this, fix $\lambda\in \Lambda$, and use
the notation of (6.1)--(6.4). Write $\lambda' = d\epsilon(\lambda)$,
$\Lambda' = {\cal F}(\lambda')$.  Clearly
$${^\vee{\frak h}}(\lambda)_n = (d\epsilon)^{-1} {^\vee{\frak g}}
(\lambda')_n$$
for every $n$.  From this it follows that 
$$\Lambda = \lambda + \sum_{n > 0} {^\vee{\frak h}}(\lambda)_n \subset
(d\epsilon)^{-1}(\lambda') + \sum_{n > 0} (d\epsilon)^{-1} {^\vee{\frak g}}
(\lambda')_n = (d\epsilon)^{-1}(\Lambda').$$
It follows that $d\epsilon(\Lambda) \subset \Lambda'$, as we wished to
show.  Notice also that if $d\epsilon$ is injective, this containment
becomes an equality:
$$\Lambda = (d\epsilon)^{-1}(\Lambda').$$

For the last assertion, we first verify that $X(\epsilon)$ is injective if
$\epsilon$ is.  So suppose
$$X(\epsilon)(y_1,\Lambda_1) = X(\epsilon)(y_2,\Lambda_2) =
(y',\Lambda').$$
This means first of all that $\epsilon(y_i) = y$, and therefore that $y_1 =
y_2$.  Next, the last observation in the preceding paragraph shows that
$\Lambda_i = (d\epsilon)^{-1}(\Lambda')$, and therefore that $\Lambda_1 =
\Lambda_2$.  This proves that $X(\epsilon)$ is injective.  Essentially the
same argument proves that $X({\cal O},\epsilon)$ is a closed immersion; we
leave the remaining details to the reader.  Q.E.D.

\medskip
{\bf Example 6.22.}  Suppose $G = GL(3,{\Bbb C})$, endowed with the inner
class of real forms including $U(2,1)$ and $U(3)$.  The based root datum of
$G$ (Definition 2.10) is the quadruple
$$({\Bbb Z}^3,\{(1,-1,0),(0,1,-1)\},{\Bbb Z}^3,\{(1,-1,0),(0,1,-1)\}),$$
with the standard pairing between the two copies of ${\Bbb Z}^3$.  The
automorphism $t$ of Proposition 2.12 associated to the inner class of real
forms acts on ${\Bbb Z}^3$ by $t(a,b,c) = (-c,-b,-a)$.  As a dual group for $G$
we can choose ${^\vee G} = GL(3,{\Bbb C})$.  Define an automorphism
$\theta$ of ${^\vee G}$ by
$$\theta  g = {\rom Ad}\pmatrix 0&0&1\cr 0&1&0\cr
1&0&0\cr\endpmatrix({^tg^{-1}}).$$
Then $\theta$ is an involutive automorphism preserving the standard Cartan
and Borel subgroups $^dT \subset {^dB}$; the induced automorphism of the
based root datum is $t$.  We can therefore construct a weak E-group ${^\vee
G^\Gamma}$ generated by ${^\vee G}$ and an element $^\vee\delta$ of order
two, subject to the relations
$$({^\vee\delta})g({^\vee\delta^{-1}}) = \theta g\qquad (g\in {^\vee G}).$$
(If we take as the distinguished set ${\cal S}$ of Definition 4.14 the
conjugacy class of $({^\vee\delta},{^dB})$, then ${^\vee G^\Gamma}$ is an
L-group for $G$.)

Consider the elements
$$\lambda = \pmatrix1&0&0\cr 0&1&0 \cr 0&0&0\cr\endpmatrix,\qquad y =
{^\vee\delta}, \qquad y\cdot\lambda = \pmatrix0&0&0\cr 0&-1&0 \cr 0&
0&-1\cr\endpmatrix.$$
Notice that $e(\lambda) = e= y^2$, and $\lambda$ commutes with $y\cdot
\lambda$; so there is a Langlands parameter $\phi = \phi(y,\lambda)$.  Since
$e(\lambda) = e$,  ${^\vee G}(\lambda) = {^\vee G}$.  The corresponding
parabolic $P(\lambda)$ is the standard parabolic in $GL(3)$ with Levi
subgroup $GL(2)\times GL(1)$, and the canonical flat is
$$\Lambda = \left\{\pmatrix1&0&a\cr 0&1&b \cr 0&0&0\cr\endpmatrix\mid a,b\in
{\Bbb C}\right\}.$$
Of course $x = (y,\Lambda)$ is a geometric parameter.  The Levi subgroup
$L(\lambda,y\cdot \lambda)$ is the diagonal subgroup; its intersection with
$K(y)$ (which is a form of $O(3)$) is
$${^\vee G}_\phi = \left\{\pmatrix z&0&0\cr 0&\pm1&0 \cr 0&0&z^{-1}
\cr\endpmatrix\mid z\in {\Bbb C}^\times \right\}.$$
The Langlands component group $A_\phi$ (Definition 5.11) therefore has
order 2.  The group $P_\theta(\Lambda)$ of Lemma 6.18 is just the standard
Borel subgroup.  The isotropy group of the geometric parameter $x$ has
unipotent radical equal to the $\theta$-invariant upper triangular unipotent
matrices; this group is 
$$K(y)\cap N_\theta(\Lambda) = \left\{\pmatrix 1&a&-a^2/2 \cr 0&1&-a
\cr 0&0&1\cr\endpmatrix\mid a\in {\Bbb C}\right\}.$$
The full stabilizer of $x$ in ${^\vee G}$ is the semidirect product of ${^\vee
G}_\phi$ with this unipotent group, namely
$${^\vee G}_x = \left\{\pmatrix z&\pm a&-z^{-1}a^2/2 \cr 0&\pm1&-z^{-1}a
\cr 0&0&z^{-1}\cr\endpmatrix\mid a\in {\Bbb C},z\in {\Bbb C}^\times\right\}.$$
\medskip

To conclude this section, we offer a variant of Proposition 6.16.  Although
the version presented first seems to be the most natural one, it has the
technical disadvantage that the group ${^\vee G}(\Lambda)$ need not be
connected.  The only subtlety in the variant is the definition
$$K(y)^0 = K(y)\cap {^\vee G}(\Lambda)_0\eqno(6.23)(a)$$
for $y\in {\cal I}(\Lambda)$ (cf. (6.10)).  In the notation of Lemma 6.12, this
is the group of fixed points of the involution $\theta_y$ on the connected
group ${^\vee G}(\Lambda)_0$.  We have
$$K(y)_0 \subset K(y)^0 \subset K(y),\eqno(6.23)(b)$$
and both of these inclusions may be proper.

\medskip
{\bf Proposition 6.24.} {\it Suppose ${^\vee G^\Gamma}$ is a weak E-group. 
Fix an orbit ${\cal O}$ of ${^\vee G}$ on the semisimple elements of
$^\vee{\frak g}$, and use the notation of (6.6), (6.10), and (6.23).  Fix a
canonical flat $\Lambda\subset {\cal O}$, and write ${\cal P}^0(\Lambda)$
for the ${^\vee G}(\Lambda)_0$ orbit of $\Lambda$ in the set of canonical
flats.  List the orbits of ${^\vee G}(\Lambda)_0$ on ${\cal I}(\Lambda)$ as
${\cal I}_1^0(\Lambda),\ldots, {\cal I}_s^0(\Lambda)$ (notation (6.10)(f) and
Lemma 6.12).  For each $j$, choose a point $y_j \in {\cal I}_j^0(\Lambda)$. 
Then conjugation by $y_j$ defines an involutive automorphism $\theta_j$ of
${^\vee G}(\Lambda)_0$ with fixed point set $K^0_j = K(y_j)^0$ (cf. (6.23)). 
>From the chosen base points $\Lambda$ and $y_j$ we get isomorphisms
$${\cal P}^0(\Lambda)\simeq {^\vee G}(\Lambda)_0/P(\Lambda),\qquad {\cal
I}_j^0(\Lambda) \simeq {^\vee G}(\Lambda)_0/K^0_j.$$
The variety $X({\cal O},{^\vee G^\Gamma})$ is the disjoint union of $s$
closed connected smooth subvarieties $X_j({\cal O},{^\vee G^\Gamma})$.  The
$j$th subvariety looks like
$$\eqalign{X_j({\cal O},{^\vee G^\Gamma})&\simeq {^\vee G} \times_{{^\vee
G}(\Lambda)_0} ( {\cal I}_j^0(\Lambda) \times {\cal P}^0(\Lambda) )\cr
&\simeq {^\vee G} \times_{{^\vee G}(\Lambda)_0} ({^\vee
G}(\Lambda)_0/K^0_j \times {^\vee G}(\Lambda)_0/P(\Lambda) )\cr
&\simeq {^\vee G} \times_{K^0_j} ({^\vee G}(\Lambda)_0/P(\Lambda)).\cr}$$
In particular, the orbits of ${^\vee G}$ on $X_j({\cal O},{^\vee G^\Gamma})$
are in one-to-one correspondence with the orbits of $K^0_j$ on the partial
flag variety ${^\vee G}(\Lambda)_0/P(\Lambda)$.  These orbits are finite in
number.}
\medskip

\noindent The proof is identical to that given for Proposition 6.16.  (It is
helpful to keep in mind that $P(\Lambda)$ is connected (Lemma 6.3(d)) and
therefore contained in ${^\vee G}(\Lambda)_0$.)  As with Proposition 6.16,
we will eventually want the stronger information about the orbit
correspondence contained in Proposition 7.14(c).
%\end
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\centerline{\bf 7. Complete geometric parameters and perverse sheaves.}
\vskip 1pc

In this section we continue our analysis of the geometric parameter spaces
described in the preceding section.  To begin, we recall some terminology
from Definition 1.22 in the introduction.  This will be applied first of all in
the setting of Definition 6.9, to the action of ${^\vee G}^{alg}$ on $X({\cal
O},{^\vee G^\Gamma})$.

\medskip
{\bf Definition 7.1.}  Suppose $Y$ is a complex algebraic variety on which
the pro-algebraic group $H$ acts with finitely many orbits.  A {\it
geometric parameter for $H$ acting on $Y$} is a (closed) point of $Y$.  Two
such parameters are called {\it equivalent} if they differ by the action of
$H$.  The set of equivalence classes of geometric parameters --- that is,
the set of orbits of $H$ on $Y$ --- is written $\Phi(Y,H)$.

Suppose $y\in Y$; write $H_y$ for the isotropy group of the action at $y$. 
The {\it (local) equivariant fundamental group at $y$} is
$$A_y^{loc} = H_y/(H_y)_0,\eqno(7.1)(a)$$
the (pro-finite) group of connected components of $H_y$.  If $y'$ is
equivalent to $y$ (that is, if it belongs to the same $H$ orbit) then we can
find $h \in H$ with $h\cdot y = y'$.  It follows that conjugation by $h$
carries $H_y$ isomorphically onto $H_{y'}$, so we get an isomorphism
$$A_y^{loc} \simeq A_{y'}^{loc}.\eqno(7.1)(b)$$
The coset $hH_y$ is uniquely determined, so this isomorphism is unique up
to inner automorphisms.

Suppose $S\in \Phi(Y,H)$ is an orbit of $H$ on $Y$.  The {\it equivariant
fundamental group of $S$} is the pro-finite group
$$A_S^{loc} = A_y^{loc}\qquad (y\in S).\eqno(7.1)(c)$$
The isomorphisms in (7.1)(b) show that $A_S^{loc}$ is well-defined up to
inner automorphism.  This means that we can safely discuss conjugacy
classes in $A_S^{loc}$, or equivalence classes of representations, but it is
dangerous to speak of particular elements as well-defined entities.

A {\it (local) complete geometric parameter for $H$ acting on $Y$} is a pair
$(y,\tau)$ with $y\in Y$ and $\tau \in \widehat{A_y^{loc}}$ an irreducible
representation.  Two such parameters $(y,\tau)$ and $(y',\tau')$ are called
{\it equivalent} if there is an element of $H$ carrying $y$ to $y'$ so that the
induced isomorphism (7.1)(b) carries $\tau$ to $\tau'$.  The set of
equivalence classes of (local) complete geometric parameters is written
$\Xi(Y,H)$.

A {\it complete geometric parameter for $H$ acting on $Y$} is a pair
$(S,\cal V)$, with $S\in \Phi(Y,H)$ an orbit of $H$ on $Y$ and ${\cal V}$ an
irreducible $H$-equivariant local system on $S$.  (By a ``local system'' we
will understand a vector bundle with a flat connection; perhaps this is not
the most reasonable definition, but we are considering only smooth
varieties and complex coefficients).
\medskip

Of course the terminology ``geometric parameter'' for a point is a little
ridiculous, but we include it to maintain consistency with the case of
Definition 6.9.

We should make a few remarks about the ``pro-algebraic'' condition.  This
means that $H$ is the limit of an inverse system $\{H_i\mid i\in I\}$ of
algebraic groups indexed by a directed set $I$.  It is harmless and
convenient to assume that all the morphisms
$$h_{ji}:H_i \rightarrow H_j\qquad(i\ge j)\eqno(7.2)(a)$$
in the inverse system are surjective.  (In our examples, the $H_i$ will all be
certain finite coverings of a fixed group $H_1$.)  Then the limit morphisms
$$h_j:H\rightarrow H_j\eqno(7.2)(b)$$
are surjective as well.  The identity component of $H$ is by definition the
inverse limit of the identity components of the $H_i$.  To say that $H$ acts
on $Y$ means that for a cofinal set $I_Y \subset I$, the various $\{H_i\mid
i\in I_Y\}$ act compatibly on $Y$; we may as well assume (by shrinking $I$)
that they all do (as will be the case in our examples).  Each $H_i$ will have
an algebraic isotropy subgroup $H_{i,y}$ at a point $y$.  These groups again
form an inverse system of algebraic groups, whose limit is what we called
$H_y$.  It is easy to check that $A_y^{loc}$ (defined as a quotient of inverse
limits) is naturally isomorphic to the inverse limit of the quotients: 
$$A_y^{loc} \simeq \lim_{\longleftarrow} H_{i,y}/(H_{i,y})_0.
\eqno(7.2)(c)$$
Again we have surjective maps
$$a_i:A_y^{loc}\rightarrow H_{i,y}/(H_{i,y})_0.\eqno(7.2)(d)$$
By a (finite-dimensional) representation of $A_y^{loc}$ we understand a
representation that factors through one of the maps $a_i$ (and therefore
automatically through all $a_j$ with $j\ge i$).  For our purposes this can be
taken as a definition; but if $A_y^{loc}$ is given the inverse limit topology,
then it is a theorem that any continuous irreducible representation has this
property.

Here are some standard and elementary facts.

\medskip
{\bf Lemma 7.3.} {\it In the setting of Definition 7.1, suppose $y\in Y$. 
Write $S = H\cdot y$ for the orbit of $y$.
\smallskip
\item {a)}  If $H$ is algebraic, connected, and simply connected, then the
local equivariant fundamental group $A_y^{loc}$ is canonically isomorphic
to the topological fundamental group $\pi_1(S;y)$ of homotopy classes of
loops in $S$ based at $y$.
\item{b)}  More generally, there is always a natural map
$$\pi_1(S;y)\rightarrow A_y^{loc}.$$
This map is surjective whenever $H$ is connected algebraic, and injective
whenever $H$ is simply connected algebraic.  In particular, any
representation of $A_y^{loc}$ gives rise to a representation of $\pi_1(S;y)$.
\item{c)}  Suppose $(\tau,V)$ is a representation of $A_y^{loc}$.  Regard
$\tau$ as a representation of $H_y$ trivial on the identity component.  Then
the induced bundle
$${\cal V} = H \times_{H_y} V \rightarrow S$$
carries an $H$-invariant flat connection.
\item{d)}  Suppose ${\cal V}$ is an $H$-equivariant vector bundle on $S$
carrying an $H$-invariant flat connection.  Write $V$ for the fiber of ${\cal
V}$ at $y$, and $\tau$ for the isotropy action of $H_y$ on $V$.  Then $\tau$
is trivial on $(H_y)_0$, and so factors to a representation of $A_y^{loc}$.
\item{e)}  The constructions of (c) and (d) establish a natural bijection
between the equivalence classes of local complete geometric parameters
for $H$ acting on $Y$, and the complete geometric parameters.
\medskip}

\noindent We will use Lemma 7.3(e) to identify the set $\Xi(Y,H)$ of
equivalence classes of local complete geometric parameters.  Thus if
$\xi\in \Xi(Y,H)$, we will write
$$\xi = (S,{\cal V}) = (S_\xi,{\cal  V}_\xi),\eqno(7.4)$$ 
and say that a pair $(y,\tau)$ in the equivalence class ``represents $\xi$.'' 
The assertions about fundamental groups are included to explain how
$H$-equivariant local systems are related to arbitrary local systems.

{\it Proof.}  We begin by defining the map in (b).  Suppose
$$\gamma:[0,1]\rightarrow S,\quad \gamma(0)=\gamma(1)=y$$
is a loop.  Identify $S$ with the quotient $H/H_y$; then $\gamma$ can be
lifted to a continuous map
$$\tilde\gamma:[0,1]\rightarrow H, \quad \tilde\gamma(0) = e.$$
The element $h = \tilde\gamma(1)$ must map to $y$, so $h\in H_y$.  Any
two liftings $\tilde\gamma$ differ by a continuous map from $[0,1]$ into
$H_y$ starting at $e$.  Such a map must take values in $(H_y)_0$, so the
class of $\overline{h} \in A_y^{loc}$ is independent of the lifting. 
Similarly, a homotopy of the loop $\gamma$ does not affect $\overline{h}$;
so we get the map in (b).  To see that the map is surjective when $H$ is
connected algebraic, fix $h\in H_y$.  A path connecting $e$ to $h$ descends
to a loop in $H/H_y$ that is the required preimage of $\overline{h}$.  For the
injectivity when $H$ is simply connected algebraic, suppose $\gamma$ is a
loop at $y$ mapping to the class of $e$.  This means that the lifting
$\tilde\gamma$ satisfies $\tilde\gamma(1) = h\in (H_y)_0$.  Choose a path
connecting $h$ to the identity in $(H_y)_0$; modifying $\tilde\gamma$ by
this path gives a new lifting $\tilde\gamma'$ satisfying
$\tilde\gamma'(1)=e$.  This new lifting is therefore a loop.  Since $H$ is
simply connected, $\tilde\gamma'$ is homotopic to a trivial map.  This
homotopy descends to a homotopy from $\gamma$ to a point in $H/H_y$. 
This proves (b), and (a) follows.  

We omit the elementary arguments for (c) and (d), but a few remarks on the
definitions in the pro-algebraic case may be helpful.  For part (c), recall
that $\tau$ is really a representation of some component group
$H_{i,y}/(H_{i,y})_0$ (cf. (7.2)).  Then the algebraic vector bundle
$${\cal V} = H_i \times_{H_{i,y}} V \rightarrow S$$
carries an action of $H_i$, and thus compatible actions of all $H_j$ with
$j\ge i$.  Therefore it is an ``$H$-equivariant vector bundle.''  Conversely, to
say that ${\cal V}$ is $H$-equivariant means that it carries an action of
some $H_i$.  Consequently the isotropy action on $V$ factors to $H_{i,y}$. 
Part (e) is immediate from (c) and (d).  Q.E.D.

\medskip
{\bf Lemma 7.5.} {\it Suppose ${^\vee G^\Gamma}$ is a weak E-group,
$\phi\in P({^\vee G^\Gamma})$ is a Langlands parameter, and $x = p(\phi) \in
X({^\vee G^\Gamma})$ is a geometric parameter (cf. Proposition 6.17).  Then
the natural inclusion of isotropy groups
$${^\vee G}_\phi \hookrightarrow {^\vee G}_x$$
induces an isomorphism of component groups
$${^\vee G}_\phi/({^\vee G}_\phi)_0 \simeq {^\vee G}_x/ ({^\vee G}_x)_0.$$
If we consider isotropy groups in the algebraic universal cover ${^\vee
G}^{alg}$ (Definition 1.16 and (5.10)), we get
$${^\vee G}^{alg}_\phi/({^\vee G}^{alg}_\phi)_0 \simeq {^\vee G}^{alg}_x/
({^\vee G}^{alg}_x)_0.$$
Similarly, in the setting (5.13),
$${^\vee G}^Q_\phi/({^\vee G}^Q_\phi)_0 \simeq {^\vee G}^Q_x/ ({^\vee
G}^Q_x)_0.$$
\medskip}

{\it Proof.}  The inclusion of isotropy groups comes from the equivariance of
the map $p$.  By Proposition 6.17, ${^\vee G}_x$ is the semidirect product of
${^\vee G}_\phi$ and a connected unipotent group $U_x$.  It follows that
$({^\vee G}_x)_0$ must be the semidirect product of $({^\vee G}_\phi)_0$
and $U_x$.  The first claim follows.  The the second is a special case of the
third, so we consider the third.  The identity component $(U_x^Q)_0$ is a
covering group of $U_x$.  Now a unipotent algebraic group is simply
connected, so  $(U_x^Q)_0$ must map isomorphically onto $U_x$, and we
identify them henceforth.  It follows easily that ${^\vee G}_x^Q$ must be the
semidirect product of ${^\vee G}_\phi^Q$ and $U_x$.  Now we can argue as
for the first claim to get the isomorphism of component groups.  Q.E.D.

\medskip
{\bf Definition 7.6.}  Suppose ${^\vee G^\Gamma}$ is a weak E-group, and
$X({^\vee G^\Gamma})$ is the corresponding set of geometric parameters
(Definition 6.9).   In order to place ourselves precisely in the setting of
Definition 7.1, we can implicitly restrict attention to a single variety
$X({\cal O},{^\vee G^\Gamma})$ (see Proposition 6.16).  For $x\in X({^\vee
G^\Gamma})$, we consider the (local) equivariant fundamental groups for the
actions of ${^\vee G}$ and ${^\vee G}^{alg}$,
$$A_x^{loc} = {^\vee G}_x/({^\vee G}_x)_0,\qquad A_x^{loc,alg} = {^\vee
G}^{alg}_x/({^\vee G}^{alg}_x)_0.$$
By Lemma 7.5, there are natural isomorphisms
$$A_x^{loc}\simeq A_\phi^{loc},\qquad A_x^{loc,alg}\simeq
A_\phi^{loc,alg}$$
for any Langlands parameter $\phi\in p^{-1}(x)$ (cf. Definition 5.11).  We
therefore refer to these groups as the {\it Langlands component group for
$x$} and the {\it universal component group for $x$}.  If $S$ is an orbit on
$X({^\vee G^\Gamma})$, we write following Definition 7.1
$$A_S^{loc} = A_x^{loc}, \qquad A_S^{loc,alg} = A_x^{loc,alg} \qquad(x\in
S);$$
again these groups are defined up to inner automorphism.

A {\it (local) complete geometric parameter for ${^\vee G^\Gamma}$} is a
pair $(x,\tau)$ with $x\in X({^\vee G^\Gamma})$ and
$\tau\in\widehat{A_x^{loc,alg}}$ an irreducible representation.  Two such
parameters are called {\it equivalent} if they are conjugate by ${^\vee
G}^{alg}$.  Lemma 7.5 provides a natural map from complete Langlands
parameters for ${^\vee G^\Gamma}$ onto (local) complete geometric
parameters, inducing a bijection on equivalence classes.  We may therefore
write $\Xi({^\vee G^\Gamma})$ for the set of equivalence classes (cf.
Definition 5.11).

As in (7.4), we identify $\Xi({^\vee G^\Gamma})$ with the set of pairs
$(S,{\cal V})$, with $S$ an orbit on $X({^\vee G^\Gamma})$ and ${\cal V}$ an
irreducible ${^\vee G}^{alg}$-equivariant local system on $S$; we call such a
pair a {\it complete geometric parameter for $X({^\vee G^\Gamma})$}.  If
$\xi \in \Xi({^\vee G^\Gamma})$, we write
$$\xi = (S_\xi,{\cal V}_\xi)$$
as in (7.4).  A (local) complete geometric parameter $(x,\tau)$ in the
corresponding equivalence class will sometimes be called a {\it
representative of $\xi$}.

All of this discussion can be extended to the setting (5.13).  We write
$$A_x^{loc,Q} = {^\vee G}^Q_x/({^\vee G}^Q_x)_0,$$
the {\it $Q$-component group for $x$}.  The set of complete geometric
parameters for ${^\vee G}^Q$ acting on $X({^\vee G^\Gamma})$ is written
$\Xi({^\vee G^\Gamma})^Q$; we call these {\it complete geometric
parameters of type $Q$}.
\medskip

We return now to the general context of Definition 7.1, and expand on the
discussion in Definition 1.22.

\medskip
{\bf Definition 7.7.}  Suppose $Y$ is a smooth complex algebraic variety on
which the pro-algebraic group $H$ acts with finitely many orbits.  Write
${\cal D}_Y$ for the sheaf of algebraic differential operators on $Y$
([BorelD], VI.1).  Define
$${\cal C}(Y,H) = \hbox{ category of $H$-equivariant constructible sheaves
of complex vector spaces on $Y$.}$$
A definition of constructible sheaves may be found in [BorelIC], V.3.

Next, set
$${\cal P}(Y,H) = \hbox{ category of $H$-equivariant perverse sheaves of
complex vector spaces on $Y$.}$$
The definition of perverse sheaves is discussed in [BBD] (see especially the
introduction.)  Again it is most convenient to take the analytic topology on
$Y$, but the equivariance condition implies that one can work algebraically.
(See [BBD], section 6.  The main point is that all the local systems involved,
which are described by Lemma 7.3, arise from representations of $\pi_1$
factoring through finite quotients.)  This will be crucial for our eventual
application of the results of [LV].

Finally, we will consider
$${\cal D}(Y,H) = \hbox{ category of $H$-equivariant regular holonomic
sheaves of ${\cal D}_Y$-modules on $Y$.}$$
In this case it is most convenient to work in the Zariski topology; then
[BorelD] is a convenient reference.  For an (algebraic) definition of regular
holonomic, we refer to [BorelD], section VII.11.  (By [BorelD], Theorem
VII.12.11, any $H$-equivariant coherent sheaf of ${\cal D}_Y$-modules on
$Y$ is automatically regular holonomic.)  

Each of these three categories is abelian, and every object has finite length. 
The corresponding Gro\-then\-dieck groups are written
$$ K{\cal C}(Y,H),\qquad K{\cal P}(Y,H),\qquad K{\cal D}(Y,H).$$
Each is a free abelian group with a basis corresponding to the (finite) set of
irreducible objects in the corresponding category.
\medskip
The next two results recall the the most important relationships among
these three categories.  The first is essentially obvious from the
definitions.  (It is the definitions that are deep.)
\medskip
{\bf Lemma 7.8} ([BBD].)  {\it In the setting of Definition 7.7, suppose $P$ is
an $H$-equivariant perverse sheaf on $Y$.  Then the cohomology sheaves
$H^iP$ are $H$-equivariant constructible sheaves on $Y$.  A short exact
sequence of perverse sheaves gives rise to a long exact sequence of
cohomology sheaves, so there is an additive map
$$\chi:{\rom Ob}\,{\cal P}(Y,H) \rightarrow K{\cal C}(Y,H),\qquad \chi(P) =
\sum (-1)^iH^iP.$$
The map $\chi$ induces an isomorphism
$$\chi:K{\cal P}(Y,H)\rightarrow K{\cal C}(Y,H)$$
of Grothendieck groups.}
\medskip
{\bf Theorem 7.9} (the Riemann-Hilbert correspondence --- see [BorelD],
Theorem VIII.14.4).  {\it In the setting of Definition 7.7, there is an
equivalence of categories (the ``de Rham functor'')
$$DR:{\cal D}(Y,H) \rightarrow {\cal P}(Y,H).$$
This induces an isomorphism
$$DR:K{\cal D}(Y,H)\rightarrow K{\cal P}(Y,H)$$
of Grothendieck groups.}
\medskip

We will use the isomorphisms of Lemma 7.8 and Theorem 7.9 to identify the
three Grothendieck groups, writing simply 
$$K(Y,H)\simeq K{\cal C}(Y,H)\simeq K{\cal P}(Y,H) \simeq K{\cal
D}(Y,H)\eqno(7.10)(a)$$
We will also need notation for the irreducible objects in the three
categories.  These irreducible objects are in each case parametrized by the
set $\Xi(Y,H)$ of complete geometric parameters (Definition 7.1).  Fix $\xi =
(S,{\cal V}) \in \Xi$.  We need a little notation from [BBD], section 1.4. 
Write
$$j=j_S:S\rightarrow \overline{S},\qquad i = i_S:\overline{S}\rightarrow
Y\eqno(7.10)(b)$$
for the inclusion of $S$ in its closure, and the inclusion of the closure in
$Y$.  Write $d=d(S)=d(\xi)$ for the dimension of $S$.  Regard the local
system ${\cal V}$ as a constructible sheaf on $S$.  Applying to this first the
functor $j_!$ of extension by zero, and then the direct image $i_*$, we get a
complex
$$\mu(\xi) = i_*j_!{\cal V} \in {\rom Ob}\,{\cal C}(Y,H),\eqno(7.10)(c)$$
the {\it extension of $\xi$ by zero}.  Its irreducibility, and the fact that
every irreducible has this form, are consequences of the adjunction
formulas relating $i_*$ and $j_!$ with the restriction functors $i^*$ and
$j^*$ ([BBD], 1.4.1).  Next, consider the complex ${\cal V}[-d]$ consisting of
the single sheaf ${\cal V}$ in degree $-d$.  This is an $H$-equivariant
perverse sheaf on $S$.  Applying to it the ``intermediate extension functor''
$j_{!*}$ ([BBD], Definition 1.4.22) followed by the direct image $i_*$, we
get a perverse sheaf on $Y$:
$$P(\xi) = i_*j_{!*}{\cal V}[-d] \in {\rom Ob}\,{\cal P}(Y,H),\eqno(7.10)(d)$$
the {\it perverse extension of $\xi$}.  Its irreducibility, and the fact that
every irreducible has this form, are consequences of Th\'eor\`eme 4.3.1 of
[BBD].  Finally, we define
$$D(\xi) = DR^{-1}(P(\xi)) \in {\rom Ob}\,{\cal D}(Y,H).\eqno(7.10)(e)$$
It is a consequence of Theorem 7.9 that every irreducible $H$-equivariant
${\cal D}_Y$-module on $Y$ is of this form.  (Of course one can define
$D(\xi)$ somewhat more directly as a ${\cal D}_Y$-module.  The starting
point is the sheaf ${\cal M}$ of germs of sections of ${\cal V}$ on $S$. 
Since ${\cal V}$ has an $H$-equivariant flat connection, ${\cal M}$ has the
structure of a an $H$-equivariant ${\cal D}_S$-module ([BorelD], section
VI.1.6).  One now applies the (subtle) notion of direct image for ${\cal
D}$-modules to get a ${\cal D}_Y$-module.  The first complication is that
the inclusion $j$ need not be affine, so there are higher direct images to
consider.  The second is that (even if $j$ is affine) the direct image will not
be irreducible; one has to pass to an appropriate subquotient.  None of this is
particularly difficult, given the machinery of [BorelD], but it is easier
simply to quote Theorem 7.9.)

The sets $\{\mu(\xi)\mid \xi \in \Xi\}$, $\{P(\xi)\mid \xi \in \Xi\}$, and
$\{D(\xi)\mid \xi \in \Xi\}$ are all bases of the Grothendieck group $K(Y,H)$. 
By Theorem 7.9 and the definitions of (7.10), the last two bases are exactly
the same.  However, the first two are {\it not} identified by the
isomorphism of Lemma 7.8.  As in Definition 1.22, we can write in $K(Y,H)$
$$\mu(\xi) = (-1)^{d(\xi)}\sum_{\gamma \in \Xi(Y,H)}
m_g(\gamma,\xi)P(\gamma) \eqno(7.11)(a)$$
with $m_g(\gamma,\xi)$ an integer.  The matrix $m_g$ is called the {\it
geometric multiplicity matrix}.  It follows easily from the definitions that
$$m_g(\xi,\xi)=1,\qquad m_g(\gamma,\xi)\ne 0\hbox{ only if } S_\gamma
\subset \left( \overline{S_\xi} - S_\xi\right)\quad(\gamma\ne
\xi).\eqno(7.11)(b)$$
It is also useful to consider the inverse matrix $c_g = m_g^{-1}$, the {\it
geometric character matrix}.  By its definition, $c_g$ satisfies
$$P(\gamma) = \sum_{\xi \in \Xi(Y,H)} (-1)^{d(\xi)} c_g(\xi,\gamma)
\mu(\xi). \eqno(7.11)(c)$$
Because of the form of the isomorphism in Lemma 7.8, we have
$$c_g(\xi,\gamma) = (-1)^{d(\xi)} \sum_i (-1)^i \hbox{(multiplicity of ${\cal
V}_\xi$ in $H^iP(\gamma)|_{S_\xi}$)}.\eqno(7.11)(d)$$
This says that the entries of $c_g$ are essentially local Euler
characteristics for intersection cohomology.  As a consequence of (7.11)(d)
(or of (7.11)(b)), 
$$ c_g(\gamma,\gamma) = 1,\qquad c_g(\xi,\gamma)\ne 0\hbox{ only if }
S_\xi \subset \left( \overline{S_\gamma} - S_\gamma\right)\quad(\xi \ne
\gamma).\eqno(7.11)(e)$$
Because of (7.11)(b), we can define the {\it Bruhat order} on $\Xi(Y,H)$ as
the smallest partial order with the property that
$$m_g(\gamma,\xi) \ne 0 \hbox{ only if } \gamma \le \xi. \eqno(7.11)(f)$$
This refines the partial preorder induced by the closure relation on the
underlying orbits.  Because the inverse of an upper triangular matrix is
upper triangular, we get the same order using $c_g$ in place of $m_g$.

To understand why we call $m_g$ a multiplicity matrix, suppose for a
moment that the orbit $S_\xi$ is affinely embedded in its closure.  (For $Y =
X({\cal O},{^\vee G^\Gamma})$ as in Definition 6.9, this turns out to be
automatic if the orbit ${\cal O}$ consists of regular elements.)  Regarded as
an element of the derived category concentrated in degree $-d$, the
constructible sheaf $\mu(\xi)[-d]$ is then perverse ([BBD], Corollary 4.1.3),
and the map of Lemma 7.8 is
$$\chi(\mu(\xi)[-d]) = (-1)^d\mu(\xi).$$ 
On the other hand, as a perverse sheaf $\mu(\xi)[-d]$ must have a finite
composition series.  The integer $m_g(\gamma,\xi)$ is just the multiplicity
of the irreducible perverse sheaf $P(\gamma)$ in this composition series. 
In particular, it is non-negative.  One can give a parallel discussion in terms
of ${\cal D}$-modules.

We will be forced occasionally to refer to certain derived categories (which
are of course central to the basic theory developed in [BBD] and [BorelIC]). 
In general an object in one of these derived categories will be a complex
${\cal S}^\bullet$ of sheaves on $Y$; we will write $H^i({\cal S}^\bullet)$
for the $i$th cohomology sheaf.  The notion of $H$-equivariance in such
categories is quite subtle, and we will simply not consider it (although a
complete treatment of some of the ``obvious'' assertions we make requires
confronting this issue).  Write
$$D^b({\cal C}(Y))\eqno(7.12)(a)$$
for the bounded derived category of sheaves of complex vector spaces on $Y$
(analytic topology) having cohomology sheaves constructible with respect to
an algebraic stratification of $Y$ (cf. [BBD], section 2.2, and [BorelIC],
section 6.3).  Perverse sheaves on $Y$ live in this derived category.  Finally,
write
$$D^b_{rh}({\cal D}(Y))\eqno(7.12)(b)$$
for the bounded derived category of sheaves of quasicoherent ${\cal
D}_Y$-modules having regular holonomic cohomology sheaves (cf. [BorelD],
Corollary VII.12.8).

\medskip
{\bf Definition 7.13.}  Suppose ${^\vee G^\Gamma}$ is a weak E-group, and
$X({^\vee G^\Gamma})$ is the corresponding set of geometric parameters
(Definition 6.9).  Define
$${\cal C}(X({^\vee G^\Gamma}),{^\vee G}^{alg})$$
to be the direct sum over semisimple orbits ${\cal O}\subset {^\vee{\frak
g}}$ of the categories ${\cal C}(X({\cal O},{^\vee G^\Gamma}),{^\vee G}^{alg})$
of Definition 7.7.  We call the objects of this category {\it ${^\vee
G}^{alg}$-equivariant constructible sheaves on $X({^\vee G^\Gamma})$}. 
Similarly we define
$${\cal P}(X({^\vee G^\Gamma}),{^\vee G}^{alg}),\qquad {\cal D}(X({^\vee
G^\Gamma}),{^\vee G}^{alg}),$$
the {\it ${^\vee G}^{alg}$-equivariant perverse sheaves} and the {\it ${^\vee
G}^{alg}$-equivariant coherent ${\cal D}$-modules} on $X({^\vee
G^\Gamma})$.  These are still abelian categories in which every object has
finite length, and we have analogues of Lemma 7.8 and Theorem 7.9.  In
particular, we can write $KX({^\vee G^\Gamma})$ for the common
Grothendieck group of the three categories.  The irreducible objects in each
category are parametrized by the set $\Xi({^\vee G^\Gamma})$ of complete
geometric parameters (Definition 7.6); we write $\mu(\xi)$, $P(\xi)$, and
$D(\xi)$ as in (7.10).  We get matrices $m_g$ and $c_g$ satisfying (7.11),
the {\it geometric multiplicity matrix} and the {\it geometric character
matrix} for ${^\vee G^\Gamma}$.

In the setting (5.13), we introduce a superscript $Q$ in this notation.  Thus
for example $KX({^\vee G^\Gamma})^Q$ is the Grothendieck group of ${^\vee
G}^Q$-equivariant perverse sheaves (or constructible sheaves, or coherent
${\cal D}$-modules) on $X({^\vee G^\Gamma})$.
\medskip

We want to use structure theorems like Proposition 6.16 to study the
geometric multiplicity matrix.  To do that, we must first understand the
behavior of the categories of Definition 7.7 under some simple
constructions.

\medskip
{\bf Proposition 7.14.}  {\it Suppose $Y$ is a smooth complex algebraic
variety on which the pro-algebraic group $H$ acts with finitely many orbits.
Suppose $G$ is a pro-algebraic group containing $H$.  Consider the induced
bundle
$$X = G \times_H Y.$$
\smallskip
\item{a)} The inclusion
$$i:Y \rightarrow X,\qquad i(y) = \hbox{ equivalence class of $(e,y)$}$$
induces a bijection
$$\Phi(Y,H) \rightarrow \Phi(X,G), \qquad S \mapsto G\cdot i(S)$$
from $H$-orbits on $Y$ to $G$-orbits on $X$.
\item{b)} If $y\in Y$, then the isotropy group of the $G$ action on $X$ at
$i(y)$ is $H_y$.  In particular, the local equivariant fundamental groups
satisfy
$$A^{loc}_y = A^{loc}_{i(y)}$$
(notation as in Definition 7.1).  Consequently there is an isomorphism
$$A^{loc}_S \simeq A^{loc}_{\Phi(i)(S)},$$
canonically defined up to inner automorphisms.
\item{c)} The maps of (a) and (b) combine to give a natural bijection
$$i: \Xi(Y,H) \rightarrow \Xi(X,G).$$
In the language of (7.4), the $H$-equivariant local system ${\cal V}$ on $S$
corresponds to
$$i({\cal V}) = G \times_H {\cal V}$$
on $G\cdot i(S)$.
\item{d)}  There are natural equivalences of categories
$${\cal C}(Y,H)\simeq {\cal C}(X,G),\qquad {\cal P}(Y,H)\simeq {\cal
P}(X,G),\qquad {\cal D}(Y,H)\simeq {\cal D}(X,G).$$
These equivalences are compatible with the parametrizations of
irreducibles by $\Xi(Y,H)\simeq \Xi(X,H)$ ((7.10) and (c) above) and with the
isomorphisms of Grothendieck groups coming from Lemma 7.8 and Theorem
7.9.
\item{e)} The geometric multiplicity and character matrices of (7.11) for
$(Y,H)$ and $(X,G)$ coincide (under the bijection of (c) above for the index
sets).
\medskip}

{\it Proof.}  Parts (a), (b), and (c) are formal; in fact similar ideas appeared
already in Lemma 6.15.  The equivalences of categories are in some sense
``obvious'' once the categories are understood, and we will just offer some
hints.  We discuss only the perverse case; the constructible case is similar
but more elementary, and the ${\cal D}$-module case is handled by Theorem
7.9.  It is helpful to introduce the intermediate space
$$Z = G \times Y,$$
on which the group $G \times H$ acts by
$$(g,h)\cdot (g',y) = (gg'h^{-1}, h\cdot y).$$
It is easy to see that $\Xi(Z,G\times H)$ is in one-to-one correspondence
with $\Xi(Y,H)$.   The groups $G$ and $H$ each act freely on $Z$ (although
the product need not).  Dividing by the $G$ action gives a smooth quotient
morphism
$$f_G : Z\rightarrow Y$$
(projection on the second factor) that respects the action of $H$.  Dividing
by the $H$ action gives a smooth quotient morphism
$$f_H: Z \rightarrow X$$
(the definition of the induced bundle) that respects the action of $G$.  We
can therefore apply Proposition 4.2.5 of [BBD] to conclude that $f^*_G[\dim
G]$ (respectively $f^*_H[\dim H]$) is a fully faithful exact functor from
${\cal P}(Y,H)$ (respectively ${\cal P}(X,G)$) into ${\cal P}(Z,G\times H)$. 
That the irreducible objects behave properly under these functors is
contained in the discussion at the bottom of page 110 in [BBD].  We leave the
rest of the argument to the reader.  Part (e) is a formal consequence of (d). 
Q.E.D.
\medskip
{\bf Proposition 7.15.}  {\it Suppose $X$ and $Y$ are smooth algebraic
varieties on which the pro-algebraic group $H$ acts with finitely many
orbits, and
$$f: X \rightarrow Y$$
is a surjective smooth equivariant morphism having connected geometric
fibers of dimension $d$.
\smallskip
\item{a)}  There is a natural inclusion 
$$f^*:\Xi(Y,H) \hookrightarrow \Xi(X,H)$$
from complete geometric parameters for $H$ acting on $Y$ to complete
geometric parameters for $H$ acting on $X$.  
\item{b)}  The functor $f^*[d]$ is a fully faithful exact functor from ${\cal
P}(Y,H)$ into ${\cal P}(X,H)$, carrying the irreducible $P(\xi)$ to $P(f^*\xi)$
(notation (7.10)(d)).
\item{c)}  The inclusion of (a) respects the geometric character matrix of
(7.11):
$$c_g(f^*\xi,f^*\gamma) = c_g(\xi,\gamma)\qquad(\xi,\gamma \in
\Xi(Y,H).$$
\medskip}

{\it Proof.}  We begin by describing the map $f^*$ on geometric parameters. 
If $S$ is an orbit of $H$ on $Y$, of dimension $d(S)$, then $f^{-1}S$ is a
smooth $H$-invariant subset of $X$, of dimension $d(S) + d$.  Since $H$ has
only finitely many orbits on $X$, there is an open orbit $f^*S$ of $H$ on
$f^{-1}S$.  Since $f$ has connected fibers, $f^*S$ is unique, and is dense in
$f^{-1}S$.  This defines an injection on the sets of orbits
$$f^*:\Phi(Y,H) \rightarrow \Phi(X,H).\eqno(7.16)(a)$$
The image of $f^*$ consists of orbits $S'\subset X$ satisfying any of the
three equivalent properties
$$S'\hbox{ is open in }f^{-1}(f(S')),\qquad \dim f(S') = \dim S' - d, \qquad
\dim H_{f(x)} = \dim H_x + d\quad (x\in S').\eqno(7.16)(b)$$
Suppose $S'$ satisfies these conditions.  A typical fiber of the map $S'
\rightarrow f(S')$ is dense in the corresponding fiber of $f$, and is
therefore connected.  Consequently
$$H_{f(x)}/H_x \hbox{ is connected}\qquad (x\in S'\in f^*(\Phi(Y,H)).
\eqno(7.16)(c)$$
For the natural map on component groups, it follows that
$$A_x^{loc} \rightarrow A_{f(x)}^{loc} \hbox{ is surjective}\qquad (x\in
S'\in f^*(\Phi(Y,H)). \eqno(7.16)(d)$$
Consequently every irreducible representation of $A_{f(x)}^{loc}$ defines by
composition an irreducible representation of $A_x^{loc}$.  This construction
clearly provides the inclusion of (a) on the level of equivalence classes of
local complete geometric parameters.  The image of $f^*$ consists of
equivalence classes of pairs $(x,\tau)$ for which $x$ and its orbit $S'$
satisfy (7.16)(b), and the representation $\tau$ is trivial on the kernel of
the map (7.16)(d).  On the level of local systems, the map $f^*$ sends
$(S,{\cal V})$ to $(S',{\cal V}')$, with
$$S' \hbox{ open in }f^{-1}S, {\cal V}' = (f^*{\cal V})\mid_{S'}.
\eqno(7.16)(e)$$
For the rest of the argument, we refer to [BBD], Proposition 4.2.5.  Q.E.D.

It is {\it not} true that the inclusion $f^*$ respects the geometric
multiplicity matrix in general.  The matrix $c_g$ for $Y$ can be regarded as
an upper left corner block of $c_g$ for $X$, but such a relationship between
two matrices is not preserved by inverting them.  (One might hope for help
from the fact, implicit in (7.11)(e), that $c_g$ is upper triangular. 
Unfortunately, the ordering of the basis needed to get upper triangular
matrices is incompatible with the ordering needed to make the matrix for
$Y$ a corner of the one for $X$.)

We conclude this section with a more general (and correspondingly weaker)
functorial relationship between different geometric parameters.  We begin
with two pairs $(X,G)$ and $(Y,H)$ as in Definition 7.7.  That is, $X$ and $Y$
are smooth algebraic varieties, on which the pro-algebraic groups $G$ and
$H$ act with finitely many orbits.  We suppose given in addition a morphism
$$\epsilon:(Y,H) \rightarrow (X,G). \eqno(7.17)(a)$$
This means that $\epsilon:Y \rightarrow X$ is a morphism of varieties,
$\epsilon: H \rightarrow G$ is a morphism of pro-algebraic groups, and
$$\epsilon(h)\cdot \epsilon(y) = \epsilon(h\cdot y) \qquad (h \in H, y \in Y).
\eqno(7.17)(b)$$
It follows immediately that $\epsilon$ induces a map from the set of orbits
of $H$ on $Y$ to the set of orbits of $G$ on $X$:
$$\Phi(\epsilon):\Phi(Y,H) \rightarrow \Phi(X,G),\qquad
\Phi(\epsilon)(H\cdot y) = G\cdot \epsilon(y) \eqno(7.17)(c)$$
(cf.~Definition 7.1).  Evidently $\epsilon$ carries the isotropy group $H_y$
into $G_{\epsilon(y)}$, so we get a homomorphism on the level of
equivariant fundamental groups
$$A^{loc}(\epsilon):A^{loc}_y \rightarrow A^{loc}_{\epsilon(y)}
\eqno(7.17)(d)$$
(Definition 7.1).

\medskip
{\bf Proposition 7.18.} {\it In the setting of (7.17), there is an exact functor
$$\epsilon^*:{\cal C}(X,G) \rightarrow {\cal C}(Y,H).$$
The stalk $(\epsilon^*C)_y$ at a point $y\in Y$ is naturally identified with
$C_{\epsilon(y)}$.  In particular, the representation of $A^{loc}_y$ on this
stalk is obtained from the representation of $A^{loc}_{\epsilon(y)}$ on
$C_{\epsilon(y)}$ by composition with the homomorphism
$A^{loc}(\epsilon)$ of (7.17)(d).

The functor $\epsilon^*$ induces a homomorphism of Grothendieck groups
$$\epsilon^*:K(X,G) \rightarrow K(Y,H)$$
(notation (7.10)(a)).}
\medskip
 
This is obvious: the functor $\epsilon^*$ is just the usual inverse image for
constructible sheaves.  This functor behaves very well on the derived
category $D^b({\cal C}(X))$, but not on the abelian subcategory of perverse
sheaves.  If $P$ is a $G$-equivariant perverse sheaf on $X$, then
$\epsilon^*P$ will be only an $H$-equivariant constructible complex on $Y$;
its ordinary cohomology sheaves and its perverse cohomology sheaves will
both live in several different degrees.  Proposition 7.18 computes the
homomorphism $\epsilon^*$ in the basis of irreducible constructible
sheaves.  We will make this explicit in Proposition 23.7.

Here is an example.  Suppose we are in the setting of (5.14).  That is, we
have an L-homomorphism $\epsilon:{^\vee H^\Gamma} \rightarrow{^\vee
G^\Gamma}$, and compatible pro-finite extensions ${^\vee H}^{Q_H}$ and
${^\vee G}^Q$.  The maps $X(\epsilon)$ of Corollary 6.21 and
$\epsilon^\bullet$ of (5.14)(c) give
$$\epsilon=(X(\epsilon),\epsilon^\bullet):(X({^\vee H^\Gamma}),{^\vee
H}^{Q_H})) \rightarrow (X({^\vee G^\Gamma}),{^\vee G}^Q) \eqno(7.19)(a)$$
as in (7.17).  In particular, we have the correspondence of orbits 
$$\Phi(\epsilon):\Phi({^\vee H^\Gamma}) \rightarrow \Phi({^\vee
G^\Gamma}) \eqno(7.19)(b)$$
and the homomorphisms of equivariant fundamental groups 
$$A^{loc}(\epsilon):A^{loc,Q_H}_y \rightarrow A^{loc,Q}_{\epsilon(y)}
\eqno(7.19)(c)$$
given by (7.17).  These are clearly identified (via Proposition 6.17 and
Lemma 7.5) with the ones constructed in Proposition 5.4 and (5.14)(d). 
Finally, we have from Proposition 7.18 a natural homomorphism
$$\epsilon^*:KX({^\vee G^\Gamma})^Q \rightarrow KX({^\vee
H^\Gamma})^{Q_H}. \eqno(7.19)(d)$$
We will see in section 26 that this last homomorphism provides the formal
part of the ``Langlands functoriality'' associated to the L-homomorphism
$\epsilon$.

%\end
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\vskip 3pc
\centerline{\bf 8. Perverse sheaves on the geometric parameter space.}
\vskip 1pc

In this section we begin to study in detail the categories of Definition 7.13. 
Specifically, we want to be able to compute the geometric multiplicity and
character matrices of (7.11).  After the formalities of Propositions 7.14 and
7.15, our main tools will be the results of [BB], [BorBIII], and [LV].  We begin
by recalling the first of these.  

Suppose $Y$ is a smooth complex algebraic variety.  Recall from Definition
7.7 the sheaf ${\cal D}_Y$ of algebraic differential operators on $Y$.  We
write
$$D_Y = \Gamma {\cal D}_Y \eqno(8.1)(a)$$
for the algebra of global sections of ${\cal D}_Y$, the {\it algebra of global
differential operators on $Y$}.  If $G$ is an algebraic group acting on $Y$,
then every element of the Lie algebra of $G$ defines a global vector field on
$Y$; that is, a first order differential operator (without constant term). 
This mapping extends to an algebra homomorphism
$$\psi_Y:U({\frak g}) \rightarrow D_Y\eqno(8.1)(b)$$
called the {\it operator representation of $U({\frak g})$} (see for example
[BorBI], section 3.1).  The kernel of $\psi_Y$ is a two-sided ideal
$$I_Y = \ker \psi_Y \subset U({\frak g}). \eqno(8.1)(c)$$
If ${\cal M}$ is any sheaf of ${\cal D}_Y$-modules, then the vector space $M
= \Gamma {\cal M}$ is in a natural way a $D_Y$-module, and therefore (via
$\psi_Y$) a module for $U({\frak g})/I_Y$.  The functor sending ${\cal M}$ to
the $U({\frak g})/I_Y$-module $M$ is called the {\it global sections
functor}.  In the other direction, if $M$ is any module for $U({\frak g})/I_Y$,
then we may form the tensor product
$${\cal M} = {\cal D}_Y \otimes_{\psi_Y(U({\frak g})/I_Y)} M.\eqno(8.1)(d)$$
This is a sheaf of ${\cal D}_Y$-modules on $Y$.  The functor sending $M$ to
${\cal M}$ is called {\it localization}.  For formal reasons, there is an
adjoint relationship
$${\rom Hom}_{{\cal D}_Y}({\cal D}_Y\otimes M,{\cal N}) \simeq {\rom
Hom}_{U({\frak g})/I_Y}(M,\Gamma{\cal N}). \eqno(8.1)(e)$$

We need to recall a little algebraic representation theory at this point.
\medskip
{\bf Definition 8.2.}  A {\it compatible pair $({\frak g},H)$} consists of a Lie
algebra ${\frak g}$ and a pro-algebraic group $H$, endowed with an
algebraic action 
$${\rom Ad}:H\rightarrow {\rom Aut}({\frak g}),$$
and an injective Lie algebra homomorphism
$$i:{\frak h} \rightarrow {\frak g}$$
compatible with the differential of ${\rom Ad}$.  A {\it compatible $({\frak
g},H)$-module} is a ${\frak g}$-module $M$ endowed with an algebraic
representation $\pi$ of $H$, satisfying the compatibility conditions
$$\pi(h)(X\cdot m) = ({\rom Ad}(h)(X))\cdot \pi(h) \cdot m $$
and 
$$d\pi(Z)\cdot m = i(Z)\cdot m $$
for $h\in H$, $X \in {\frak g}$, $m \in M$, and $Z\in {\frak h}$.  We allow $M$
to be infinite-dimensional; to say that the representation of $H$ is
algebraic means that each vector belongs to a finite-dimensional
$H$-invariant subspace on which $H$ acts algebraically in the obvious
sense.  Fix an ideal $I \subset U({\frak g})$.  The category of finite length
compatible $({\frak g},H)$-modules annihilated by $I$ is denoted 
$${\cal F}({\frak g},H,I).$$
\medskip
\noindent This idea is essentially due to Harish-Chandra; the definition was
formalized by Lepowsky. 
 
\medskip
{\bf Theorem 8.3} (Beilinson-Bernstein localization theorem --- see [BB],
[BorBI], Theorem 3.8, and [BorBIII], Theorem 1.9).  {\it Suppose $G$ is a
complex connected algebraic group, and $Y$ is a complete homogeneous
space for $G$.
\smallskip
\item{a)}  The operator representation $\psi_Y:U({\frak g}) \rightarrow
D_Y$ of (8.1)(b) is surjective.
\item{b)}   The global section and localization functors provide an
equivalence of categories between quasicoherent sheaves of ${\cal
D}_Y$-modules on $Y$ and modules for $U({\frak g})/I_Y$.
\item{c)}  Suppose $({\frak g},H)$ is a compatible pair (Definition 8.2), and
that $H$ acts (compatibly) with finitely many orbits on $Y$.  Then the
functors in (b) provide an equivalence of categories between ${\cal D}(Y,H)$
(Definition 7.7) and  ${\cal F}({\frak g},H,I_Y)$ (Definition 8.2).
\medskip}
To understand this result, one should keep in mind that the complete
homogeneous spaces for an algebraic group $G$ are precisely the quotients
$G/P$, with $P$ a parabolic subgroup.  The unipotent radical of $G$ must act
trivially on such a homogeneous space, so there is essentially no loss of
generality in assuming $G$ to be reductive.  In part (b), what is being
established is an analogue of the relationship between modules and
quasicoherent sheaves on affine algebraic varieties ([Hartshorne], Corollary
II.5.5).  The case of projective space (with $G$ equal to $GL(n)$) is treated
in [BorelD], VII.9. Once (b) is proved, part (c) is nearly formal; the only point
is to show that an $H$-equivariant coherent ${\cal D}_Y$-module is
automatically regular holonomic ([BorelD], Theorem VII.12.11).

For the rest of this section, we will fix
$${^\vee G^\Gamma},\qquad {\cal O}\subset {^\vee{\frak g}}\eqno(8.4)(a)$$
a weak E-group and a semisimple orbit in its Lie algebra.  Fix a canonical
flat $\Lambda\subset {\cal O}$, and define
$$e(\Lambda)\in {^\vee G},\qquad P(\Lambda) \subset {^\vee G}(\Lambda)
\eqno(8.4)(b)$$
as in (6.6).  Recall (Lemma 6.3) that $P(\Lambda)$ is a connected parabolic
subgroup of the reductive group ${^\vee G}(\Lambda)$.  As in Proposition
6.24, write
$${\cal P}^0(\Lambda) = {^\vee G}(\Lambda)_0\cdot \Lambda \subset {\cal O}
\eqno(8.4)(c)$$
for the indicated orbit of $\Lambda$ in the variety of all canonical flats in
${\cal O}$.  Proposition 6.5 identifies ${\cal P}^0(\Lambda)$ naturally with
the variety of conjugates of $P(\Lambda)$ in ${^\vee G}(\Lambda)_0$ (by
sending each flat in ${\cal P}^0(\Lambda)$ to its stabilizer).  In particular, 
$${\cal P}^0(\Lambda) \simeq {^\vee
G}(\Lambda)_0/P(\Lambda).\eqno(8.4)(d)$$
This space is a complete homogeneous space for ${^\vee G}(\Lambda)_0$,
and we will apply Theorem 8.3 to it.  Recall the orbit decomposition
$${\cal I}(\Lambda) = {\cal I}^0_1(\Lambda) \cup \ldots \cup {\cal
I}^0_s(\Lambda) \eqno(8.4)(e)$$
of Proposition 6.24, and choose a representative $y_j\in {\cal
I}^0_j(\Lambda)$ for each orbit.  Define $K^0_j$ as in Proposition 6.24, so
that
$${\cal I}^0_j(\Lambda) \simeq {^\vee G}(\Lambda)_0/K^0_j.\eqno(8.4)(f)$$
Recall from (5.10) that we write $K_j^{0,alg}$ for the preimage of $K_j^0$
in ${^\vee G}^{alg}$

\medskip
{\bf Theorem 8.5.} {\it Suppose ${^\vee G^\Gamma}$ is a weak E-group and
${\cal O}\subset {^\vee{\frak g}}$ a semisimple orbit in its Lie algebra.  Use
the notation of (8.4).  Then the category of ${^\vee G}^{alg}$-equivariant
perverse sheaves on $X({\cal O},{^\vee G}^{alg})$ (Definitions 6.9, 7.7, and
7.13) is equivalent to the direct sum of the categories ${\cal F}({^\vee{\frak
g}}(\Lambda),K_j^{0,alg},I_{{\cal P}^0(\Lambda)})$ (Definition 8.2) of
compatible modules of finite length annihilated by the kernel $I_{{\cal
P}^0(\Lambda)}$ of the operator representation on ${\cal P}^0(\Lambda)$
(see (8.1)).}
\medskip

\noindent This is an immediate consequence of Proposition 6.24, Proposition
7.14, Theorem 7.9, and Theorem 8.3.

Next, we want to reduce (that is, to reduce the calculation of the geometric
multiplicity and character matrices) to the case of regular $\Lambda$.  For
that we need a kind of ``translation functor'' for these categories.

\medskip
{\bf Definition 8.6.}  In the setting of (8.5), suppose ${\cal O}'$ is another
orbit of semisimple elements.  A {\it translation datum from ${\cal O}$ to
${\cal O}'$} is a ${^\vee G}$-conjugacy class ${\cal T}$ of pairs
$(\Lambda_1,\Lambda_1')$, subject to the following conditions.
\smallskip
\item{(a)} $\Lambda_1\subset {\cal O}$ and $\Lambda_1'\subset {\cal O}'$
are canonical flats.
\item{(b)} $P(\Lambda_1') \subset P(\Lambda_1)$, and $e(\Lambda_1) =
e(\Lambda_1')$.
\smallskip

There are several other ways of specifying the same data.  If we fix a
canonical flat $\Lambda \subset {\cal O}$, then it is equivalent to specify a
single canonical flat $\Lambda' \subset {\cal O}'$, subject to the conditions
\smallskip
\item{(c)} $P(\Lambda') \subset P(\Lambda)$, and $e(\Lambda) =
e(\Lambda')$.
\smallskip
\noindent (Necessarily $\Lambda'$ will be contained in ${\frak
p}(\Lambda)$.)  In this case two canonical flats define the same translation
datum if and only if they are conjugate by $P(\Lambda)$.

If we fix a point $\lambda\in {\cal O}$, then it is equivalent to specify
another point $\lambda' \in {\cal O}'$, subject to 
\smallskip
\item{(d)} $P(\lambda') \subset P(\lambda)$, $L(\lambda') \subset
L(\lambda)$, and $e(\lambda) = e(\lambda')$.
\smallskip
\noindent (Necessarily $\lambda'$ will belong to ${\frak l}(\lambda)$.)  In
this case two points $\lambda_1',\lambda_2'$ define the same translation
family if and only if they are conjugate by $L(\lambda)$.

Finally, we could fix a pair $(\lambda,T)$, with $T$ a maximal torus and
$\lambda \in {\cal O}\cap {\frak t}$.  Then the translation datum is
determined by the choice of another point $\lambda' \in {\cal O}'\cap {\frak
t}$, subject to
\smallskip
\item{(e)} For every root $\alpha$ of $T$ in ${^\vee G}$,
$$\alpha(\lambda) \in {\Bbb N}-\{0\} \Rightarrow \alpha(\lambda') \in
{\Bbb N}-\{0\},$$
and
$$\lambda - \lambda' \in X_*(T).$$
\smallskip
\noindent (Of course the lattice on the right here is the kernel of the
normalized exponential mapping $e$ on ${\frak t}$.)  Two such points
$\lambda_1',\lambda_2'$ determine the same translation datum if and only
if they are conjugate by the Weyl group of $T$ in $L(\lambda)$.)
\medskip

{\bf Lemma 8.7.}  {\it Suppose ${^\vee G^\Gamma}$ is a weak E-group and
${\cal O}\subset {^\vee{\frak g}}$ a semisimple orbit in its Lie algebra. 
Then there is a regular semisimple orbit ${\cal O}'\subset {^\vee{\frak g}}$
and a translation datum ${\cal T}$ from ${\cal O}$ to ${\cal O}'$ (Definition
8.6)}
\medskip

{\it Proof.}  We can use the last description (e) of translation data in
Definition 8.6.  With that notation, choose a set $R^+$ of positive roots for
$T$ in ${^\vee G}$ containing the roots of $T$ in $N(\lambda)$ (notation (6.1)
and (6.2)).  Let $\mu$ be any element of $X_*(T)$ on which the positive
roots take strictly positive values; for example, the sum of the positive
coroots.  Then $\lambda' = \lambda + \mu$ is a regular semisimple element
satisfying the conditions of Definition 8.6(e), so the corresponding
conjugacy class
$${\cal T} = {^\vee G}\cdot ({\cal F}(\lambda),{\cal F}(\lambda'))$$
is the translation datum we need.  Q.E.D.

\medskip
{\bf Proposition 8.8.} {\it Suppose ${^\vee G^\Gamma}$ is a weak E-group,
${\cal O}$, ${\cal O}'$ are semisimple orbits in its Lie algebra, and ${\cal
T}$ is a translation datum from ${\cal O}$ to ${\cal O}'$.  
\smallskip
\item{a)}  The set ${\cal T}$ is the graph of a ${^\vee G}$-equivariant
smooth proper morphism
$$f_{\cal  T}: {\cal F}({\cal O'})\rightarrow {\cal F}({\cal O})$$
(notation (6.10)).
\item{b)}  The morphism in (a) induces an equivariant smooth proper
morphism
$$f_{\cal T}: X({\cal O}',{^\vee G^\Gamma}) \rightarrow X({\cal O},{^\vee
G^\Gamma}).$$
Consequently (Proposition 7.15) the category of ${^\vee
G}^{alg}$-equivariant perverse sheaves on $X({\cal O},{^\vee G^\Gamma})$ is
equivalent to a full subcategory of ${^\vee G}^{alg}$-equivariant perverse
sheaves on $X({\cal O}',{^\vee G^\Gamma})$.  This inclusion respects the
geometric character matrix.
\smallskip
\noindent Fix a pair $(\Lambda, \Lambda') \in {\cal T}$.
\smallskip
\item{c)} The objects of (8.4) for $\Lambda$ and $\Lambda'$ are related by
$$e(\Lambda) = e(\Lambda'),\quad {^\vee G}(\Lambda)_0 = {^\vee
G}(\Lambda')_0, \quad {\cal I}(\Lambda) = {\cal I}(\Lambda').$$
The morphism of (a) restricts to a smooth ${^\vee
G}(\Lambda)_0$-equivariant proper morphism
$$f_{\cal T}:{\cal P}^0(\Lambda')\rightarrow {\cal P}^0(\Lambda).$$
\item{d)}  The kernel $I_{{\cal P}^0(\Lambda)}$ of the operator
representation of $U({^\vee{\frak g}}(\Lambda))$ contains the kernel
$I_{{\cal P}^0(\Lambda')}$.  Consequently (for any $y\in {\cal I}(\Lambda)$)
the category ${\cal F}({^\vee{\frak g}}(\Lambda),K^{0,alg}(y),I_{{\cal
P}^0(\Lambda)})$ (Definition 8.2) is a full subcategory of ${\cal
F}({^\vee{\frak g}}(\Lambda'),K^{0,alg}(y),I_{{\cal P}^0(\Lambda')})$.
\item{e)}  The inclusions of categories in (b) and (d) correspond under the
equivalences of Theorem 8.5.
\medskip}

{\it Proof.}  Part (a) is a consequence of Definition 8.6(b) and (6.10)(b). 
(That the morphism is projective follows from the fact that $P(\Lambda')$
is parabolic in ${^\vee G}(\Lambda)$ (Lemma 6.3), and therefore also in
$P(\Lambda)\subset {^\vee G}(\Lambda)$.)  The second condition in
Definition 8.6(b) implies that ${\cal C}({\cal O}) = {\cal C}({\cal O}')$
(notation 6.10(c)).  Now (b) follows from the description of the spaces as
fiber products in Proposition 6.16.  Part (c) is immediate from the
definitions; the last statement is just (a) applied to the group ${^\vee
G}(\Lambda)_0$.  Part (d) is just the obvious statement that the kernel of
the operator representation on a homogeneous space grows as the isotropy
group grows.  (It is instructive to find a description of the kernel in terms
of the isotropy subgroup.  Halfway through this exercise you will no longer
believe that the preceding assertion is obvious, but in the end you will see
that it is after all.)  Notice that the inclusion of categories in (d) is really
that: an object of the smaller category is by definition an object of the
larger one as well.  Part (e) is an immediate consequence of the dense
thicket of definitions from [BBD] and [BorelD] that it conceals; we will not
attempt even to sketch an argument.  Q.E.D.

Lemma 8.7 and Proposition 8.8 reduce the calculation of the geometric
character matrix --- and therefore also its inverse, the multiplicity matrix
--- to the case of a regular orbit.  We recall now very briefly how that case
is treated in [LV].  Suppose we are in the setting (8.4).  Write $\Xi({\cal
O},{^\vee G^\Gamma})$ for the corresponding set of complete geometric
parameters (Definition 7.6).  By Propositions 6.24 and 7.14, there is a
natural one-to-one correspondence
$$\Xi({\cal O},{^\vee G^\Gamma}) = \coprod_j \Xi({\cal P}^0(\Lambda),
K_j^{0,alg}).\eqno(8.9)(a)$$
This decomposition respects the geometric multiplicity and character
matrices.  (In particular, if $\xi$ and $\xi'$ correspond to different values
of $j$ on the right, then $m_g(\xi,\xi') = c_g(\xi,\xi') = 0.$)

Suppose now that ${\cal O}$ is regular.  Then the groups $L(\lambda)$ of
Lemma 6.3 are all maximal tori, so $P(\Lambda)$ is a Borel subgroup of
${^\vee G}(\Lambda)_0$, and the variety ${\cal P}^0(\Lambda)$ may be
identified with the variety ${\cal B}$ of Borel subgroups of ${^\vee
G}(\Lambda)_0$.  The parameter set corresponding to $j$ on the right in
(8.9)(a) is now
$${\cal D}_j = \Xi({\cal B},K_j^{0,alg}).\eqno(8.9)(b)$$
This is almost precisely the setting of Definition 1.1 in [LV].  (The only
difference, aside from a harmless change of ground field, is that our group
$K_j^{0,alg}$ is a covering of the fixed point group of an involution, rather
than the fixed point group itself.  The arguments of [LV] carry over to this
setting unchanged.)  For every pair $(\xi,\xi')$ of elements of ${\cal D}_j$,
Theorem 1.11 of [LV] provides a polynomial
$$P_g(\xi,\xi') \in {\Bbb Z}[u],\eqno(8.9)(c)$$
the {\it Kazhdan-Lusztig polynomial}.  The algorithm of [LV] for computing
this polynomial involves an action of the Hecke algebra associated to the
Weyl group of ${^\vee G}(\Lambda)_0$ on the free ${\Bbb
Z}[u,u^{-1}]$-module with basis ${\cal D}_j$.  Theorem 1.12 of [LV] asserts
that these polynomials compute the geometric character matrix:
$$c_g(\xi,\xi') = (-1)^{d(\xi) - d(\xi')}P_g(\xi,\xi')(1).\eqno(8.9)(d)$$
To be a little more precise: (7.11)(d) says that $c_g(\xi,\xi')$ is an Euler
characteristic.  The theorem in [LV] says that the cohomology groups in the
Euler characteristic vanish in every other degree, and that the dimensions of
the remaining ones are given by the coefficients of $P_g(\xi,\xi')$.  We will
recall some additional details in section 16, in connection with the proof of
Theorem 1.24.
%\end
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\centerline{\bf 9. The Langlands classification for tori.}
\vskip 1pc

In this section we establish Theorem 1.18 (the bijection between
irreducible representations of strong real forms and complete geometric
parameters on an L-group) in the special case of a torus.  We will treat the
general case of Theorem 1.18 by reduction to this special case.  (Of course
such an approach seems unlikely to succeed for groups over other local
fields, where not all representations are constructed from characters of
tori.) 

So suppose that $T$ is an algebraic torus; that is, a connected complex
reductive abelian algebraic group.  As in Definition 2.10, we write
$$X_*(T) = {\rom Hom}_{alg}({\Bbb C}^\times,T),\eqno(9.1)(a)$$
the {\it lattice of (rational) one-parameter subgroups of $T$}.  We can use
the exponential map to identify the Lie algebra of ${\Bbb C}^\times$ with
${\Bbb C}$.  A one-parameter subgroup is determined by its differential,
which is a complex-linear map from ${\Bbb C}$ to ${\frak t}$.  Such a map is
in turn determined by its value at $1\in {\Bbb C}$; so $X_*(T)$ is identified
with a lattice in ${\frak t}$.  As is well-known ([AV2], Proposition 2.2)
$$X_*(T) \simeq (1/2\pi i)\ker(\exp) = \ker(e) \subset {\frak
t}.\eqno(9.1)(b)$$
Here $\exp$ refers to the exponential mapping in $T$, and $e$ is as in
(6.2)(a).  We can recover $T$ from $X_*(T)$ by the natural isomorphism
$$ X_*(T) \otimes_{\Bbb Z} {\Bbb C}^\times \simeq T,\quad \phi\otimes z
\mapsto \phi(z).\eqno(9.1)(c)$$
On the level of Lie algebras, this gives
$$X_*(T) \otimes_{\Bbb Z} {\Bbb C} \simeq {\frak t}, \quad \phi\otimes z
\mapsto d\phi(z)\eqno(9.1)(d)$$
$${\frak t}^* \simeq {\rom Hom}_{\Bbb Z}(X_*(T),{\Bbb C}),\quad \lambda
\mapsto (\phi \mapsto \lambda(d\phi(1))).\eqno(9.1)(e)$$
Dually, write
$$X^*(T) = {\rom Hom}_{alg}(T,{\Bbb C}^\times),\eqno(9.2)(a)$$
the {\it lattice of (rational) characters of $T$}.  A character is determined
by its differential, which is a linear functional on ${\frak t}^*$; the
resulting lattice may be identified as
$$X^*(T) = \{\,\lambda\in {\frak t}^*\mid \lambda(X_*(T)) \subset {\Bbb
Z}\,\}.\eqno(9.2)(b)$$
This exhibits the natural pairing between $X_*(T)$ and $X^*(T)$ as the
restriction of the pairing between ${\frak t}$ and ${\frak t}^*$.  The
analogue of (9.1)(c) is
$$T\simeq {\rom Hom}_{\Bbb Z}(X^*(T),{\Bbb C}^\times),\quad t\mapsto
(\tau\mapsto \tau(t)).\eqno(9.2)(c)$$
Similarly,
$${\frak t}\simeq {\rom Hom}_{\Bbb Z}(X^*(T),{\Bbb C}), \quad X\mapsto
(\tau\mapsto d\tau(X))\eqno(9.2)(d)$$
$$X^*(T) \otimes_{\Bbb Z} {\Bbb C} \simeq {\frak t}^*,\quad \tau\otimes
z\mapsto (X\mapsto zd\tau(X)).\eqno(9.2)(e)$$

By the remark after Definition 4.2, a dual group ${^\vee T}$ for $T$ is
determined up to unique isomorphism.  Explicitly, the formulas in (9.1)(c)
and (9.2)(c) show that we may choose
$${^\vee T} = X^*(T) \otimes_{\Bbb Z} {\Bbb C}^\times \quad\hbox{ or } \quad
{^\vee T} = {\rom Hom}_{\Bbb Z}(X_*(T),{\Bbb C}^\times).\eqno(9.3)(a)$$
In any case Definition 4.2, (9.1)(d)--(e), and (9.2)(d)--(e) provide natural
isomorphisms
$${^\vee {\frak t}} \simeq {\frak t}^*,\qquad {^\vee {\frak t}}^* \simeq
{\frak t}.\eqno(9.3)(b)$$

\medskip
{\bf Proposition 9.4} (Langlands --- see [Langlands], Lemma 2.8, or [AV2],
Theorem 4.4.) {\it Suppose $T$ is an algebraic torus defined over ${\Bbb R}$. 
Fix an L-group $({^\vee T^\Gamma},{\cal D})$ for $T$ (Definition 4.6).  Then
there is a natural bijection
$$\phi \rightarrow \pi(\phi)$$
from the set $\Phi(T/{\Bbb R}) = \Phi({^\vee T^\Gamma})$ of equivalence
classes of Langlands parameters (Definitions 5.2 and 5.3) onto the set
$\Pi(T({\Bbb R}))$ of equivalence classes of irreducible representations
(that is, continuous homomorphisms into ${\Bbb C}^\times$) of $T({\Bbb
R})$.  In this bijection, the differential $d\pi(\phi)$ (an element of ${\frak
t}^*$) is identified by (9.3)(b) with the parameter $\lambda(\phi)\in
{^\vee{\frak t}}$ (cf. (5.8)(a)).  The trivial character of $T({\Bbb R})$ is
attached to the parameters $\phi({^\vee\delta},0)$ with ${^\vee\delta}\in
{\cal D}$ (notation (5.8)(a)); these constitute a single equivalence class.}
\medskip

\noindent Notice that the last assertion shows clearly the dependence of the
classification on the L-group structure (and not just the weak E-group).

This beautiful result is {\it not} as it stands a special case of Theorem
1.18, for it mentions neither strong real forms of $T({\Bbb R})$ nor
complete geometric parameters. We will analyze each of these ideas in turn
and compare them to get that result.  Our main tool is the following
elementary version of Pontriagin duality.

\medskip
{\bf Lemma 9.5.}  {\it Suppose $T$ is an algebraic torus; use the notation of
(9.2).  There is a natural inclusion-reversing bijection between algebraic
subgroups of $T$ and sublattices (by which we mean simply subgroups) of
$X^*(T)$.  To an algebraic subgroup $S\subset T$ corresponds the sublattice
$$L = \{\,\tau\in X^*(T)\mid \tau(s) = 1, \hbox{ all } s\in S\,\}.\eqno(a)$$
Dually, to a sublattice $L$ corresponds the algebraic subgroup
$$S = \{\,s\in T\mid \tau(s) = 1, \hbox{ all } \tau\in L\,\}.\eqno(b)$$

Suppose the subgroups $S_1\subset S_2$ correspond to sublattices $L_1
\supset L_2$.  Then restriction of characters from $T$ to $S_1$ defines a
natural isomorphism
$${\rom Hom}_{alg}(S_1/S_2,{\Bbb C}^\times) \simeq L_2/L_1.\eqno(c)$$

Suppose the subgroup $S$ corresponds to the sublattice $L$.  Then the
identity component $S_0\subset S$ corresponds to the lattice
$$L^0 = \{\,\tau\in X^*(T)\mid n\tau \in L \hbox{ for some positive integer
$n$}\,\}\supset L.\eqno(d)$$

Suppose $\theta$ is an algebraic automorphism of $T$, and $a$ is the
transpose automorphism of $X^*(T)$ (so that $\tau(\theta t) = (a\tau)(t)$
for all $\tau \in X^*(T)$ and $t\in T$).  Then the algebraic subgroup
$T^\theta$ of fixed points of $\theta$ corresponds to the sublattice
$$(1-a)X^*(T) = \{\,\tau - a\tau\mid \tau\in X^*(T)\,\}.\eqno(e)$$
Dually, the sublattice $X^*(T)^a$ of fixed points of $a$ corresponds to the
(connected) subgroup
$$(1-\theta)T = \{\,t(\theta t)^{-1}\mid t\in T\,\}.\eqno(f)$$}
\medskip
This is well-known and elementary; we omit the proof.

We turn now to the description of complete geometric parameters for tori.
\medskip
{\bf Proposition 9.6.}  {\it Suppose ${^\vee T^\Gamma}$ is a weak E-group
for the algebraic torus $T$.  Write $a$ for the corresponding automorphism
of the based root datum (Proposition 4.4) and $\theta$ for the automorphism
of ${^\vee T}$ defined by any element of ${^\vee T^\Gamma} - {^\vee T}$. 
Suppose $x\in X({^\vee T^\Gamma})$ is a geometric parameter (Definition
6.9).
\smallskip
\item{a)} The stabilizer ${^\vee T}_x$ of $x$ for the action of ${^\vee T}$ on
geometric parameters is the group ${^\vee T}^{\theta}$ of fixed points of
$\theta$.
\item{b)} The Langlands component group $A_x^{loc}$ is 
$${^\vee T}^{\theta}/{^\vee T}^{\theta}_0 = {^\vee T}^{\theta} /(1+\theta)
{^\vee T} $$.
\item{c)}  The group of characters of the Langlands component group is
given by a natural isomorphism
$$\widehat{A}_x^{loc} \simeq X_*(T)^{-a}/(1-a)X_*(T).$$
\medskip}

\noindent Since $T$ is a torus, geometric parameters are in one-to-one
correspondence with Langlands parameters (even before passing to
equivalence classes) by Proposition 6.17.  

{\it Proof.}  Part (a) is immediate from Proposition 6.16 (or from Corollary
5.9, or from the definitions directly).  Part (b) follows from (a).  Part (c)
follows from (b) using Lemma 9.5(c), (e) and (f).  Q.E.D.

We now extend this result to algebraic coverings.  Suppose $\widetilde
{^\vee T}$ is a finite covering of an algebraic torus ${^\vee T}$.  The lattice
of rational characters of $\widetilde {^\vee T}$ must contain $X^*({^\vee
T})$ as a sublattice of finite index:
$$X^*({^\vee T}) \subset X^*(\widetilde {^\vee T}).\eqno(9.7)(a)$$
We may therefore regard $X^*(\widetilde {^\vee T})$ as a lattice in the
rational vector space generated by $X^*({^\vee T})$:
$$ X^*(\widetilde {^\vee T}) \subset X^*({^\vee T})\otimes_{\Bbb Z} {\Bbb
Q}.\eqno(9.7)(b)$$
Conversely, any lattice in this vector space containing $X^*({^\vee T})$ as a
subgroup of finite index is the character lattice of a finite covering of
${^\vee T}$.  It follows that the group on the right (which is no longer
finitely generated, and so not really a ``lattice'') is the group of rational
characters of the pro-algebraic group ${^\vee T}^{alg}$.  Extending the
notation of (9.2), we write
$$X^*({^\vee T}^{alg}) = X^*({^\vee T})\otimes_{\Bbb Z} {\Bbb Q}.
\eqno(9.7)(c)$$
Equivalently,
$$X^*({^\vee T}^{alg}) = X_*(T)\otimes_{\Bbb Z} {\Bbb Q}. \eqno(9.7)(d)$$
\medskip
{\bf Proposition 9.8.}  {\it Suppose we are in the setting of Proposition 9.6;
use the notation there and in (9.7).  Write as usual ${^\vee T}_x^{alg}$ for the
preimage of ${^\vee T}_x$ in ${^\vee T}^{alg}$, and $({^\vee T}_x^{alg})_0$ for
its identity component.
\smallskip
\item{a)} The group of rational characters of ${^\vee T}^{alg}$ trivial on
${^\vee T}_x^{alg}$ is identified via (9.7)(d) with $(1-a)X_*(T)$.
\item{b)} The group of rational characters of ${^\vee T}^{alg}$ trivial on
$({^\vee T}_x^{alg})_0$ is identified via (9.7)(d) with $(X_*(T)\otimes_{\Bbb
Z} {\Bbb Q})^{-a}.$
\item{c)} The group of characters of the universal component group
(Definition 7.6) is given by a natural isomorphism
$$\widehat{A}_x^{loc,alg} \simeq (X_*(T)\otimes_{\Bbb Z} {\Bbb Q})^{-a}
/(1-a)X_*(T).$$
\medskip}

The proof is parallel to that of Proposition 9.6, so we omit it.  (For (c) we
are using the fact that any rational character of a pro-algebraic subgroup of
a reductive abelian pro-algebraic group extends to the whole group.  This is
a formal consequence of the corresponding fact for algebraic groups.)

Next, we consider the notion of strong real form in the case of tori.  We
make use of the following elementary result, which is in some sense dual to
Lemma 9.5.

\medskip
{\bf Lemma 9.9.}  {\it Suppose $T$ is an algebraic torus; use the notation
(9.1).  Write $T^{fin}$ for the subgroup of $T$ consisting of elements of
finite order.  Define 
$${\frak t}_{\Bbb Q} = X_*(T) \otimes_{\Bbb Z} {\Bbb Q} \subset {\frak t}$$
(cf. (9.1)(d)).
\smallskip
\item{a)} The normalized exponential mapping $e$, given by $e(\tau) =
\exp(2\pi i\tau)$ (cf. (6.2)(a)) defines an isomorphism
$${\frak t}/X_*(T) \simeq T.$$
\item{b)} The preimage of $T^{fin}$ under the isomorphism in (a) is
precisely ${\frak t}_{\Bbb Q}$, so we have a natural isomorphism
$${\frak t}_{\Bbb Q}/X_*(T) \simeq T^{fin}.$$
\item{c)} Suppose $\sigma$ is an antiholomorphic automorphism of $T$. 
Then the isomorphism of (a) carries $-d\sigma$ to $\sigma$:
$$e(-d\sigma(\tau)) = \sigma(e(\tau)).$$
\item{d)}  In the setting of (c), write $a$ for the automorphism of $X_*(T)$
induced by $\sigma$ (Proposition 2.12).  Write $a$ also for the
corresponding (${\Bbb Q}$-linear) automorphism of ${\frak t}_{\Bbb Q}$. 
Then the isomorphism of (b) carries $-a$ to $\sigma$:
$$e(-a(\tau)) = \sigma(e(\tau)).$$
In particular, $a = d\sigma$ on ${\frak t}_{\Bbb Q}$. 
\medskip}

{\it Proof.}  Part (a) is immediate from (9.1)(b).  Part (b) follows from (a). 
Part (c) is clear.  For (d), one must examine the definition of $a$ in the proof
of Proposition 2.12, and use the fact that if $z\in {\Bbb C}^\times$ has
finite order, then $\overline{z} = z^{-1}$.  Q.E.D.

\medskip
{\bf Proposition 9.10.} {\it Suppose $T^\Gamma$ is a weak extended group
for $T$ (Definition 2.13).  Write $\sigma$ for the antiholomorphic involution
on $T$ defined by conjugation by any element of $T^\Gamma - T$.  Consider
the three subgroups
$$\eqalign{T^{-\sigma,fin} &= \{\,t\in T\mid t\sigma(t) \hbox{ has finite
order}\,\}\cr
T^{-\sigma} &= \{\,t\in T\mid t\sigma(t) = 1\,\}\cr
T^{-\sigma}_0 &= \hbox{ identity component of } T^{-\sigma} =
\{\,s\sigma(s^{-1})\mid s\in T\,\}.\cr}$$
\smallskip
\item{a)} Left multiplication defines a natural simply transitive action of
$T^{-\sigma,fin}$ on the set of strong real forms of $T^\Gamma$ (Definition
2.13).
\item{b)} Two strong real forms are equivalent precisely when they lie in
the same coset of $T^{-\sigma}_0$.  Consequently there is a natural simply
transitive action of the quotient $T^{-\sigma,fin}/T^{-\sigma}_0$ on the set
of equivalence classes of strong real forms.  
\item{c)}  The mapping $\tau \mapsto e(\tau/2)$ (cf. Lemma 9.9)  maps the
$-1$ eigenspace ${\frak t}_{\Bbb Q}^{-a}$ into $T^{-\sigma,fin}$.  The
preimage of the subgroup $T^{-\sigma}$ is the lattice $X_*(T)^{-a}$.  The
preimage of the subgroup $T^{-\sigma}_0$ is the lattice $(1-a)X_*(T)$.
\item{d)}  After composition with the quotient, the map of (c) becomes a
surjection onto $T^{-\sigma,fin}/T^{-\sigma}_0$.
\item{e)}  There are natural isomorphisms
$${\frak t}_{\Bbb Q}^{-a}/(1-a)X_*(T) \simeq
T^{-\sigma,fin}/T^{-\sigma}_0$$
$$X_*(T)^{-a}/(1-a)X_*(T) \simeq T^{-\sigma}/T^{-\sigma}_0.$$
\medskip}

{\it Proof.}   The first thing to prove is the equivalence of the two
definitions of $T^{-\sigma}_0$.  The Lie algebra of $\{s\sigma(s^{-1})\}$ is
$\{\tau -d\sigma(\tau)\}$.  Since $\sigma$ has order two, this is equal to
the $-1$-eigenspace of $d\sigma$, which in turn is the Lie algebra of
$T^{-\sigma}$.  Since $\{s\sigma(s^{-1})\}$ is clearly connected, the claim
follows.  

Now consider (a).  Suppose $t \in T$, and $\delta$ is a strong real form. 
Then
$$(t\delta)^2 = t(\delta t \delta^{-1})\delta^2 = t\sigma(t)\delta^2.$$
Since $\delta^2$ is assumed to have finite order, it follows that $t\delta$
is a strong real form if and only if $t\in T^{-\sigma,fin}$.  A similar
calculation shows that
$$s(t\delta)s^{-1} = t(s\sigma(s^{-1}))\delta,$$
from which (b) follows.

For (c), suppose that $\tau \in {\frak t}_{\Bbb Q}^{-a}$.  Then 
$$e(\tau/2)\sigma(e(\tau/2)) = e(\tau/2)e(-a\tau/2) = e(\tau/2)e(\tau/2) =
e(\tau).$$
Here we use successively Lemma 9.9(d) and the assumption on $\tau$.  Now
the first two claims of (c) are clear.  

For the rest of the argument, we will need to use the fact that
$$T = T^{-\sigma}_0 T^{\sigma}_0.\eqno(9.11)$$
(The two factors may have a non-trivial intersection; the claim is simply
that every element of $T$ is a product.)  This follows from the
corresponding assertion on the Lie algebra, which in turn follows from the
fact that $\sigma$ has order 2.

For the third part of (c), suppose that $\tau \in {\frak t}_{\Bbb Q}^{-a}$ and
that $e(\tau/2) = s\sigma(s^{-1})$.  Because of (9.11), we may as well
assume that $s \in T^{-\sigma}_0$.  By Lemma 9.9(c), $s = e(\beta)$, with
$\beta$ in the $+1$-eigenspace of $d\sigma$.  The assumption on $s$ says
(in light of Lemma 9.9(a) and (c)) that for some $\gamma\in X_*(T)$,
$$\tau/2 = \beta/2 +d\sigma(\beta/2) +\gamma = \beta + \gamma.$$
The first conclusion is that $\beta\in {\frak t}_{\Bbb Q}$.  By Lemma 9.9(c),
$\beta$ is in the $+1$-eigenspace of $a$.  Consequently $\tau/2$ is the
projection of $\gamma$ on the $-1$-eigenspace of $a$.  That is, 
$$\tau/2 = (\gamma-a\gamma)/2,$$
which gives the third part of (c).

For (d), (9.11) and Lemma 9.9(c) imply that every coset in
$T^{-\sigma,fin}/T^{-\sigma}_0$ has a representative $t = e(\tau/2)$ with
$\tau$ in the $-1$-eigenspace of $d\sigma$.  As in (a) we compute
$t\sigma(t) = e(\tau)$; so $\tau$ must actually belong to the
$-1$-eigenspace of $d\sigma$ on ${\frak t}_{\Bbb Q}$.  By Lemma 9.9(d),
$\tau$ belongs to ${\frak t}^{-a}_{\Bbb Q}$, as we wished to show.

Part (e) is an immediate consequence of (c) and (d).  Q.E.D.

\medskip
{\bf Corollary 9.12.} {\it Suppose ${^\vee T^\Gamma}$ is a weak E-group for
the algebraic torus $T$ (Definition 4.3), and $T^\Gamma$ is a weak extended
group for the corresponding real form (Definition 2.13).  Suppose $x\in
X({^\vee T^\Gamma})$ is a geometric parameter (Definition 6.9).  Then there
is a natural simply transitive action of the group ${\widehat A}^{loc, alg}$
(of characters of the universal component group) on the set of equivalence
classes of strong real forms of $T^\Gamma$.}
\medskip
{\it Proof.}  Combine Proposition 9.8(c) with Proposition 9.10(d).  Q.E.D.

{\it Proof of Theorem 1.18 for tori.}  The extra datum ${\cal W}$ needed to
make an extended group for a torus from a weak extended group is precisely
an equivalence class of strong real forms (cf. Definition 1.12).  The action of
Corollary 9.12 therefore gives a natural bijection between strong real forms
and characters of the universal component group (sending the distinguished
class of strong real forms to the trivial character).  In conjunction with
Proposition 9.4, this gives Theorem 1.18 for tori.  Q.E.D.
%\end
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\centerline{\bf 10. Covering groups and projective representations.}
\vskip 1pc

In order to deduce Theorem 1.18 from the special case established in the
last section, we will need to exploit a relationship between characters of
tori in $G$ and representations of $G$.  This relationship is most natural
when it is formulated in terms of certain coverings of the tori related to
``$\rho$-shifts'' for $G$ (see for example Theorem 1.37 or Theorem 6.8 in
[Orange]).  We will therefore need for tori a version of Theorem 1.18 that
describes representations of such coverings.  It is just as easy to treat
coverings of general groups; in any case this will be necessary when we
discuss endoscopy.

\medskip
{\bf Definition 10.1.}  Suppose $G^\Gamma$ is a weak extended group for $G$
(Definition 2.13).  A connected finite covering group
$$1\rightarrow F\rightarrow {\widetilde G} \rightarrow G\rightarrow 1$$
is said to be {\it distinguished} if the following two conditions are
satisfied. 
\smallskip
\item{(a)}  For every $x\in G^\Gamma-G$, the conjugation action $\sigma_x$
of $x$ on $G$ lifts to an automorphism ${\widetilde \sigma}_x$ of
${\widetilde G}$.
\smallskip
\noindent It is equivalent to require this condition for a single $x$.  As $x$
varies, ${\widetilde \sigma}_x$ changes by inner automorphisms; so its
restriction ${\widetilde \sigma}_Z$ to $Z({\widetilde G})$ is independent of
the choice of $x$.  Necessarily ${\widetilde \sigma}_Z$ preserves the
subgroup $F$ of $Z({\widetilde G})$.  We can now formulate the second
condition on ${\widetilde G}$.
\smallskip
\item{(b)} The automorphism ${\widetilde \sigma}_Z$ sends every element
of $F$ to its inverse.
\smallskip
\noindent Write $G^{sc}$ for the simply-connected covering group of $G$,
and $\pi_1(G)$ for the kernel of the covering map.  This covering
automatically satisfies the condition analogous to (a) above, so that we can
define an involutive automorphism ${\sigma}^{sc}_Z$ of $Z(G^{sc})$, which
preserves the subgroup $\pi_1(G)$.  We may regard $F$ as a quotient of
$\pi_1(G)$:
$$1\rightarrow K \rightarrow \pi_1(G) \rightarrow F \rightarrow 1.$$
Then condition (a) amounts to the requirement that $K$ should be preserved 
by ${\sigma}^{sc}_Z$, and condition (b) to
$$K\supset \{\,y\cdot{{\sigma}^{sc}_Z}(y)\mid y\in \pi_1(G)\,\}.$$

The {\it canonical covering} $G^{can}$ is the projective limit of all the
distinguished coverings of $G$.  This is a pro-algebraic group, of which each
finite-dimensional representation factors to some distinguished finite
cover of $G$.  We write (in analogy with (5.10))
$$1\rightarrow \pi_1(G)^{can} \rightarrow G^{can} \rightarrow
G\rightarrow 1.$$
The group $\pi_1(G)^{can}$ depends on the inner class of real forms under
consideration.  It is a pro-finite abelian group, the inverse limit of certain
finite quotients of $\pi_1(G)$.  
\medskip
{\bf Lemma 10.2.}  {\it Suppose $G$ is a connected complex reductive
algebraic group, and ${^\vee G}$ is a dual group (Definition 4.2).
\smallskip
\item{a)} The group of complex characters of $\pi_1(G)$ is naturally
isomorphic to the center of ${^\vee G}$:
$$Z({^\vee G}) \simeq {\rom Hom}(\pi_1(G),{\Bbb C}^\times).$$
Write this isomorphism as $z \mapsto \chi_z$.
\smallskip
Suppose $G$ is endowed with an inner class of real forms.  Write
$\sigma_Z^{sc}$ for the induced action of any of these real forms on
$\pi_1(G)$.  Dually, write $a$ for the automorphism of the based root datum
of $G$ defined by the inner class of real forms (Definition 2.12), and
$\theta_Z$ for the automorphism of $Z({^\vee G})$ induced by any
automorphism of ${^\vee G}$ corresponding to $a$ (cf. Proposition 4.4).
\smallskip
\item{b)} The automorphisms $\sigma_Z^{sc}$ and $\theta_Z$ composed
with inversion are transposes of each other with respect to the
isomorphism in (a).  That is, if $z \in Z({^\vee G})$ and $p\in \pi_1(G)$, then
$$\chi_z(p) = \chi_{\theta_Z(z)^{-1}}(\sigma_Z^{sc}(p)).$$
\item{c)} The character $\chi_z$ is trivial on the subgroup
$\{p\sigma_Z^{sc}(p)\}$ of $\pi_1(G)$ if and only if $\theta_Z(z) = z$.
\item{d)}  The group of continuous characters of $\pi_1(G)^{can}$ (Definition
10.1) is naturally isomorphic to the group of elements of finite order in
$Z({^\vee G})^{\theta_Z}$.
\medskip}

{\it Proof.}  Part (a) is well-known and elementary (see for example [AV2],
Proposition 10.1).  Part (b) is immediate from the definitions, and (c)
follows from (b).  Part (d) is a consequence of (c) and the description of
$\pi_1(G)^{can}$ in Definition 10.1.  (Recall that a continuous character of a
pro-finite group factors through any large enough finite quotient.)  Q.E.D.
\medskip
{\bf Definition 10.3} (cf. Definition 2.13).  Suppose $G^\Gamma$ is a weak
extended group for $G$; use the notation of Definition 10.1 and Proposition
10.2.  If $\delta$ is a strong real form of $G^\Gamma$ (Definition 2.13),
write $G({\Bbb R},\delta)^{can}$ for the preimage of $G({\Bbb R},\delta)$ in
$G^{can}$.  There is a short exact sequence
$$1\rightarrow \pi_1(G)^{can} \rightarrow G({\Bbb R},\delta)^{can}
\rightarrow G({\Bbb R},\delta)\rightarrow 1.$$

A {\it canonical projective representation of a strong real form of
$G^\Gamma$} is a pair $(\pi,\delta)$, subject to
\smallskip
\item{(a)} $\delta$ is a strong real form of $G^\Gamma$; and
\item{(b)} $\pi$ is an admissible representation of $G({\Bbb
R},\delta)^{can}$.
\smallskip
\noindent Equivalence is defined as in Definition 2.13.  Suppose $z\in
Z({^\vee G})^{\theta_Z}$.  We say that $(\pi,\delta)$ is {\it of type $z$} if
the restriction of $\pi$ to $\pi_1(G)^{can}$ is a multiple of $\chi_z$
(Proposition 10.2(a)).  (Notice that every irreducible canonical projective
representation has a unique type.)  Finally, define
$$\Pi^z(G^\Gamma) = \Pi^z(G/{\Bbb R})$$
to be the set of infinitesimal equivalence classes of irreducible canonical
projective representations of type $z$.
\medskip
With this definition, we can formulate a mild generalization of Theorem
1.18.  (It is important to remember that the coverings appearing in
Definition 10.3 are all linear.  Extending the Langlands classification to
non-linear coverings is a much more difficult and interesting problem.)
\medskip
{\bf Theorem 10.4.} {\it Suppose $(G^\Gamma,{\cal W})$ is an extended group
for $G$ (Definition 1.12), and $({^\vee G^\Gamma},{\cal D})$ is an E-group for
the corresponding inner class of real forms (Definition 4.6).  Write $z$ for
the second invariant of the E-group (Definition 4.6).  Then there is a natural
bijection between the set $\Pi^z(G/{\Bbb R})$ of equivalence classes of
irreducible canonical projective representations of strong real forms of $G$
of type $z$ (Definition 10.3), and the set $\Xi^z(G/{\Bbb R})$ of complete
geometric parameters for ${^\vee G^\Gamma}$ (Definition 7.6 and Definition
5.11).}
\medskip
We will give the proof in the next four sections.  For the moment, we record
only the corresponding generalization of Proposition 9.4.  It is helpful first
to describe the rational characters of the pro-algebraic group $T^{can}$
when $T$ is an algebraic torus defined over ${\Bbb R}$.  A calculation
analogous to (9.7) gives
$$X^*(T^{can}) = \{\,\lambda \in X_*({^\vee T})\otimes_{\Bbb Z} {\Bbb Q}
\mid \lambda - \theta\lambda \in X_*({^\vee T})\}.\eqno(10.5)(a)$$
Following Lemma 9.9, we write
$${^\vee{\frak t}}_{\Bbb Q} = X_*({^\vee T})\otimes_{\Bbb Z} {\Bbb
Q}.\eqno(10.5)(b)$$
Now Lemma 9.9(b) says that the normalized exponential mapping $e$
provides an isomorphism
$${^\vee{\frak t}}_{\Bbb Q}/X_*({^\vee T}) \simeq {^\vee
T}^{fin}.\eqno(10.5)(c)$$
Clearly $X^*(T^{can})$ is precisely the inverse image of the $\theta$-fixed
elements ${^\vee T}^{fin}$ under this map; so 
$$X^*(T^{can}) /X^*(T) \simeq ({^\vee T}^{fin})^\theta. \eqno(10.5)(d)$$
Of course the group on the left may be identified with the characters of
$\pi_1(T)^{can}$, and we recover the isomorphism in Lemma 10.2(d).

\medskip
{\bf Proposition 10.6} ([AV2], Theorem 5.11.)  {\it Suppose $T$ is an
algebraic torus defined over ${\Bbb R}$.  Fix an E-group $({^\vee
T^\Gamma},{\cal W})$ for $T$ with second invariant $z$ (Definition 4.6). 
Then there is a natural bijection
$$\phi\rightarrow \pi(\phi)$$
from the set $\Phi({^\vee T^\Gamma})$ of equivalence classes of Langlands
parameters (Definition 5.2) onto the set $\Pi^z(T({\Bbb R}))$ of equivalence
classes of irreducible canonical projective representations of $T({\Bbb R})$
of type $z$ (Definition 10.3).  In this bijection, the differential $d\pi(\phi)$
(an element of ${\frak t}^*$) is identified by (9.3)(b) with the parameter
$\lambda(\phi) \in {^\vee {\frak t}}$ (cf. (5.8)(a)).

Suppose $\lambda\in {^\vee{\frak t}}_{\Bbb Q}$  satisfies $e(\lambda) = z$. 
By (10.5), $\lambda$ may be identified with a rational character of
$T^{can}$ of type $z$.  By restriction to $T({\Bbb R})^{can}$, $\lambda$
defines an irreducible canonical projective representation of $T({\Bbb R})$
of type $z$.  Its Langlands parameters are the various
$\phi({^\vee\delta},\lambda)$ (notation (5.8)(a)) with ${^\vee\delta} \in
{\cal D}$; these constitute a single equivalence class.}
\medskip
As in Proposition 9.4, the last assertion is included primarily to show how
the classification theorem forces the choice of the class ${\cal D}$ in
addition to the weak E-group structure.

\medskip
{\bf Corollary 10.7.}  {\it Theorem 10.4 is true if $G$ is an algebraic torus.}
\medskip
Just as in the argument at the end of section 9, we need only apply Corollary
9.12 and Proposition 10.6.

We record some notation based on Theorem 10.4, extending (1.19) in the
introduction.

\medskip
{\bf Definition 10.8.}  In the setting of Theorem 10.4, suppose $\xi\in 
\Xi^z(G/{\Bbb R})$ is a complete geometric parameter.  A representative for
the corresponding equivalence class of representation and strong real form
will be written as
$$(\pi(\xi),\delta(\xi)) \in \Pi^z(G/{\Bbb R})$$
(Definition 10.3).  If we regard $\xi$ as represented by a complete Langlands
parameter $(\phi,\tau)$ (Definition 5.11), we may write instead
$(\pi(\phi,\tau),\delta(\phi,\tau))$.

Conversely, suppose $(\pi,\delta) \in \Pi^z(G/{\Bbb R})$.  The corresponding
complete geometric parameter $\xi$ will be written as
$$\xi(\pi,\delta) = (S(\pi,\delta),{\cal V}(\pi,\delta)).$$
Here $S$ is an orbit of $^\vee G$ on the geometric parameter space, and
${\cal V}$ is a ${^\vee G}^{alg}$-equivariant local system on $S$.  A
representative for the corresponding equivalence class of complete
Langlands parameters is $(\phi(\pi,\delta),\tau(\pi,\delta))$ (Definition
5.11).  More generally, we may use invariants previously attached to
complete geometric parameters as if they were attached directly to
representations.  For example, we write $d(\pi)$ or $d(\pi,\delta)$ for the
dimension of the orbit $S(\pi,\delta)$ (cf.~(7.10)), and $e(\pi)$ or
$e(\pi,\delta)$ for the Kottwitz sign attached to $\xi(\pi,\delta)$
(Definition 15.8 below).

Suppose now that $\phi\in \Phi^z(G/{\Bbb R})$ (Definition 5.3), and let
$S_\phi\subset X({^\vee G^\Gamma})$ be the corresponding orbit
(Proposition 6.17).  The {\it L-packet attached to $\phi$}  is the set of
irreducible (canonical projective) representations of strong real forms
parametrized by complete geometric parameters supported on $S$:
$$\eqalign{\Pi^z(G/{\Bbb R})_\phi &= \{\,(\pi(\xi),\delta(\xi))\mid \xi =
(S_\phi,{\cal V})\,\}\cr
&= \{\,(\pi,\delta)\mid \phi(\pi,\delta)\ \hbox{\rom is equivalent to\
}\phi\,\}.\cr}$$ 
\medskip

Perhaps the least satisfactory aspect of Theorem 10.4 is the appearance of
the algebraic universal covering ${^\vee G}^{alg}$.  (We can avoid coverings
of $G$, at least in the final result, by using the L-group instead of another
E-group.)  It is therefore of interest to understand exactly what coverings
are needed to classify what representations.  This information will be
important in our treatment of endoscopy in section 26 as well.

\medskip
{\bf Lemma 10.9.}  {\it Suppose we are in the setting of Lemma 10.2.
\smallskip
\item{a)} The group of continuous characters of $\pi_1({^\vee G})^{alg}$ is
naturally isomorphic to the group $Z(G)^{fin}$ of elements of finite order in
$Z(G)$.  Write $\chi_z$ for the character corresponding to $z$.
\item{b)}  In the setting of Theorem 10.4, write $z_0\in Z(G)^{fin}$ for the
second invariant of the extended group $(G^\Gamma,{\cal W})$ (cf.~(3.5)). 
Suppose that the complete geometric parameter $\xi$ corresponds to the
irreducible canonical projective representation $(\pi(\xi),\delta(\xi))$.  Set
$z_1 = \delta^2 \in Z(G)^{fin}$.  Then $\pi_1({^\vee G})^{alg}$ acts on $\xi$
by the character $\chi_{z_0z_1}$.
\medskip}

{\it Proof.}  Part (a) is almost identical to Lemma 10.2.  For (b), the
definitions underlying Theorem 10.4 (see particularly section 13) reduce it
to the case of a torus, where it follows by inspection of the proof of
Corollary 9.12.  We omit the details.

Here is an important case of the situation considered in (5.13).

\medskip
{\bf Definition 10.10.}  Suppose ${^\vee G^\Gamma}$ is a weak E-group, and
$Q$ is any quotient of $\pi_1({^\vee G})^{alg}$:
$$1\rightarrow K_Q \rightarrow \pi_1({^\vee G})^{alg} \rightarrow Q
\rightarrow 1 \eqno(10.10)(a)$$
(We assume that $K_Q$ is closed; equivalently, that $Q$ is the inverse limit
of its finite quotients.)  Define
$${^\vee G}^Q = {^\vee G}^{alg}/K_Q, \eqno(10.10)(b)$$
a pro-finite covering of ${^\vee G}$:
$$1 \rightarrow Q \rightarrow {^\vee G}^Q \rightarrow {^\vee G}\rightarrow
1. \eqno(10.10)(c)$$
(Notice in particular that ${^\vee G}^{\pi_1({^\vee G})^{alg}}$ is just ${^\vee
G}^{alg}$.)  Recall from Definition 7.6 the notion of a complete geometric
parameter of type $Q$; this may be regarded as a complete geometric
parameter $(S,{\cal V})$ with the property that $K_Q$ acts trivially on
${\cal V}$, so
$$\Xi({^\vee G^\Gamma})^Q \subset \Xi({^\vee G^\Gamma}). \eqno(10.10)(c)$$
In the setting of Theorem 10.4, we may write $\Xi^z(G/{\Bbb R})^Q$ for
$\Xi({^\vee G^\Gamma})^Q$.

Suppose $(G^\Gamma,{\cal W})$ is an extended group with second invariant
$z_0$ (cf.~(3.5)), and $J \subset Z(G)^{fin}$ is any subgroup.  A {\it strong
real form of $G$ of type $J$} is an element $\delta\in G^\Gamma-G$ such
that $\delta^2 \in z_0J$.  (The shift by $z_0$ guarantees that the real forms
in ${\cal W}$ are of type $J$ for every $J$.)  We write 
$$\Pi(G/{\Bbb R})_J = \{\,(\pi,\delta) \in  \Pi(G/{\Bbb R})\mid \delta^2 \in
z_0J\,\}$$
for the set of equivalence classes of irreducible representations of strong
real forms of $G$ of type $J$, and extend other notation analogously.

\medskip
{\bf Theorem 10.11.} {\it In the setting of Theorem 10.4, suppose $Q$ is the 
quotient of $\pi_1({^\vee G})^{alg}$ by a closed subgroup $K_Q\subset
\pi_1({^\vee G})^{alg}$.  Define $J \subset Z(G)^{fin}$ to consist of all those
elements for which the corresponding character (Lemma 10.9)(a)) is trivial
on $K_Q$:
$$J = \{\,z\in Z(G)^{fin}\mid \chi_z(k) = 1,{\rom \ all\ } k\in K_Q\,\}.$$
Then the bijection of Theorem 10.4 restricts to a bijection 
$$\Xi^z(G/{\Bbb R})^Q \leftrightarrow \Pi^z(G/{\Bbb R})_J.$$}
\medskip

This is immediate from Theorem 10.4 and Lemma 10.9.  Notice that Lemma
10.9(a) provides a natural identification of $J$ with the group
$\widehat{Q}$ of characters of $Q$, and conversely.  The conclusion of
Theorem 10.11 may therefore be written as
$$\Xi^z(G/{\Bbb R})^Q \leftrightarrow \Pi^z(G/{\Bbb R})_{\widehat{Q}},
\eqno(10.12)(a)$$
or as
$$ \Pi^z(G/{\Bbb R})_J\leftrightarrow \Xi^z(G/{\Bbb R})^{\widehat{J}}.
\eqno(10.12)(b)$$

%\end
%\input mssymb.tex
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%\hsize=27pc
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\vskip 3pc
\centerline{\bf 11. The Langlands classification without L-groups.}
\vskip 1pc

In this section we recall the ``elementary'' version of the Langlands
classification of representations, in which L-groups do not appear (Theorem
11.14 below).  Because some of the groups we consider (such as $G({\Bbb
R})^{can}$) are not precisely groups of real points of connected algebraic
groups, we need to formulate this result in a slightly more general setting. 
With possible generalizations in mind, we allow even nonlinear groups.  The
class of groups we consider is essentially the one in section 5 of [Springer]. 
(The only difference is that Springer allows $G$ to be disconnected, and
imposes an additional technical hypothesis that is empty if $G$ is
connected.)  We refer the reader to [Springer] for basic structural facts and
further references.

We will be quoting a number of technical results, particularly from [Green]. 
Unfortunately our hypotheses in this section are weaker than those of
[Green], where it is assumed that $G_{\Bbb R}$ is linear.  We offer two
possible responses to this problem.  The first is that we will quote only
results that can be extended routinely to the present setting. The second is
that all of the groups considered elsewhere in this paper {\it are} linear.

So suppose $G$ is a connected reductive complex algebraic group defined
over ${\Bbb R}$; and suppose $G_{\Bbb R}$ is a real Lie group endowed with
a homomorphism
$$G_{\Bbb R} \rightarrow G({\Bbb R})\eqno(11.1)(a)$$
having finite kernel and cokernel.  We use the homomorphism to identify the
Lie algebra ${\frak g}_{\Bbb R}$ with ${\frak g}({\Bbb R})$, and its
complexification with ${\frak g}$.  A {\it Cartan subgroup} of $G_{\Bbb R}$
is by definition the centralizer in $G_{\Bbb R}$ of a Cartan subalgebra of
${\frak g}_{\Bbb R}$.  (Such a subgroup $T_{\Bbb R}$ is the preimage of the
real points $T({\Bbb R})$ of a unique maximal torus defined over ${\Bbb R}$:
$$ T_{\Bbb R} \rightarrow T({\Bbb R}).\eqno(11.1)(b)$$
(We should remark that $T_{\Bbb R}$ may be non-abelian under these
hypotheses, although this does not happen inside our canonical coverings.) 
Using such homomorphisms, we can pull extensions of $G({\Bbb R})$ and its
subgroups back to $G_{\Bbb R}$, getting for example a central extension
$$ 1 \rightarrow \pi_1(G)^{can} \rightarrow G_{\Bbb R}^{can} \rightarrow
G_{\Bbb R} \rightarrow 1. \eqno(11.1)(c)$$
We may therefore speak of a ``canonical projective representation of
$G_{\Bbb R}$ of type $z$'' as in Definition 10.3, and we write
$$\Pi^z(G_{\Bbb R}) =\{\hbox{ infinitesimal equivalence classes of such
irreducible representations }\}. \eqno(11.1)(d)$$ 
(Here $z$ is an element of finite order in $Z({^\vee G})^{\theta_Z}$ if we
have a dual group ${^\vee G}$ available; otherwise we can just think of $z$
directly as a character of $\pi_1(G)^{can}$ as in Lemma 10.2(d).)  If
$H_{\Bbb R}$ is a subgroup of $G_{\Bbb R}$, we write $H_{\Bbb R}^{can}$ or
$H_{\Bbb R}^{can,G}$ for its preimage in $G_{\Bbb R}^{can}$.  This is a
central extension
$$ 1 \rightarrow \pi_1(G)^{can} \rightarrow H_{\Bbb R}^{can,G} \rightarrow
H_{\Bbb R} \rightarrow 1. \eqno(11.1)(e)$$

\medskip
{\bf Definition 11.2} (cf. [Under], Definition 2.6, and [AV2], Definition 8.18). 
In the setting of (11.1), suppose $T_{\Bbb R}$ is a Cartan subgroup of
$G_{\Bbb R}$.  Fix an element $z\in Z({^\vee G})^{\theta_Z}$ of finite order. 
We may choose as a dual group for $T$ a maximal torus ${^\vee T} \subset
{^\vee G}$; then
$$Z({^\vee G})^{\theta_Z} \subset {^\vee T}^{\theta_Z},$$
and this embedding is independent of all choices.  It therefore makes sense
to regard $z$ as an element of finite order in ${^\vee T}^{\theta_Z}$.  Recall
the element $z(\rho)\in {^\vee T}^{\theta_Z}$ constructed in Definition 4.9. 
(Actually $z(\rho)$ even belongs to $Z({^\vee G})^{\theta_Z}$, but this fact
is a bit of an accident in the present context, and is better ignored.)  In any
case $zz(\rho) \in {^\vee T}^{\theta_Z}$ is an element of finite order, so it
makes sense to speak of canonical projective representations of $T_{\Bbb
R}$ of type $zz(\rho)$.

A {\it $G_{\Bbb R}$-limit character of $T_{\Bbb R}$ of type $z$} is a triple
$\Lambda = (\Lambda^{can},R^+_{i{\Bbb R}},R^+_{\Bbb R})$ subject to the
following conditions.  The first term is an irreducible canonical projective
representation
$$\Lambda^{can} \in \Pi^{zz(\rho)}(T_{\Bbb R}).\eqno(11.2)(a)$$
This means that $\Lambda^{can}$ is an irreducible representation of
$T_{\Bbb R}^{can,T}$ and that the restriction of $\Lambda^{can}$ to
$\pi_1(T)^{can}$ is a multiple of $\chi^T_{zz(\rho)}$.  Write
$$\lambda = d\Lambda^{can} \in {\frak t}^*.\eqno(11.2)(b)$$
Next, 
$$R^+_{\Bbb R},\qquad R^+_{i{\Bbb R}}\eqno(11.2)(c)$$
are positive root systems for the real and imaginary roots of $T$ in $G$. 
Finally, we assume that
$$\langle \alpha,\lambda\rangle \ge 0,\qquad(\alpha\in R^+_{i{\Bbb
R}}).\eqno(11.2)(d)$$
If these inequalities are strict (for example if $\lambda$ is regular), the
choice of $R^+_{i{\Bbb R}}$ is forced by (11.2)(d), and we may sometimes
omit it in writing the limit character.

The limit character is called {\it $G$-regular} if $\langle\alpha
,\lambda\rangle\ne 0$ for any root $\alpha$ of $T$ in $G$.

Attached to each limit character $\Lambda$ there is a {\it standard limit
representation} $M(\Lambda)$, defined by a procedure outlined in section 8
of [AV2] or section 2 of [Under].  We define our standard limit
representations as in [Langlands] (using for example real parabolic induction
with non-negative continuous parameter), so that the Langlands
subquotients appear as {\it quotients.}  Thus we define
$$\pi(\Lambda) = \hbox{ largest completely reducible quotient of
}M(\Lambda), \eqno(11.2)(e)$$
the {\it Langlands quotient of $M(\Lambda)$.}  Occasionally we will need the
standard representation having a Langlands subrepresentation; this is
written
$$\tilde{M}(\Lambda) \supset \pi(\Lambda). \eqno(11.2)(f)$$
This representation has exactly the same composition factors and
multiplicities as $M(\Lambda)$.

The infinitesimal character of $M(\Lambda)$ or $\pi(\Lambda)$ is given in
the Harish-Chandra para\-metrization (see for example [Green], Corollary
0.2.10, or [Hump], p. 130) by the weight $\lambda = d\Lambda^{can} \in
{\frak t}^*$.  In particular, the infinitesimal character is regular if and only
if the limit character is $G$-regular.
\medskip

The definition of limit character given here looks somewhat different from
the one in [Under], so we will explain briefly their relationship.  It is
convenient to fix a maximal compact subgroup $K_{\Bbb R}$ of $G_{\Bbb R}$
so that the corresponding Cartan involution preserves $T_{\Bbb R}$.  This
gives rise to a direct product decomposition
$$T_{\Bbb R} = (T_{\Bbb R}\cap K_{\Bbb R})(A_{\Bbb R})\eqno(11.3)(a)$$
with the second factor a vector group.  This decomposition in turn leads to
$$T_{\Bbb R}^{can} = (T_{\Bbb R}\cap K_{\Bbb R})^{can}(A_{\Bbb
R}).\eqno(11.3)(b)$$
The first step is to choose a positive root system
$R^+$ for $T$ in $G$ containing the systems of positive real and imaginary
roots already fixed.  We may assume that the set of non-imaginary roots in
$R^+$ is preserved by complex conjugation; then
$$\rho(R^+) = \rho(R^+_{i{\Bbb R}})\hbox{ on the Lie algebra of $T_{\Bbb
R}\cap K_{\Bbb R}$.}\eqno(11.3)(c)$$
Now it follows from (10.5) that $\rho(R^+)$ may be regarded as a rational
character of $T^{can,T}$ of type $z(\rho)$.   Consequently
$$\Lambda^{can}\otimes\rho(R^+) \in \Pi^z(T({\Bbb R}))\eqno(11.3)(d)$$
may be regarded as a character of $T_{\Bbb R}^{can,T}$ of type $z$.  Since
the type is in $Z({^\vee G})^{\theta_Z}$, we can now replace $T_{\Bbb
R}^{can,T}$ by  $T_{\Bbb R}^{can,G}$.  After tensoring with the sum of the
negative compact imaginary roots and an appropriate character of the vector
group $A_{\Bbb R}$, we therefore get a character $\tilde \Lambda \in
\Pi^z(T({\Bbb R}))$ satisfying
$$d\tilde \Lambda = \lambda + \rho(R^+_{i{\Bbb R}}) - 2\rho(R^+_{i{\Bbb
R},compact})\eqno(11.3)(e)$$
It is not difficult to show that $\tilde \Lambda$ is independent of the
choice of $R^+$; the argument is contained in Lemma 11.5 below.  The pair
$(\tilde \Lambda,\lambda)$ is what is called $(\Gamma,\overline \gamma)$
in Definition 2.4 of [Under].  Write $L_{\Bbb R}$ for the centralizer of
$A_{\Bbb R}$ in $G_{\Bbb R}$.  The standard representation $M(\Lambda)$
may be constructed by parabolic induction from a limit of discrete series
representation on $L_{\Bbb R}^{can}$ having lowest $L_{\Bbb R}\cap K_{\Bbb
R}$-type of highest weight $\tilde \Lambda$.

As is explained in [AV2], the dependence of $M(\Lambda)$ on $R^+_{\Bbb R}$
is very mild.  We recall the result.  Suppose $(R^+_{\Bbb R})'$ is another set
of positive real roots for $T$ in $G$. Define
$${\frak n}(R^+_{\Bbb R},(R^+_{\Bbb R})') = {\rom span}(X_\alpha\mid
\alpha\in R^+_{\Bbb R} - (R^+_{\Bbb R})').\eqno(11.4)(a)$$
The Cartan subgroup $T_{\Bbb R}$ acts on ${\frak n}(R^+_{\Bbb R},(R^+_{\Bbb
R})')$, and the determinant of this action is a real-valued character of
$T_{\Bbb R}$.  We may therefore define a character $\tau(R^+_{\Bbb
R},(R^+_{\Bbb R})')$ taking values in $\{\pm 1\}$ by
$$\tau(R^+_{\Bbb R},(R^+_{\Bbb R})')(t) = {\rom sgn}\left(\det({\rom Ad}(t)
\hbox{ on }{\frak n}(R^+_{\Bbb R},(R^+_{\Bbb R})')\right).\eqno(11.4)(b)$$

\medskip
{\bf Lemma 11.5} ([AV2], Lemma 8.24).  {\it Suppose $\Lambda =
(\Lambda^{can},R^+_{i{\Bbb R}},R^+_{\Bbb R})$ is a limit character of
$T_{\Bbb R}$ (Definition 11.2) and $(R^+_{\Bbb R})'$ is another set of
positive real roots of $T$ in $G$.  Choose sets of positive roots $R^+\supset
R^+_{i{\Bbb R}}, R^+_{\Bbb R}$ and $(R^+)' \supset R^+_{i{\Bbb R}}, (R^+_{\Bbb
R})'$ for $T$ in $G$ as in (11.3), and use them to construct characters
$\tilde \Lambda$ and ${\tilde\Lambda}'$.
\smallskip
\item{a)}  $\rho(R^+) = \rho((R^+)')\otimes \tau(R^+_{\Bbb R},(R^+_{\Bbb
R})')$  on $(T_{\Bbb R}\cap K_{\Bbb R})^{can,T}$.
\item{b)} $\tilde\Lambda = {\tilde\Lambda}'\otimes \tau(R^+_{\Bbb
R},(R^+_{\Bbb R})')$.
\item{c)} Write
$$\Lambda' = (\Lambda^{can}\otimes \tau(R^+_{\Bbb R},(R^+_{\Bbb R})')
,R^+_{i{\Bbb R}},(R^+_{\Bbb R})').$$
Then there is an isomorphism of standard limit representations
$M(\Lambda) \simeq M(\Lambda').$
\medskip}

{\it Proof.}  With notation analogous to (11.4), it is easy to check that (as
rational characters of $T^{can,T}$)
$$\rho(R^+) = \rho((R^+)')\otimes (\det({\rom Ad} \hbox { on } {\frak
n}(R^+,(R^+)'))).$$
That is, 
$$\rho(R^+)(t) = \rho((R^+)')(t)\prod_{\alpha\in R^+\atop \alpha\notin (R^+)'}
\alpha(t).$$
Assume now that $t \in (T_{\Bbb R}\cap K_{\Bbb R})^{can,T}$.  The roots
(like all characters of $T_{\Bbb R}$) have absolute value $1$ on the compact
group $T_{\Bbb R}\cap K_{\Bbb R}$; so the real roots contribute exactly
$\tau(R^+_{\Bbb R},(R^+_{\Bbb R})')(t)$ to the product.  Therefore
$$\rho(R^+)(t) = \rho((R^+)')(t)\tau(R^+_{\Bbb R},(R^+_{\Bbb
R})')(t)\prod_{\alpha\in R^+\ {\rom complex}\atop \alpha\notin (R^+)'}
\alpha(t).$$ 
The roots in this last product occur in complex conjugate pairs, since the
non-imaginary roots in $R^+$ and $(R^+)'$ were assumed to be stable under
complex conjugation.  Such a pair contributes $|\alpha(t)|^2 = 1$ to the
product.  This proves (a).  Part (b) is immediate from (a) and the definition
of $\tilde\Lambda$.   For (c), part (b) shows that the two limit characters
correspond to exactly the same limit character in the sense of [Under], so
they define the same representation.  Q.E.D.

\medskip
{\bf Definition 11.6.}  In the setting of (11.1) and Definition 11.2, suppose
$(\Lambda^{can},R^+_{i{\Bbb R}},R^+_{\Bbb R})$ is a limit character of
$T_{\Bbb R}$, and $((\Lambda^{can})',(R^+_{i{\Bbb R}})',(R^+_{\Bbb R})')$ is a
limit character of $T_{\Bbb R}'$.  We say that these limit characters are
{\it equivalent} if there is a $g \in G_{\Bbb R}$ that conjugates $T_{\Bbb
R}'$ to $T_{\Bbb R}$ and $(R^+_{i{\Bbb R}})'$ to $R^+_{i{\Bbb R}}$, and has the
following additional property.  Write $(R^+_{\Bbb R})'' = {\rom
Ad}(g)(R^+_{\Bbb R})'$ (a set of positive real roots for $T$ in $G$), and
$$(\Lambda^{can})'' = {\rom Ad}(g)(\Lambda^{can})',$$
a canonical projective representation of $T_{\Bbb R}$.  Then our final
requirement is
$$(\Lambda^{can})'' = \Lambda^{can}\otimes \tau(R^+_{\Bbb R},(R^+_{\Bbb
R})'').$$
Because of Lemma 11.5(c), equivalent limit characters define equivalent
standard limit representations.
\medskip
With this (rather subtle) notion of equivalence in hand, we can formulate an
important special case of the Langlands classification.

\medskip
{\bf Theorem 11.7} ([Langlands]) {\it Suppose $G_{\Bbb R}$ is a real
reductive group as in (11.1).  Then the infinitesimal equivalence classes of
irreducible admissible representations of $G_{\Bbb R}$ with regular
infinitesimal character are in one-to-one correspondence with the
equivalence classes of $G$-regular limit characters, as follows.
\smallskip
\item{a)}  Suppose $\Lambda$ is a $G$-regular limit character of a Cartan
subgroup $T_{\Bbb R}$ (Definition 11.2).  Then the corresponding standard
representation $M(\Lambda)$ is non-zero, and has a unique irreducible
quotient.  In particular, the Langlands quotient representation
$\pi(\Lambda)$ is irreducible.
\item{b)}  Suppose $\pi$ is an irreducible canonical projective
representation of $G_{\Bbb R}$ of type $z$ (Definition 11.2), having regular
infinitesimal character.  Then there is a $G$-regular limit character
$\Lambda$ of type $z$ with the property that $\pi$ is infinitesimally
equivalent to $\pi(\Lambda)$.
\item{c)}  Suppose $\Lambda$ and $\Lambda'$ are $G$-regular limit
characters, and that $\pi(\Lambda)$ is infinitesimally equivalent to
$\pi(\Lambda')$.  Then $\Lambda$ is equivalent to $\Lambda'$ (Definition
11.6).
\medskip}

This formulation of Langlands' result incorporates the work of several other
people.  We mention in particular Harish-Chandra's result on the
irreducibility of tempered induction at regular infinitesimal character, and
Mili\v ci\'c' observation that the Langlands quotient (originally defined as
the image of an intertwining operator) is simply a unique irreducible
quotient.

Of course we need a result for singular infinitesimal character as well. 
Before beginning the technical preliminaries to its formulation, let us
consider what goes wrong with Theorem 11.7 when we drop the regularity
hypotheses.  First, the standard limit representation $M(\Lambda)$ may be
zero. The simplest example occurs with $G_{\Bbb R} = SU(2)$,  and $T_{\Bbb
R}$ a compact torus.  In that case $z(\rho) = 1$ (since half the sum of the
positive roots is already a rational character of the algebraic torus $T$), so
all the coverings under consideration are trivial.  We can take
$\Lambda^{can}$ to be the trivial character of $T_{\Bbb R}$, and
$R^+_{i{\Bbb R}}$ arbitrary.  (There are no real roots.)  The corresponding
standard representation must have infinitesimal character zero by
Definition 11.2; but no representation of $SU(2)$ has infinitesimal
character zero.  (Harish-Chandra's parametrization assigns the infinitesimal
character $\rho$ to the trivial representation.)

Next, the standard representation may be non-zero, but it may have several
distinct irreducible quotients.  The first example occurs with $G_{\Bbb R} =
SL(2,{\Bbb R})$, and $T_{\Bbb R}\simeq {\Bbb R}^\times$ the diagonal
subgroup.  Again all the coverings are trivial.   We take $\Lambda^{can}$ to
be the trivial character of $T_{\Bbb R}$, and $R^+_{\Bbb R}$.  Because the
restriction to $T_{\Bbb R}\cap K_{\Bbb R} = \{\pm 1\}$ of the rational
character $\rho$ is non-trivial, we find that the character $\tilde\Lambda$
of (11.3) is the signum character of ${\Bbb R}^\times$.  The standard
representation is induced from $\tilde\Lambda$, and is therefore equal to
the reducible unitary principal series representation of $SL(2,{\Bbb R})$: a
direct sum of two limits of discrete series.  The Langlands quotient is
therefore also equal to this direct sum.

We will see that part (b) of Theorem 11.7 remains valid for singular
infinitesimal characters.  This already provides a kind of counterexample to
(c):  in the preceding example, the two constituents of the Langlands
quotient must themselves be (irreducible) Langlands quotients of some
other standard limit representations.  In fact the situation is even a little
worse: there can be two non-zero standard limit representations having
isomorphic and irreducible Langlands quotients, with inequivalent limit
characters.  The simplest example occurs with $G_{\Bbb R} = GL(2,{\Bbb
R})$, $T_{\Bbb R} = $ split Cartan subgroup, and $(T_{\Bbb R})' = $
fundamental Cartan subgroup.  The standard representation we want is an
irreducible unitary principal series whose restriction to $SL(2,{\Bbb R})$ is
reducible; this representation is also a limit of (relative) discrete series. 
We will write down one of the limit characters, just as an example of the
coverings involved.  A dual group is $GL(2,{\Bbb C})$, and the element
$z(\rho)$ is $-I$.  The complex torus $T$ consists of diagonal matrices in
$GL(2,{\Bbb C})$, so
$$T \simeq \{\,(z_1,z_2)\mid z_i\in {\Bbb C}^\times\,\}\eqno(11.8)(a)$$
The $T$-canonical covering of $T$ (Definition 10.1) turns out to be the
projective limit of all the finite covers, since $T$ is split.  A character of
type $z(\rho)$ factors through the double cover
$$\tilde T = \{\,(z_1,z_2,w)\mid z_i,w\in {\Bbb C}^\times, w^2 =
z_1z_2^{-1}\,\}.\eqno(11.8)(b)$$
(This is the ``square root of $\rho$ double cover'' considered in Definition
1.33 of [Orange], or Definition 8.3.11 of [AV2].)  Its pullback to $T_{\Bbb R}$
is
$$\tilde T_{\Bbb R} = \{\,(x_1,x_2,w)\mid x_i \in {\Bbb R}^\times,w\in
{\Bbb C}^\times, w^2 = x_1x_2^{-1}\,\}.\eqno(11.8)(c)$$
We take $\Lambda^{can}$ to be the character
$$\Lambda^{can}(x_1,x_2,w) = |w| {\rom sgn}(x_1)/w,\eqno(11.8)(d)$$
which takes values in $\{\pm1,\pm i\}$.  Let $R^+_{\Bbb R}$ be the positive
root corresponding to the upper triangular matrices.  Then the rational
character $\rho$ on $\tilde T$ is $\rho(z_1,z_2,w) = w.$  The character
$\tilde \Lambda$ of (11.3) from which we induce is now easily computed to
be
$$\tilde\Lambda(x_1,x_2) = {\rom sgn}(x_1).\eqno(11.8)(e)$$
Now it is clear that $M(\Lambda)$ is an irreducible unitary principal series
whose restriction to $SL(2,{\Bbb R})$ is the reducible principal series of
the preceding example.  We leave to the reader the construction of the
corresponding limit character on $(T_{\Bbb R})'$.

In order to extend Theorem 11.7 to singular infinitesimal character, we need
to restrict the class of limit characters in order to avoid such phenomena. 
It is a remarkable fact (related to the simple nature of the Galois group of
${\Bbb C}/{\Bbb R}$) that nothing essentially more complicated than these
examples can happen.  The problem of vanishing is particularly simple.

\medskip
{\bf Proposition 11.9.}  {\it In the setting of Definition 11.2, suppose
$\Lambda = (\Lambda^{can},R^+_{i{\Bbb R}},R^+_{\Bbb R})$ is a limit
character.  Then $M(\Lambda) = 0$ if and only if there is a simple root
$\alpha\in R^+_{i{\Bbb R}}$ such that $\alpha$ is compact, and $\langle
\alpha,\lambda\rangle = 0.$}
\medskip

\noindent (Notice that the first condition provides a subgroup of $G_{\Bbb
R}$ locally isomorphic to $SU(2)$, and the second says that, along that
subgroup, our limit character looks like the one in the first example after
Theorem 11.7.)

{\it Proof.}  The standard limit representation may be constructed by
induction from a parabolic subgroup $P_{\Bbb R} = L_{\Bbb R}N_{\Bbb R}$, so
it will be zero if and only if the inducing representation is zero.  This
provides a reduction to the case $L_{\Bbb R} = G_{\Bbb R}$; that is, to the
case of (relative) limits of discrete series.  That the condition in the
proposition implies that the standard limit representation is zero is easy. 
For example, it is immediate from one of the two character identities
established in [Schmid].  Another approach (cf. [Green], Proposition 8.4.3) is
to construct the standard limit representation using cohomological
parabolic induction.  Then an induction by stages argument reduces one to
the case of $SU(2)$.

The converse is not quite so easy.  The argument is a special case of the one
we will give for Theorem 11.14 below, and the reader may refer to that for
some additional details.  We are still assuming that $L_{\Bbb R} = G_{\Bbb
R}$; that is, that $R^+_{i{\Bbb R}}$ is a full set of positive roots of $T$ in
$G$.  Fix a dominant regular rational character $\mu$ of $T$, and consider
the limit character
$$(\Lambda^{can}\otimes \mu,R^+_{i{\Bbb R}},R^+_{\Bbb R}).$$
We denote this by the symbol $\Lambda + \mu$.  The corresponding weight is
$\lambda + d\mu$, which is dominant regular by (11.2)(d) and the choice of
$\mu$.  Then $M(\Lambda + \mu)$ is a (non-zero irreducible) discrete series
representation.  The standard limit representation $M(\Lambda)$ is obtained
from $M(\Lambda +\mu)$ by applying a Jantzen-Zuckerman translation
functor (namely the one denoted $\psi_{\lambda + \mu}^{\lambda}$ in
[Green]).  By [Green], Corollary 7.3.23, it follows that $M(\Lambda) = 0$ if
and only if there is a root $\alpha$ in $\tau(M(\Lambda+\mu))$ (the
Borho-Jantzen-Duflo $\tau$-invariant) satisfying
$\langle\alpha,\lambda\rangle = 0.$  Now the $\tau$-invariant is a subset
of the set of simple roots for the system of $\lambda$-integral roots.  By
(11.2)(d), a simple $\lambda$-integral singular root $\alpha$ must be
simple in all of $R^+$.  By hypothesis, $\alpha$ must be noncompact.  By
[Green], Corollary 8.4.7, $\alpha$ does not lie in the $\tau$-invariant; so
$M(\Lambda) \ne 0$, as we wished to show.  Q.E.D.

We want now a condition analogous to the one in Proposition 11.9 that will
rule out the other bad examples given after Theorem 11.7.  In each example,
there was a real root $\alpha$ orthogonal to $\lambda$.  Assuming that such
roots do not exist certainly eliminates these phenomena, but it also
eliminates limit characters that are needed for the classification.  The
prototypical example is the spherical representation $\pi'$ of $SL(2,{\Bbb
R})$ of infinitesimal character zero.  In the notation of the second example
after Theorem 11.7, $\pi'$ is the unique Langlands quotient of --- in fact is
equal to --- the standard limit representation $M(\Lambda')$, with
$(\Lambda')^{can}$ the signum character of $T_{\Bbb R}$.  Furthermore
$\pi'$ occurs in no other standard limit representation.  To get a good
classification theorem, we need to keep the limit character $\Lambda'$ but
discard $\Lambda$.  What distinguishes these two limit characters is their
behavior on the non-identity component of $T_{\Bbb R}$.  The appropriate
generalization of this is provided by the ``parity condition.''

\medskip
{\bf Definition 11.10} ([IC1], Proposition 4.5, or [Green], Definition 8.3.11).
Suppose $G_{\Bbb R}$ is as in (11.1), $T_{\Bbb R}$ is a Cartan subgroup, and
$\alpha$ is a real root of $T$ in $G$.  Choose a root subgroup homomorphism
$$\phi_\alpha:SL(2) \rightarrow G\eqno(11.10)(a)$$
defined over ${\Bbb R}$, so that $\phi_\alpha(\hbox{diagonal matrices})
\subset T$.  Since $SL(2)$ is simply connected, this lifts to 
$$\phi_\alpha^{can}:SL(2) \rightarrow G^{can}.\eqno(11.10)(b)$$
Since $SL(2,{\Bbb R})$ is connected, this map restricts to
$$\phi_\alpha^{can}: SL(2,{\Bbb R}) \rightarrow \hbox{ image of }G_{\Bbb
R}^{can} \subset G({\Bbb R})^{can}.\eqno(11.10)(c)$$
The mapping (11.1)(a) therefore induces an extension of $SL(2,{\Bbb R})$. 
Passing to its identity component, we get
$$\tilde\phi_\alpha^{can}:\widetilde {SL}(2,{\Bbb R}) \rightarrow G_{\Bbb
R}^{can},\eqno(11.10)(d)$$
with
$$\widetilde {SL}(2,{\Bbb R})\rightarrow SL(2,{\Bbb R})\eqno(11.10)(e)$$
a connected cover.  Define
$$\tilde m = \exp\pmatrix0&\pi\cr -\pi&0\cr\endpmatrix \in \widetilde{SL}(2,{\Bbb
R}),\qquad {\tilde m}_\alpha = \tilde\phi_\alpha^{can}(\tilde m) \in G_{\Bbb
R}^{can}.\eqno(11.10)(f)$$
(If $G_{\Bbb R}$ is linear, then $\widetilde {SL}(2,{\Bbb R}) = SL(2,{\Bbb
R})$, and $\tilde m = -I$.  In general $\tilde m$ is a generator for
$Z(\widetilde {SL}(2,{\Bbb R}))$, and it may have any (finite) order.)  The
homomorphism $\phi_\alpha$ is not quite uniquely determined by the
conditions we have imposed on it.  It is easy to check that changing the
choice replaces ${\tilde m}_\alpha$ by ${\tilde m}_\alpha^{\pm 1}$.

Suppose $\Lambda = (\Lambda^{can},R^+_{i{\Bbb R}},R^+_{\Bbb R})$ is a limit
character of $T_{\Bbb R}$.  Choose any set $R^+$ of positive roots for $T$ in
$G$.  As in (11.3), regard $\Lambda^{can}\otimes\rho$ as a representation of
$T_{\Bbb R}^{can,G}$.  It may then be applied to the element ${\tilde
m}_\alpha$ defined above.  We say that $\alpha$ {\it satisfies the parity
condition for $\Lambda$} if the eigenvalues of the operator
$(\Lambda^{can}\otimes \rho)({\tilde m}_\alpha)$ are contained in the set
$$-\exp\left(\pm i\pi\langle \alpha^\vee,\lambda +\rho - \rho(R^+_{\Bbb
R})\rangle\right). \eqno(11.10)(g)$$
(Because of the $\pm$ in the exponent, this condition is unchanged if we
replace ${\tilde m}_\alpha$ by ${\tilde m}_\alpha^{-1}$.)  Here it is
important to understand that $\rho(R^+_{\Bbb R})$ refers to the set of
positive real roots fixed in $\Lambda$, but that $\rho$ refers to the
arbitrary set $R^+$ of positive roots.  We do not assume that $R^+_{\Bbb R}$
is contained in $R^+$.  If $G_{\Bbb R}$ is linear (so that ${\tilde m}_\alpha$
is a central element of order 2 in $T_{\Bbb R}^{can,G}$) this condition
amounts to
$$\langle \alpha^\vee,\lambda +\rho - \rho(R^+_{\Bbb R})\rangle = n\in
{\Bbb Z},\hbox{ and }(\Lambda^{can}\otimes \rho)({\tilde m}_\alpha) =
(-1)^{n+1}. \eqno(11.10)(h)$$
\medskip

Since the notion of limit character we are using is a little different from
that in [IC1] and [Green], we need to check that this definition of the parity
condition agrees with the (more complicated) one there.  The first thing to
notice is that our condition (11.10)(g) does not depend on the choice of
positive root system $R^+$.  The reason is this.  Write $\alpha^\vee \in
X_*(T)$ for the coroot corresponding to $\alpha$.  By a calculation in
$SL(2)$, the image of ${\tilde m}_\alpha$ in $G$ is
$$m_\alpha = \alpha^\vee(-1) \in T.\eqno(11.11)(a)$$
It follows that if $\gamma$ is any rational character of $T$, then
$$\gamma(m_\alpha) = (-1)^{\langle \alpha^\vee, \gamma\rangle}.
\eqno(11.11)(b)$$
In particular, if $R^+$ and $(R^+)'$ are two systems of positive roots,
$$\left(\rho(R^+)\otimes\rho((R^+)')^{-1}\right)(m_\alpha) = (-1)^{\langle
\alpha^\vee, \rho(R^+) - \rho((R^+)')\rangle}.\eqno(11.11)(c)$$
This equality shows that the condition at (11.10)(g) does not depend on
$R^+$.  (A similar argument shows that the parity condition behaves well
under equivalence of limit characters.)  To complete the comparison with
[IC1] or [Green], one can use the prescription in (11.3) for translating limit
characters from one form to the other.  The argument is not trivial, but it is
straightforward and dull, so we omit it.

Here is an analogue of Proposition 11.9.

\medskip
{\bf Proposition 11.12} (The Hecht-Schmid character identity --- see
[Schmid] and [IC1], Proposition 4.5.) {\it In the setting of Definition 11.2,
suppose there is a real root $\alpha$ such that $\alpha$ satisfies the parity
condition for $\Lambda$ (Definition 11.10), and
$\langle\alpha,\lambda\rangle = 0$.  Then the standard limit representation
$M(\Lambda)$ is isomorphic to a direct sum of one or two standard limit
representations attached to limit characters on a Cartan subgroup $(T_{\Bbb
R})'$ having a strictly smaller split part.}
\medskip

{\bf Definition 11.13} ([Under], Definition 2.6.)  In the setting of Definition
11.2, the limit character $\Lambda = (\Lambda^{can},R^+_{i{\Bbb
R}},R^+_{\Bbb R})$ is called {\it final} if it satisfies the following two
conditions.
\smallskip
\item{(a)}  Suppose $\alpha$ is a simple root in $R^+_{i{\Bbb R}}$, and
$\langle\alpha,\lambda \rangle = 0$.  Then $\alpha$ is noncompact.
\item{(b)}   Suppose $\alpha$ is a real root, and $\langle\alpha,\lambda
\rangle = 0$.  Then $\alpha$ does not satisfy the parity condition for
$\Lambda$ (Definition 11.10).  

\noindent We write
$$L^z(G_{\Bbb R}) = \{\hbox{ equivalence classes of final limit characters of
type $z$ }\}.$$
The standard limit representations attached to final limit characters are
called {\it final standard limit representations}.

\medskip
{\bf Theorem 11.14} ([KZ]; see [Under], Theorem 2.6).  {\it Suppose $G_{\Bbb
R}$ is a real reductive group as in (11.1).  Then the set $\Pi^z(G_{\Bbb R})$
of infinitesimal equivalence classes of irreducible admissible
representations of $G_{\Bbb R}$ is in one-to-one correspondence with the
set $L^z(G_{\Bbb R})$ of equivalence classes of final limit characters, as
follows.
\smallskip
\item{a)}  Suppose $\Lambda$ is a final limit character of a Cartan subgroup
$T_{\Bbb R}$ (Definition 11.13).  Then the corresponding standard
representation $M(\Lambda)$ is non-zero, and has a unique irreducible
quotient.  In particular, the Langlands quotient representation
$\pi(\Lambda)$ is irreducible.
\item{b)}  Suppose $\pi$ is an irreducible canonical projective
representation of $G_{\Bbb R}$ of type $z$ (Definition 11.2).  Then there is a
final limit character $\Lambda$ of type $z$ with the property that $\pi$ is
infinitesimally equivalent to $\pi(\Lambda)$.
\item{c)}  Suppose $\Lambda$ and $\Lambda'$ are final limit characters, and
that $\pi(\Lambda)$ is infinitesimally equivalent to $\pi(\Lambda')$.  Then
the limit characters are equivalent (Definition 11.6). 
\smallskip

Any standard limit representation of $G_{\Bbb R}$ (Definition 11.2) is
isomorphic to a direct sum of final standard limit representations.}
\medskip
{\it Proof.}  (The result in [KZ] is for linear groups, and is in any case
formulated in a very different way.  Since [Under] contains no proof, we
outline the argument.)  The last claim is clear from Propositions 11.9 and
11.12.  For the rest, one can use the Jantzen-Zuckerman translation
principle just as it is used in the proof of Proposition 11.9.  Here is a
sketch.  (Another development of the translation principle, emphasizing the
connection with the geometric version in section 8, may be found in section
16.)  Fix a maximal torus $T_a$ of $G$ and a weight
$$\lambda_a \in {\frak t}_a^*.\eqno(11.15)(a)$$
We study representations of infinitesimal character $\lambda_a$.  Fix also
a system of positive roots $R^+_a$ for $T_a$ in $G$, chosen so that
$$\langle\alpha,\lambda_a\rangle > 0 \Rightarrow \alpha \in R^+_a.
\eqno(11.15)(b)$$
Fix a regular dominant rational character $\mu_a \in X^*(T_a)$.  By
(11.15)(b),
$$\lambda'_a = \lambda_a + \mu_a\eqno(11.15)(c)$$
is a $G$-regular weight, and satisfies the analogue of (11.15)(b).  As in the
proof of Proposition 11.9, we consider the translation functor
$$\psi = \psi_{\lambda_a + \mu_a}^{\lambda_a} \eqno(11.15)(d)$$
from modules with the (regular) infinitesimal character $\lambda_a +
\mu_a$ to modules with infinitesimal character $\lambda_a$.  (The functor
is defined by tensor product with the finite-dimensional representation of
lowest weight $-\mu_a$, followed by projection on the infinitesimal
character $\lambda_a$.)  Finally, define
$$\Delta^0_a = \hbox{ simple roots for $R^+_a$ orthogonal to
$\lambda_a$.}\eqno(11.15)(e)$$
Any irreducible representation $\pi'$ of infinitesimal character
$\lambda'_a$ has a Borho-Jantzen-Duflo $\tau$-invariant, $\tau(\pi')$
which is a subset of the simple roots for the positive $\lambda'_a$-integral
roots (cf. [Green], Definition 7.3.8.)  This set of simple roots includes
$\Delta^0_a$, so we can define
$$\tau^0(\pi') = \tau(\pi') \cap \Delta^0_a.\eqno(11.15)(f)$$

Here are the general facts we need from the translation principle. 

\medskip
{\bf Proposition 11.16} ([Green], Corollary 7.3.23; cf. also [Dix], Proposition
7.7).  {\it Suppose $G_{\Bbb R}$ is a real reductive group as in (11.1), and
suppose we are in the setting (11.15).  Then the translation functor $\psi$
of (11.15)(d) is an exact functor from canonical projective representations
of $G_{\Bbb R}$ of type $z$ and infinitesimal character $\lambda'_a$ to
canonical projective representations of type $z$ and infinitesimal character
$\lambda_a$.  It has the following additional properties.
\smallskip
\item{a)}  Suppose $\pi'$ is an irreducible representation of infinitesimal
character $\lambda'_a$.  Then $\psi(\pi')$ is irreducible or zero.  The first
possibility occurs if and only if $\tau^0(\pi')$ (cf. (11.15)(f)) is empty.
\item{b)}  Suppose $\pi$ is an irreducible representation of infinitesimal
character $\lambda_a$.  Then there is a unique irreducible representation
$\pi'$ with $\psi(\pi') \simeq \pi$.
\medskip}

In order to use this result to reduce Theorem 11.14 to the special case of
Theorem 11.7, we need to know two more specific things: how the
translation functor affects standard limit representations, and how to
compute $\tau^0(\pi')$ in terms of the classification of Theorem 11.7. 
Suppose then that $\Lambda' = ((\Lambda')^{can},R^+_{i{\Bbb R}},R^+_{\Bbb
R})$ is a $G_{\Bbb R}$-limit character of $T_{\Bbb R}$ of infinitesimal
character $\lambda'_a$.  Write $\lambda'$ for the differential of
$\Lambda'$.  Since $\lambda'$ and $\lambda'_a$ are regular and define the
same infinitesimal character,  there is a unique isomorphism
$$i(\lambda'_a,\lambda'): T_a \rightarrow T\eqno(11.17)(a)$$
induced by an element of $G$ and carrying $\lambda'_a$ to $\lambda'$.  We
also write $i(\lambda'_a,\lambda')$ for the induced isomorphism from  the
root system of $T_a$ in $G$ to that of $T$, and so on.  Put
$$R^+ = i(\lambda'_a,\lambda')(R^+_a),\eqno(11.17)(b)$$
a system of positive roots for $T$ in $G$.  Because of (11.15)(b) and
(11.2)(d), we have
$$R^+ \supset R^+_{i{\Bbb R}}.\eqno(11.17)(c)$$
Similarly, define
$$\mu = i(\lambda'_a,\lambda')(\mu_a) \in X^*(T). \eqno(11.17)(d)$$
Then we can define
$$\Lambda' - \mu = ((\Lambda')^{can}\otimes \mu^{-1},R^+_{i{\Bbb
R}},R^+_{\Bbb R}).\eqno(11.17)(e)$$
Because of (11.15)(b) and (11.17)(c), this is a $G_{\Bbb R}$-limit character
of $T_{\Bbb R}$ of infinitesimal character $\lambda_a$.

\medskip
{\bf Proposition 11.18} {\it Suppose we are in the setting of (11.15), and
$\Lambda'$ is a $G_{\Bbb R}$-limit character of infinitesimal character
$\lambda'_a$.  Use the notation of (11.17).
\smallskip
\item{a)}  The translation functor $\psi$ satisfies
$$\psi(M(\Lambda')) \simeq M(\Lambda' - \mu).$$
\item{b)} Write $\Delta^0$ for the image of $\Delta^0_a$ under the bijection
$i(\lambda'_a,\lambda')$.  Then the $\tau$ invariant $\tau^0(\pi(\Lambda'))$
corresponds to the subset of roots in $\Delta^0$ satisfying one of the
following three conditions:
\itemitem{i)} $\alpha$ is compact imaginary;
\itemitem{ii)} $\alpha$ is complex, and its complex conjugate $\overline
\alpha$ is positive; or
\itemitem{iii)} $\alpha$ is real and satisfies the parity condition
(Definition 11.10).
\item{c)}  Suppose $\tau^0(\pi(\Lambda'))$ is empty.  Then
$\psi(\pi(\Lambda'))$ is the unique irreducible quotient of
$M(\Lambda'-\mu)$.  That is,
$$\psi(\pi(\Lambda')) = \pi(\Lambda' - \mu).$$
\medskip}

{\it Proof.}  Part (a) is proved for linear groups in [Green], Proposition 7.4.1;
the argument is unchanged in general.  For (b), the $\tau$-invariant depends
only on the restriction of a representation to the identity component (in
fact only on its annihilator in the enveloping algebra); so we may assume
$G_{\Bbb R}$ is connected.  Then the result is proved in [IC1], Theorem 4.12. 
(For linear groups, this is [Green], Theorem 8.5.18.)  

For (c), suppose that $\pi_1$ is an irreducible quotient of $M(\Lambda)$.  By
Proposition 11.16, there is an irreducible representation $\pi'_1$ of
infinitesimal character $\lambda'_a$ with $\psi(\pi'_1) \simeq \pi_1$.  The
translation functor $\psi$ has a two-sided adjoint
$$\phi = \psi^{\lambda_a + \mu_a}_{\lambda_a}\eqno(11.19)(a)$$
([Green], Proposition 4.5.8).  Consequently
$${\rom Hom}(M(\Lambda'-\mu),\pi_1) \simeq {\rom Hom}(\psi(M(\Lambda'))
,\psi(\pi'_1)) \simeq {\rom Hom}(M(\Lambda'),
\phi\psi(\pi'_1)).\eqno(11.19)(b)$$
(The ${\rom Hom}$ spaces are infinitesimal homomorphisms; that is, $({\frak
g},K_{\Bbb R})$-module maps for an appropriate maximal compact $K_{\Bbb
R}$.)  Now it is a formal consequence of the adjointness of $\phi$ and $\psi$
that any irreducible subrepresentation $\pi'_2$ of $\phi\psi(\pi'_1)$ must
be a preimage of $\pi_1$ under $\psi$.  By Proposition 11.16(b),
$${\rom Hom}(\pi'_2,\phi\psi(\pi'_1)) =\cases{\Bbb C},&if
\pi'_2\simeq\pi'_1\\ 
0&otherwise.\cr\endcases\eqno(11.19)(c)$$
On the other hand, (11.19)(b) implies that the unique irreducible quotient
$\pi'$ of $M(\Lambda')$ must be a composition factor of $\phi\psi(\pi'_1)$. 
But the only composition factor of $\phi\psi(\pi'_1)$ having $\tau^0$ empty
is $\pi'_1$ itself ([Green], Proposition 7.3.2(b) and Corollary 7.3.21). 
Consequently $\pi'_1\simeq \pi'$, and $\pi_1 \simeq \psi(\pi')$.  The last
group in (11.19)(b) is therefore
$${\rom Hom}(M(\Lambda'), \phi\psi(\pi')).\eqno(11.19)(d)$$
The image $P'$ of a non-zero map in this space must have $\pi'$ as its
unique irreducible quotient (since $P'$ is a quotient of $M(\Lambda')$) and
as its unique irreducible subrepresentation (by (11.19)(c)).  But $\pi'$
occurs only once as a composition factor of $M(\Lambda')$, so it follows
that $P' = \pi'$.  Consequently
$${\rom Hom}(M(\Lambda'), \phi\psi(\pi')) \simeq {\rom Hom}(\pi',
\phi\psi(\pi')) \simeq {\Bbb C}.\eqno(11.19)(e)$$
In conjunction with (11.19)(b), this shows that $\psi(\pi')$ occurs exactly
once as a quotient of $M(\Lambda)$, as we wished to show.  Q.E.D.

Propositions 11.16 and 11.18 explain rather explicitly how to describe
representations of infinitesimal character $\lambda_a$ in terms of
representations of (regular) infinitesimal character $\lambda'_a$.  The
proof of Theorem 11.14 now comes down to checking that these
constructions are compatible with the parametrization in terms of final
limit characters.  Here is the result we need.

\medskip
{\bf Proposition 11.20.}  {\it In the setting (11.15), there is a bijection from
the set
\smallskip
$$\{\hbox{\rom equivalence classes of limit characters $\Lambda'$ of
infinitesimal character $\lambda'_a$ with $\tau^0(\pi(\Lambda'))$ empty
}\}$$
\smallskip
\noindent onto the set
\smallskip
$$\{\hbox{\rom equivalence classes of final limit characters $\Lambda$ of
infinitesimal character $\lambda_a$}\}.$$
\smallskip
In the notation of (11.17), this bijection sends the class of $\Lambda'$ to
the class of $\Lambda'-\mu$.
\medskip}

{\it Proof.} Suppose first that $\tau^0(\pi(\Lambda'))$ is empty; we want to
show that $\Lambda = \Lambda' - \mu$ is final (Definition 11.13).  First,
Proposition 11.18 guarantees that $M(\Lambda) \ne 0$.  By Proposition 11.9,
it follows that $\Lambda$ satisfies condition (11.13)(a).  To verify
condition (11.13)(b), we need to understand a little about the root system
$$R^0 = \{\,\alpha\in R(G,T)\mid \langle\alpha,\lambda\rangle =
0\,\}.\eqno(11.21)(a)$$
First, it has $(R^0)^+ = R^+ \cap R^0$ (notation (11.17)(c)) as a set of
positive roots, with $\Delta^0$ the corresponding simple roots.  Define
$$\Delta^0_{\Bbb R} = \Delta^0 \cap R_{\Bbb R},\qquad R^0_{\Bbb R} =
R^0\cap R_{\Bbb R}.\eqno(11.21)(b)$$
What we need to show is that
$$\hbox{$\Delta^0_{\Bbb R}$ is a set of simple roots for $R^0_{\Bbb
R}$.}\eqno(11.21)(c)$$
Assume for a moment that we have established this.  From Definition 11.10
one can check that
$$\hbox{if $\alpha$ and $\beta$ in $R^0_{\Bbb R}$ do not satisfy the parity
condition, then neither does $s_\alpha(\beta)$.}\eqno(11.21)(d)$$
(For linear groups one can say even more: because of (11.11)(a), the set of
coroots for roots in $R^0_{\Bbb R}$ not satisfying the parity condition is
closed under addition (cf. [Green], Lemma 8.6.3).  In the non-linear case, one
can reduce immediately to the case of split simple groups of rank 2.  These
can be treated by hand; the calculation is very easy except in type $B_2$.) 
By Proposition 11.18(b)(iii) and the hypothesis on $\Lambda'$, the roots in
$\Delta^0_{\Bbb R}$ do not satisfy the parity condition.  By (11.21)(c) and
(d), it follows that no root in $R^0_{\Bbb R}$ satisfies the parity condition. 
This is (11.13)(b).

To prove (11.21)(c), observe first that
$$\hbox{ if $\alpha \in (R^0)^+$, then $\langle -\overline\alpha ,\lambda
\rangle \ge 0$.}\eqno(11.22)(a)$$
For it is enough to prove this for $\alpha\in \Delta^0$. If $\alpha$ is real or
imaginary, the inner product is zero.  If $\alpha$ is complex, then the
hypothesis on $\Lambda'$ and Proposition 11.18(b)(ii) guarantee that
$-\overline\alpha \in R^+$; so the inequality follows from (11.15)(b) (and
(11.17)).  Now define
$$\eqalignno{R^{00} &= \{\,\alpha\in R^0\mid -\overline\alpha\in R^0\,\}\cr
&=\{\,\alpha\in R\mid \langle\alpha,\lambda\rangle = \langle
-\overline\alpha,\lambda\rangle = 0\,\}.&(11.22)(b)\cr}$$
This is a root system; we define
$$(R^{00})^+ = R^{00}\cap R^+,\qquad \Delta^{00} = \Delta^0 \cap
R^{00}.\eqno(11.22)(c)$$
Because of (11.22)(a),
$$\hbox{$\Delta^{00}$ is a set of simple roots for $(R^{00})^+$.}
\eqno(11.22)(d)$$
Now $\alpha\mapsto -\overline\alpha$ is an involutive automorphism of
$R^{00}$, so we can apply to it Lemma 8.6.1 of [Green].  Suppose $\alpha$ is
a complex simple root for $(R^{00})^+$.  By (11.22)(d), $\alpha\in \Delta^0$;
so by Proposition 11.18(b)(ii) and the hypothesis on $\Lambda'$,
$-\overline\alpha \in (R^{00})^+$.  Lemma 8.6.1 of [Green] therefore implies
that
$$\hbox{the set of non-real roots in $(R^{00})^+$ is stable under
$\alpha\mapsto -\overline\alpha$.}\eqno(11.22)(e)$$
Now (11.21)(c) is precisely the conclusion of Lemma 8.6.2 of [Green].

We have therefore shown that the map in Proposition 11.20 is well-defined. 
To see that it is surjective, assume that
$$\hbox{$\Lambda = (\Lambda^{can},R^+_{i{\Bbb R}},R^+_{\Bbb R})$ is a final
limit character of $T_{\Bbb R}$ of infinitesimal character
$\lambda_a$.}\eqno(11.23)(a)$$
Write $R$ for the set of roots of $T$ in $G$.  Define
$$R^0 = \{\,\alpha \in R\mid \langle\alpha,\lambda\rangle =
0\,\},\eqno(11.23)(b)$$
$$R^{00} = \{\,\alpha\in R^0\mid -\overline\alpha\in
R^0\,\}.\eqno(11.23)(c)$$
Choose a set of positive roots $(R^{00})^+$ for $R^{00}$ with the property
that
$$\hbox{the set of non-real roots in $(R^{00})^+$ is stable under
$\alpha\mapsto -\overline\alpha$.}\eqno(11.23)(d)$$
(This is certainly possible.)  Possibly after modifying this system by
reflections in imaginary roots, we may assume that
$$R^+_{i{\Bbb R}}\cap R^{00} \subset (R^{00})^+.\eqno(11.23)(e)$$
Now define a positive root system for $R^0$ by
$$(R^0)^+ = (R^{00})^+ \cup \{\,\alpha\in R^0\mid \langle
-\overline\alpha,\lambda\rangle > 0\,\}.\eqno(11.23)(f)$$
Write
$$\Delta^0 = \hbox{ simple roots for $(R^0)^+$.}\eqno(11.23)(g)$$

Since $\lambda$ and $\lambda_a$ define the same infinitesimal character,
there is a unique isomorphism
$$i(\lambda_a,\lambda;\Delta^0_a,\Delta^0):T_a \rightarrow T
\eqno(11.24)(a)$$
induced by an element of $G$, carrying $\lambda_a$ to $\lambda$ and
$\Delta^0_a$ to $\Delta^0$.  Just as in (11.17), we now define
$$R^+ = i(\lambda_a,\lambda;\Delta^0_a,\Delta^0)(R^+_a),\eqno(11.24)(b)$$
a system of positive roots for $T$ in $G$.  Because of (11.23)(e) and (f), and
(11.2)(d), we have
$$R^+ \supset R^+_{i{\Bbb R}}.\eqno(11.24)(c)$$
Put
$$\mu = i(\lambda_a,\lambda;\Delta^0_a,\Delta^0)(\mu_a) \in X^*(T).
\eqno(11.24)(d)$$
Then we can define
$$\Lambda' = (\Lambda^{can}\otimes \mu,R^+_{i{\Bbb R}},R^+_{\Bbb
R}).\eqno(11.24)(e)$$
Because of (11.23)(f) and (11.24)(c), this is a $G_{\Bbb R}$-limit character
of $T_{\Bbb R}$ of infinitesimal character $\lambda'_a$.  We are again in
the situation (11.17), and $\Lambda = \Lambda'-\mu$.  We need to check that
$\tau^0(\pi(\Lambda'))$ is empty.  So fix a root $\alpha\in \Delta^0$.  If
$\alpha$ is imaginary, then (11.24)(c) implies that it is simple in
$R^+_{i{\Bbb R}}$; so since $\Lambda$ is final, $\alpha$ is noncompact.  If
$\alpha$ is complex, then (11.23)(d) and (f) imply that $-\overline\alpha$ is
positive.  If $\alpha$ is real, then since $\Lambda$ is final, $\alpha$ does
not satisfy the parity condition.  By Proposition 11.18(b), $\alpha$ is not in
the $\tau$-invariant.  This proves the surjectivity of the map in Proposition
11.20.

For the injectivity, suppose $\Lambda'_1$ and $\Lambda'_2$ both map to the
equivalence class of $\Lambda$.  By Proposition 11.18(c),
 $$\psi(\pi(\Lambda'_1)) \simeq \pi(\Lambda) \simeq
\psi(\pi(\Lambda'_2)).$$
By Proposition 11.16(b), it follows that $\pi(\Lambda'_1) \simeq
\pi(\Lambda'_1)$.  By Theorem 11.7(c), $\Lambda'_1$ is equivalent to
$\Lambda'_2$.  Q.E.D.

Theorem 11.14 is a formal consequence of Proposition 11.16, Proposition
11.18, Proposition 11.20, and Theorem 11.7 (applied to the infinitesimal
character $\lambda'_a$ in the setting (11.15)).  Q.E.D.
%\end




 
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\centerline{\bf 12. Langlands parameters and Cartan subgroups.}
\vskip 1pc

In this section we show how to reformulate Theorem 10.4 in terms of
Cartan subgroups of extended groups and E-groups.  We begin by recasting
Theorem 11.14 in the language of extended groups.

{\bf Definition 12.1.}  Suppose $G^\Gamma$ is a weak extended group for
$G$, ${^\vee G}$ is a dual group for $G$, and $z\in Z({^\vee G})^{\theta_Z}$
(cf. Definition 10.3).  A {\it Cartan subgroup of $G^\Gamma$} is a weak
extended group $T^\Gamma \subset G^\Gamma$ with identity component a
maximal torus $T\subset G$.   The conjugation action of any element $\delta
\in T^\Gamma - T$ on $T$ is a real form of $T$, independent of the choice of
$\delta$; we write $T({\Bbb R})$ for the group of real points.  

A {\it $G$-limit character of $T^\Gamma$ of type $z$} is a pair
$(\delta,\Lambda)$ subject to the following conditions.  First,
$$\hbox{$\delta\in T^\Gamma - T$ is a strong real form of $G$}
\eqno(12.1)(a)$$
(Definition 2.13).  Thus $T({\Bbb R})$ is a Cartan subgroup of $G({\Bbb
R},\delta)$.  We can therefore impose the second requirement
$$\hbox{$\Lambda$ is a $G({\Bbb R},\delta)$-limit character of $T({\Bbb
R})$ of type $z$} \eqno(12.1)(b)$$
(Definition 11.2).  We say that $(\delta,\Lambda)$ is {\it final} if $\Lambda$
is (Definition 11.13).  We say that $(\delta,\Lambda)$ is {\it equivalent} to
the $G$-limit character $(\delta',\Lambda')$ of $(T')^\Gamma$ if there is an
element $g\in G$ that conjugates $\delta'$ to $\delta$, so that $g\cdot
\Lambda'$ and $\Lambda$ are equivalent as limit characters of $G({\Bbb
R},\delta)$ (Definition 11.6).  Write 
$$L^z(G/{\Bbb R}) = \{\hbox{ equivalence classes of final limit characters
$(\delta,\Lambda)$ of type $z$ }\}. \eqno(12.1)(c)$$
\medskip

This definition can be formulated a little more directly, by incorporating
some of the earlier definitions to which it refers.  We can say that a limit
character of $T^\Gamma$ of type $z$ is a quadruple
$$(\delta,\Lambda^{can},R^+_{i{\Bbb R}},R^+_{\Bbb R}),\eqno(12.2)(a)$$
subject to the conditions below.  First,
$$\hbox{ $\delta\in T^\Gamma - T$, and $\delta^2 \in Z(G)$.}
\eqno(12.2)(b)$$
Write $T^{can,T}$ for the corresponding canonical cover (Definition 10.1),
and $T^{can,T}({\Bbb R})$ for the preimage of $T({\Bbb R})$ in this cover 
(Definition 10.3):
$$ 1 \rightarrow \pi_1(T)^{can} \rightarrow T({\Bbb R})^{can,T} \rightarrow
T({\Bbb R}) \rightarrow 1. \eqno(12.2)(c)$$
Then $zz(\rho)\in {^\vee T}^{\theta_Z}$ (Definition 11.2) defines a character
$\chi^T_{zz(\rho)}$ of $\pi_1(T)^{can}$.  The second condition is
$$\hbox{ $\Lambda^{can}$ is a character of $T({\Bbb R})^{can,T}$, and
$\Lambda^{can}|_{\displaystyle\pi_1(T)^{can}} = \chi^T_{zz(\rho)}$.}
\eqno(12.2)(d)$$
(Notice that this condition does not involve $\delta$.)  Write $R_{i{\Bbb R}}$
for the set of roots assuming imaginary values on ${\frak t}({\Bbb R})$, and
$R_{\Bbb R}$ for those assuming real values.  Then
$$\hbox{$R^+_{i{\Bbb R}}$ is a set of positive roots for $R_{i{\Bbb R}}$, and
$R^+_{\Bbb R}$ for $R_{\Bbb R}$.}\eqno(12.2)(e)$$
Finally, write $\lambda = d\Lambda^{can} \in {\frak t}^*$.  Then we require
$$\langle \alpha,\lambda\rangle \ge 0,\qquad(\alpha\in R^+_{i{\Bbb
R}}).\eqno(12.2)(f)$$
We leave to the reader the straightforward task of formulating similarly
explicit descriptions of the notion of final and the equivalence relation on
limit characters.

\medskip
{\bf Theorem 12.3.} {\it Suppose $G^\Gamma$ is a weak extended group for
$G$ (Definition 2.13).  Then there is a natural bijection between the set
$\Pi^z(G/{\Bbb R})$ of equivalence classes of irreducible canonical
projective representations of type $z$ of strong real forms of $G$ 
(Definition 10.3) and the set $L^z(G/{\Bbb R})$ of equivalence classes of
final limit characters of type $z$ of Cartan subgroups in $G^\Gamma$
(Definition 12.1).}
\medskip

This is just a reformulation of Theorem 11.14.

We would like a parallel description of the Langlands parameters, in terms
of tori in the dual group.

\medskip
{\bf Definition 12.4.}  Suppose ${^\vee G^\Gamma}$ is a weak E-group
(Definition 4.3).  A {\it Cartan subgroup of ${^\vee G^\Gamma}$} is a weak
E-group ${^d{T}}^\Gamma \subset {^\vee G^\Gamma}$ such that the identity
component ${^d{T}}$ is a maximal torus in ${^\vee G}$, and the inclusion of
${^d{T}}^\Gamma$ in ${^\vee G^\Gamma}$ is an L-homomorphism (Definition
5.1).  Conjugation by any element $y$ of ${^d{T}}^\Gamma - {^d{T}}$ defines
an involutive automorphism $\theta$ of ${^d{T}}$, which is independent of
$y$.  Write
$$A({^d{T}}^\Gamma) = {^d{T}}^\theta/({^d{T}}^\theta)_0\eqno(12.4)(a)$$
for the group of connected components of the fixed points of $\theta$.  We
call this group the {\it Langlands component group for ${^d{T}}^\Gamma$}. 
As in (5.10) we can form such groups as ${^d{T}}^{alg,{^\vee G}}$ (which is a
quotient of the algebraic universal covering ${^d{T}}^{alg,{^d{T}}}$).  Define
$$A({^d{T}}^\Gamma)^{alg,{^\vee G}}  = ({^d{T}}^\theta)^{alg,{^\vee
G}}/(({^d{T}}^\theta)^{alg,{^\vee G}})_0,\eqno(12.4)(b)$$
the {\it universal component group for ${^d{T}}^\Gamma$ with respect to
${^\vee G^\Gamma}$}.

Since ${^d{T}}^\Gamma$ is a weak E-group in its own right, we can speak of
Langlands parameters for ${^d{T}}^\Gamma$ (Definition 5.2).  A Langlands
parameter $\phi$ is said to be {\it ${^\vee G}$-regular} if for every root
$\alpha$ of ${^d{T}}$ in ${^\vee G}$, we have
$$\alpha(\lambda(\phi)) \ne 0\eqno(12.4)(c)$$
(notation (5.8)).  In general the roots that fail to satisfy (12.4)(c) are called
{\it $\lambda(\phi)$-singular} or {\it $\phi$-singular}.

A {\it complete Langlands parameter for ${^d{T}}^\Gamma$ with respect to
${^\vee G^\Gamma}$} is a pair $(\phi,\tau_1)$ with $\phi$ a Langlands
parameter for ${^d{T}}^\Gamma$, and $\tau_1$ an irreducible representation
of $A({^d{T}}^\Gamma)^{alg,{^\vee G}}$ (cf. Definition 5.12).  Such a
parameter is said to be {\it equivalent} to a complete parameter
$(\phi',\tau_1')$ for $({^d{T}}^\Gamma)'$ if the triple $({^d{T}}^\Gamma, \phi,
\tau_1)$ is conjugate by ${^\vee G}$ to $(({^d{T}}^\Gamma)', \phi', \tau_1')$.
\medskip

To formulate a definition analogous to that of ``final,'' we need a little
preliminary notation.  Suppose we are in the setting of Definition 12.4. 
Since $\theta$ extends (although not uniquely) to an automorphism of
${^\vee G}$, it must permute the roots and coroots of ${^d{T}}$ in ${^\vee G}$. 
We write
$${^d{R}} = R({^\vee G},{^d{T}})\eqno(12.5)(a)$$
$${^d{R}}_{\Bbb R} = \{\,\alpha \in {^d{R}} \mid \theta\alpha =
-\alpha\,\}\eqno(12.5)(b)$$
$${^d{R}}_{i{\Bbb R}} = \{\,\alpha \in {^d{R}} \mid \theta\alpha =
\alpha\,\},\eqno(12.5)(c)$$
the {\it real} and {\it imaginary roots of ${^d{T}}^\Gamma$}.  Suppose now
that $\alpha$ is a real root.  The corresponding coroot $\alpha^\vee$ is a
homomorphism from ${\Bbb C}^\times$ to ${^d{T}}$ (or even into
${^d{T}}^{alg,{^\vee G}}$).  Since $\alpha$ is real, it satisfies
$$\theta(\alpha^\vee(z)) = \alpha^\vee(z^{-1}).\eqno(12.6)(a)$$
In particular, we get a distinguished element of order two
$$m_\alpha = \alpha^\vee(-1) \in {^d{T}}^\theta.\eqno(12.6)(b)$$
Similarly, we write
$$m^{alg}_\alpha = \alpha^\vee(-1) \in ({^d{T}}^\theta)^{alg,{^\vee
G}}.\eqno(12.6)(c)$$
These elements define elements of order two in the component groups of
Definition 12.4, which we write as
$$\overline{m^{alg}_\alpha} \in A({^d{T}}^\Gamma)^{alg,{^\vee
G}}\qquad(\alpha\in {^d{R}}_{\Bbb R}).\eqno(12.6)(d)$$

Next, suppose $\alpha$ is an imaginary root, and $y\in {^d{T}}^\Gamma -
{^d{T}}$.  Choose root vectors $X_{\pm \alpha} \in {^\vee {\frak g}}$.  Since
$\theta$ fixes $\alpha$, ${\rom Ad}(y)$ must send each root vector to a
multiple of itself:
$${\rom Ad}(y)X_{\pm\alpha} = z_{\pm \alpha}X_\alpha.\eqno(12.7)(a)$$
The bracket of these two root vectors is a multiple of the derivative of the
coroot $\alpha^\vee$, and is therefore fixed by $\theta = {\rom Ad}(y)$.  It
follows that
$$z_\alpha = (z_{-\alpha})^{-1}.\eqno(12.7)(b)$$
Similarly, if $\alpha$, $\beta$ and $\alpha+\beta$ are all imaginary roots, 
$$z_{\alpha+\beta} = z_\alpha z_\beta.\eqno(12.7)(c)$$
Now $y^2$ belongs to ${^d{T}}$, so $\alpha(y^2)$ is defined.  Clearly
$$(z_\alpha)^2 = \alpha(y^2).\eqno(12.7)(d)$$
We say that $\alpha$ is {\it $y$-compact} if $z_\alpha = 1$, and {\it
$y$-noncompact} if $z_\alpha = -1$.  As a consequence of (12.7)(d), every
imaginary root in the centralizer of $y^2$ is either compact or noncompact.

\medskip
{\bf Definition 12.8.}  Suppose $\phi$ is a Langlands parameter for
${^d{T}}^\Gamma$.  As in (5.8), we associate to $\phi$ elements $y = y(\phi)
\in {^d{T}}^\Gamma - {^d{T}}$ and $\lambda = \lambda(\phi) \in {^d{{\frak
t}}}$.  An imaginary root $\alpha$ is called {\it $\phi$-compact}
(respectively {\it $\phi$-noncompact}) if it is $y(\phi)$-compact
(respectively $y(\phi)$-noncompact) in the sense of (12.7).  By Proposition
5.6(b) and (12.7)(d), every $\phi$-singular imaginary root is either
$\phi$-compact or $\phi$-noncompact.

Suppose next that $(\phi,\tau_1)$ is a complete Langlands parameter for
${^d{T}}^\Gamma$ with respect to ${^\vee G^\Gamma}$ (Definition 12.4).  We
say that the pair $(\phi,\tau_1)$ is {\it final} if it satisfies the following
two conditions.
\smallskip
\item{(a)}  Every $\phi$-singular imaginary root of ${^d{T}}$ in ${^\vee G}$
is $\phi$-compact.
\item{(b)}  If $\alpha$ is a $\phi$-singular real root of ${^d{T}}^\Gamma$ in
${^\vee G^\Gamma}$, then
$$\tau_1(\overline{m_\alpha^{alg}}) = 1$$
(notation 12.6)(c).)  
\medskip

{\bf Theorem 12.9.} {\it Suppose ${^\vee G^\Gamma}$ is a weak E-group
(Definition 4.3).  Then there is a one-to-one correspondence between the set
$\Xi({^\vee G^\Gamma})$ of equivalence classes of complete Langlands
parameters for ${^\vee G^\Gamma}$ (Definition 5.11) and the set of
equivalence classes of final complete Langlands parameters for Cartan
subgroups of ${^\vee G^\Gamma}$ (Definition 12.8).}
\medskip

{\bf Proof.}  Suppose $(\phi,\tau)$ is a complete Langlands parameter for
${^\vee G^\Gamma}$.  We want to construct a Cartan subgroup
${^d{T}}^\Gamma$ containing the image of $\phi$.  To do this, use the
notation of Corollary 5.9.  There we defined a Levi subgroup
$L(\lambda,y\cdot\lambda)$ of ${^\vee G}$, on which conjugation by $y$
acts as an involutive automorphism $\theta_y$.  We apply to this subgroup
the following well-known facts.  (The slightly awkward hypotheses on $K$
will allow us to apply the result to coverings.)

\medskip
{\bf Lemma 12.10}  {\it Suppose $G$ is a connected complex reductive
algebraic group, and $\theta$ is an involutive automorphism of $G$.  Suppose
$K$ is an algebraic subgroup of $G$ having the same identity component as
the group of fixed points of $\theta$, with the property that the
automorphisms of $G$ in ${\rom Ad}(K)$ all commute with $\theta$.  Then
there is a $\theta$-stable maximal torus $T \subset G$, determined up to
conjugation by $K_0$ by either of the following properties.  Write ${\frak
a}$ for the $-1$ eigenspace  of $\theta$ on ${\frak t}$, and $M$ for the
centralizer of ${\frak a}$ in $G$.
\smallskip
\item{a)}  The Lie algebra ${\frak a}$ is a maximal semisimple abelian
subalgebra in the $-1$ eigenspace of $\theta$ on ${\frak g}$.
\item{b)}  Every root of $T$ in $M$ is compact (cf. (12.7); the roots in $M$
are precisely the imaginary roots of $T$ in $G$).
\smallskip
In addition, $T$ has the following properties.
\smallskip
\item{c)}  $T\cap K$ meets every connected component of $K$.  That is, the
natural map $(T\cap K)/(T\cap K)_0 \rightarrow K/K_0$ is surjective.
\item{d)}  The kernel $T\cap K_0$ of the map in (c) is generated by $(T\cap
K)_0$ and the elements $m_\alpha \in T$ attached to real roots $\alpha$
(cf. (12.6)).
\smallskip
In particular, the characters of the component group $K/K_0$ are in
one-to-one correspondence with the characters of $(T\cap K)/(T\cap K)_0$
trivial on all the elements $m_\alpha$ (for $\alpha$ real).}
\medskip

A torus with the properties in the lemma will be called a {\it maximally
$\theta$-split torus for $G$}.   We postpone a discussion of the proof for a
moment, and continue with the proof of Theorem 12.9.  So choose a
maximally $\theta_y$-split torus 
$${^d{T}}\subset L(\lambda,y\cdot \lambda).\eqno(12.11)(a)$$
Since $L(\lambda,y\cdot\lambda)$ is a Levi subgroup of ${^\vee G}$, it
follows that ${^d{T}}$ is a maximal torus in ${^\vee G}$.  Furthermore the Lie
algebra ${^d{\frak t}}$ must contain the central elements $\lambda$ and
$y\cdot\lambda$ of ${\frak l}(\lambda,y\cdot\lambda)$, so
$$\phi|_{{\Bbb C}^\times} :{\Bbb C}^\times \rightarrow {^d{T}}.$$
Define
$${^d{T}}^\Gamma = \hbox{ group generated by $y$ and
${^d{T}}$,}\eqno(12.11)(b)$$
so that
$$\phi:W_{\Bbb R} \rightarrow {^d{T}}^\Gamma.\eqno(12.11)(c)$$
Since $\theta_y$ preserves ${^d{T}}$, the element $y$ normalizes ${^d{T}}$. 
Since $y^2$ belongs to ${^d{T}}$ (cf. Proposition 5.6(b)), it follows that
${^d{T}}$ has index two in ${^d{T}}^\Gamma$.  It follows easily that ${^d{T}}$
is a weak E-group, and so a Cartan subgroup of ${^\vee G^\Gamma}$
(Definition 12.4).  

A real or imaginary root is $\lambda$-singular if and only if it is
$y\cdot\lambda$-singular.  It follows that the real or imaginary
$\phi$-singular roots are exactly the real or imaginary roots of ${^d{T}}$ in
$L(\lambda,y\cdot\lambda)$.  By Lemma 12.10(b), this means that
$$\hbox{ every $\phi$-singular imaginary root of ${^d{T}}$ in ${^\vee G}$ is
$\phi$-compact. }\eqno(12.11)(d)$$

As in Corollary 5.9, write $K(y)$ for the centralizer of $y$ in ${^\vee G}$. 
The component group of which the datum $\tau$ is a character is
$$A_\phi^{loc,alg} = [K(y) \cap L(\lambda,y\cdot\lambda)]^{alg}/\left([K(y)
\cap L(\lambda,y\cdot\lambda)]^{alg}\right)_0$$
(Corollary 5.9(c)).  Lemma 12.10(c) (applied to the group
$L(\lambda,y\cdot\lambda)^{alg,{^\vee G}}$) provides a surjective map
$$A({^d{T}}^\Gamma)^{alg,{^\vee G}} \rightarrow
A_\phi^{loc,alg}.\eqno(12.11)(e)$$
We use this map to pull $\tau$ back to a character $\tau_1$ of
$A({^d{T}}^\Gamma)^{alg,{^\vee G}}$.  By Lemma 12.10(d),
$$\tau_1(m_\alpha^{alg}) = 1\qquad\hbox{($\alpha$ real and
$\phi$-singular).}\eqno(12.11)(f)$$

By (12.11)(d) and (12.11)(f), the pair $(\phi,\tau_1)$ is a final complete
Langlands parameter for ${^d{T}}^\Gamma$ with respect to ${^\vee
G^\Gamma}$.  The only choice involved is that of the maximal torus $T$, and
Lemma 12.10 guarantees that $T$ is unique up to conjugation by the
centralizer of $\phi$ in ${^\vee G}$.  It follows easily that the map from
parameters for ${^\vee G^\Gamma}$ to parameters for Cartan subgroups is
well-defined on equivalence classes.

This argument can be reversed without difficulty, to recover a unique
complete parameter for ${^\vee G^\Gamma}$ from a complete final
parameter for a Cartan subgroup.  It follows that the correspondence is
bijective.  Q.E.D.

\medskip
{\bf Sketch of proof of Lemma 12.10.}  That conditions (a) and (b) are
equivalent is fairly easy; the main point is the uniqueness of $T$ up to
$K_0$-conjugacy .  This is proved in [KR], Theorem 1 (or [Vust], p.323).  Part
(c) is [KR], Proposition 1 (or [Vust], Proposition 7).

For (d), consider the Levi subgroup
$$L = \hbox{ centralizer of $(T\cap K)_0$ in $G$.}$$
As the centralizer of a torus, $L$ is connected.  The roots of $T$ in $L$ are
precisely the real roots of $T$ in $G$.  Since $(T\cap K)_0$ is also a torus in
$K_0$, it follows that $L\cap K_0$ is also connected.  It follows that
$$T\cap K_0 = T\cap (L\cap K_0) = T \cap (L\cap K)_0.$$
It therefore suffices to prove (d) for the subgroup $L$ instead of for all of
$G$; that is, under the assumption that all roots are real.  It is enough to
prove (d) for a finite cover of $G$, and then for each factor of a
$\theta$-stable direct product decomposition.  After such reductions, we
may assume that $G$ is simple and simply connected, and that $\theta$ acts
by $-1$ on ${\frak t}$.  Then $T^\theta$ consists precisely of the elements
of order 2 in $T$.  Since $G$ is simply connected, $X_*(T)$ is generated by
the coroot lattice.  Now it follows from Lemma 9.9 and the definitions that
the elements $m_\alpha$ generate the full group of elements of order 2 in
$T$.  Consequently
$$T\cap K_0 \subset T^\theta = \hbox{ group generated by the
$m_\alpha$.}$$
The other containment (that is, that $m_\alpha\in K_0$) follows from a
standard calculation in $SL(2)$.  Q.E.D.

%\end\input mssymb.tex
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\vskip 3pc
\centerline{\bf 13. Pairings between Cartan subgroups and the proof of
Theorem 10.4.}
\vskip 1pc

Theorem 12.3 describes representations in terms of Cartan subgroups of
$G^\Gamma$,  and Theorem 12.9 describes Langlands parameters in terms of
Cartan subgroups of ${^\vee G^\Gamma}$.  To complete the proof of Theorem
10.4, we need to relate Cartan subgroups of $G^\Gamma$ and ${^\vee
G^\Gamma}$.  We begin by doing this for maximal tori.  The following lemma
more or less restates the definition of dual group; we leave its proof to the
reader.

\medskip
{\bf Lemma 13.1.}  {\it Suppose $G$ is a complex connected reductive
algebraic group, and ${^\vee G}$ is a dual group for $G$ (Definition 4.2).  Fix
maximal tori $T\subset G$ and ${^d{T}}\subset {^\vee G}$.  Write ${^\vee T}$
for a dual torus to $T$ (cf. (9.3)).  Suppose $\Delta\subset X^*(T)$ is a set of
simple roots for $T$ in $G$, and ${^d{\Delta}}\subset X^*({^d{T}})$ a set of
simple roots for ${^d{T}}$ in ${^\vee G}$.  Write $\Delta^\vee\subset X_*(T)$
and ${^d{\Delta}}^\vee\subset X_*({^d{T}})$ for the corresponding sets of
simple coroots.
\smallskip
\item{a)} There is a natural isomorphism
$$\zeta(\Delta,{^d{\Delta}}): {^\vee T} \rightarrow {^d{T}}.$$
\item{b)} The induced map on one-parameter subgroup lattices
$$\zeta_*(\Delta,{^d{\Delta}}):(X_*({^\vee T}) = X^*(T)) \rightarrow
X_*({^d{T}})$$
carries $\Delta$ onto ${^d{\Delta}}^\vee$.
\item{c)} The induced map (in the other direction) on character lattices
$$\zeta^*(\Delta,{^d{\Delta}}): X^*({^d{T}}) \rightarrow (X^*({^\vee T}) =
X_*(T))$$
carries ${^d{\Delta}}$ onto $\Delta^\vee$.
\item{d)} If sets of simple roots are not fixed, we obtain a finite family of
natural isomorphisms
$$\zeta: {^\vee T} \rightarrow {^d{T}}.$$
Any two of these isomorphisms differ by composition with the action of a
unique element of the Weyl group of ${^d{T}}$ in ${^\vee G}$.  
\smallskip
The characteristic property of the isomorphism $\zeta(\Delta,{^d{\Delta}})$
is that the induced maps described in (b) and (c) implement the natural
isomorphism from the dual
$${^\vee{\Psi}_0}(G) = (X_*(T),\Delta^\vee,X^*(T),\Delta)$$
of the based root datum for $G$, onto the based root datum
$$\Psi_0({^\vee G}) = (X^*({^d{T}}), {^d{\Delta}}, X_*({^d{T}}),
{^d{\Delta}}^\vee). $$
\medskip}

With the r\^oles of $G$ and ${^\vee G}$ reversed, (a) provides also an
isomorphism
$$\zeta({^d{\Delta}},\Delta): {^\vee {^d{T}}} \rightarrow T.$$
It coincides with the one obtained from $\zeta(\Delta,{^d{\Delta}})$ by
applying the dual torus functor (cf. (9.3)).  Using that functor, we can also
think of the various natural isomorphisms of (d) as differing by elements of
the Weyl group of $T$ in $G$.

\medskip
{\bf Definition 13.2} (cf. [AV2], Definition 9.11).  Suppose $G^\Gamma$ is a
weak extended group, and ${^\vee G^\Gamma}$ is a weak E-group for
$G^\Gamma$ (Definition 4.3).  Fix Cartan subgroups $T^\Gamma\subset
G^\Gamma$ and ${^d{T}}^\Gamma \subset {^\vee G^\Gamma}$ (Definitions
12.1 and 12.4).  Write $\sigma$ for the real form of $T$ defined by any
element of $T^\Gamma - T$, and $\theta$ for the involutive automorphism
of ${^d{T}}$ defined by any element of ${^d{T}}^\Gamma - {^d{T}}$.  By
Proposition 2.12, $\sigma$ defines an involutive automorphism
$$a_T \in {\rom Aut}(\Psi_0(T)) = {\rom Aut}(X^*(T), X_*(T)).$$
That is, $a_T$ gives an automorphism of each of the lattices $X^*(T)$ and
$X_*(T)$, respecting the pairing into ${\Bbb Z}$.  Similarly (Proposition 2.11
or Proposition 4.4) the involution $\theta$ defines an involutive
automorphism
$$a_{^d{T}} \in {\rom Aut}({^\vee{\Psi_0({^d{T}}}})) = {\rom
Aut}(X_*({^d{T}}),X^*({^d{T}})).$$

A {\it weak pairing between $T^\Gamma$ and ${^d{T}}^\Gamma$} is an
identification of ${^d{T}}^\Gamma$ with a weak E-group for $T^\Gamma$,
subject to one additional condition that we now describe.  According to
Definition 4.3, the identification in question amounts to an isomorphism
$$\zeta: {^\vee T}\rightarrow {^d{T}}\eqno(13.2)(a)$$
of the dual torus ${^\vee T}$ for $T$ (cf. (9.3)) with ${^d{T}}$.  Such an
identification is just an isomorphism
$$\zeta:(X^*(T), X_*(T)) \rightarrow (X_*({^d{T}}),X^*({^d{T}}))
\eqno(13.2)(b)$$
In order to make ${^d{T}}^\Gamma$ an E-group for $T^\Gamma$, this
isomorphism must satisfy
$$\hbox{$\zeta$ carries $a_T$ to $a_{^d{T}}$.}\eqno(13.2)(c)$$
The additional requirement that we impose for a weak pairing is
$$\hbox {$\zeta$ is one of the natural isomorphisms of Lemma 13.1(d).}
\eqno(13.2)(d)$$
This condition brings the groups $G^\Gamma$ and ${^\vee G}^\Gamma$ into
the definition.
\medskip
{\bf Lemma 13.3.}  {\it Suppose we are in the setting of Definition 13.2; use
the notation there.
\smallskip
\item{a)}  The isomorphism $\zeta$ induces an identification of Weyl groups
$$\zeta: W(G,T) \rightarrow W({^\vee G},{^d{T}}).$$
\item{b)}  The isomorphism of (a) carries the action of the involution
$\sigma$ on $W(G,T)$ to the action of $\theta$ on $W({^\vee G},{^d{T}})$.  In
particular, we have an isomorphism
$$\zeta: W(G,T)^\sigma \rightarrow W({^\vee G},{^d{T}})^\theta.$$
\item{c)}  Suppose $\zeta'$ is another isomorphism from ${^\vee T}$ to
${^d{T}}$ (Lemma 13.1(d)).  Then $\zeta'$ is a weak pairing between the
Cartan subgroups if and only if $\zeta' = w\circ \zeta$ for some $w \in
W({^\vee G},{^d{T}})^\theta$.
\medskip}

This is obvious.

\medskip
{\bf Proposition 13.4} ([AV2], Lemma 9.16).  {\it Suppose $G^\Gamma$ is a
weak extended group, and ${^\vee G^\Gamma}$ is a weak E-group for
$G^\Gamma$ (Definition 4.3).
\smallskip
\item{a)}  Suppose $T^\Gamma$ is a Cartan subgroup of $G^\Gamma$
(Definition 12.1).  Then there is a Cartan subgroup ${^d{T}}^\Gamma$ of
${^\vee G^\Gamma}$ and a weak pairing between $T^\Gamma$ and
${^d{T}}^\Gamma$ (Definition 13.2).  The Cartan subgroup ${^d{T}}^\Gamma$
is unique up to conjugation by ${^\vee G}$.
\item{b)}  Suppose ${^d{T}}^\Gamma$ is a Cartan subgroup of ${^\vee
G^\Gamma}$ (Definition 12.4).  Then there is a Cartan subgroup $T^\Gamma$
of $G^\Gamma$ and a weak pairing between $T^\Gamma$ and
${^d{T}}^\Gamma$ (Definition 13.2).  The Cartan subgroup $T^\Gamma$ is
unique up to conjugation by $G$.
\medskip}

{\it Proof.}  For (a), let ${^d{T}}$ be any maximal torus in ${^\vee G}$, and $y$
an element of ${^\vee G^\Gamma} - {^\vee G}$ normalizing ${^d{T}}$.  (To find
such an element $y$, start with any $y_0$ in ${^\vee G^\Gamma} - {^\vee G}$,
and modify it by an element of ${^\vee G}$ conjugating ${\rom
Ad}(y_0)({^d{T}})$ back to ${^d{T}}$.)  Then conjugation by $y$ defines an
automorphism $\theta_y$ of ${^d{T}}$, and so an automorphism $a_y$ of
${^\vee \Psi}_0({^d{T}})$.  

Fix one of the natural isomorphisms $\zeta$ from ${^\vee T}$ to ${^d{T}}$
(Lemma 13.1).  This isomorphism carries the automorphism $a_T$ to an
involutive automorphism $a_{^d{T}}$ of ${^\vee \Psi}_0({^d{T}})$.  Since
${^\vee G^\Gamma}$ is an E-group for $G^\Gamma$, it follows from
Definition 4.3 that $a_y$ and $a_{^d{T}}$ must differ by the action of an
element $w\in W({^\vee G},{^d{T}})$:
$$a_{^d{T}} = w\circ a_y.$$
Now let $n \in {^\vee G}$ be any representative of $w$, and set $y' = ny$. 
Then $y' \in {^\vee G^\Gamma} - {^\vee G}$ still normalizes ${^d{T}}$, and the
conjugation action of $y'$ defines the involutive automorphism $\theta$ of
${^d{T}}$ corresponding to $a_{^d{T}}$.  The group ${^d{T}}^\Gamma$
generated by $y'$ and ${^d{T}}$ is therefore a Cartan subgroup of ${^\vee
G^\Gamma}$ paired with $T^\Gamma$.

For the uniqueness, suppose $({^d{T}}_1^\Gamma)'$ is another Cartan
subgroup paired by $\zeta_1$ with $T^\Gamma$.  After conjugating by
${^\vee G}$, we may assume that ${^d{T}}_1 = {^d{T}}$ and that $\zeta_1 =
\zeta$.  Then the automorphism of ${^d{T}}$ defined by any element $y$ of
${^d{T}}^\Gamma - {^d{T}}$ coincides with the automorphism defined by any
element $y'$ of $({^d{T}}^\Gamma)' - {^d{T}}$.  It follows that $y{^d{T}} =
y'{^d{T}}$, so that ${^d{T}}^\Gamma = ({^d{T}}^\Gamma)'$ as we wished to
show.

The proof of (b) is formally identical, and we omit it.  Q.E.D.

Our next goal is to reduce the ambiguity in the notion of weak pairing
described in Lemma 13.3(c).

\medskip
{\bf Definition 13.5.}  Suppose $(G^\Gamma,{\cal W})$ is an extended group
for $G$ (Definition 1.12).  A {\it based Cartan subgroup of $(G^\Gamma,{\cal
W})$} is a quadruple ${\cal T}^\Gamma = (T^\Gamma,{\cal
W}(T^\Gamma),R^+_{i{\Bbb R}},R^+_{\Bbb R})$, subject to the following
conditions.
\smallskip
\item{(a)} $T^\Gamma$ is a Cartan subgroup of $G^\Gamma$ (Definition
12.1).
\item{(b)} ${\cal W}(T^\Gamma)$ is an extended group structure on
$T^\Gamma$ (Definition 1.12).  That is, ${\cal W}(T^\Gamma)$ is a
$T$-conjugacy class of elements $\delta \in T^\Gamma - T$, with the
property that $\delta^2$ has finite order.
\item{(c)} ${\cal W}(T^\Gamma)$ is a subset of ${\cal W}$, in the sense that
for every $\delta\in {\cal W}(T^\Gamma)$ there are $N$ and $\chi$ so that
$(\delta,N,\chi)\in {\cal W}$.
\item{(d)}  $R^+_{i{\Bbb R}}$ is a set of positive imaginary roots of $T$ in
$G$, and $R^+_{\Bbb R}$ is a set of positive real roots.
\item{(e)} The extended group structure and the positive imaginary roots are
compatible in the following sense.  Fix $\delta\in {\cal W}(T^\Gamma)$, and
$N$ and $\chi$ so that $(\delta,N,\chi)\in {\cal W}$.  Write $G({\Bbb R}) =
G({\Bbb R},\delta)$ for the corresponding (quasisplit) real form of $G$. 
Suppose $\Lambda^{can}$ is an  irreducible canonical projective
representation of $T({\Bbb R})^{can,T}$ of type $zz(\rho)$, and that
$$\Lambda = (\Lambda^{can},R^+_{i{\Bbb R}},R^+_{\Bbb R}))$$
is a $G({\Bbb R})$-limit character of type $z$ (Definition 11.2).  Then what
we require is that the corresponding standard limit representation
$M(\Lambda)$ admit a Whittaker model of type $\chi$ (Definition 3.1).
\smallskip
Two based Cartan subgroups are called {\it equivalent} if they are conjugate
by $G$.

\medskip
{\bf Proposition 13.6.}  {\it Suppose $T^\Gamma$ is a Cartan subgroup of the
extended group $(G^\Gamma,{\cal W})$, and $(R^+_{i{\Bbb R}},R^+_{\Bbb R})$
are systems of imaginary and real positive roots.  Then there is a unique
extended group structure ${\cal W}(T^\Gamma)$ on $T^\Gamma$ making the
quadruple $(T^\Gamma,{\cal W}(T^\Gamma),R^+_{i{\Bbb R}},R^+_{\Bbb R})$ a
based Cartan subgroup.}
\medskip

The proof is fairly long, so we postpone it to the next section.

There is an analogue of Definition 13.5 for E-groups.  The key to it is the
notion of ``special'' systems of positive imaginary roots formulated in
section 6 of [AV2].

\medskip
{\bf Definition 13.7.}  Suppose $({^\vee G^\Gamma},{\cal S})$ is an E-group
for $G$ (Definition 4.14).  A {\it based Cartan subgroup of $({^\vee
G^\Gamma},{\cal S})$} is a quadruple ${^d{\cal T}^\Gamma} =
({^d{T}}^\Gamma, {\cal S}({^d{T}}^\Gamma), {^d{R}}^+_{i{\Bbb
R}},{^d{R}}^+_{\Bbb R})$, subject to the following conditions.
\smallskip
\item{(a)} ${^d{T}}^\Gamma$ is a Cartan subgroup of ${^\vee G^\Gamma}$
(Definition 12.4).
\item{(b)} ${\cal S}({^d{T}}^\Gamma)$ is an E-group structure on
${^d{T}}^\Gamma$ (Definition 4.14).  That is, ${\cal S}({^d{T}}^\Gamma)$ is a
${^d{T}}$-conjugacy class of elements ${^\vee{\delta}} \in T^\Gamma - T$,
with the property that ${^\vee{\delta}}^2$ has finite order.
\item{(c)} ${\cal S}({^d{T}}^\Gamma)$ is a subset of ${\cal S}$, in the sense
that for every ${^\vee{\delta}}\in {\cal S}({^d{T}}^\Gamma)$ there is a Borel
subgroup ${^d{B}}$ so that $({^\vee{\delta}},{^d{B}})\in {\cal S}$.
\item{(d)}  ${^d{R}}^+_{i{\Bbb R}}$ is a set of positive imaginary roots of
${^d{T}}$ in ${^\vee G}$, and ${^d{R}}^+_{\Bbb R}$ is a set of positive real
roots (cf. (12.5)).
\item{(e)} The E-group structure and the positive imaginary roots are
compatible in the following sense.  Fix ${^\vee{\delta}}\in {\cal
S}({^d{T}}^\Gamma)$, and define ${^d{\cal B}}$ to be the set of Borel
subgroups ${^d{B}}$ so that $({^\vee{\delta}},{^d{B}})\in {\cal S})$.  (The set
${^d{\cal B}}$ is a single orbit under the action of the centralizer $K$ of
${^\vee{\delta}}$ in ${^\vee G}$.)  Then what we require is that
${^d{R}}^+_{i{\Bbb R}}$ be special with respect to ${^d{\cal B}}$ ([AV2],
Definition 6.29).
\smallskip
Two based Cartan subgroups are called {\it equivalent} if they are conjugate
by ${^\vee G}$.
\medskip

It is possible to give a more geometric definition of ``special'' than the one
in [AV2], and in fact this is crucial for the proof given there of some
technical results we will use.

Here is the analogue of Proposition 13.6.

\medskip
{\bf Proposition 13.8.}  {\it Suppose ${^d{T}}^\Gamma$ is a Cartan subgroup
of the E-group $({^\vee G^\Gamma},{\cal S})$, and $({^d{R}}^+_{i{\Bbb
R}},{^d{R}}^+_{\Bbb R})$ are systems of imaginary and real positive roots. 
Then there is a unique E-group structure ${\cal S}({^d{T}}^\Gamma)$ on
${^d{T}}^\Gamma$ making the quadruple $({^d{T}}^\Gamma,{\cal
W}({^d{T}}^\Gamma),R^+_{i{\Bbb R}},R^+_{\Bbb R})$ a based Cartan subgroup.}
\medskip

Again we postpone the proof until the next section.  

\medskip
{\bf Definition 13.9.}  Suppose $(G^\Gamma,{\cal W})$ is an extended group
for $G$ (Definition 1.12), and $({^\vee G^\Gamma},{\cal S})$ is a
corresponding E-group (Definition 4.14).  Fix based Cartan subgroups ${\cal
T}^\Gamma$ for $G^\Gamma$ (Definition 13.5) and ${^d{\cal T}}^\Gamma$ for
${^\vee G^\Gamma}$ (Definition 13.7).  A {\it pairing between ${\cal
T}^\Gamma$ and ${^d{\cal T}}^\Gamma$}  is a weak pairing
$$\zeta: {^\vee{T}} \rightarrow {^d{T}} \eqno(13.9)(a)$$
(Definition 13.2), subject to the additional conditions
$$\hbox{ $\zeta$ carries the positive imaginary roots $R^+_{i{\Bbb R}}$ onto
the positive real coroots ${^d{R}}^{\vee,+}_{\Bbb R}$}\eqno(13.9)(b)$$
(cf. Lemma 13.1(b)), and
$$\hbox{ $\zeta$ carries the positive real roots $R^+_{{\Bbb R}}$ onto the
positive imaginary coroots ${^d{R}}^{\vee,+}_{i{\Bbb R}}$.}\eqno(13.9)(c)$$
In particular, this identifies $({^d{T}}^\Gamma,{\cal S}({^d{T}}^\Gamma))$ as
an E-group of $(T^\Gamma,{\cal W}(T^\Gamma))$.
\medskip

{\bf Proposition 13.10.}  {\it Suppose $(G^\Gamma,{\cal W})$ is an extended
group for $G$ (Definition 1.12), and $({^\vee G^\Gamma},{\cal S})$ is a
corresponding E-group (Definition 4.14).
\smallskip
\item{a)} Suppose ${\cal T}^\Gamma$ is a based Cartan subgroup for
$G^\Gamma$ (Definition 13.5).  Then there is a based Cartan subgroup
${^d{\cal T}^\Gamma}$ for ${^\vee G^\Gamma}$ and a pairing between them
(Definition 13.9).  ${^d{\cal T}^\Gamma}$ is uniquely determined up to
conjugation by ${^\vee G^\Gamma}$.
\item{b)} Suppose ${^d{\cal T}^\Gamma}$ is a based Cartan subgroup of
${^\vee G^\Gamma}$ (Definition 13.7).  Then there is a based Cartan subgroup
${\cal T}^\Gamma$ for $G^\Gamma$ and a pairing between them.  ${\cal
T}^\Gamma$ is unique up to conjugation by $G$.
\item{c)} Suppose $\zeta$ is a pairing between based Cartan subgroups as
above.  If $\zeta'$ is another isomorphism from ${^\vee T}$ to ${^d{T}}$, then
$\zeta'$ is a pairing if and only if $\zeta' = w\circ\zeta$ for some $w\in
W({^\vee G},{^d{T}})^\theta$ preserving ${^d{R}}^+_{i{\Bbb R}}$ and
${^d{R}}^+_{\Bbb R}$.  
\item{d)} Suppose $w$ is as in (c), and use the isomorphism of Lemma
13.3(b) to identify $w$ with an element of $W(G,T)^\sigma$.  If $\sigma'$ is
any real form of $G$ extending $\sigma$ on $T$, then $w$ has a
representative in $G({\Bbb R},\sigma')$.
\medskip}

{\it Proof.}  For (a), use Proposition 13.4(a) to find a Cartan subgroup
${^d{T}}^\Gamma$ of ${^\vee G^\Gamma}$ and a weak pairing $\zeta$ between
it and $T^\Gamma$.  By (13.2)(c), $\zeta$ carries the positive imaginary
roots $R^+_{i{\Bbb R}}$ onto some set ${^d{R}}^{\vee,+}_{\Bbb R}$ of positive
real coroots for ${^d{T}}$; and similarly for real roots.  Proposition 13.8
then guarantees that we can use these sets of positive roots to construct a
based Cartan subgroup ${^d{\cal T}^\Gamma}$. The uniqueness of its
equivalence class follows from the uniqueness in Proposition 13.4(a),
Lemma 13.3(c), and the uniqueness in Proposition 13.8.  Part (b) is proved in
exactly the same way.  Part (c) is clear from Lemma 13.3(c) and Definition
13.9.  Part (d) follows from the description of real Weyl groups in [IC4],
Proposition 4.16.  Q.E.D.

We are near the end of the maze leading to Theorem 10.4 now.  Before we
pull all the pieces together, we need one more definition.

\medskip
{\bf Definition 13.11.}  Suppose ${\cal T}^\Gamma$ is a based Cartan
subgroup of an extended group $(G^\Gamma,{\cal W})$ (Definition 3.5).  A
$G$-limit character of $T^\Gamma$ (Definition 12.1) is said to be {\it
compatible with ${\cal T}^\Gamma$} if the corresponding systems of
positive real and imaginary roots agree.

Suppose ${^d{\cal T}^\Gamma}$ is a based Cartan subgroup of the E-group
$({^\vee G^\Gamma},{\cal S})$ (Definition 13.7).  A  complete Langlands
parameter for ${^d{T}}^\Gamma$ (Definition 12.4) is said to be {\it
compatible with ${^d{\cal T}^\Gamma}$} if
$$\langle \alpha,\lambda \rangle \ge 0 \qquad (\alpha \in {^d{R}}^+_{\Bbb
R}).\eqno(13.11)$$
Here $\lambda$ is constructed from the Langlands parameter as in (5.8).  We
do {\it not} assume any relationship between $\lambda$ and the positive
imaginary roots.
\medskip

{\bf Proposition 13.12.}  {\it Suppose ${\cal T}^\Gamma$ is a based Cartan
subgroup for an extended group $(G^\Gamma,{\cal W})$, ${^d{\cal
T}^\Gamma}$ is a based Cartan subgroup for an E-group $({^\vee
G^\Gamma},{\cal S})$, and $\zeta$ is a pairing between them (Definition
13.9).  Write $z$ for the second invariant of the E-group (Definition 4.14). 
Use $\zeta$ to identify $({^d{T}^\Gamma},{\cal S}({^d{T}^\Gamma}))$ as an
E-group for $T^\Gamma$ (of second invariant $zz(\rho)$).  Then the
correspondence of Corollary 10.7 induces a bijection between the set of
limit characters of $T^\Gamma$ compatible with ${\cal T}^\Gamma$ and the
set of complete Langlands parameters for ${^d{T}^\Gamma}$ compatible with
${^d{\cal T}^\Gamma}$ (Definition 13.11).  This bijection identifies final
limit characters with final limit parameters.

A little more precisely, each Langlands parameter $\phi$ for
${^d{T}^\Gamma}$ gives rise to a canonical projective character
$\Lambda^{can}(\phi)$ of $T({\Bbb R})$ of type $zz(\rho)$ (Proposition 10.6). 
Each irreducible representation $\tau_1$ of $A({^d{T}^\Gamma})^{alg}$ gives
rise to a $T$-conjugacy class in $T^\Gamma - T$, say with a representative
$\delta(\tau_1)$ (Corollary 9.12).  We have
\smallskip
\item{a)} The element $\delta(\tau_1)$ defines a strong real form of $G$
--- that is, $\delta(\tau_1)^2$ belongs to $Z(G)$ --- if and only if $\tau_1$
factors through the quotient group $A({^d{T}^\Gamma})^{alg,{^\vee G}}$.
\item{b)} The positivity requirement (12.2)(f) on $\Lambda^{can}(\phi)$ is
equivalent to the compatibility requirement (13.11).
\item{c)} A simple imaginary root $\alpha$ for $T$ is noncompact with
respect to $\delta(\tau_1)$ if and only the corresponding simple real root
${^d{\alpha}}$ for ${^d{T}}$ satisfies
$$\tau_1(\overline{m^{alg}_{^d{\alpha}}}) = 1.$$
\item{d)} A real root $\beta$ for $T$ satisfies the parity condition for
$\Lambda^{can}(\phi)$ if and only if the corresponding imaginary root
${^d{\beta}}$ for ${^d{T}}$ is $\phi$-noncompact (Definition 12.8).
\medskip}

{\it Proof.}  Because of the definitions, it suffices to prove the assertions
(a)--(d).  For (a), fix an element $\delta \in {\cal W}(T^\Gamma)$.  The
element $\delta(\tau_1)$ is obtained by multiplying $\delta$ by an
appropriate element $t = e(\tau/2)$, with $\tau\in {\frak t}^{-a}_{\Bbb Q}$
(Proposition 9.10); here $\tau$ represents $\tau_1$ in the isomorphism of
Proposition 9.8(c).  One can check easily that $\tau_1$ factors through
$A({^d{T}^\Gamma})^{alg,{^\vee G}}$ if and only if the roots of $T$ in $G$ take
integer values on the element $\tau$.  (To see what the root lattice has to
do with the problem, recall that $\pi_1({^d{T}})$ may be identified with
$X_*({^d{T}}) \simeq X^*(T)$.  The inclusion of ${^d{T}}$ in ${^\vee G}$
induces a surjection $\pi_1({^d{T}})\rightarrow \pi_1({^\vee G})$, and the
kernel of this map is the lattice of roots of $T$ in $G$.)  On the other hand,
we compute
$$\delta(\tau_1)^2 = t\sigma(t)\delta^2 = e(\tau)\delta^2 = e(\tau)z(\rho)$$
(cf. proof of Proposition 9.10).  So $\delta(\tau_1)$ is a strong real form if
and only if $e(\tau) \in Z(G)$.  Of course this condition is also equivalent to 
the roots of $T$ in $G$ taking integral values on $\tau$, proving (a).

Part (b) is clear (see Proposition 10.6).  For (c), we can use the notation of
(a).  Inspecting the definitions in section 9 and at (12.6), we find that
$$\tau_1(\overline{m^{alg}_{^d{\alpha}}}) = \exp(2\pi i\alpha(\tau)) =
(-1)^{\alpha(\tau/2)}.$$
So the condition on $\tau_1$ is equivalent to 
$$\hbox{ $\alpha(\tau/2)$ is an even integer.}$$
On the other hand, every simple imaginary root $\alpha$ of $T$ in $G$ is
noncompact with respect to $\delta$ (see the proof of Proposition 13.6 in
section 14).  Consequently $\alpha$ is nocompact with respect to
$\delta(\tau_1) = t\delta$ if and only if $\alpha(t) = 1$.  Since
$t=e(\tau/2)$, this is equivalent to $\alpha(\tau/2)$ being an even integer.

The proof of (d) is similar; since we have referred to [AV2] for the
construction of $\Lambda(\phi)$, we omit the details.  Q.E.D.

\medskip
{\bf Proposition 13.13.} {\it Suppose $(G^\Gamma,{\cal W})$ is an extended
group for $G$, and $({^\vee G^\Gamma},{\cal S})$ is an E-group for the
corresponding inner class of real forms.  Write $z$ for the second invariant
of the E-group (Definition 4.14).  Then the various correspondences of
Proposition 13.12 induce a bijection from the set of equivalence classes of
final limit characters of type $z$ of Cartan subgroups of $G^\Gamma$
(Definition 12.1) and equivalence classes of final complete limit parameters
of Cartan subgroups of ${^\vee G^\Gamma}$ (Definition 12.8).}
\medskip

{\it Proof.}  Suppose $(\phi,\tau_1)$ is a complete limit parameter for
${^d{T}}^\Gamma$.  To construct the corresponding limit character, we need
to make ${^d{T}^\Gamma}$ a based Cartan subgroup; so we must choose
certain sets of positive roots.  Let ${^d{R}^+_{\Bbb R}}$ be a set of positive
real roots for ${^d{T}}$ making $\lambda(\phi)$ dominant (cf. (12.5) and
Definition 13.11); and let ${^d{R}^+_{i{\Bbb R}}}$ be an arbitrary set of
positive imaginary roots.  Let ${^d{\cal T}^\Gamma}$ be the corresponding
based Cartan subgroup (Proposition 13.8).   Fix a based Cartan subgroup
${\cal T}^\Gamma$ for $G^\Gamma$ and a pairing 
$$\zeta:{^\vee{T}} \rightarrow {^d{T}}\eqno(13.14)$$
with ${^d{\cal T}^\Gamma}$.  Proposition 13.12 gives a final limit character
$(\delta(\tau_1),\Lambda(\phi))$ of  $T^\Gamma$.  We must show that the
equivalence class of this character is independent of the two positive root
systems we chose.

Consider first a second system $({^d{R}^+_{\Bbb R}})'$ of positive real roots
making $\lambda(\phi)$ dominant.  These two positive systems differ by a
Weyl group element $w$ that is a product of simple reflections in real roots
vanishing on $\lambda$.  Let $n$ be a representative of $w$.  Changing the
sytem of positive real roots changes the based Cartan subgroup ${^d{\cal
T}^\Gamma}$ by conjugation by $n$, and replaces $\zeta$ by $w\circ\zeta $.  
(Evidently this implies that the final limit character $(\delta(\tau_1),
\Lambda(\phi))$ is unchanged.)  We claim that $w$ has a representative $n$
fixing $(\phi,\tau_1)$.  It is enough to check this for a simple reflection
$s_\alpha$, with $\alpha$ a real root vanishing on $\lambda$.  Choose a root
subgroup $\phi_\alpha:SL(2) \rightarrow {^\vee G}$ as usual (cf. Definition
11.10)) with the additional property that $\phi_\alpha$ carries the inverse
transpose involution on $SL(2)$ to the action of $\theta_y$.  We may take 
$$n_\alpha = \phi_\alpha\pmatrix0&1\cr -1 &0\cr\endpmatrix$$
as a representative of $s_\alpha$.  Evidently $n_\alpha$ belongs to $K(y)
\cap L(\lambda,y\cdot \lambda)$ (cf. Corollary 5.9); that is, to the
centralizer of $\phi$. To compute its effect on $\tau_1$, recall that
$\tau_1$ is a character of (a certain quotient of)
$({^d{T}^\theta})^{alg,{^\vee G}}$.  If $t$ is in this group, then $\alpha(t) =
\pm 1$ since $\alpha$ is real.  Consequently
$$s_\alpha(t) = t\alpha^\vee(\alpha(t^{-1})) = t\alpha^\vee(\pm 1) = t \hbox{
or } tm_\alpha^{alg}.$$
Since $(\phi,\tau_1)$ is final, $\tau_1(m_\alpha^{alg}) = 1$; so we find that
$s_\alpha(\tau_1) = \tau_1$, as we wished to show.

Next, consider a second choice $({^d{R}^+_{i{\Bbb R}}})'$ of positive
imaginary roots.  This differs from the original by an element $w$ in the
Weyl group of the imaginary roots.  To get the corresponding based Cartan
subgroup $({^d{\cal T}^\Gamma})'$, we must also conjugate ${\cal
S}({^d{T}^\Gamma})$ by a representative $n$ of $w$ (because of Definition
13.7(e)).  This conjugation has the effect of multiplying ${\cal
S}({^d{T}^\Gamma})$ by an element $t = n\theta(n^{-1})$; this element
satisfies $\theta(t) = t^{-1}$.  Back in $G$, we can get the new based Cartan
subgroup just by replacing the set of positive real roots by the new set
$(R^+_{\Bbb R})'$ corresponding to $({^d{R}^+_{i{\Bbb R}}})'$.  The pairing
$\zeta$ is then unchanged.  The canonical character $(\Lambda^{can})'$
attached to these new choices differs from $\Lambda^{can}$ only because of
the change in the E-group structure on ${^d{T}^\Gamma}$.  The proof of
Lemma 9.28 of [AV2] shows that the effect of this change is to twist
$\Lambda^{can}$ by the character $\tau(R^+_{\Bbb R},(R^+_{\Bbb R})')$ of
(11.4).  By Definition 11.6 and Definition 12.1, the equivalence class of the
limit character is unchanged.

That this construction actually depends only on the equivalence class of
$(\lambda,\tau_1)$ is obvious.  That it is surjective is clear from
Proposition 13.12.  That the inverse correspondence is well-defined may be
proved in exactly the same way.  Q.E.D.

Theorem 10.4 is a consequence of Proposition 13.13, Theorem 12.3, and
Theorem 12.9. 

%\end

%\input mssymb.tex
%\font\Bbb=cmr10
%\font\frak=cmr10
%\hsize=27pc
%\vsize=42pc
%\hoffset=3.5pc
%\voffset=5.5pc
%\pageno=88
\vskip 3pc
\centerline{\bf 14. Proof of Propositions 13.6 and 13.8.}
\vskip 1pc

Evidently part of what a based Cartan subgroup provides is some
distinguished extensions of its real form to all of $G$.  We begin by studying
such extensions.

\medskip
{\bf Lemma 14.1.}  {\it Suppose $T$ is a maximal torus in the connected
reductive algebraic group $G$, and $\sigma$ is a real form of $T$ permuting
the roots of $T$ in $G$.  Suppose that $\sigma_1$ and $\sigma_2$ are two
real forms of $G$ extending $\sigma$, and that they determine the same
sets of compact (and noncompact) imaginary roots.  Then $\sigma_1$ and
$\sigma_2$ are conjugate by an element of $T$.}
\medskip
{\it Proof.}  Choose a set $R^+$ of positive roots of $T$ in $G$ having the
property that $\sigma$ preserves the non-imaginary positive roots.  (One
way to do this is to order the roots using first their restrictions to the
$+1$-eigenspace of $\sigma$ on the rational span of the coroots.)  In the
argument below, we will usually write $\beta$ for an imaginary root,
$\gamma$ for a non-imaginary root, and $\alpha$ for a general root.  For any
root in $R^+$, we have either
$$\hbox{ $\beta$ is imaginary, and $\sigma\beta = -\beta$}\eqno(14.2)(a)$$
or
$$\hbox{ $\gamma$ is not imaginary, and $\sigma\gamma$ is positive.}
\eqno(14.2)(b)$$
For simple roots in $R^+$, one can say even more. 

\medskip
{\bf Lemma 14.3.} {\it In the setting of Lemma 14.1, suppose $R^+$ is a set
of positive roots for $T$ in $G$ such that $\sigma$ preserves the
non-imaginary positive roots.  Then every simple root of $R^+$ falls into
exactly one of the following three categories.
\smallskip
\item{a)} $\beta$ is imaginary, and $\sigma\beta = -\beta$.
\item{b)} $\gamma$ is not imaginary, and
$$\sigma\gamma = \gamma + \sum_{\beta \ {\rom simple\ imaginary}}
n_\beta(\gamma) \beta $$
with $n_\beta(\gamma)$ a non-negative integer.
\item{c)} $\gamma$ is not imaginary, and there is a distinct non-imaginary
root $\gamma'$ so that
$$\sigma\gamma = \gamma' + \sum_{\beta \ {\rom simple\ imaginary}}
n_\beta(\gamma) \beta $$
with $n_\beta(\gamma)$ a non-negative integer.  In this case
$$\sigma\gamma' = \gamma + \sum_{\beta \ {\rom simple\ imaginary}}
n_\beta(\gamma) \beta.$$
\medskip}

{\it Proof.}  Suppose $\gamma$ is a non-imaginary simple root.  Then
$\sigma\gamma$ is a positive root by (14.2)(b), so it can be expressed as a
sum of simple roots with non-negative coefficients.  In this expression
there must appear at least one non-imaginary simple root $\gamma' =
\gamma'(\gamma)$, for otherwise $\sigma\gamma$ (and therefore
$\gamma$) would be imaginary.  Thus
$$\sigma\gamma = \gamma' + \hbox{ (other non-imaginary simple roots)
}+\sum n_\beta\beta.$$
Now apply $\sigma$ to this expression.  On the right we get at least one
non-imaginary simple root from the first term, more from the second if it
is non-zero, and various imaginary simple roots.  On the left we get
$\gamma$, since $\sigma$ is an involution.  It follows that the second term
is zero, and that $\sigma\gamma'$ involves $\gamma$.  The remaining
assertions of the lemma are now clear.  Q.E.D.

We continue now with the proof of Lemma 14.1.  Fix root vectors
$\{X_\alpha\}$ for all the roots.  As usual we may choose these so that
$$[X_\alpha,X_{-\alpha}] = H_\alpha \eqno(14.4)(a)$$
(the usual coroot), and
$$[X_\alpha,X_{\alpha'}] = q(\alpha,\alpha')X_{\alpha+\alpha'}
\eqno(14.4)(b)$$
(with $q(\alpha,\alpha')$ rational) whenever $\alpha+ \alpha'$ is a root.
Define complex constants $c_i(\alpha)$ by
$$\sigma_i(X_\alpha) = c_i(\alpha) X_{\sigma\alpha}.\eqno(14.4)(c)$$
Since $\sigma_1$ and $\sigma_2$ agree on $T$, they differ by the adjoint
action of an element $s$ of $T$:
$$c_2(\alpha)/c_1(\alpha) = \alpha(s).\eqno(14.4)(d)$$
In particular,
$$\hbox{ $c_2/c_1$ extends to a multiplicative character of the root
lattice.}\eqno(14.4)(e)$$


Replacing $\sigma_1$ by $t\sigma_1 t^{-1}$ replaces $c_1(\alpha)$ by
$$[(\sigma\alpha)(t)/{\overline{\alpha(t)}}]c_1(\alpha).\eqno(14.5)(a)$$
It follows that $t$ conjugates $\sigma_1$ to $\sigma_2$ if and only if 
$$(\sigma\alpha)(t)/{\overline{\alpha(t)}} = c_2(\alpha)/c_1(\alpha)
\eqno(14.5)(b)$$
for every root $\alpha$.  Because of (14.4)(e), it suffices to verify (14.5)(b)
for any set of roots $\alpha$ generating the root lattice.  What we must
show is that our hypotheses on $\sigma_i$ provide enough control on $c_i$
to guarantee the existence of an element $t$ satisfying (14.5)(b).  We choose
a generating set $S$ of roots as follows: $S$ is the union of
$$\eqalign{S_1 &= \hbox{simple imaginary roots;}\cr
S_2 &= \hbox{simple roots as in Lemma 14.3(b);}\cr
S_3 &= \hbox{one simple root $\gamma$ from each pair
$(\gamma,\gamma')$ as in Lemma 14.3(c); and}\cr
S_4 &= \hbox{the roots $\sigma\gamma$, with $\gamma \in
S_3$.}\cr}\eqno(14.6)$$
Lemma 14.3 shows that $S$ is actually a basis of the root lattice.  We may
therefore choose $t$ so that $\alpha(t)$ has any value we specify for
$\alpha\in S$; we must show that this can be done so that (14.5)(b) holds
for $\alpha\in S$.

Applying the antiholomorphic involution $\sigma_i$ to (14.4)(c), we find
that
$$\overline{c_i(\alpha)}c_i(\sigma\alpha) = 1.\eqno(14.7)(a)$$
Applying $\sigma_i$ to (14.4)(b) gives
$$c_i(\alpha)c_i(\alpha') = q(\alpha,\alpha')c_i(\alpha+\alpha')
\eqno(14.7)(b)$$
whenever $\alpha+\alpha'$ is a root.  

Suppose now that $\beta$ is an imaginary root.  Then $\sigma H_\beta =
-H_\beta$.  Applying $\sigma_i$ to (14.4)(a) gives
$$c_i(\beta)c_i(-\beta) = 1.\eqno(14.8)(a)$$
On the other hand, (14.7)(a) gives
$$\overline{c_i(\beta)}c_i(-\beta) = 1.\eqno(14.8)(b)$$
Consequently
$$\hbox{$c_i(\beta)$ is real, and $c_i(-\beta) = c_i(\beta)^{-1}$.}
\eqno(14.8)(c)$$
Calculation in $SL(2)$ shows that
$$\hbox{$c_i(\beta)$ is positive if and only if $\beta$  is noncompact for
$\sigma_i$.} \eqno(14.8)(d)$$
The hypothesis of Lemma 14.1 therefore guarantees that
$c_1(\beta)/c_2(\beta)$ is positive for every imaginary root $\beta$.  We
therefore require of our element $t$ that 
$$\hbox{ $\beta(t)$ is a square root of $c_1(\beta)/c_2(\beta)$}
\eqno(14.8)(e)$$
for every $\beta$ in $S_1$.  Condition (14.5)(b) follows for such $\beta$,
and then for all imaginary roots by (14.4)(e).

Suppose next that $\gamma$ belongs to $S_2$.  The requirement (14.5)(b) on
$t$ may be written as
$$(c_2/c_1)(\gamma) =
(\gamma(t)/{\overline{\gamma(t)}})(\sigma\gamma-\gamma)(t).\eqno(14.9)(a)$$
This is the same as
$$\gamma(t)/{\overline{\gamma(t)}} =
(c_2/c_1)(\gamma)(\gamma-\sigma\gamma)(t).\eqno(14.9)(b)$$
We can choose $\gamma(t)$ so that this equation is satisfied if and only if
the right side has absolute value 1.  To see that this is the case, divide
(14.7)(a) for $\sigma_2$ by the same equation for $\sigma_1$, to obtain
$$\eqalign{ 1 &=  \overline{(c_2/c_1)(\gamma)}
(c_2/c_1)(\sigma\gamma)\cr
&= \overline{(c_2/c_1)(\gamma)} (c_2/c_1)(\gamma)
(c_1/c_2)(\gamma-\sigma\gamma)\cr
&= |(c_2/c_1)(\gamma)|^2
(c_1/c_2)(\gamma-\sigma\gamma).\cr}\eqno(14.9)(c)$$
Now Lemma 14.3 shows that $\gamma-\sigma\gamma$ is a sum of
imaginary roots.  It therefore follows from (14.8) that the last factor on the
right here is the positive real number $(\gamma-\sigma\gamma)(t)^2$. 
Consequently
$$1 = |(c_2/c_1)(\gamma)|^2 (\gamma- \sigma\gamma)(t)^2.
\eqno(14.9)(d)$$
This guarantees the existence of a solution $\gamma(t)$ to (14.9)(b).

Finally, suppose $\gamma\in S_3$.  We require of $t$ that
$$\gamma(t) = 1, \qquad \sigma\gamma(t) = (c_2/c_1)(\gamma).
\eqno(14.10)$$
Then (14.5)(b) is automatically satisfied for $\alpha = \gamma$.  To check
it for $\alpha = \sigma\gamma$, just apply (14.7)(a).  This completes the
construction of $t$ satisfying (14.5)(b), and so the proof of Lemma 14.1. 
Q.E.D.

{\it Proof of Proposition 13.6.}  The real form $\sigma_T$ of $T$ (defined by
conjugation by any element $y$ of $T^\Gamma - T$) is an antiholomorphic
involution preserving the roots.  It may therefore be extended to a quasisplit
real form $\sigma_G$ of $G$, with the property that every simple root of
$T$ in $R^+_{i{\Bbb R}}$ is noncompact.  (A proof of this fact may be found
in [IC4], Lemma 10.9.)  Clearly $\sigma_G$ belongs to the inner class
defined by $G^\Gamma$, so by Proposition 2.12 there is a triple
$(\delta_1,N_1,\chi_1) \in {\cal W}$ with ${\rom Ad}(\delta_1) = \sigma_G$. 
In particular, $\delta_1$ and $y$ both act by $\sigma_T$ on $T$, so
$\delta_1 \in yT = T^\Gamma - T$.   If we write ${\cal W}_1(T^\Gamma)$ for
the $T$-conjugacy class of $\delta_1$, then conditions (a) -- (d) of
Definition 13.5 are satisfied.  The interesting and subtle point is arranging
(e).  This is essentially in [Kostant]; we sketch the argument.

Fix $(\delta_1,N,\chi_1)$ as above, and write $G({\Bbb R},\delta_1)$ for the
corresponding real form.  Define $A_{\Bbb R} \subset T({\Bbb R})$ as in
(11.3) (the identity component of the split part of $T({\Bbb R})$), and put
$$M = \hbox{ centralizer of $A_{\Bbb R}$ in $G$.}$$
Fix a $\sigma_G$-stable parabolic subgroup $P=MU$ of $G$ with Levi
subgroup $M$.  After replacing $(N,\chi_1)$ by a conjugate under $G({\Bbb
R},\delta_1)$, we may assume that
$$\hbox{$PN$ is open in $G$.}$$
It follows that $N_M = N\cap M$ is a maximal unipotent subgroup of $M$
normalized by $\delta_1$, and that $\chi_{M,1} = \chi_1 |_{N_M({\Bbb R})}$
is a non-degenerate unitary character (Definition 1.12).  We apply to this
situation the following lemma.

\medskip
{\bf Lemma 14.11.}  {\it Suppose $\sigma$ is a quasisplit real form of a
complex connected reductive algebraic group $G$, and $P=MU$ is a
$\sigma$-stable Levi decomposition of a parabolic subgroup of $G$. 
Suppose $(N,\chi)$ is the data for a Whittaker model for $G$ (Definition 3.1). 
Suppose also that $PN$ is open in $G$, so that if we write $N_M = N\cap M$,
$\chi_M = \chi|_{N_M({\Bbb R})}$, then $(N_M,\chi_M)$ is the data for a
Whittaker model for $M$.

Suppose $\pi_M$ is a finite length admissible Hilbert space representation
of $M({\Bbb R})$, and $\pi = {\rom Ind}_{P({\Bbb R})}^{G({\Bbb R})} \pi_M$. 
Then $\pi$ admits a Whittaker model of type $\chi$ if and only if $\pi_M$
admits a Whittaker model of type $\chi_M$.}
\medskip

{\it Proof.}  The ``only if'' assertion is a special case of Theorem 1 of
[Hashizume].  The ``if'' part is a formal consequence of the ``only if,''
Harish-Chandra's subquotient theorem, and the fact that a principal series
representation has exactly one Whittaker model of a given type ([Kostant],
Theorem 6.6.2).  Q.E.D.

We continue with the argument for Proposition 13.6.  Now the standard limit
representations in Definition 13.5(e) may be constructed by induction from
limits of discrete series on $M$ (see the discussion at (11.2)).  By Lemma
14.11, they will admit Whittaker models of type $\chi_1$ if and only if
these limits of discrete series admit Whittaker models of type
$\chi_{M,1}$.  Because all of the limits of discrete series in question
correspond to a single system of positive imaginary roots, it is possible to
pass from any one to any other by tensoring with finite-dimensional
representations of $M({\Bbb R})$ (more precisely, of its canonical covering). 
This does not affect the existence of Whittaker models (cf. [Kostant], proof
of Theorem 6.6.2).

We may therefore confine our attention to a single limit of discrete series
representation $\pi_M$ of $M({\Bbb R})$.  By the construction of
$\sigma_G$, every simple imaginary root of $T$ in $M$ is noncompact.  It
follows that the annihilator of $\pi_M$ in the enveloping algebra of ${\frak
m}$ is a minimal primitive ideal ([GKDim], Theorem 6.2).  By Theorem 6.7.2
of [Kostant], there is a non-degenerate unitary character $\chi_M$ of
$N_M({\Bbb R})$ so that $\pi_M$ admits a Whittaker model of type $\chi_M$. 
It is clear from Definition 3.1 that there is a non-degenerate unitary
character $\chi$ of $N({\Bbb R})$ restricting to $\chi_M$ on $M$.  By Lemma
14.11, our standard limit representations all admit Whittaker models of
type $\chi$.  By Lemma 3.2, there is a $t\in G$ normalizing $N$, so that
${\rom Ad}(t)$ is defined over ${\Bbb R}$, and $t\cdot \chi_1 = \chi$. 
Consider now the triple
$$(t\delta_1 t^{-1}, N,t\cdot \chi_1) = (\delta,N,\chi).$$
Since ${\cal W}$ is a $G$-orbit, this triple belongs to ${\cal W}$.  Since
${\rom Ad}(t)$ is defined over ${\Bbb R}$, conjugation by $\delta$ defines the
same real form $\sigma_G$ as $\delta_1$.  Define
$${\cal W}(T^\Gamma) = \hbox{ $T$-conjugacy class of $\delta$.}$$
This satisfies the requirements of Definition 13.5.

For the uniqueness, suppose ${\cal W}'(T^\Gamma)$ is another extended
group structure on $T^\Gamma$ satisfying the requirements of Definition
13.5.  Fix a triple $(\delta', N',\chi')\in {\cal W}$ so that $\delta'$ belongs to
${\cal W}'(T^\Gamma)$.  The argument in the first half of the proof may be
reversed to deduce from the assumed existence of Whittaker models
(Definition 13.5)(e)) that every simple root of $R^+_{i{\Bbb R}}$ must be
noncompact for the real form $\sigma'_G$ defined by conjugation by
$\delta'$.  By the construction of $\sigma_G$, it follows that $\sigma_G$
and $\sigma'_G$ define exactly the same sets of compact and noncompact
imaginary roots.  By Lemma 14.1, they are conjugate by an element of $T$. 
Since ${\cal W}'(T^\Gamma)$ is a $T$-conjugacy class, we may therefore
assume that $\delta'$ is chosen so that
$$\hbox{ the conjugation action $\sigma'_G$ of $\delta'$ is equal to
$\sigma_G$.}\eqno(14.12)(a)$$
This means that the real form $G({\Bbb R})$ defined by $\delta'$ coincides
with the one defined by $\delta$.  The groups $N$ and $N'$ are therefore
conjugate by $G({\Bbb R})$; after applying such a conjugation to $N'$ (which
does not change $\delta'$) we may assume that $N = N'$.  Fix  a maximal
torus $T_s\subset M$ normalizing $N$ and defined over ${\Bbb R}$; we write
$$B_s = T_sN,\qquad B_s({\Bbb R}) = T_s({\Bbb R})N_s({\Bbb R}).
\eqno(14.12)(b)$$
Of course $T_s$ contains $Z(G)$, so the set of elements of $B_s$ for which
the conjugation action on $G$ is defined over ${\Bbb R}$ may be decomposed
as
$$(B_s/Z(G))({\Bbb R}) = (T_s/Z(G))({\Bbb R})N_s({\Bbb R}).
\eqno(14.12)(c)$$

Recall now that $(\delta,N,\chi)$ and $(\delta',N,\chi')$ both belong to
${\cal W}$, and so are conjugate by an element $t$ of $G$.  This element
normalizes $N$, and so belongs to $B_s$.  By (14.12), the element may be
chosen to belong to $T_s$ (where it must represent a class in
$(T_s/Z(G))({\Bbb R})$).  In particular,
$$t\cdot\chi = \chi',\qquad t\delta t^{-1} = \delta'.\eqno(14.13)$$
On the other hand, Definition 13.5(e) and Lemma 14.11 provide a single limit
of discrete series representation for $M({\Bbb R})$ admitting Whittaker
models of types $\chi_M$ and $\chi'_M$.   To this situation we can apply

\medskip
{\bf Lemma 14.14.}  {\it Suppose $G({\Bbb R})$ is a quasisplit real form of a
complex connected  reductive algebraic group, and $\pi$ is an irreducible
representation in the limits of (relative) discrete series.  If $\pi$ admits
Whittaker models of types $\chi$ and $\chi'$, then $\chi$ and $\chi'$ must
be conjugate under $G({\Bbb R})$.}
\medskip

{\it Proof.}  Fix a Borel subgroup $B_s = T_s N$ defined over ${\Bbb R}$, and
a maximally compact maximal torus $T({\Bbb R})$.  We may as well assume
that $\chi$ and $\chi'$ are non-degenerate unitary characters of $N({\Bbb
R})$.  Consider the group $({\rom Ad}\, G)({\Bbb R})$ of inner automorphisms
of $G$ defined over ${\Bbb R}$.  It contains the image $({\rom Ad}(G({\Bbb
R}))$ of $G({\Bbb R})$ as a subgroup of finite index.   Define
$$Q(G({\Bbb R})) = ({\rom Ad}\, G)({\Bbb R})/({\rom Ad}(G({\Bbb R})).
\eqno(14.15)(a)$$
This finite group acts on the set of equivalence classes of representations
of $G({\Bbb R})$, and on the set of conjugacy classes of Whittaker models.  

It follows easily from the essential uniqueness of $B_s$ that every coset in
$Q(G({\Bbb R}))$  meets $T_s$.  It therefore follows from Lemma 3.2 that
$$\hbox{ $Q(G({\Bbb R}))$ acts simply transitively on conjugacy classes of
Whittaker models for $G({\Bbb R})$.} \eqno(14.15)(b)$$
At the same time, the existence of representatives in $T_s$ shows that
$$\hbox{ $Q(G({\Bbb R}))$ acts trivially on the set of principal series
representations for $G({\Bbb R})$.} \eqno(14.15)(c)$$

On the other hand, the essential uniqueness of the maximally compact torus
$T$ shows that every coset in $Q(G({\Bbb R}))$ meets the normalizer of $T$
in $G$.  Define
$$W_2(G,T) = \hbox{ Weyl group of $T$ in $({\rom Ad}\, G)({\Bbb R})$.} $$
Since $T({\Bbb R})$ is connected, it follows that
$$Q(G({\Bbb R})) \simeq W_2(G,T)/W(G({\Bbb R}),T({\Bbb R})). $$
>From this it follows that 
$$\hbox{ $Q(G({\Bbb R}))$ acts without fixed points on limits of discrete
series representations of $G({\Bbb R})$.} \eqno(14.15)(d)$$

To prove the lemma, realize $\pi$ as a quotient of a principal series
representation $\rho$.  By (14.15)(b), there is an element $q\in Q(G({\Bbb
R}))$ with $q\cdot \chi$ is conjugate by $G({\Bbb R})$ to $\chi'$.  It follows
formally that $\pi' = q\cdot\pi$  has a Whittaker model of type $\chi'$.  By
(14.15)(c), $q\cdot \rho$ is equivalent to $\rho$, so $\pi'$ is also a quotient
of $\rho$.  The quotient representations $\pi$ and $\pi'$ of $\rho$ now both
have Whittaker models of type $\chi'$.  Because $\rho$ has a unique
Whittaker model ([Kostant], Theorem 6.6.2) it follows that $\pi$ must be
equivalent to $\pi'$.  By (14.15)(d), $q=1$, so $\chi$ is conjugate to $\chi'$. 
Q.E.D.

Returning to the proof of Proposition 13.6, we deduce that $\chi_M$ is
conjugate to $\chi'_M$ by an element of $M({\Bbb R})$; by Lemma 3.2, we
may as well choose this element $t_M$ in $T_s({\Bbb R})$.  We can replace
$\chi'$ by $t_M\cdot \chi'$, and $t$ by $tt_M$; then (14.13) remains true,
and we have in addition that $\chi_M = \chi'_M$.  From this we deduce that
${\rom Ad}(t)$ acts trivially on each of the simple restricted root spaces in
${\frak m}$.  Consequently $t$ is central in $M$, so it belongs also to the
maximal torus $T$ of $M$.  The second equation of (14.13) now shows that
$\delta$ and $\delta'$ are conjugate by $T$, as we wished to show.  Q.E.D.

We turn now to the proof of Proposition 13.8.  Formally the argument is
quite similar, and most of the subtleties have been dealt with in [AV2].  We
will therefore omit some details.  Again the first point is to understand
extensions of involutions from maximal tori to reductive groups.

\medskip
{\bf Lemma 14.16.} {\it Suppose $T$ is a maximal torus in the connected
reductive algebraic group $G$, and $\theta$ is a (holomorphic) involutive
automorphism of $T$ permuting the roots of $T$ in $G$.  Suppose $\theta_1$
and $\theta_2$ are two involutive automorphisms of $G$ extending
$\theta$, and that they determine the same sets of compact (and
noncompact) imaginary roots.  Then $\theta_1$ and $\theta_2$ are
conjugate by an element of $T$.}
\medskip

\noindent (Recall from (12.5) -- (12.7) that a root is called imaginary if it
is fixed by $\theta$, and compact if the corresponding root vector is also
fixed.)

The proof is exactly parallel to (and perhaps slightly simpler than) that of
Lemma 14.1, so we omit it.

{\it Proof of Proposition 13.8.}  By Lemma 9.17 of [AV2], there is an element
${^\vee{\delta}_1} \in {\cal S}\cap {^d{T}}^\Gamma$.  Write $K_1$ for the
centralizer of ${^\vee{\delta}_1}$ in ${^\vee G}$, and 
$${^d{\cal B}_1} = \{\,{^d{B}}| ({^\vee{\delta}_1}, {^d{B}}) \in {\cal S}\,\} $$
This is evidently a single $K_1$-orbit of Borel subgroups.  By Proposition
6.30 of [AV2], there is a set ${^d{R}}^+_{i{\Bbb R},1}$ of positive imaginary
roots special with respect to ${^d{\cal B}_1}$.   Now there is a unique
element $w$ of the Weyl group of imaginary roots carrying
${^d{R}}^+_{i{\Bbb R},1}$ to ${^d{R}}^+_{i{\Bbb R}}$.  The action of $w$ on
${^d{T}}$ (as a product of reflections in imaginary roots) commutes with the
action of ${^\vee{\delta}_1}$ on ${^d{T}}$.  Choose an element $n$ of the
normalizer of ${^d{T}}$ in ${^\vee G}$ representing $w$, and define
${^\vee{\delta}} = n({^\vee{\delta}_1})n^{-1}.$  Clearly ${^\vee{\delta}}$
belongs to ${\cal S}$.  The action of ${^\vee{\delta}}$ on ${^d{T}}$ agrees
with that of ${^\vee{\delta}_1}$ (since the latter commutes with $w$); so
${^\vee{\delta}} = {^\vee{\delta}_1}t$ for some $t \in {^d{T}}$; so
${^\vee{\delta}}$ belongs to ${\cal S}\cap {^d{T}}^\Gamma$.  The
corresponding set ${^d{\cal B}}$ of Borel subgroups is obtained from
${^d{\cal B}_1}$ by conjugating by $n$; so it follows immediately from the
definition that ${^d{R}}^+_{i{\Bbb R}} = w({^d{R}}^+_{i{\Bbb R},1})$ is special
with respect to ${^d{\cal B}}$.  The ${^d{T}}$-conjugacy class ${\cal
S}({^d{T}}^\Gamma)$ of ${^\vee{\delta}}$ therefore satisfies the
requirements of the proposition.

For the uniqueness, suppose ${\cal S}'({^d{T}}^\Gamma)$ is another E-group
structure satisfying the requirements of the proposition.  Fix
${^\vee{\delta}'} \in {\cal S}'({^d{T}}^\Gamma)$; what we are trying to show
is that ${^\vee{\delta}'}$ is conjugate to ${^\vee{\delta}}$ by ${^d{T}}$. 
Write $\theta$ and $\theta'$ for the involutive automorphisms of ${^\vee
G}$ defined by conjugation by ${^\vee{\delta}}$ and ${^\vee{\delta}'}$.  By
Proposition 6.30(a) of [AV2], every simple root in ${^d{R}}^+_{i{\Bbb R}}$ is
noncompact with respect both to $\theta$ and to $\theta'$.  By Lemma
14.16, it follows that $\theta$ and $\theta'$ are conjugate by ${^d{T}}$. 
After replacing ${^\vee{\delta}'}$ by a conjugate, we may therefore assume
that $\theta = \theta'$.  

Now ${^\vee{\delta}}$ and ${^\vee{\delta}'}$ belong to ${\cal S}$, which is a
single orbit of ${^\vee G}$.  It follows that there is an element $x$ of
${^\vee G}$ conjugating ${^\vee{\delta}}$ to ${^\vee{\delta}'}$.   We can
multiply $x$ on the right or left by elements of $K$ without affecting this
property; what we need to show is that this may be done so as to put $x$ in
${^d{T}}$.  The automorphism ${\rom Ad}(x)$ commutes with $\theta$, so the
coset $xZ({^\vee G})$ belongs to the group $\overline K$ of fixed points of
$\theta$ in ${\rom Ad}({^\vee G})$.  Now ${\rom Ad}(x)$ carries ${^d{T}}$ to
another $\theta$-stable maximal torus ${^d{T}'}$, so these two tori are
conjugate by $\overline K$.  We apply to ${\rom Ad}({^\vee G})$ the following
easy lemma.

\medskip
{\bf Lemma 14.17.}  {\it Suppose $G$ is a connected reductive algebraic
group, and $\theta$ is an involutive automorphism of $G$.  Write $K$ for the
group of fixed points of $\theta$.  Suppose $T$ and $T'$ are $\theta$-stable
maximal tori in $G$, conjugate by $K$.  Then they are conjugate by the
identity component $K_0$.}
\medskip

\noindent We omit the elementary proof.

In our situation, we conclude that there is an element $x_0 \in K$
conjugating ${^d{T}'}$ to ${^d{T}}$.  Multiplying $x$ by $x_0$ on the left, we
may therefore assume that ${^d{T}'} = {^d{T}}$; that is, that $x$ normalizes
${^d{T}}$.  In particular, the action of $x$ defines an element $w \in W({^\vee
G},{^d{T}})$ commuting with $\theta$.  Proposition 3.12 of [IC4] describes
the Weyl group elements commuting with $\theta$.  By Proposition 4.16 of
[IC4], we may modify $x$ by a representative in $K$ of an appropriate Weyl
group element, and arrange for $w$ to be in the imaginary Weyl group.  Since
$x$ carries ${^\vee{\delta}}$ to ${^\vee{\delta}'}$, it follows that
$w({^d{R}}^+_{i{\Bbb R}})$ must (like ${^d{R}}^+_{i{\Bbb R}}$) be special with
respect to ${^d{\cal B}'}$.  By Proposition 6.30(c) of [AV2], it follows that
$w$ has a representative in $K$.  After multiplying $x$ by the inverse of
such a representative, we get $x \in {^d{T}}$, as we wished to show.  Q.E.D.
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\centerline{\bf 15. Multiplicity formulas for representations.}
\vskip 1pc
Our next goal is Theorem 1.24 of the introduction, relating the geometric
invariants discussed in sections 7 and 8 to representation theory.  We begin
by discussing a little more carefully the definition of the
representation-theoretic multiplicity and character matrices (cf. (1.21)). 
For the same reasons as in section 11, we work at first in the setting of
(11.1).  Recall from Definition 11.13 the set $L^z(G_{\Bbb R})$ of
equivalence classes of final limit characters of type $z$.  For $\Lambda \in
L^z(G_{\Bbb R})$, write
$$M(\Lambda) = \hbox{ standard limit representation attached to $\Lambda$
} \eqno(15.1)(a)$$
(cf. (11.2)), and
$$\pi(\Lambda) = \hbox{ Langlands quotient of $M(\Lambda)$. }
\eqno(15.1)(b)$$
(We leave open the question of which form of the representation to use ---
Harish-Chandra module or some topological version.  Several reasonable
possibilities are discussed below.)   By Theorem 11.14, $\pi(\Lambda)$ is
irreducible, and this correspondence establishes a bijection
$$L^z(G_{\Bbb R}) \leftrightarrow \Pi^z(G_{\Bbb R}),\qquad \Lambda
\leftrightarrow \pi(\Lambda). \eqno(15.1)(c)$$

There are several ways to fit these representations into a nice abelian
category. By far the simplest approach, due to Harish-Chandra, is to choose
a maximal compact subgroup $K_{\Bbb R}$ of $G_{\Bbb R}$. (This is unique
up to conjugation by $G_{\Bbb R}$.)  One can then consider
$${\cal M}^z({\frak g},K_{\Bbb R}), \eqno(15.2)(a)$$
the category of finite-length canonical projective $({\frak g},K_{\Bbb
R})$-modules of type $z$, or {\it Harish-Chandra modules}.  (Such a module
is a representation simultaneously of ${\frak g}$ and of $K^{can,G}_{\Bbb
R}$ (cf. (11.1)(e)), with $\pi_1(G)^{can}$ acting by the character
parametrized by $z$.)   Sometimes it is convenient to obscure the choice of
$K_{\Bbb R}$, writing instead
$${\cal M}^z_{HC}(G_{\Bbb R})\eqno(15.2)(b)$$
for this category.  Alternatively, one can consider the category 
$${\cal M}^z_{\infty}(G_{\Bbb R})\eqno(15.2)(c)$$
for which a typical object is the space of smooth vectors in a finite-length
canonical projective representation of $G_{\Bbb R}$ of type $z$ on a Hilbert
space.  That this is a nice category is a deep theorem of Casselman and
Wallach; in fact they show that it is equivalent (by taking $K_{\Bbb
R}$-finite vectors) to ${\cal M}^z({\frak g},K_{\Bbb R})$.  Instead of smooth
vectors, one can consider distribution vectors, analytic vectors, or
hyperfunction vectors; all of these choices lead to equivalent categories
(designated with a subscript $-\infty$, $\omega$, or $-\omega$).  For our
purposes in this paper, the category of Harish-Chandra modules is
sufficient; but the aesthetic advantages of the other possibilities (such as
the elimination of the choice of $K_{\Bbb R}$) are significant.  Recall also
that the critical Definition 13.5(e) really makes sense only on ${\cal
M}^z_{\infty}$ (cf. Definition 3.1).

In any case, the categories are all canonically equivalent, so we can safely
define
$$K\Pi^z(G_{\Bbb R}) = \hbox{ Grothendieck group of ${\cal
M}^z_{HC}(G_{\Bbb R})$.} \eqno(15.2)(d)$$

Suppose $\Theta$ and $\Lambda$ belong to $L^z(G_{\Bbb R})$.  As in (1.21),
define
$$m_r(\Theta,\Lambda) = \hbox{ multiplicity of $\pi(\Theta)$ in
$M(\Lambda)$.} \eqno(15.3)(a)$$
We call $m_r$ the {\it representation-theoretic multiplicity matrix}.  In the
Grothendieck group this definition amounts to
$$M(\Lambda) = \sum_{\Theta \in L^z(G_{\Bbb R})}
m_r(\Theta,\Lambda)\pi(\Theta). \eqno(15.3)(b)$$
Entries of $m_r$ corresponding to limit characters of distinct infinitesimal
character are zero, so $m_r$ is ``block-diagonal'' with finite blocks.  In an
appropriate ordering of the basis, each block is upper triangular with one's
on the diagonal ([Green], Lemma 6.6.6).  Consequently the multiplicity matrix
is invertible; its inverse $c_r$ is called the {\it representation-theoretic
character matrix}.  Explicitly,
$$\pi(\Theta) = \sum_{\Lambda \in L^z(G_{\Bbb R})} c_r(\Lambda,\Theta)
M(\Lambda) \eqno(15.3)(c)$$
in $K\Pi^z(G_{\Bbb R})$.  In particular, the final standard limit
representations $M(\Theta)$ form a basis of the Grothendieck group.  (The
distribution characters of the standard representations are relatively
simple.  The equation (15.3)(c) therefore provides a fairly good formula for
the distribution character of the irreducible representation $\pi(\Theta)$;
hence the term ``character matrix.''  In the case of regular infinitesimal
character, the standard representation $M(\Lambda)$ is characterized by the
appearance of one particular term in the formula for its distribution
character (on the Cartan subgroup $T_{\Bbb R}$ corresponding to
$\Lambda$).  The entries $c_r(\Lambda,\Theta)$ may therefore be
interpreted as certain coefficients in the character formula for
$\pi(\Theta)$.)  In analogy with (7.11)(f), we define the {\it Bruhat order on
$L^z(G_{\Bbb R})$} to be the smallest partial order with the property that
$$m_r(\Theta,\Lambda) \ne 0 \hbox{ only if } \Theta \le  \Lambda.
\eqno(15.3)(d)$$
We could replace $m_r$ by $c_r$ without changing the order.  The Bruhat
order makes tempered final limit characters minimal, since the
corresponding standard representations are irreducible.  (Of course there
are non-tempered minimal elements as well.)

We will need to recall a little about the Kazhdan-Lusztig algorithm for
computing the matrices $m_r$ and $c_r$ (for linear groups).  Just as in the
geometric case (Proposition 8.8) the first step is a reduction to the regular
case using a translation principle.  We begin with an easy and well-known
result.

\medskip
{\bf Lemma 15.4.}  {\it Suppose $G$ is a complex connected reductive
algebraic group, and ${^\vee G}$ is a dual group for $G$.  Write ${\frak g}$
for the Lie algebra of $G$, and
$${\frak Z}({\frak g}) = \hbox{ center of $U({\frak g})$.}$$
Then there is a natural one-to-one correspondence between the set of
algebra homomorphisms
$$\chi: {\frak Z}({\frak g}) \rightarrow {\Bbb C}$$
and the set of semisimple orbits ${\cal O}$ of ${^\vee G}$ on ${^\vee {\frak
g}}$.}
\medskip

{\it Proof.}  Fix maximal tori $T\subset G$ and ${^d{T}}\subset {^\vee G}$. 
Harish-Chandra's theorem (see [Hump]) parametrizes the characters of
${\frak Z}({\frak g})$ by orbits of the Weyl group on the dual ${\frak t}^*$ of
the Lie algebra of $T$.  By (9.3)(b), ${\frak t}^*$ is naturally isomorphic to
the Lie algebra ${^\vee{\frak t}}$ of the dual torus ${^\vee T}$.  By Lemma
13.1, there is an isomorphism of ${^\vee T}$ with ${^d{T}}$ determined
uniquely up to the action of $W$; so Weyl group orbits on the respective Lie
algebras are canonically identified.  But every semisimple orbit of ${^\vee
G}$ on ${^\vee {\frak g}}$ meets ${^d{\frak t}}$ in a unique Weyl group orbit. 
Q.E.D.

We sometimes write
$$\chi_{\cal O}: {\frak Z}({\frak g}) \rightarrow {\Bbb C}\eqno(15.5)(a)$$
for the character corresponding to the semisimple orbit ${\cal O}$ in the
bijection of Lemma 15.4.  If $T$ is a maximal torus in $G$ and $\lambda\in
{\frak t}^*$, then we may also write
$$\chi_\lambda: {\frak Z}({\frak g}) \rightarrow {\Bbb C}\eqno(15.5)(b)$$
using Harish-Chandra's theorem more directly.  The various definitions in
(15.2) may be restricted to a single infinitesimal character, as in
$$L^z({\cal O},G_{\Bbb R}) = \{\,\Lambda\in L^z(G_{\Bbb R})\mid
\pi(\Lambda) \hbox{ has infinitesimal character $\chi_{\cal O}$ }\}
\eqno(15.5)(c)$$
$${\cal M}^z_{HC}({\cal O},G_{\Bbb R}) = \hbox{ Harish-Chandra modules of
generalized infinitesimal character $\chi_{\cal O}$} \eqno(15.5)(d)$$
$$K\Pi^z({\cal O},G_{\Bbb R}) = \hbox{ Grothendieck group of ${\cal
M}^z_{HC}({\cal O},G_{\Bbb R})$.} \eqno(15.5)(e)$$
The set $L^z({\cal O},G_{\Bbb R})$ is finite, so $K\Pi^z({\cal O},G_{\Bbb R})$
is a lattice of finite rank.

It is convenient to include here the notation we will use in the extended
group setting, even though we have a little more work to do with $G_{\Bbb
R}$.   So suppose for a moment that $(G^\Gamma,{\cal W})$ is an extended
group for $G$.  As in Lemma 1.15, choose a set $\{\,\delta_s\mid s\in
\Sigma\,\}$ of representatives for the equivalence classes of strong real
forms of $G$.  Recall from Definition 12.1 that $L^z(G/{\Bbb R})$ is the set
of $G$-conjugacy classes of pairs $(\Lambda,\delta)$, with $\delta$ a
strong real form of $G$ (Definition 1.13) and $\Lambda \in L^z(G({\Bbb
R},\delta))$ a final limit character.  As in Lemma 1.15, there is a natural
identification
$$L^z(G/{\Bbb R}) \simeq \coprod_{s \in \Sigma} L^z(G({\Bbb R},\delta_s)).
\eqno(15.6)(a)$$
Theorem 12.3 provides a bijection between these parameters and the set
$\Pi^z(G/{\Bbb R})$ of Definition 10.3; and the proof of Lemma 1.15 shows
that
$$\Pi^z(G/{\Bbb R}) \simeq \coprod_{s \in \Sigma} \Pi^z(G({\Bbb
R},\delta_s)). \eqno(15.6)(b)$$
Just as in (15.5), we can restrict attention to a single infinitesimal
character corresponding to a ${^\vee G}$-orbit ${\cal O}$, writing for
example $L^z({\cal O},G/{\Bbb R})$.   Finally, let $({^\vee G^\Gamma},{\cal
S})$ be an E-group for $G^\Gamma$ with second invariant $z$.  Theorem 10.4
identifies $\Pi^z(G/{\Bbb R})$ with
$$\Xi^z(G/{\Bbb R}) = \Xi({^\vee G^\Gamma}).\eqno(15.6)(c)$$
This set also decomposes according to the semisimple orbits of ${^\vee G}$
on ${^\vee {\frak g}}$, and we have
$$\Xi^z({\cal O},G/{\Bbb R}) = \{\hbox{ complete geometric parameters for
${^\vee G}^{alg}$ acting on $X({\cal O},{^\vee G^\Gamma})$ }\}
\eqno(15.6)(d)$$
It is clear from the definitions that Theorem 10.4 provides an identification
$$\Pi^z({\cal O},G/{\Bbb R}) \simeq \Xi^z({\cal O},G/{\Bbb R}).
\eqno(15.6)(e)$$

In analogy with Definition 7.13, we can form the direct sum of abelian
categories
$${\cal M}^z_{HC}(G/{\Bbb R}) = \bigoplus_{s \in \Sigma} {\cal
M}^z_{HC}(G({\Bbb R},\delta_s)). \eqno(15.7)(a)$$
(Recall that the notation conceals a choice of maximal compact subgroup for
each strong real form.)  Every object in this category has finite length, and
we have irreducible and standard representations parametrized by
$L^z(G/{\Bbb R})$ or $\Xi^z(G/{\Bbb R})$.  For $\xi \in \Xi^z({\cal O},G/{\Bbb
R})$, we may write as in (15.3)
$$M(\xi) = \sum_{\gamma\in \Xi^z({\cal O},G/{\Bbb R})}
m_r(\gamma,\xi)\pi(\gamma) \eqno(15.7)(b)$$
and so forth; this identity is in the Grothendieck group
$$K\Pi^z({\cal O},G/{\Bbb R}) = \bigoplus_{s \in \Sigma} K\Pi^z({\cal
O},G({\Bbb R},\delta_s)). \eqno(15.7)(c)$$

In order to define the pairing of Theorem 1.24, we need one more definition.

\medskip
{\bf Definition 15.8.}  Suppose ${^\vee G^\Gamma}$ is a weak E-group. 
Recall from Definition 4.9 the element $z(\rho) \in Z({^\vee G}^{alg})$; it has
order 2.  Because it is canonically defined, it is fixed by the conjugation
action of any element of ${^\vee G^\Gamma}$.  If $\xi = (\phi,\tau) \in
\Xi({^\vee G^\Gamma})$, it follows that $z(\rho)$ belongs to ${^\vee
G}^{alg}_\phi$.  Write $\overline{z(\rho)}$ for its image in the universal
component group $A^{loc,alg}_\phi$; this is a central element of order 2.  It
therefore acts by $\pm 1$ in the irreducible representation $\tau$, and we
define
$$e(\xi) = \tau(\overline{z(\rho)}) = \pm 1.$$
We also recall from (1.22) the notation
$$d(\xi) = \dim S_\xi,$$
with $S_\xi$ the ${^\vee G}$-orbit on $X({^\vee G^\Gamma})$ corresponding
to $\xi$ (Definition 7.6).

\medskip

{\bf Lemma 15.9.}  {\it In the setting of Theorem 10.4, suppose $\xi\in
\Xi^z(G/{\Bbb R})$ corresponds to an irreducible representation $\pi$ of a
real form $G({\Bbb R},\delta)$.  Then the sign $e(\xi)$ of Definition 15.8 is
equal to the sign $e(G({\Bbb R},\delta))$ defined in [Kott1].  In particular, it
is equal to 1 if $G({\Bbb R},\delta)$ is quasisplit.}
\medskip

We will give a proof at the end of section 17.

\medskip
{\bf Definition 15.11.}  Suppose we are in the setting (15.6).  The
Grothendieck groups $K\Pi^z({\cal O},G/{\Bbb R})$ (cf. (15.7)(c)) and
$KX({\cal O},{^\vee G^\Gamma})$ (Definition 7.13) are both free ${\Bbb
Z}$-modules on bases parametrized by $\Xi^z(G/{\Bbb R})$.  It therefore
makes sense to define the {\it canonical perfect pairing}
$$\langle,\rangle:K\Pi^z({\cal O},G/{\Bbb R})\times KX({\cal O},{^\vee
G^\Gamma}) \rightarrow {\Bbb Z}$$
by the requirement
$$\langle M(\xi),\mu(\gamma) \rangle = e(\xi) \delta_{\xi,\gamma}.$$
The last term on the right is a Kronecker delta.
\medskip

We can now restate Theorem 1.24 using E-groups instead of L-groups.

\medskip
{\bf Theorem 15.12.} {\it Suppose $(G^\Gamma,{\cal W})$ is an extended
group for $G$ (Definition 1.12), and $({^\vee G^\Gamma},{\cal S})$ is an
E-group for the corresponding inner class of real forms, having second
invariant $z$ (Definition 4.14).  Fix a semisimple orbit ${\cal O}$ of ${^\vee
G}$ on ${^\vee {\frak g}}$.  Write $K\Pi^z({\cal O},G/{\Bbb R})$ for the
Grothendieck group of canonical projective representations of type $z$ and
infinitesimal character $\chi_{\cal O}$ of strong real forms of $G$ (cf.
(15.7)(c)), and $KX({\cal O},{^\vee G^\Gamma})$ for the Grothendieck group of
${^\vee G}^{alg}$-equivariant constructible sheaves on the geometric
parameter space $X({\cal O},{^\vee G^\Gamma})$ (Definitions 6.9 and 7.13). 
Fix complete geometric parameters $\xi,\gamma \in \Xi^z({\cal O},G/{\Bbb
R})$ (cf. (15.6)(d)), and write $\pi(\xi), P(\gamma)$ for the corresponding
irreducible representation and perverse sheaf (Theorem 10.4 and Definition
7.13).  Then the canonical pairing of Definition 15.11 satisfies
$$\langle \pi(\xi),P(\gamma) \rangle = e(\xi)(-1)^{d(\xi)}
\delta_{\xi,\gamma}.$$
Here $e(\xi)$ is defined in (15.8).}
\medskip

We will prove this result in the next two sections.  The following
relationship between the geometric and representation-theoretic
multiplicity matrices is just a reformulation.

\medskip
{\bf Corollary 15.13.}  {\it In the setting of Theorem 15.12, fix $\eta$ and
$\gamma$ in $\Xi^z(G/{\Bbb R})$.  Then the geometric and
representation-theoretic multiplicity and character matrices (cf. (7.11) and
(15.3)) satisfy
\smallskip
\item{a)} $c_g(\eta,\gamma)(-1)^{d(\eta) - d(\gamma)} =
m_r(\gamma,\eta)$.
\item{b)}  $c_r(\eta,\gamma)(-1)^{d(\eta) - d(\gamma)} =
m_g(\gamma,\eta)$.
\medskip}

{\it Proof.}  Using (15.3)(c) and (7.11)(c), we can rewrite the pairing in
Theorem 15.12 in terms of standard representations and elementary
constructible sheaves.  The result is
$$\eqalign{\langle \pi(\xi),P(\gamma) \rangle &=\langle \sum_{\xi'}
c_r(\xi',\xi)M(\xi'),\sum_{\gamma'}
c_g(\gamma',\gamma)\mu(\gamma')(-1)^{d(\gamma')}\rangle\cr
&= \sum_{\xi',\gamma'}c_r(\xi',\xi)c_g(\gamma',\gamma)(-1)^{d(\gamma')}
\langle  M(\xi'),\mu(\gamma') \rangle.\cr}$$
By Definition 15.11, only the terms with $\gamma' = \xi'$ contribute; we get
$$\sum_\eta c_r(\eta,\xi)c_g(\eta,\gamma)(-1)^{d(\eta)}e(\eta).$$
Now the central element $z(\rho)$ clearly acts by $e(\gamma)$ on all the
stalks of the perverse sheaf $P(\gamma)$.   So whenever the second factor
is non-zero, we must have  $e(\eta) = e(\gamma)$.  This leads to
$$e(\gamma)(-1)^{d(\gamma)} \sum_\eta
c_r(\eta,\xi)c_g(\eta,\gamma)(-1)^{d(\eta) - d(\gamma)}.$$
Comparing with the formula in Theorem 15.12, we find
$$\sum_\eta c_r(\eta,\xi)c_g(\eta,\gamma)(-1)^{d(\eta) - d(\gamma)} =
\delta_{\xi,\gamma}.$$
Since the inverse of the matrix $c_r(\gamma,\xi)$ is by definition
$m_r(\gamma,\xi)$, it follows that
$$m_r(\gamma,\eta) = c_g(\eta,\gamma)(-1)^{d(\eta) - d(\gamma)},$$
which is (a).

For (b), write
$$c_g(\eta,\gamma) = m_r(\gamma,\eta)(-1)^{d(\eta) - d(\gamma)}.$$
Multiply by $m_g(\xi,\eta)$ and sum over $\eta$.  Since $c_g$ is by
definition the inverse of $m_g$, we get $\delta_{\xi,\gamma}$ on the left. 
The right side can be written as
$$(-1)^{d(\gamma)-d(\xi)}\sum_\eta
m_r(\gamma,\eta)m_g(\xi,\eta)(-1)^{d(\xi) - d(\eta)}.$$
Since the left side is zero unless $\gamma = \xi$, we can drop the sign in
front of the summation.  Since $c_r$ is by definition the inverse of $m_r$,
it follows that
$$c_r(\eta,\xi) = m_g(\xi,\eta)(-1)^{d(\xi) - d(\eta)}.$$
This is (b).  Q.E.D.

We leave to the reader the straightforward verification that the argument
given for Corollary 15.13(a) can be reversed, so that
$$\hbox{Theorem 15.12 is equivalent to Corollary 15.13(a).} \eqno(15.14)$$
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\centerline{\bf 16. The translation principle, the Kazhdan-Lusztig
algorithm, and Theorem 1.24.}
\vskip 1pc
In this section we will begin the proof of Theorem 15.12 (which is
essentially Theorem 1.24).  The first step is a reduction to the case of
regular infinitesimal character.  For this, we reformulate the translation
principle for representations so as to emphasize the connection with the
geometric translation principle of Proposition 8.8.

\medskip
{\bf Definition 16.1.}  In the setting of Lemma 15.4, suppose ${\cal O}$ and
${\cal O}'$ are semisimple orbits of ${^\vee G}$ on ${^\vee {\frak g}}$, and
${\cal T}$ is a translation datum from ${\cal O}$ to ${\cal O}'$ (Definition
8.6).  Fix  maximal tori $T \subset G$ and ${^d{T}} \subset {^\vee G}$, and
construct $\mu = \lambda' - \lambda \in X_*({^d{T}})$ as in Definition
8.6(e); this element is defined up to the Weyl group.  By Lemma 13.1(b),
$\mu$ corresponds to a weight (also called $\mu$) in $X^*(T)$, defined up to
the action of the Weyl group.  By the Cartan-Weyl theory, there is a unique
finite-dimensional irreducible algebraic representation $F_{\cal T}$ of $G$
having extremal weight $-\mu$.  Write $P_{\cal O}$ for the functor of
projection on the generalized infinitesimal character $\chi_{\cal O}$; this
is defined on all representations of ${\frak g}$ on which ${\frak Z}({\frak
g})$ acts in a locally finite way.

The {\it Jantzen-Zuckerman translation functor}
$$\psi_{\cal T}: {\cal M}^z({\cal O}',G_{\Bbb R}) \rightarrow {\cal M}^z ({\cal
O},G_{\Bbb R})$$
is defined by
$$\psi_{\cal T}(M) = P_{\cal O}(M\otimes F_{\cal T}).$$
(Of course it is also defined on ${\cal M}^z({\cal O}',G/{\Bbb R})$.)  In the
notation of (11.15), this is $\psi^{\lambda_a}_{\lambda_a+\mu_a}$.  Our
present hypotheses are somewhat weaker.  Because of the definition of
translation datum, this functor is a translation ``to the wall.''  A complete
development of the theory also requires the adjoint translation functor
(``away from the wall'')
$$\phi_{\cal T}: {\cal M}^z({\cal O},G_{\Bbb R}) \rightarrow {\cal M}^z ({\cal
O}',G_{\Bbb R})$$
defined by
$$\phi_{\cal T}(M) = P_{{\cal O}'}(M\otimes F^*_{\cal T}).$$
\medskip

The most basic properties of these functors (that they are well-defined,
covariant, exact, and adjoint to each other) are established in [Green],
Proposition 4.5.8.  Before we recall anything deeper, it may be helpful to
reformulate the definition of translation datum so as to put Definition 16.1
in a more familiar setting.

\medskip
{\bf Lemma 16.2.} {\it Suppose $G$ is a complex connected reductive
algebraic group, ${^\vee G}$ is a dual group, and ${\cal O}$ and ${\cal O}'$ are
semisimple orbits of ${^\vee G}$ on ${^\vee {\frak g}}$.  Then a translation
datum ${\cal T}$ from ${\cal O}$ to ${\cal O}'$ (Definition 8.6) may be
identified with a $G$-conjugacy class of triples $(T,\lambda,\lambda')$
subject to the conditions below.
\smallskip
\item{i)}  $T$ is a maximal torus in  $G$, and $\lambda$ and $\lambda'$
belong to ${\frak t}^*$.
\item{ii)}  The infinitesimal characters $\chi_\lambda$ and
$\chi_\lambda'$ are equal to $\chi_{\cal O}$ and $\chi_{{\cal O}'}$
respectively (notation (15.5)).
\item{iii)}  Suppose $\alpha$ is a root of $T$ in ${\frak g}$, and $\langle
\lambda,\alpha^\vee \rangle$ is a positive integer.  Then $\langle
\lambda',\alpha^\vee \rangle$ is a positive integer.
\item{iv)}  The weight $\mu = \lambda' - \lambda$ belongs to $X^*(T)$.
\smallskip
Any triple $(T,\lambda,\lambda')$ satisfying (i)--(iv) determines a unique
translation datum.} 
\medskip

This is clear from Definition 8.6 and Lemma 13.1.  In the language of
Definition 4.5.7 of [Green], our $({\frak t},\lambda,\lambda')$ corresponds to
$({\frak h}^s,\xi+\overline{\lambda_2},\xi+\overline{\lambda_1})$. 
(Various extra complications appear in [Green] because the
finite-dimensional representations are not assumed to come from an
ambient algebraic group.  On the other hand, the definition in [Green] does
not require any condition like (iii) of Lemma 16.2; such conditions appear
only when there are theorems to be proved.)  Our functor $\psi_{\cal T}$ is
called $\psi^{\xi + \lambda_1}_{\xi + \lambda_2}$ in [Green], and our
$\phi_{\cal T}$ is $\psi^{\xi + \lambda_2}_{\xi + \lambda_1}$.

In the setting of Definition 16.1, suppose $\Lambda'=((\Lambda^{can})',
R^+_{i{\Bbb R}},R^+_{\Bbb R})$ is a limit character of infinitesimal
character ${\cal O}'$ (cf.~(11.2)).  {\it When $G$ is linear} we can define a
new limit character
$$\Lambda = \psi_{\cal T}(\Lambda') \eqno(16.3)(a)$$
as follows (cf.~(11.17)).  First, the Cartan subgroup $T_{\Bbb R}$ and the
positive root systems are unchanged.  Identify the translation datum ${\cal
T}$ as in Lemma 16.2, and choose a triple $(T,\lambda,\lambda')$ with $T$
the complexification of $T_{\Bbb R}$ and $\lambda'$ the differential of
$(\Lambda')^{can}$; this is possible by our hypothesis on the infinitesimal
character of $\Lambda'$.  By the definition of a translation datum (see
Lemma 16.2(iii)), the stabilizer of $\lambda'$ in the Weyl group $W(G,T)$
also stabilizes $\lambda$.  It follows that $\lambda$ is uniquely
determined by $T$ and $\lambda'$; so the weight $\mu = \lambda' - \lambda
\in X^*(T)$ is well-defined.  (This step is not symmetric in $\Lambda$ and
$\Lambda'$; we cannot reverse it to define ``$\phi_{\cal T}(\Lambda)$.'') 
Via the map (11.1)(b), we can regard $\mu$ as a character of $T_{\Bbb R}$;
so we can define
$$\Lambda^{can} = ((\Lambda)^{can})' \otimes \mu^{-1}.\eqno(16.3)(b)$$
The positivity assumption in Lemma 16.2(iii) guarantees that $\Lambda$
inherits from $\Lambda'$ the requirement (11.2)(d) in the definition of a
limit character.   (This is where the linearity of $G$ enters:  it forces the
imaginary roots to be integral, so that Lemma 16.2(iii) imposes some
restrictions on their positivity.  This point introduces substantial
difficulties in the detailed character theory of non-linear groups.  In section
11, we avoided these problems by assuming $\mu$ dominant.)

\medskip
{\bf Theorem 16.4.}  {\it Suppose $G$ is a complex connected reductive
algebraic group, and $G_{\Bbb R}$ is a linear real Lie group as in (11.1).  Fix
infinitesimal characters ${\cal O}$ and ${\cal O}'$ (Lemma 15.4) and a
translation datum ${\cal T}$ from ${\cal O}$ to ${\cal O}'$.  
\smallskip
\item{a)}  Suppose $\Lambda'$ is a limit character  with infinitesimal
character ${\cal O}'$ write $\Lambda = \psi_{\cal T}(\Lambda')$ (cf.~(16.3)). 
Then 
$$\psi_{\cal T}(M(\Lambda')) = M(\Lambda)$$
(notation (11.2) and (16.1).)
\item{b)}  Suppose $\pi' \in \Pi^z(G_{\Bbb R})$ is an irreducible canonical
projective representation of infinitesimal character ${\cal O}'$.  Then
$\psi_{\cal T}(\pi')$ is irreducible or zero.
\item{c)} In the setting of (a), suppose in addition that $\Lambda'$ is final
and that $\psi_{\cal T}(\pi(\Lambda')) \ne 0$.  Then $\Lambda$ is also final,
and 
$$\psi_{\cal T}(\pi(\Lambda')) = \pi(\Lambda).$$
In this way $\psi_{\cal T}$ defines a bijection from a subset of $\Pi^z({\cal
O}',G_{\Bbb R})$ onto $\Pi^z({\cal O},G_{\Bbb R})$.
\medskip}

This is another version of the results in Propositions 11.16, 11.18, and
11.20; the arguments and references given there apply here as well.   Write
$$\psi^{-1}_{\cal T}:\Pi^z({\cal O},G_{\Bbb R}) \rightarrow \Pi^z({\cal
O}',G_{\Bbb R}) \eqno(16.5)(a)$$
for the injective map inverting the correspondence of (c).  This is not so
easy to describe explicitly; one characterization is
$$\psi^{-1}_{\cal T}(\pi) = \hbox{ unique irreducible subrepresentation of
$\phi_{\cal T}(\pi)$.} \eqno(16.5)(b)$$
Because of Theorem 11.14, there is a corresponding injective map on
equivalence classes of final limit characters
$$\phi_{\cal T}: L^z({\cal O},G_{\Bbb R}) \rightarrow L^z({\cal O}',G_{\Bbb
R}). \eqno(16.5)(c)$$
As a consequence of Theorem 16.4, this map respects the
representation-theoretic multiplicity matrix:
$$m_r(\phi_{\cal T}(\Theta),\phi_{\cal T}(\Lambda)) =
m_r(\Theta,\Lambda)\qquad(\Theta,\Lambda\in L^z({\cal O},G_{\Bbb R})).
\eqno(16.5)(d)$$ 
Because of Lemma 8.7, this reduces the calculation of the multiplicity
matrix $m_r$ to the case of regular infinitesimal character.  In connection
with Theorem 1.24, we need to know that this reduction is compatible with
the corresponding one on the geometric side.

\medskip
{\bf Proposition 16.6.}  {\it In the setting of (15.6), suppose ${\cal T}$ is a
translation datum from ${\cal O}$ to ${\cal O}'$.  Then the diagram
$$ \matrix L^z({\cal O},G/{\Bbb R})&\buildrel{\phi_{\cal
T}}\over{\longrightarrow} &L^z({\cal O}',G/{\Bbb R})\cr \downarrow
&{}&\downarrow\cr \Xi^z({\cal O},{^\vee G^\Gamma})&\buildrel{f^*_{\cal
T}}\over{\longrightarrow}  &\Xi^z({\cal O}',{^\vee G^\Gamma})\cr\endmatrix$$
commutes.  Here the vertical arrows are the bijections of Theorem 10.4.
\medskip}

{\it Proof.}  The map in the top row has been described rather explicitly in
terms of Cartan subgroups of $G^\Gamma$ in the course of the proof of
Proposition 11.20.  (The hypotheses there were slightly different, but the
same ideas apply.)  The sets in the bottom row have been parametrized in
terms of Cartan subgroups of ${^\vee G^\Gamma}$ (Theorem 12.9).  The
vertical maps are defined in terms of Cartan subgroups (Propositions 13.12
and 13.13).  All that remains is to check that the geometrically defined map
in the bottom row (cf.~(7.16)) can be computed in terms of Cartan
subgroups.  This is an elementary exercise, and we leave it to the reader. 
(Notice that there are no perverse sheaves in this calculation --- just
homogeneous spaces and representations of component groups.)  Q.E.D.

To get the reduction of Theorem 15.12 to the case of regular infinitesimal
character, we use the equivalent form Corollary 15.13(a) (cf.~(15.14)).  For
that, we need only compare Proposition 7.15(c) with (16.5)(d).  (The orbit
correspondence of Proposition 7.15 does not preserve dimensions of orbits,
but it changes all of them by the same constant $d$.  Since only differences
of dimensions appear in Corollary 15.13(a), this suffices.)

For the balance of this section, we may therefore assume that
$$\hbox{ ${\cal O} \subset {^\vee {\frak g}}$ is a regular semisimple orbit of
${^\vee G}$.} \eqno(16.7)(a)$$
In order to describe the proof of Theorem 15.12, we need to recall in some
detail the structure of the Kazhdan-Lusztig algorithm that computes the
various character matrices.   Suppose $({^d{T}}_1,\lambda_1)$ and
$({^d{T}}_2,\lambda_2)$ are pairs with ${^d{T}}_i$ a maximal torus in
${^\vee G}$ and $\lambda_i\in {^d{T}}_i \cap {\cal O}$.  Just as in (11.17),
there is a unique isomorphism
$$j(\lambda_1,\lambda_2): {^d{T}}_1 \rightarrow {^d{T}}_2 \eqno(16.7)(b)$$
induced by an element of ${^\vee G}$ and carrying $\lambda_1$ to
$\lambda_2$.  (It is the assumption that ${\cal O}$ is regular that makes
the isomorphism unique.)  Define
$$({^\vee{T}}_{\cal O},\lambda_{\cal O}) = \lim \,({^d{T}},\lambda),
\eqno(16.7)(c)$$
the projective limit taken over pairs as above using the isomorphisms of
(16.7)(b).  Then any such pair is canonically isomorphic to $({^d{T}}_{\cal
O},\lambda_{\cal O})$, say by
$$j(\lambda_{\cal O},\lambda): {^\vee{T}}_{\cal O} \rightarrow {^d{T}}
\eqno(16.7)(d)$$

Write 
$$T_{\cal O} = \hbox{ dual torus to ${^\vee{T}}_{\cal O}$. }\eqno(16.8)(a)$$
The dual of the Lie algebra of $T_{\cal O}$ may be identified with
${^\vee{\frak t}}_{\cal O}$, so we may write
$$\lambda_{\cal O} \in {\frak t}^*_{\cal O}. \eqno(16.8)(b)$$
By inspection of the proof of Lemma 15.4, it is clear that we may identify
$$(T_{\cal O},\lambda_{\cal O}) = \lim \,(T,\lambda). \eqno(16.8)(c)$$
The projective limit is taken over pairs $(T,\lambda)$ with $T$ a maximal
torus in $G$, and $\lambda \in {\frak t}^*$ a weight defining the
infinitesimal character $\chi_{\cal O}$, using isomorphisms analogous to
those of (16.7)(b).  In particular, we get isomorphisms
$$i(\lambda_{\cal O},\lambda): T_{\cal O} \rightarrow T \eqno(16.8)(d)$$ 
as in (16.7)(d).

The torus ${^\vee{T}}_{\cal O}$ inherits a root system $R({^\vee G},{^\vee
{T}}_{\cal O}) \subset X^*({^\vee{T}}_{\cal O})$, a Weyl group $W({^\vee
G},{^\vee{T}}_{\cal O})$, and so on.  Similarly (using (16.8)(c)) we can define
$R(G,T_{\cal O})$.  The identification (16.8)(a) makes $R(G,T_{\cal O})$ and
$R({^\vee G},{^\vee T}_{\cal O})$ into dual root systems, so that the roots
for one may be regarded as coroots for the other.  We define the set of {\it
${\cal O}$-integral coroots}
$$R^\vee({\cal O}) = \{ \, \alpha^\vee \in R({^\vee G},{^\vee {T}}_{\cal O}) |
\alpha^\vee(\lambda_{\cal O}) \in {\Bbb Z}\,\}. \eqno(16.9)(a)$$
 Equivalently,
$$R^\vee({\cal O}) = \{ \, \alpha^\vee \in R^\vee(G,T_{\cal O}) |
\lambda_{\cal O}(\alpha^\vee) \in {\Bbb Z}\,\}. \eqno(16.9)(b)$$
Clearly an isomorphism $j(\lambda_{\cal O},\lambda)$ as in (16.7)(d)
identifies $R^\vee({\cal O})$ with the set of roots of ${^d{T}}$ in ${^\vee
G}(\lambda)$ (cf.~(6.2)(b)).  In $G$, $i(\lambda_{\cal O},\lambda)$ identifies
$R({\cal O})$ with the set of $\lambda$-integral roots of $T$ in $G$.  There
is a natural positive root system
$$R^+({\cal O}) = \{\,\alpha \in R({\cal O}) | \alpha^\vee(\lambda_{\cal O}) >
0\,\}; \eqno(16.9)(c)$$
$j(\lambda_{\cal O},\lambda)$ identifies the positive coroots with the
roots of ${^d{T}}$ in the Borel subgroup $P(\lambda)$ of ${^\vee G}(\lambda)$ 
(cf.~(6.2)(e)).  We write
$$R(\lambda) = i(\lambda_{\cal O},\lambda)(R({\cal O})) = \{\,\alpha\in
R(G,T) | \alpha^\vee(\lambda) \in {\Bbb Z}\,\},\quad R^+(\lambda) =
i(\lambda_{\cal O},\lambda)(R^+({\cal O})),\eqno(16.9)(d)$$
and similarly for $R^\vee$.  Define
$$\Delta({\cal O}) = \hbox{ set of simple roots of $R^+({\cal O})$},\qquad
W({\cal O}) = \hbox{ Weyl group of $R({\cal O})$.} \eqno(16.9)(e)$$
We can regard $W({\cal O})$ as a group of automorphisms of $T_{\cal O}$ or
of ${^\vee T}_{\cal O}$.  The corresponding set of simple reflections is
written
$$S({\cal O}) = \{\,s_\alpha | \alpha \in \Delta({\cal O})\,\}.
\eqno(16.9)(f)$$

Of course $(W({\cal O}),S({\cal O}))$ is a Coxeter group.  We may therefore
attach to it a {\it Hecke algebra} ${\cal H}({\cal O})$ (cf.~[KL0]).  The Hecke
algebra is a free ${\Bbb Z}[u^{1/2},u^{-1/2}]$-algebra with basis
$$\{\,T_w|w\in W({\cal O})\,\}. \eqno(16.10)(a)$$
It is characterized by the relations
$$T_xT_y = T_{xy}\qquad (x,y \in W({\cal O}), l(x) + l(y) = l(xy))
\eqno(16.10)(b)$$
and
$$(T_s + 1)(T_s - u) = 0 \qquad(s \in S({\cal O})). \eqno(16.10)(c)$$
(Here $l(x)$ is the length function on the Coxeter group $(W({\cal O}),S({\cal
O}))$; it need not be the restriction of any length function on the larger Weyl
group $W(G,T_{\cal O})$.)   It follows that the operators $T_s$ generate the
Hecke algebra.  The specialization to $u^{1/2}=1$ of ${\cal H}({\cal O})$ ---
that is, its quotient by the ideal generated by $u^{1/2}-1$ --- is naturally
isomorphic to the group algebra of $W({\cal O})$.

This is the basic structure required for the Kazhdan-Lusztig algorithms, and
we will make no explicit use of any more natural descriptions of it.  As a
hint about the geometry underlying the algorithms, however, we mention a
(well-known) geometric description of $W({\cal O})$.  Recall the map $e$
from ${^\vee{\frak g}}$ to ${^\vee G}$ (cf. (6.2)(a)).  We use the notation of
(6.10).  Thus ${\cal F}({\cal O})$ is the set of canonical flats in ${\cal O}$,
${\cal C}({\cal O})$ is the corresponding conjugacy class in ${^\vee G}$, and
$e$ is a smooth projective algebraic morphism from ${\cal F}({\cal O})$ to
${\cal C}({\cal O})$.  Define
$$Z({\cal O},{^\vee G}) = {\cal F}({\cal O}) \times_{{\cal C}({\cal O})} {\cal
F}({\cal O}) \eqno(16.11)(a)$$
This definition is analogous to that of the geometric parameter space in
Proposition 6.16, but it is substantially simpler.  An analysis along the lines
of that proposition shows that there is a natural bijection
$$\{\hbox{ ${^\vee G}$-orbits on $Z({\cal O})$ }\}\leftrightarrow W({\cal O}).
\eqno(16.11)(b)$$
Here the diagonal orbit (which is just ${\cal F}({\cal O})$) corresponds to
the identity element of $W({\cal O})$.  The length function on $W({\cal O})$
corresponds to the dimension of orbits (shifted by the dimension of ${\cal
F}({\cal O})$).  Now one can realize the group algebra of $W({\cal O})$ (and
even the Hecke algebra ${\cal H}({\cal O})$) as an algebra of
correspondences, endowed with a natural geometric action on $KX({\cal
O},{^\vee G^\Gamma})$.  For a sketch of this, we refer to the end of [LV]. 
(Our construction of geometric translation functors in section 8 was of
essentially the same nature; recall that a translation datum was a very
special kind of ${^\vee G}$-orbit on ${\cal F}({\cal O}) \times_{{\cal C}({\cal
O})} {\cal F}({\cal O}')$.)  
 
We return now to our description of the Kazhdan-Lusztig algorithms.
Suppose $G_{\Bbb R}$ is a linear Lie group as in (15.1)--(15.2).  Define
$${\cal K}\Pi^z({\cal O},G_{\Bbb R}) = K\Pi^z({\cal O},G_{\Bbb R})
\otimes_{\Bbb Z} {\Bbb Z}[u^{1/2},u^{-1/2}],\eqno(16.12)(a)$$
(cf. (15.7)), the free ${\Bbb Z}[u^{1/2},u^{-1/2}]$-module with basis the
irreducible (or standard) representations of $G_{\Bbb R}$.  We may refer to
this as the {\it mixed Grothendieck group of representations}. (In a sense
the definition is misleading: an ordinary Grothendieck group should be
naturally the specialization to $u=1$ of a mixed one, but a mixed one does
not arise naturally by extension of scalars from an ordinary one.  In our case
there is little to be done, because of the lack of a category of ``mixed
representations.'')  In the setting of (15.6), we will write
$${\cal K}\Pi^z({\cal O},G/{\Bbb R}) = K\Pi^z({\cal O},G/{\Bbb R})
\otimes_{\Bbb Z} {\Bbb Z}[u^{1/2},u^{-1/2}],\eqno(16.12)(b)$$
Similarly, we define
$${\cal K}X({\cal O},{^\vee G^\Gamma}) = KX({\cal O},{^\vee G^\Gamma})
\otimes_{\Bbb Z} {\Bbb Z}[u^{1/2},u^{-1/2}],\eqno(16.12)(c)$$
the {\it mixed Grothendieck group of perverse sheaves}.  In this case a more
geometric interpretation is available ([LV], Definition 2.2).  Finally, we will
need an analogue of the definition in Corollary 1.26:
$$\overline{\cal K}\Pi^z({\cal O},G/{\Bbb R}) = \hbox{ ${\Bbb
Z}[u^{1/2},u^{-1/2}]$-linear combinations of elements of $\Pi^z({\cal
O},G/{\Bbb R})$;} \eqno(16.12)(d)$$
here infinite linear combinations are allowed.  (Infinite combinations can
arise only for non-semisimple groups, because of the possibility of
infinitely many strong real forms.)  We call $\overline{\cal K}$ the {\it
formal mixed Grothendieck group of representations}.

\medskip
{\bf Proposition 16.13.}  {\it In the setting (16.7) -- (16.12), there are
natural actions of the Hecke algebra ${\cal H}({\cal O})$ on the mixed
Grothendieck groups ${\cal K}\Pi^z({\cal O},G_{\Bbb R})$ (or ${\cal
K}\Pi^z({\cal O},G/{\Bbb R})$, or $\overline{\cal K}\Pi^z({\cal O},G/{\Bbb
R})$) and ${\cal K}X({\cal O},{^\vee G^\Gamma})$.}
\medskip

{\it Proof.}  By (8.9), the ${\Bbb Z}[u^{1/2},u^{-1/2}]$-module ${\cal
K}X({\cal O},{^\vee G^\Gamma})$ is a direct sum of copies of Hecke algebra
modules constructed in [LV].  Explicit formulas for the action of the
generators $T_s$ appear in Lemma 3.5 of [LV].  For ${\cal K}\Pi^z({\cal
O},G_{\Bbb R})$, the action is constructed in [IC4], section 12.  Explicit
formulas may be found there or in [IC3], Definition 6.4.  (There is a small but
dangerous subtlety concealed in the details omitted here.  The basis for
${\cal K}\Pi^z({\cal O},G_{\Bbb R})$ used in the references corresponds after
Beilinson-Bernstein localization to a local system on an orbit, placed in
degree 0.  This differs by something like $(-1)^d$, with $d$ the orbit
dimension, from a standard representation.  One effect appears in the
formula for $C_r$ in Proposition 16.20 below, which contains a sign absent
from the references.  Of course the sign could be absorbed in the definition
of the representation-theoretic Kazhdan-Lusztig polynomials, and this is
probably where it belongs.)  Q.E.D.

In addition to the Hecke algebra module structure, there are two more
structures that are needed for the Kazhdan-Lusztig algorithms.

\medskip
{\bf Definition 16.14} (see [KL0]). {\it Verdier duality} is the unique algebra
automorphism
$$D:{\cal H}({\cal O}) \rightarrow {\cal H}({\cal O})$$
satisfying
$$Du^{1/2} = u^{-1/2}, DT_s = u^{-1}(T_s + (1-u)) \qquad (s \in S({\cal O})).$$
The terminology arises from the geometric interpretation of ${\cal H}({\cal
O})$.  Using the defining relations (16.10), one can easily verify that $D$ is
also characterized by
$$Du^{1/2} = u^{-1/2}, DT_w = (T_{w^{-1}})^{-1}\qquad(w\in W({\cal O})).$$
Another useful characterization is
$$Du^{1/2} = u^{-1/2}, D(T_s+1) = u^{-1}(T_s + 1) \qquad (s \in S({\cal O})).$$
It follows from either characterization that $D^2 = 1$.  If ${\cal M}$ is a
module for ${\cal H}({\cal O})$, then a {\it Verdier duality for ${\cal M}$} is
a ${\Bbb Z}$-linear involution
$$D_{\cal M}:{\cal M} \rightarrow {\cal M}$$ 
satisfying
$$D_{\cal M}(a\cdot m) = (Da)\cdot (D_{\cal M}m)\qquad (a\in {\cal H}({\cal
O}), m \in {\cal M}).$$
Again several equivalent characterizations are possible, notably
$$D_{\cal M}(u^{1/2}\cdot m) = u^{-1/2}(D_{\cal M} m),\quad D_{\cal
M}((T_s+1)\cdot m) = u^{-1}(T_s+1)\cdot (D_{\cal M} m) \qquad (m \in {\cal
M}, s\in S({\cal O}).$$

\medskip

{\bf Proposition 16.15} ([LV], Theorem 1.10, and [IC3], Lemma 6.8).  {\it In
the setting of Proposition 16.13, there is a natural Verdier duality $D_X$ on
the Hecke algebra module ${\cal K}X({\cal O},{^\vee G^\Gamma})$.  It is
characterized uniquely by the requirement that the matrix of $D_X$ with
respect to the basis $\{\mu(\xi)\}$ of (7.10)(c) be upper triangular in the
Bruhat order (cf. (7.11)(f)) with $u^{-d(\xi)}$ on the diagonal.}
\medskip

The duality $D_X$ is constructed in [LV] directly from Verdier duality for
complexes of sheaves.  (The powers of $u$ on the diagonal here differ from
those in [IC3] by a constant.  This modifies $D_X$ by a power of $u$, but has
no other effect.)  Since we are interested in using this duality to compute
the geometric character matrix, there is a small problem of circularity: the
characterization of $D_X$ involves the Bruhat order, which in turn is
defined using the geometric character matrix.  This problem is circumvented
in [IC3] by defining in elementary terms a weaker preorder relation, then
proving a stronger uniqueness result for $D_X$, involving only the weaker
preorder.  In any case, the proof of the uniqueness of $D_X$ provides an
algorithm for computing it. 
  
To formulate an analogous result for the mixed Grothendieck group of
representations, we need a function on $L^z(G_{\Bbb R})$ analogous to the
function $d(\xi)$ (dimension of the orbit) on $\Xi({^\vee G^\Gamma})$.  In the
cases of interest Theorem 10.4 provides a bijection between these two
sets, so we could simply use $d(\xi)$; but this is a little unsatisfactory
aesthetically.  What we want is essentially the ``integral length'' of [Green],
Definition 8.1.4; we will normalize it in a slightly different way, however. 

\medskip
{\bf Definition 16.16.}  Let $G_{\Bbb R}$ be a linear Lie group as in (11.1). 
Fix a Langlands decomposition $M_{\Bbb R}A_{\Bbb R}N_{\Bbb R}$ of a
minimal parabolic subgroup of $G_{\Bbb R}$, and let $B_M$ be a Borel
subgroup of $M$.  Define
$$c_0(G_{\Bbb R}) = 1/2( \dim B_M). \eqno(16.16)(a)$$

Suppose that $\Lambda$ is a $G_{\Bbb R}$-limit character of a Cartan
subgroup $T_{\Bbb R}$, with differential $\lambda \in {\frak t}_{\Bbb R}$
(Definition 11.2).  Assume that $\Lambda$ has infinitesimal character
corresponding to ${\cal O}$ as in (16.7), so that we have a well-defined set
$R^+(\lambda)$ of positive integral roots (cf. (16.9)(d)).  Choose a Cartan
involution $\theta$ of $G_{\Bbb R}$ preserving $T_{\Bbb R}$.  Then $\theta$
acts on the roots of $T$ in $G$, and this action preserves the integral roots. 
(For this we need the linearity of $G_{\Bbb R}$.)  We can therefore define
the {\it integral length of $\Lambda$} by
$$l^I(\Lambda) = -1/2\left(|\{\,\alpha\in R^+(\lambda)|\theta\alpha \in
R^+(\lambda)\,\}| + \dim (T^{\theta}_{\Bbb R})\right) + c_0(G_{\Bbb R}).
\eqno(16.16)(b)$$
Clearly this differs by a constant (depending on ${\cal O}$ and $G_{\Bbb R}$)
from the definition in [IC4], Definition 12.1 or [Green], Definition 8.1.4.  It is
also evident from the discussion in [IC4] that $l^I$ takes non-positive
integral values.  In the setting of (15.6), we will write
$$l^I(\xi) = l^I(\Lambda) \eqno(16.16)(c)$$
whenever $\xi \in \Xi^z(G/{\Bbb R})$ corresponds to $\Lambda \in
L^z(G({\Bbb R},\delta_s))$.
\medskip

{\bf Proposition 16.17} ([IC4], Lemma 12.14).  {\it In the setting of
Proposition 16.13, there is  natural Verdier duality $D_\Pi$ on the Hecke
algebra modules ${\cal K}\Pi^z({\cal O},G/{\Bbb R})$.  It is characterized
uniquely by the requirement that the matrix of $D_\Pi$ with respect to the
basis $\{M(\Lambda)\}$  of (15.1)(a) be upper triangular in the Bruhat order
(cf. (15.3)(d)) with $u^{-l^I(\Lambda)}$ on the diagonal.}
\medskip

The construction of $D_\Pi$ in [IC4] is a very complicated reduction to the
case of integral infinitesimal character.  In that case the
Beilinson-Bernstein localization theory provides a further reduction to the
case of Proposition 16.15.  Again it is important to establish a stronger
uniqueness theorem (not involving the Bruhat order) so that $D_\Pi$ is
computable.

We now have all the ingredients needed to define the Kazhdan-Lusztig
polynomials.

\medskip
{\bf Proposition 16.18} ([IC3], Theorem 7.1).  {\it In the setting
(16.7)--(16.12), there is for every geometric parameter $\gamma \in
\Xi({\cal O},{^\vee G^\Gamma})$ an element
$$C_g(\gamma) = \sum_\xi P_g(\xi,\gamma)(u) \mu(\xi)
\qquad(P_g(\xi,\gamma) \in {\Bbb Z}[u^{1/2},u^{-1/2}]) $$
of ${\cal K}X({\cal O},{^\vee G^\Gamma})$ characterized by the following
properties.
\smallskip
\item{i)} $D_XC_g(\gamma) = u^{-d(\gamma)}C_g(\gamma)$ (cf. Proposition
16.15).
\item{ii)} $P_g(\gamma,\gamma) = 1$.
\item{iii)} $P_g(\xi,\gamma) \ne 0$ only if $\xi\le \gamma$ in the Bruhat
order (cf. (7.11)(f)).
\item{iv)} If $\xi \ne \gamma$, then $P_g(\xi,\gamma)$ is a polynomial in
$u$ of degree at most $1/2(d(\gamma) - d(\xi) -1)$.
\medskip}

As in Proposition 16.15, there is actually a stronger uniqueness theorem,
and an algorithm for computing the polynomials.  We call the $P_g$ {\it
geometric Kazhdan-Lusztig polynomials}.  Here is the main result of [LV].

\medskip
{\bf Theorem 16.19} ([LV], Theorem 1.12).  {\it In the setting of Proposition
16.18, fix $\gamma,\xi \in \Xi({\cal O},{^\vee G^\Gamma})$.  Recall from
(7.10)--(7.11) the perverse sheaf $P(\gamma)$ and the local system ${\cal
V}_\xi$ on $S_\xi$.
\smallskip
\item {a)} $H^{-d(\gamma) + i}P(\gamma) = 0$ if $i$ is odd.
\item {b)} The multiplicity of ${\cal V}_\xi$ in $H^{-d(\gamma) +
i}P(\gamma)|_{S_\xi}$ is the coefficient of $u^{i/2}$ in $P_g(\xi,\gamma)$.
\item {c)}  $c_g(\xi,\gamma) = (-1)^{d(\xi) - d(\gamma)}
P_g(\xi,\gamma)(1).$
\item{d)} The specialization $C_g(\gamma)(1) \in KX({\cal O},{^\vee
G^\Gamma})$ is equal to $(-1)^{d(\gamma)}P(\gamma)$.
\medskip}

Here the first two assertions are in [LV] (although our present definition of
$P(\gamma)$, which follows [BBD], differs by a degree shift from the one in
[LV]).  The third assertion is immediate from the second and (7.11)(d), and
the last follows from (7.11)(c).

On the representation-theoretic side, things are formally quite similar.

\medskip
{\bf Proposition 16.20} ([IC4], Lemma 12.15).  {\it Suppose we are in the
setting (16.7)--(16.12) (so that in particular $G_{\Bbb R}$ is a linear group). 
Then there is for every element $\Theta \in L^z({\cal O},G_{\Bbb R})$ an
element
$$C_r(\Theta) = \sum_\Lambda (-1)^{l^I(\Theta) - l^I(\Lambda)}
P_r(\Lambda,\Theta)(u) M(\Lambda) \qquad(P_r(\Lambda,\Theta) \in {\Bbb
Z}[u^{1/2},u^{-1/2}]) $$
of ${\cal K}\Pi^z({\cal O},G_{\Bbb R})$ characterized by the following
properties.
\smallskip
\item{i)} $D_\Pi C_r(\Theta) = u^{-l^I(\Theta)}C_r(\Theta)$ (cf. Proposition
16.17).
\item{ii)} $P_r(\Theta,\Theta) = 1$.
\item{iii)} $P_r(\Lambda,\Theta) \ne 0$ only if $\Lambda\le \Theta$ in the
Bruhat order (cf. (15.3)(d)).
\item{iv)} If $\Lambda \ne \Theta$, then $P_r(\Lambda,\Theta)$ is a
polynomial in $u$ of degree at most $1/2(l^I(\Theta) - l^I(\Lambda) -1)$.
\medskip}

Again there is a better uniqueness theorem and an algorithm for computing
the polynomials.  We call the $P_r$ {\it representation-theoretic
Kazhdan-Lusztig polynomials}.  To state a result completely analogous to
Theorem 16.19, we need to compute entries of the representation-theoretic
character matrix as Euler characteristics.

\medskip
{\bf Proposition 16.21.}  {\it Suppose $\Lambda = (\Lambda^{can},
R^+_{i{\Bbb R}},R^+_{\Bbb R})$ is a $G_{\Bbb R}$-limit character of $T_{\Bbb
R}$ of infinitesimal character ${\cal O}$ (cf. Definition 11.2 and (16.7)). 
Choose a Cartan involution $\theta$ for $G_{\Bbb R}$ preserving $T_{\Bbb
R}$.  Write $\lambda$ for the differential of $\Lambda^{can}$, so that
$R^+(\Lambda)$ is the corresponding set of positive integral roots (cf.
(16.9)(d)).  After replacing $\Lambda$ by an equivalent limit character,
assume that $R^+_{\Bbb R}(\lambda) \subset R^+_{\Bbb R}$ (Definition 11.6). 
Fix a system of positive roots
$$R^+ \supset R^+(\lambda) \cup R^+_{\Bbb R}$$
for $T$ in $G$.  Define
$$b = 1/2|\{\,\alpha\in R^+ - R^+(\lambda) \mid \theta\alpha \in R^+\,\}|,$$
a non-negative integer.  Let ${\frak n}$ be the nilpotent subalgebra spanned
by the negative root vectors for $R^+$, so that $\Lambda^{can}\otimes
\rho({\frak n})$ is a character of $T_{\Bbb R}^{can,G}$ of type $z$.  Recall
from Definition 11.2 the dual standard representation ${\tilde M}(\Lambda)$.
\smallskip
\item{a)}  The weight $\Lambda^{can}\otimes \rho({\frak n})$ occurs in
$H_i({\frak n},{\tilde M}(\Lambda))$ exactly once, in degree
$b-l^I(\Lambda)$.
\item{b)}  If $\Theta$ is another standard limit character not equivalent to
$\Lambda$, then $\Lambda^{can}\otimes \rho({\frak n})$ does not occur in
$H_i({\frak n},{\tilde M}(\Theta))$.
\item{c)}  For any standard limit character $\Theta$, the
representation-theoretic character matrix is given by
$$c_r(\Lambda,\Theta) = (-1)^{\l^I(\Lambda) + b}\sum_i
(-1)^i(\hbox{multiplicity of $\Lambda^{can}\otimes \rho({\frak n})$ in
$H_i({\frak n},\pi(\Theta))$}).$$ 
\medskip}

\noindent We use here the Harish-Chandra module version of the standard
representations (cf. (15.2), taking for $K_{\Bbb R}$ the fixed points of the
Cartan involution $\theta$ chosen at the beginning of the proposition.  The
Lie algebra ${\frak t}$ and the group $T_{\Bbb R}\cap K_{\Bbb R}$ both act
on the Lie algebra homology groups, and the multiplicities are to be
interpreted in the category of $({\frak t}, T_{\Bbb R}\cap K_{\Bbb
R})$-modules.  It is not difficult to reformulate the result without a choice
of $\theta$, using instead the complex conjugation on $T$ coming from
$T_{\Bbb R}$.  In this form it is probably true for the various smooth forms
of the standard modules discussed in (15.2), but we do not know how to
prove it.
\smallskip
{\it Proof.}  Since $G_{\Bbb R}$ is assumed to be linear, all the imaginary
roots are integral.  The set of roots appearing in the definition of $b$
therefore consists of certain pairs ${\alpha,\theta\alpha}$ of complex
roots.  Consequently $b$ is an integer.

In the case of integral infinitesimal character, parts (a) and (b) are
Corollary 4.7 of [IC3].  The proof given there carries through essentially
without change in general.  (It consists mostly of references to [Green],
where there is no assumption of integrality.)  Part (c) is a formal
consequence of (a) and (b), together with the fact that the Euler
characteristic of ${\frak n}$ homology is a well-defined map from the
Grothendieck group $K\Pi^z(G_{\Bbb R})$ to $K\Pi^z(T_{\Bbb R})$.

\medskip
{\bf Theorem 16.22.}  {\it In the setting of Proposition 16.20, fix $\Lambda$
and $\Theta$ in $L^z({\cal O},G_{\Bbb R})$, and choose ${\frak n}$ as in
Proposition 16.21.
\item{a)} $\Lambda^{can}\otimes \rho({\frak n})$ does not occur in
$H_{b-l^I(\Theta) +i}({\frak n},\pi(\Theta))$ if $i$ is odd.
\item{b)} The multiplicity of $\Lambda^{can}\otimes \rho({\frak n})$ in
$H_{b-l^I(\Theta) +i}({\frak n},\pi(\Theta))$ is the coefficient of $u^{i/2}$
in $P_r(\Lambda,\Theta)$.
\item{c)} $c_r(\Lambda,\Theta) = (-1)^{l^I(\Lambda) -
l^I(\Theta)}P_r(\Lambda,\Theta)(1).$
\item{d)} The specialization $C_r(\Theta)(1) \in K\Pi^z({\cal O}, G_{\Bbb
R})$ is equal to $\pi(\Theta)$.
\medskip}

We postpone a discussion of the proof to the next section.

\medskip
{\bf Definition 16.23.} Suppose we are in the setting (15.6).  The mixed
Grothendieck groups ${\cal K}\Pi^z({\cal O},G/{\Bbb R})$ and  ${\cal
K}X({\cal O},{^\vee G^\Gamma})$ (cf. (16.12)) are both free ${\Bbb
Z}[u^{1/2},u^{-1/2}]$-modules on bases parametrized by $\Xi^z({\cal
O},G/{\Bbb R})$.  It therefore makes sense to define the {\it canonical
perfect pairing}
$$\langle,\rangle: {\cal K}\Pi^z({\cal O},G/{\Bbb R}) \times {\cal K}X({\cal
O},{^\vee G^\Gamma}) \rightarrow {\Bbb Z}[u^{1/2},u^{-1/2}]$$
to be the ${\Bbb Z}[u^{1/2},u^{-1/2}]$-linear map satisfying
$$\langle M(\xi), \mu(\gamma)\rangle = e(\xi)\delta_{\xi,\gamma}
u^{1/2(d(\gamma) + l^I(\xi))}$$
(cf. Definition 15.8, Definition 16.16)  Notice that the specialization to
$u=1$ of this pairing is the canonical pairing of Definition 15.11.
\medskip
{\bf Theorem 16.24.}  {\it Suppose we are in the setting (15.6).  With
respect to the pairing of Definition 16.23, the elements $C_r(\xi)$ and
$C_g(\gamma)$ of Propositions 16.18 and 16.20 satisfy
$$\langle C_r(\xi), C_g(\gamma)\rangle = e(\xi)\delta_{\xi,\gamma}
u^{1/2(d(\gamma) + l^I(\xi))}.$$
The geometric and representation-theoretic Kazhdan-Lusztig matrices are
essentially inverse transposes of each other:
$$\sum_\eta (-1)^{l^I(\xi) - l^I(\eta)}P_r(\eta,\xi)P_g(\eta,\gamma) =
\delta_{\xi,\gamma}.$$
\medskip}
We will discuss the proof of the first assertion in section 17.  The second is
a formal consequence, as in the proof of Corollary 15.13 from Theorem
15.12.

Theorem 15.12 follows from Theorem 16.24 by specializing to $u=1$ (see
Theorem 16.19(d) and Theorem 16.22(d)).  Q.E.D.
%\end\input mssymb.tex
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\vskip 3pc
\centerline{\bf 17. Proof of Theorems 16.22 and 16.24.}
\vskip 1pc
When the infinitesimal character is integral, Theorem 16.22 is the main
result (Theorem 7.3) of [IC3]; it is a more or less straightforward
consequence of [BB] and [LV].  The proofs in [IC3] can be modified easily to
cover the case when the simple root system $\Delta({\cal O})$ of (16.9) is
contained in a set of simple roots for $R(G,T_{\cal O})$.  Unfortunately this
is not always the case.  That the general case can be treated has been known
to various experts for many years, but there does not seem to be an account
of it in print.  The geometric part of the argument for the case of Verma
modules may be found in the first chapter of [Lbook].  The outline below is
gleaned from conversations with Bernstein, Brylinski, Kashiwara, and
Lusztig; it is due to them and to Beilinson.  To simplify the notation, we
take the central element $z$ in $Z({^\vee G})^{\theta_Z}$ to be trivial; this
changes nothing.

To begin, we must choose a system of positive roots
$$R^+(G,T_{\cal O}) \supset R^+({\cal O}) \eqno(17.1)(a)$$
for the root system $R(G,T_{\cal O})$.  There is no distinguished choice for
this positive system, and in fact the argument will use several different
ones.  We write
$$\rho_{\cal O} = 1/2\sum_{\alpha \in R^+(G,T_{\cal O})} \alpha.
\eqno(17.1)(b)$$
Now classical intertwining operator methods (as for example in [SV]) show
that $c_r(\Lambda,\Theta)$ is unchanged by a small modification of
$\Theta$ that does not affect the integral roots.  (Implicit here is the
assertion that it is possible to make a corresponding modification of
$\Lambda$.)  After making such a modification, we may assume that the
infinitesimal character is {\it rational}; that is, (in the notation of (16.8))
that there is a positive integer $n$ with 
$$n(\lambda_{\cal O}-\rho_{\cal O}) = \gamma_{\cal O} \in X^*(T_{\cal O}).
\eqno(17.1)(c)$$

Define
$${\cal B} = \hbox{ variety of Borel subalgebras of ${\frak g}$.}
\eqno(17.2)(a)$$
If ${\frak b} \in {\cal B}$ is any Borel subalgebra with Cartan subalgebra
${\frak t}$, then there is a natural isomorphism $j:{\frak t}_{\cal O}
\rightarrow {\frak t}$ carrying $R^+(G,T_{\cal O})$ to the roots of ${\frak
t}$ in ${\frak g}/{\frak b}$.  Using these isomorphisms, and the character
$\gamma_{\cal O}$ of $T_{\cal O}$, we can define an algebraic line bundle
$${\cal L} \rightarrow {\cal B}. \eqno(17.2)(b)$$
(If the infinitesimal character is integral, then ${\cal L}$ has non-trivial
sections.)  The complexification $K_{\Bbb C}$ of a maximal compact
subgroup $K_{\Bbb R}$ of $G_{\Bbb R}$ acts algebraically on ${\cal B}$ and
${\cal L}$, with a finite number of orbits on ${\cal B}$.  We make the
multiplicative group ${\Bbb C}^\times$ act on ${\cal L}$ one fiber at a time,
by
$$z\cdot \xi = z^n\xi \qquad (z\in {\Bbb C}^\times, \xi \in {\cal L}).
\eqno(17.2)(c)$$
Then the product group $H = K_{\Bbb C} \times {\Bbb C}^\times$ acts on
$${\cal L}^\times = {\cal L} - \hbox{ zero section }, \eqno(17.2)(d)$$
with finitely many orbits corresponding precisely to the orbits of $K_{\Bbb
C}$ on ${\cal B}$.  

It is convenient to develop this situation a little more generally at first.  So
suppose $Y$ is a smooth complex algebraic variety, and ${\cal L}$ is an
algebraic line bundle.  Write ${\cal L}^\times$ for ${\cal L}$ with the zero
section removed:
$$\pi:{\cal L}^\times \rightarrow Y \eqno(17.3)(a)$$
a principal ${\Bbb C}^\times$ bundle.  We make ${\Bbb C}^\times$ act on
${\cal L}^\times$ as in (17.2)(c):
$$z\cdot \xi = z^n\xi \qquad (z\in {\Bbb C}^\times, \xi \in {\cal L}^\times).
\eqno(17.3)(b)$$
We may therefore speak of {\it ${\Bbb C}^\times$-equivariant ${\cal
D}_{{\cal L}^\times}$-modules} on ${\cal L}^\times$. If ${\cal M}$ is such a
module, then its direct image on $Y$ is graded by the action of ${\Bbb
C}^\times$:
$$\pi_*({\cal M}) = \sum_{k \in {\Bbb Z}} \pi_*({\cal M})(k) \eqno(17.3)(c)$$
We say that ${\cal M}$ is {\it genuine} if the group of $n$th roots of unity in
${\Bbb C}^\times$ acts by the inverse of the tautological character:
$$\omega\cdot m = \omega^{-1}m \qquad (\omega^n = 1, m\in {\cal M})
\eqno(17.3)(d)$$
Evidently this is equivalent to 
$$\pi_*({\cal M}) = \sum_{k \equiv -1\pmod{n}} \pi_*({\cal M})(k)
\eqno(17.3)(e)$$


One can define a sheaf of algebras ${\cal D}_Y({\cal L}^{1/n})$ on $Y$, as
follows.  Informally, ${\cal D}_Y({\cal L}^{1/n})$ is the algebra of
differential operators on sections of the $(1/n)$th power of ${\cal L}$.  The
difficulty is that there is usually no such line bundle.  Now if $k$ is an
integer, the space of sections of ${\cal L}^{\otimes k}$ may be identified
with the space of functions on ${\cal L}^\times$ homogeneous of degree
$-k$ in the fiber variable; that is, with functions satisfying
$$f(z\cdot \xi) = z^{-nk}f(\xi). \eqno(17.4)(a)$$
Write $E$ for the (Euler) vector field on ${\cal L}^\times$, induced by the
vector field $z{d\over dz}$ on ${\Bbb C}^\times$ and the action (17.2)(c). 
This is a globally defined vector field on ${\cal L}^\times$.  Clearly
$$\{\hbox{ functions homogeneous of degree $-k$ }\} = \{\hbox{ functions
killed by $E+nk$ }\}. \eqno(17.4)(b)$$
It follows easily that the differential operators on sections of ${\cal
L}^{\otimes k}$ may be identified with the differential operators on ${\cal
L}$ commuting with $E$, modulo the ideal generated by $E+nk$.  This
suggests defining (over every open set $U \subset Y$)
$${\cal D}_Y({\cal L}^{1/n})(U) = ({\cal D}_{{\cal
L}^\times}(\pi^{-1}U))^E/(E+1) \eqno(17.4)(c)$$
Here on the right we are dividing by the ideal generated by $E+1$.

\medskip
{\bf Proposition 17.5.}  {\it In the setting (17.3)-(17.4), there is a natural
equivalence of categories
$$(\hbox{ genuine ${\Bbb C}^\times$-equivariant ${\cal D}_{{\cal
L}^\times}$-modules on ${\cal L}^\times$ }) \leftrightarrow (\hbox{ ${\cal
D}_Y({\cal L}^{1/n})$-modules on $Y$ }). $$
In the notation of (17.3)(e), the equivalence is
$${\cal M} \leftrightarrow \pi_*({\cal M})(-1).$$
\medskip}

This is easy general nonsense.

We now return to the setting of (17.1) and (17.2).  The action of $G$ on
${\cal B}$ lifts to ${\cal L}^\times$, and so defines an operator
representation
$$\psi_{{\cal L}^\times}:U({\frak g}) \rightarrow D_{{\cal  L}^\times}.
\eqno(17.6)(a)$$
The action of $G$ commutes with that of ${\Bbb C}^\times$, so the image of
$U({\frak g})$ commutes with $E$.  By (17.4)(c), we get
$$\psi_{\cal B}({\cal L}^{1/n}):U({\frak g}) \rightarrow D_{\cal B}({\cal 
L}^{1/n}). \eqno(17.6)(b)$$
We write
$$I_{\cal B}({\cal L}^{1/n}) = \ker \psi_{\cal B}({\cal L}^{1/n}) \subset
U({\frak g}).\eqno(17.6)(c)$$

\medskip
{\bf Theorem 17.7} (Beilinson-Bernstein localization theorem --- see [BB]). 
{\it Suppose we are in the setting (17.1)--(17.2) (so that in particular the
weight $\lambda_{\cal O}$ is regular).  Use the notation of (17.3)--(17.4)
and (17.6).
\smallskip
\item{a)}  The operator representation (17.6)(b) is surjective, with kernel
equal to the ideal generated by the kernel of the infinitesimal character
$\chi_{\cal O}$.
\item{b)}  The global sections and localization functors provide an
equivalence of categories between quasicoherent sheaves of ${\cal D}_{\cal
B}({\cal L}^{1/n})$-modules on ${\cal B}$ and ${\frak g}$-modules of
infinitesimal character $\chi_{\cal O}$.
\medskip}

{\bf Corollary 17.8.}  {\it In the setting of Theorem 17.7, there is a
(contravariant) equivalence of categories between finite-length $({\frak
g},K_{\Bbb C})$-modules of infinitesimal character $\chi_{\cal O}$, and
genuine $H$-equivariant perverse sheaves on ${\cal L}^\times$.
\medskip}

Here $H= K_{\Bbb C} \times {\Bbb C}^\times$ acts on ${\cal L}^\times$ as in
(17.2)(c), and ``genuine'' is explained in (17.3)(d).  To get this, we first apply
Proposition 17.5 to pass to genuine $H$-equivariant coherent ${\cal
D}_{{\cal L}^\times}$-modules, then use the Riemann-Hilbert
correspondence of Theorem 7.9.  To get the equivalence to be contravariant,
we must use the solution sheaf version of the Riemann-Hilbert
correspondence rather than the DeRham functor; this is the approach used in
[IC3].

According to (7.10), the genuine irreducible $H$-equivariant perverse
sheaves on ${\cal L}^\times$ are parametrized by genuine $H$-equivariant
local systems on orbits; so we must study such local systems.

\medskip
{\bf Lemma 17.9.} {\it In the setting of Theorem 17.7, there is a natural
bijection between genuine $H$-equivariant local systems on $H$-orbits on
${\cal L}^\times$, and equivalence classes of limit characters in $L({\cal
O},G_{\Bbb R})$  (cf.~(15.5)).  Suppose the limit character $\Lambda$
corresponds to a local system on an orbit $S$.  In the notation of Definition
16.16 and Propostion 16.21, the codimension of the orbit $S$ is equal to $b -
l^I(\Lambda)$. 
\medskip}

{\it Proof.}  Fix a $K_{\Bbb C}$-orbit $S_0$ on ${\cal B}$, and write $S$ for
its preimage in ${\cal L}^\times$ (an orbit of $H$). We can find a Borel
subalgebra
$${\frak b} = {\frak t} + {\frak n} \in S_0 \eqno(17.10)(a)$$
with the property that ${\frak t}$ is both $\theta$-stable and defined over
${\Bbb R}$.  The stabilizer $B_K$ of ${\frak b}$ in $K$ has a Levi
decomposition
$$B_K = T_K N_K;\eqno(17.10)(b)$$
the first factor is reductive, with compact real form
$$T_{K,{\Bbb R}} = T_K \cap K_{\Bbb R} = T_{\Bbb R} \cap K_{\Bbb R}.
\eqno(17.10)(c)$$
There is a natural map from $B_K$ into the Borel subgroup $B$
corresponding to ${\frak b}$ (induced by restricting (11.1)(a) to $K_{\Bbb
R}$ and complexifying).  On the other hand, the line bundle ${\cal L}$ gives a
character $\gamma$ of $B$, and so of $B_K$.  The stabilizer in $H$ of any
point $\xi$ of ${\cal L}^\times$ over ${\frak b}$ is then 
$$\widetilde{B_K} = \{\,(b,z)\mid b\in B_K, z \in {\Bbb C}^\times,
\gamma(b) = z^{-n}\,\}. \eqno(17.10)(d)$$
By projection on the first factor, we see that $\widetilde{B_K}$ is a central
extension of $B_K$ by the $n$th roots of one.  Projection on the second
factor defines a genuine character of $\widetilde{B_K}$ of differential
equal to $\rho -\lambda$.  Tensoring with this character therefore defines a
bijection
$$(\hbox{ characters of $B_K$ with differential $\lambda - \rho$ })
\leftrightarrow (\hbox{ genuine characters of $\widetilde{B_K}/(
\widetilde{B_K})_0$ }).\eqno(17.10)(e)$$
The objects on the right are essentially the genuine geometric parameters
for the orbit $S$.  Using (17.10)(b) and (17.10)(c), we may identify the
characters on the left with characters of $T_{\Bbb R}$ with differential
$\lambda - \rho$.
(The point is that $T_{\Bbb R}$ is a direct product of $T_{K,{\Bbb R}}$ with
a vector group, so that characters with specified differential are
determined by their restrictions to $T_{K,{\Bbb R}}$.)  Such characters in
turn give rise (by twisting by $\rho$ as in (11.3)) to limit characters of
differential $\lambda$, and so to classes in $L({\cal O},G_{\Bbb R})$.  We
leave to the reader the easy verification that this construction establishes
the bijection we need.  The last assertion is an elementary calculation,
which we omit.  Q.E.D

\medskip
{\bf Proposition 17.11.}  {\it In the setting of Theorem 17.7, suppose $V$ is
a finite-length Harish-Chandra module for $G_{\Bbb R}$ of infinitesimal
character $\chi_{\cal O}$.  Let $P$ be the corresponding perverse sheaf on
${\cal L}^\times$ (Theorem 17.7).  Fix an orbit $S$ and a genuine
$H$-equivariant irreducible local system ${\cal V}$ on $S$.  Choose
corresponding $T_{\Bbb R}$, ${\frak n}$, and limit character $\Lambda$ as
in the proof of Lemma 17.9.  Then the multiplicity of ${\cal V}$ in $H^iP|_S$
is equal to the multiplicity of the character $\Lambda^{can}\otimes
\rho({\frak n})$ in $H_{i+\dim {\cal L}^\times}({\frak n},V)$.}
\medskip

The proof is parallel to that of Proposition 4.1 of [IC3], and we omit it.

The main assertion in Theorem 16.22 is (b); for (a) is a special case, and (c)
and (d) follow from (b) and Proposition 16.21.  So we need to calculate
certain Lie algebra homology groups with coefficients in irreducible
representations.  Proposition 17.11 reduces this to calculating the stalks of
some irreducible perverse sheaves.  Assembling all of this (and the
dimension calculation in Lemma 17.9) we see that Theorem 16.22 is
equivalent to

\medskip
{\bf Theorem 17.12.}  {\it In the setting of Theorem 17.7, suppose
$P(\Theta)$ is the irreducible perverse sheaf on ${\cal L}^\times$
corresponding to the limit character $\Theta \in L({\cal O},G_{\Bbb R})$. 
Write $d$ for the dimension of the underlying orbit. Suppose $\Lambda$ is
another limit character, corresponding to the local system ${\cal V}$ on the
orbit $S$.  Then the multiplicity of ${\cal V}$ in $H^{-d+i}P(\Theta)|_S$ is
the coefficient of $u^{i/2}$ in $P_r(\Lambda,\Theta)$.}
\medskip

The analogy with Theorem 16.19 is now immediately apparent.

To prove Theorem 17.12, we can proceed as follows.  The corresponding
calculation of Lie algebra homology groups is carried out in [Green], under
the hypothesis that certain representations $U_\alpha(X)$ are completely
reducible (cf.~[IC3], section 7).  Here $X$ is an irreducible representation of
infinitesimal character $\chi_{\cal O}$, and $\alpha \in \Delta({\cal O})$ is
a simple integral root (cf.~(16.9)).  The idea is to prove that complete
reducibility more or less geometrically.  Write $P$ and $U_\alpha(P)$ for
the perverse sheaves corresponding to $X$ and $U_\alpha(X)$ (Corollary
17.8).  We may assume by induction that Theorem 17.12 is true for $P$.  We
would like to find a geometric description of $U_\alpha(P)$.  Unfortunately,
no such description is available in general.  To get one, we must first
replace our choice of positive root system in (17.1) by a new one
$$(R^+)'(G,T_{\cal O}) \supset R^+({\cal O})\eqno(17.13)(a)$$
with the additional property that
$$\hbox{ $\alpha$ is a simple root in $(R^+)'(G,T_{\cal O})$.}
\eqno(17.13)(b)$$
Such a choice certainly exists; the difficulty is that we cannot make one
choice for all $\alpha$.  This change replaces all the nilpotent algebras
${\frak n}$ whose homology we considered by slightly different algebras
${\frak n}'$; but Proposition 4.3 of [IC3] guarantees that the homology
changes only by a shift in degree.  Next, we must replace the infinitesimal
character $\lambda_{\cal O}$ by some
$$\lambda_{{\cal O}'} = \lambda_{\cal O} + \mu, \eqno(17.13)(c)$$
with $\mu \in X^*(T_{\cal O})$.  What we require of the new infinitesimal
character is
$$\hbox{ $\lambda_{{\cal O}'}$ is regular and dominant for $R^+({\cal O})$,
and $\alpha^\vee(\lambda_{{\cal O}'}) = 1$.}  \eqno(17.13)(d)$$
(To get the second condition, we may have to replace $G$ by some finite
cover.)  The translation principle identifies the Lie algebra homology groups
we want with some corresponding ones for representations of infinitesimal
character attached to ${\cal O}'$. 

After making all of these adjustments, we are reduced to the case when our
simple integral root $\alpha$ is simple in $R^+(G,T)$, and the coroot
$\alpha^\vee$ is one on $\lambda_{\cal O}$.  Because of (17.1)(c), we have
$$\alpha^\vee(\gamma_{\cal O}) = 0 \eqno(17.14)(a)$$
Write
$$ {\cal P}_\alpha = \hbox{ variety of parabolic subalgebras of type
$\alpha$} \eqno(17.14)(b)$$
(as for example in [LV], (3.2)).  The condition (17.14)(a) is what is needed for
the weight $\gamma_{\cal O}$ to define an equivariant algebraic line bundle
$${\cal L}_\alpha \rightarrow {\cal P}_\alpha. \eqno(17.14)(c)$$  
The natural projection $\pi_\alpha$ from ${\cal B}$ to ${\cal P}_\alpha$
pulls ${\cal L}_\alpha$ back to ${\cal L}$; so it defines a (proper
equivariant) morphism
$$\pi_\alpha:{\cal L}^\times \rightarrow {\cal L}^\times_\alpha.
\eqno(17.14)(d)$$
The fibers of $\pi_\alpha$ are one-dimensional projective lines.  The {\it
geometric $U_\alpha$ functor} on the derived category of $H$-equivariant
constructible sheaves on ${\cal L}^\times$ is given by 
$$U_{\alpha,g}(C) = \pi_\alpha^*(\pi_\alpha)_*(C) \eqno(17.14)(e)$$
(cf.~[LV], Definition 3.1, and [Lbook], Lemma 1.19).

Suppose now that $P$ is an irreducible genuine $H$-equivariant perverse
sheaf on ${\cal L}^\times$ as above.  By the (very deep!) decomposition
theorem of Beilinson, Bernstein, Deligne, and Gabber ([BBD], Theorem 6.2.5),
the direct image $(\pi_\alpha)_*P$ (in an appropriate derived category) is a
direct sum of irreducible (genuine $H$-equivariant) perverse sheaves on
${\cal L}^\times_\alpha$, with various degree shifts.  By Proposition 7.15,
$\pi_\alpha^*[1]$ carries irreducible perverse sheaves on ${\cal
L}^\times_\alpha$ to irreducible perverse sheaves on ${\cal L}^\times$.  It
follows that $U_{\alpha,g}(P)$ is a direct sum in the derived category of
irreducible perverse sheaves, with degree shifts.  Since we know the
cohomology sheaves of $P$ by induction, and since they vanish in every other
degree, it is possible to compute the cohomology sheaves of
$U_{\alpha,g}(P)$ completely.  The result agrees with the calculation of the
Lie algebra homology of $U_\alpha(X)$ made in [Green] (compare the
argument at (7.9) -- (7.12) in [IC3]).  It follows first of all that
$U_\alpha(P)$ and $U_{\alpha,g}(P)$ have the same image in the
Grothendieck group.  Now an equivariant perverse sheaf in our setting that is
{\it not} completely reducible must have some stalks of its homology
sheaves strictly smaller than those for the direct sum of its composition
factors.  (This is elementary; it is also not difficult to prove a
corresponding statement on the level of Harish-Chandra modules.) 
Consequently $U_\alpha(P)$ is completely reducible, as we wished to show. 
Q.E.D. 

We turn now to the proof of Theorem 16.24.  Of course this is essentially
taken from [IC4].  The idea is to define a certain natural dual of the Hecke
algebra module ${\cal K}X({\cal O},{^\vee G^\Gamma})$, then to identify this
dual with ${\cal K}\Pi^z({\cal O},G/{\Bbb R})$ (or rather an appropriate
completion) using the pairing of Definition 16.23.  The dual module has a
basis dual to the basis $\{C_g(\gamma)\}$; an element of this basis must
for formal reasons satisfy the requirements of Definition 16.20
characterizing the $C_r(\xi)$.  The first point is therefore to define a dual
Hecke module.

\medskip
{\bf Definition 17.15.}  In the setting of (16.10), suppose ${\cal M}$ is a
module for ${\cal H}({\cal O})$.  Set
$${\cal M}^* = {\rom Hom}_{{\Bbb Z}[u^{1/2},u^{-1/2}]}({\cal M},{\Bbb
Z}[u^{1/2},u^{-1/2}]).\eqno(17.15)(a)$$
Any ${\Bbb Z}[u^{1/2},u^{-1/2}]$-linear map $A$ on ${\cal M}$ defines a
${\Bbb Z}[u^{1/2},u^{-1/2}]$-linear map $A^t$ on ${\cal M}^*$, by the
requirement
$$(A^t\mu)(m) = \mu(Am) \qquad (m \in {\cal M},\mu\in {\cal
M}^*).\eqno(17.15)(b)$$
As usual, we have $(AB)^t = B^tA^t$.  Since the Hecke algebra is
non-commutative, we cannot make ${\cal M}^*$ into a module for it just by
transposing the action on ${\cal M}$; we must twist by an
anti-automorphism of ${\cal H}({\cal O})$.  Such an anti-automorphism may
be defined by sending $T_w$ to $(-u)^{l(w)}T_w^{-1}$.  (The inverse of $T_s$
may be computed from (16.10)(c); it is
$$T_s^{-1} = u^{-1}(T_s + (1-u)).\eqno(17.15)(c)$$
The invertibility of the other $T_w$ then follows from (16.10)(b).)  We can
therefore make ${\cal M}^*$ a module for the Hecke algebra by defining
$$T_w\cdot \mu = (-u)^{l(w)}(T_w^{-1})^t\cdot \mu. \eqno(17.15)(d)$$ 
Using (17.15)(b)--(d), we can get three equivalent formulations of this
definition:
$$\eqalign{(T_s\cdot \mu)(m) &= \mu((-T_s-1+u)\cdot m),\hbox{ or }\cr
 ((T_s+1)\cdot \mu)(m) &= -\mu((T_s-u)\cdot m),\hbox{ or } \cr
((T_s-u)\cdot \mu)(m) &= -\mu((T_s+1)\cdot m).\cr}  \eqno(17.15)(e)$$

Suppose that ${\cal M}$ admits a Verdier duality $D_{\cal M}$.  We can
define a ${\Bbb Z}$-linear involution $D_{{\cal M}^*}$ on ${\cal M}^*$ by
$$(D_{{\cal M}^*}\mu)(m) = \overline{ \mu(D_{\cal M} m)} \eqno(17.15)(f)$$
Here bar denotes the automorphism of ${\Bbb Z}[u^{1/2},u^{-1/2}]$ defined by
$\overline{u^{1/2}} = u^{-1/2}$.  It is straightforward to check that
$D_{{\cal M}^*}$ is a Verdier duality for ${\cal M}^*$ (Definition 16.14).
\medskip

{\bf Proposition 17.16.}  {\it Suppose we are in the setting of (15.6); use the
notation of (16.12).  Then the pairing of Definition 16.23 extends to
$$\langle,\rangle: \overline{\cal K}\Pi^z({\cal O},G/{\Bbb R}) \times {\cal
K}X({\cal O},{^\vee G^\Gamma}) \rightarrow {\Bbb Z}[u^{1/2},u^{-1/2}]$$
On this level it provides an identification
$$\overline{\cal K}\Pi^z({\cal O},G/{\Bbb R}) \simeq {\rom Hom}_{{\Bbb
Z}[u^{1/2},u^{-1/2}]}({\cal K}X({\cal O},{^\vee G^\Gamma}),{\Bbb
Z}[u^{1/2},u^{-1/2}]) = {\cal K}X({\cal O},{^\vee G^\Gamma})^*.$$
The Hecke algebra actions provided by Proposition 16.13 and Definition
17.15 are identified by this isomorphism.}
\medskip

{\it Proof.}  Because the representation-theoretic character matrix is a
direct sum of invertible finite blocks, the formal mixed Grothendieck group
$\overline{\cal K}\Pi^z({\cal O},G/{\Bbb R})$ may be identified with formal
sums of standard representations.  Now the first assertion is obvious from
Definition 16.23: the dual of a free module is a direct product over a dual
basis.  For the second, we must show (according to Definition 17.15(e))
$$\langle (T_s - u)M(\xi_1) ,\mu(\xi_2) \rangle = -\langle M(\xi_1),(T_s +
1)\mu(\xi_2)\rangle \eqno(17.17)$$
for every $\xi_1,\xi_2 \in \Xi^z({\cal O},G/{\Bbb R})$, and every simple
reflection $s = s_{\alpha}\in S({\cal O})$ (cf.~(16.9)(f)).  The proof of this is
essentially identical to the proof of Proposition 13.10(b) in [IC4].  We will
examine carefully one easy case and one difficult case.

So suppose $\xi_1$ corresponds to a complete Langlands parameter
$(\phi_1,\tau_1)$ for a Cartan subgroup ${^d{T}_1^\Gamma}$ of ${^\vee
G^\Gamma}$ (Definition 12.4 and Theorem 12.9).  Choose a compatible based
Cartan subgroup structure
$${^d{\cal T}_1^\Gamma} = ({^d{T}}_1^\Gamma,{\cal
S}({^d{T}}_1^\Gamma),{^d{R}}^+_{i{\Bbb R},1},{^d{R}}^+_{{\Bbb R},1})
\eqno(17.18)(a)$$
(Definition 13.11).  Choose a based Cartan subgroup
$${\cal T}_1^\Gamma = (T_1^\Gamma,{\cal W}(T_1^\Gamma),R^+_{i{\Bbb
R},1},R^+_{{\Bbb R},1}) \eqno(17.18)(b)$$
for $G^\Gamma$ and a pairing
$$\zeta_1 :{^\vee T}_1 \rightarrow {^d{T}}_1 \eqno(17.18)(c)$$
between them (Definition 13.9 and Proposition 13.10(b)).  Construct a limit
character $(\delta_1,\Lambda_1)$ for $T^\Gamma_1$ (Definition 12.1) as in
Proposition 13.12.  By the definition in Proposition 13.13, the standard
representation $M(\xi_1)$ is attached to the real form $\delta_1$ of $G$
and the limit character $\Lambda_1$.  The parameter $\lambda_1 \in
{^d{T}}_1$ attached to $\phi_1$ (Proposition 5.6) belongs to ${\cal O}$, so
(16.7)(d) provides natural isomorphisms
$$j(\lambda_{\cal O},\lambda_1) : {^\vee T}_{\cal O} \rightarrow  {^d{T}}_1,
\qquad i(\lambda_{\cal O},\lambda_1): T_{\cal O} \rightarrow T_1.
\eqno(17.18)(d)$$
Under these maps, the simple root $\alpha \in \Delta({\cal O})$ of (17.17)
corresponds to
$$\alpha_1 \in \Delta(\lambda_1) \eqno(17.18)(e)$$
(a root of $T_1$ in $G$), and $\alpha^\vee$ to a root
$$^d{\alpha}_1 \in {^d{\Delta}}(\lambda_1) \eqno(17.18)(f)$$
of ${^d{T}}_1$ in ${^\vee G}$.

For our easy case, assume that $\alpha_1$ is a compact imaginary root of
$T_1$ in $G({\Bbb R},\delta_1)$, and that $\xi_1 = \xi_2$.  By [IC4],
Definition 12.3,
$$T_sM(\xi_1) = uM(\xi_1). \eqno(17.19)(a)$$
By Proposition 13.12(c), $^d{\alpha}_1$ is a real root failing to satisfy the
parity condition.  By [LV], Lemma 3.5(e),
$$T_s \mu(\xi_2) = -\mu(\xi_2). \eqno(17.19)(b)$$
It follows from (17.19) that both sides of (17.17) are zero. 

For a hard case, assume that $\alpha_1$ is noncompact imaginary in
$G({\Bbb R},\delta_1)$, and that the reflection $s_{\alpha_1}$ does {\it not}
belong to the Weyl group of $T_1({\Bbb R})$ in $G({\Bbb R},\delta_1)$.  (In
the terminology of [Green], $\alpha_1$ is ``type I.''  This case arises in
$SL(2,{\Bbb R})$ but not in $PGL(2,{\Bbb R})$.  Our assumption on $\xi_2$
will be formulated after (17.20).)  Define $G(\alpha_1)$ to be the subgroup
generated by $T_1$ and the image of the root subgroup $\phi_{\alpha_1}$
(see (11.10)).  The assumption that $\alpha_1$ is imaginary implies that
$G(\alpha_1)$ is normalized by $T_1^\Gamma$, and in fact
$$G(\alpha_1)^\Gamma = G(\alpha_1)T_1^\Gamma \eqno(17.20)(a)$$
is a weak extended group.  By the proof of Proposition 13.6 (after (14.10)),
$\alpha_1$ is also noncompact imaginary with respect to the real forms in
${\cal W}(T_1^\Gamma)$.  We can therefore find an element
$$\delta_1^0 \in  {\cal W}(T_1^\Gamma) \eqno(17.20)(b)$$
that acts as $\delta_1$ does on $G(\alpha_1)$.  Up to center, the common
real form $G(\alpha_1,\delta_1) = G(\alpha_1,\delta_1^0)$ is locally
isomorphic to $SL(2,{\Bbb R})$, and $T_1({\Bbb R})$ corresponds to a
compact Cartan subgroup.  Now let $T_2({\Bbb R})$ be another Cartan 
subgroup of $G(\alpha_1,\delta_1^0)$, corresponding to a split Cartan
subgroup of $SL(2,{\Bbb R})$.  Define
$$T_2^\Gamma = \hbox{ group generated by $T_2$ and $\delta_1^0$,} \qquad 
{\cal W}(T_2^\Gamma) = {\rom Ad}(T_2)\cdot \delta_1^0. \eqno(17.20)(c)$$
This is a Cartan subgroup of $G^\Gamma$, the {\it Cayley transform of
$T_1^\Gamma$ through $\alpha_1$}.  (Compare [Green], Definition 8.3.4.)  We
want to define on $T_2^\Gamma$ a structure of based Cartan subgroup.  To
do that, we first fix an isomorphism
$$c:T_1 \rightarrow T_2 \eqno(17.20)(d)$$
given by conjugation by an element of $G(\alpha_1)$.  This choice carries
$\alpha_1$ to a real root $\alpha_2$ of $T_2$ in $G$.  It identifies the
imaginary roots in $T_2$ with the imaginary roots in $T_1$ orthogonal to
$\alpha_1$, so we can define
$$R^+_{i{\Bbb R},2} = c(R^+_{i{\Bbb R},1}) \cap R_{i{\Bbb R},2}.
\eqno(17.20)(e)$$
On the other hand, $c^{-1}$ carries the real roots of $T_2$ orthogonal to
$\alpha_2$ onto the real roots of $T_1$.  We choose a set of positive real
roots subject to the two conditions
$$R^+_{{\Bbb R},2} \supset c(R^+_{{\Bbb R},1}),\qquad \hbox{ $\alpha_2$ is
simple in $R^+_{{\Bbb R},2}$.} \eqno(17.20)(f)$$
This is certainly possible.  We claim that 
$${\cal T}_2^\Gamma = (T_2^\Gamma,{\cal W}(T_2^\Gamma),R^+_{i{\Bbb
R},2},R^+_{{\Bbb R},2}) \eqno(17.20)(g)$$
is a based Cartan subgroup of $G^\Gamma$.  The only difficult condition is
(e) of Definition 13.5.  To check it, suppose $\Lambda_1^0$ is a limit
character for $G({\Bbb R},\delta_1^0)$ as in Definition 13.5(e).  Form the
Cayley transform of this limit character through the noncompact root
$\alpha_1$ ([Green], Definition 8.3.6).  This is a limit character
$\Lambda_2^0$ for the Cartan subgroup $T_2({\Bbb R})$ of $G({\Bbb
R},\delta_1^0)$, again of the type considered in Definition 13.5(e).  By a
theorem of Hecht and Schmid ([Green], Proposition 8.4.5), the standard
representation $M(\Lambda_2^0)$ contains $M(\Lambda_1^0)$ as a
subrepresentation.  Since the latter is assumed to admit a Whittaker model,
the former must as well; so ${\cal T}_2^\Gamma$ is a based Cartan
subgroup.  We define
$$\Lambda_2 = (\Lambda_2^{can},R^+_{i{\Bbb R},2},R^+_{{\Bbb R},2})
\eqno(17.20)(h)$$
to be the Cayley transform of $\Lambda_1$ through the root $\alpha_1$
([Green], Definition 8.3.6).

We assume (back in (17.17)) that $\xi_2$ is the geometric parameter
corresponding to $(\Lambda_2,\delta_1)$.  Definition 12.3(c) of [IC4]
expresses $(T_s-u)M(\xi_1)$ in terms of standard representations.  The
coefficient  of $M(\xi_2)$ is 1.  By Definition 16.23, the left side of (17.17)
in this case is therefore $e(\xi_2)u^{1/2(d(\xi_2) +  l^I(\xi_2))}$.

To continue with our calculation, we need to describe a based Cartan
subgroup for ${^\vee G^\Gamma}$ paired with ${\cal T}_2^\Gamma$.  Since 
$\alpha_1$ was assumed to be real, the corresponding root ${^d\alpha_1}$
for ${^d{T}_1}$ is a real root satisfying the parity condition (Proposition
13.12(c)).  We may therefore proceed in analogy with (17.20) to construct a
Cayley transform ${^d{T}^\Gamma_2}$ of ${^d{T}^\Gamma_1}$ through
${^d\alpha_1}$.  Specifically , the map $\phi_1$ of (17.18) provides an
element
$$y \in {^d{T}_1^\Gamma} - {^d{T}_1},\eqno(17.21)(a)$$
and we can choose
$${^\vee{\delta}_1} \in {\cal S}({^d{T}^\Gamma_1}) \eqno(17.21)(b)$$
acting on ${^\vee G}({^d\alpha_1})$ as $y$ does.  We define
$${\cal S}({^d{T}^\Gamma_2}) = {\rom Ad}({^d{T}_2})\cdot {^\vee{\delta}_1},
\eqno(17.21)(c)$$
and fix an isomorphism $^d c$ between the maximal tori inner for  ${^\vee
G}({^d\alpha_1})$.  As in (17.20)(e), this choice provides a noncompact
imaginary root ${^d\alpha_2}$, and a natural choice $^d R^+_{{\Bbb R},2}$ of
positive real roots.  It also carries $\lambda_1 \in {^d{\frak t}_1}$ to
$\lambda_2 \in {^d{\frak t}_2}$, and the pair $(y,\lambda_2)$ defines a
Langlands parameter
$$\phi_2:W_{\Bbb R} \rightarrow {^d{T}_2^\Gamma}. \eqno(17.21)(d)$$
We want to define a pairing between $T_2$ and ${^d{T}_2}$.  We have a
pairing $\zeta_1$ from (17.18)(c), and the map $c$ of (17.20)(d) defines an
isomorphism $^\vee c$ from $^\vee T_2$ to $^\vee T_1$.  Put
$$\zeta_2 = {^d c}\circ \zeta_1 \circ {^\vee c}:{^\vee T_2} \rightarrow {^d
T_2}, \eqno(17.21)(e)$$ 
Finally, let $^d R^+_{i{\Bbb R},2}$ be the set of roots corresponding to
$R^+_{{\Bbb R},2}$ under this pairing.  By construction and (17.20)(f),
$$R^+_{i{\Bbb R},2} \supset {^d c}(R^+_{i{\Bbb R},1}),\qquad \hbox{
$^d\alpha_2$ is simple in $R^+_{i{\Bbb R},2}$.} \eqno(17.21)(f)$$
We claim that $R^+_{i{\Bbb R},2}$ is special ([AV2], Definition 6.29.)  This
follows from the appendix to [AV2] in essentially the  same way as we
checked Definition 13.5(e) after (17.20)(g).  Granted this, it is now clear
that 
$${^d{\cal T}_2^\Gamma} = ({^d{T}}_2^\Gamma,{\cal
S}({^d{T}}_2^\Gamma),{^d{R}}^+_{i{\Bbb R},2},{^d{R}}^+_{{\Bbb R},2})
\eqno(17.21)(g)$$
is a based Cartan subgroup (Definition 13.7) paired with ${\cal
T}_2^\Gamma$ by $\zeta_2$.  The strong real form $\delta_1$ of $T_2$
(cf.~(17.20)) defines a character $\tau_2$ of the canonical component group
for ${^d{T}_2^\Gamma}$ with respect to ${^\vee G^\Gamma}$ (Definition 12.4
and Proposition 13.12(a)).  By Proposition 13.12, the pair $(\phi_2,\tau_2)$
corresponds to the complete geometric parameter $\xi_2$.  

By inspection of the definitions (particularly the proofs in section 9) one
can find the following more direct description of $\tau_2$.  (Additional
details may be found in [Green], section 8.3).  Write
$$A_i = ({^d{T}_i^y})^{alg,{^\vee G}}/({^d{T}_i^y})^{alg,{^\vee G}}_0
\eqno(17.22)(a)$$
By calculation in $SL(2)$, one sees that
$$({^d{T}_1^y})^{alg,{^\vee G}}_0 \subset ({^d{T}_2^y})^{alg,{^\vee G}}_0.
\eqno(17.22)(b)$$
(That is, the compact part of the more compact Cartan subalgebra is larger.) 
Set
$$A_{12} = ({^d{T}_1^y} \cap {^d{T}_2^y})^{alg,{^\vee
G}}/({^d{T}_1^y})^{alg,{^\vee G}}_0. \eqno(17.22)(c)$$
By definition $A_{12}$ is a subgroup of $A_1$; in fact it is precisely the
kernel of the character defined by the real root $^d\alpha_1$ (which takes
values in $\{\pm 1\}$).  Similarly, the natural map from $A_{12}$ to $A_2$
is surjective; the kernel is generated by the element
$\overline{m^{alg}_{^d\alpha_1}}$.  Now $\tau_1$ is a representation of
$A_1$.  We may therefore restrict it to a representation $\tau_{12}$ of
$A_{12}$.  Because of Proposition 13.12(c), $\tau_{12}$ descends to a
representation of $A_2$, and this is the one we want.

We can now consider the right side of (17.17).  Lemma 3.5(c2) of [LV]
expresses $(T_s + 1)\mu(\xi_2)$ in terms of other extensions by zero of
sheaves on orbits.  The coefficient of $\mu(\xi_1)$ is $-1$.  By Definition
16.23, the right side of (17.17) is therefore $e(\xi_1)u^{1/2(d(\xi_1) + 
l^I(\xi_1))}$.  To compare this with the formula for the left side obtained
after (17.20), notice first that $\tau_1$ and $\tau_2$ agree on the central
element $z(\rho)$ by (17.22).  By Definition 15.8, $e(\xi_1) = e(\xi_2)$. 
Next, the geometric description of orbits in [LV] shows that $d(\xi_1) =
d(\xi_2) +1$.  Finally, it is elementary to show from Definition 16.16  and
(17.20) that $l^I(\xi_1) = l^I(\xi_2) -1$.  It follows that the exponents of
$u$ on the two sides of (17.17) agree as well.

This completes our hard case; the similar arguments for other cases are left
to the reader.  Q.E.D.

\medskip
{\bf Corollary 17.23.} {\it The identification
$$\overline{\cal K}\Pi^z({\cal O},G/{\Bbb R}) \simeq  {\cal K}X({\cal
O},{^\vee G^\Gamma})^*$$
of Proposition 17.16 identifies the Verdier duality $D_\Pi$ of Proposition
16.17 with the ``conjugate transpose'' of $D_X$ (Definition 17.15). 
Explicitly,
$$\langle D_\Pi M, \mu \rangle = \overline{\langle M, D_X \mu \rangle}$$
for $M \in \overline{\cal K}\Pi^z({\cal O},G/{\Bbb R})$ and $\mu \in {\cal
K}X({\cal O},{^\vee G^\Gamma})$.}
\medskip

{\it Proof.}  Proposition 17.16 guarantees that the conjugate transpose of
$D_X$ is a Verdier duality.  Proposition 16.17 provides a characterization of
$D_\Pi$ in terms of its action on the basis $\{M(\xi)\}$.  Definition 16.23
allows us to check very easily that the conjugate transpose of  $D_X$ has
the required characteristic properties.  (We refer to [IC4], Lemma 13.4 for
more details.)  Q.E.D.

To complete the proof of Theorem 16.24, notice that Proposition 16.18
guarantees that the elements $C_g(\xi)$ form a basis of ${\cal K}X({\cal
O},{^\vee G^\Gamma})$.  By Proposition 17.16, we may therefore define
elements $C'_r(\xi)$ of ${\cal K}\Pi^z({\cal O},G/{\Bbb R})$ by the
requirement
$$\langle C'_r(\xi),C_g(\gamma) \rangle =
e(\xi)\delta_{\xi,\gamma}u^{1/2(d(\gamma) + l^I(\xi))}.$$
By Corollary 17.23 and Proposition 16.18(i), these elements satisfy
Proposition 16.20(i).  Slightly more calculation (using also Definition 16.23)
shows that the remaining conditions (ii)--(iv) of Proposition 16.18
guarantee those in Proposition 16.20.  By Proposition 16.20, $C'_r(\xi) =
C_r(\xi)$, as we wished to show.  (We refer to  Lemma 13.7 of [IC4] for more
details.)  Q.E.D.
 
We conclude this section with the proof of Lemma 15.9.  Applying the
translation principle as in Proposition 16.6, we can reduce to the case when
$\xi$ has regular infinitesimal character.  We may therefore place ourselves
in the setting of (12.4)--(12.6), with $\xi$ corresponding to some complete
Langlands parameter $(\phi,\tau_1)$ for ${^d{T}}^\Gamma$.  We can choose a
set $^d R^+$ of positive roots for ${^d{T}}$ in ${^\vee G}$ with the property
that
$${^d{R}^+} - {^d{R}^+_{\Bbb R}} \hbox{ is preserved by $\theta$.}
\eqno(17.24)(a)$$
For every root $\alpha$, we can define an element
$$m_\alpha = \alpha^\vee (-1) \eqno(17.24)(b)$$
as in (12.6).  For any positive root system, we have by (4.9)(b)
$$z(\rho) = \prod_{\alpha \in {^d{R}^+}} m_\alpha. \eqno(17.24)(c)$$
By (17.24),
$$m_\alpha m_{\theta \alpha} = m_\alpha (\theta m_ \alpha) \in
{^d{T}^\theta_0}. \eqno(17.24)(d)$$
Since the character $\tau_1$ is trivial on the identity component of
${^d{T}^\theta}$, it follows from (17.24)(c)--(d) that
$$\tau_1(z(\rho)) = \prod_{\alpha \in {^d{R}^+_{\Bbb R}}} \tau_1(m_\alpha).
\eqno(17.24)(e)$$
Proposition 13.12(c) allows us to translate this back to $G({\Bbb R},\delta)$
and a Cartan subgroup $T({\Bbb R})$ dual to ${^d{T}^\Gamma}$.  Here is the
result.  Let $\chi$ be the character of the lattice of imaginary roots for
$T({\Bbb R})$ in $G({\Bbb R},\delta)$ which is 1 on noncompact simple roots
and $-1$ on compact simple roots.  Then
$$e(\xi) = \prod_{\beta \in R^+_{i{\Bbb R}}} \chi(\beta). \eqno(17.24)(f)$$
At this point there are several ways to proceed.  For one of them, let
$A({\Bbb R})$ be the maximal ${\Bbb R}$-split torus in $T({\Bbb R})$, and
$M({\Bbb R})$ its centralizer in $G({\Bbb R},\delta)$.  We have shown that
$e(\xi)$ may be calculated in $M({\Bbb R})$; since the same is true of
Kottwitz's sign ([Kott1], Corollary (6) on page 295) we may assume $M = G$. 
Let $\chi_0$ be the character of the imaginary root lattice that is $-1$ on
each simple root.  Evidently $\chi\chi_0$ is $-1$ exactly on the noncompact
roots, so 
$$e(\xi) = (-1)^{\rom (number\ of\ noncompact\ positive\ roots)}\cdot e_0,$$
where $e_0$ does not depend on the real form.  The exponent of $-1$ here is
just $q(G({\Bbb R},\delta))$, half the dimension of the symmetric space
attached to $G({\Bbb R})$; so
$$e(\xi) = (-1)^{q(G({\Bbb R},\delta))} \cdot e_0.$$
On the other hand, if $G({\Bbb R},\delta')$ is quasisplit, then we can choose
the positive root system so that all simple imaginary roots are noncompact. 
It follows from (17.24)(f) that $e(\xi) = 1$ in that case.  Comparing with the
last formula gives $e_0 = (-1)^{q(G({\Bbb R},\delta'))}$, so finally
$$e(\xi) = (-1)^{q(G({\Bbb R},\delta)) - q(G({\Bbb R},\delta'))},
\eqno(17.24)(g)$$
which agrees with Kottwitz's definition.  Q.E.D.
%\end\input mssymb.tex
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\vskip 3pc
\centerline{\bf 18. Strongly stable characters and Theorem 1.29.}
\vskip 1pc
We begin by recalling Langlands' notion of stable characters.  This seems to
make sense only for linear groups, so we will not try to define it in the
setting of (11.1).  Suppose therefore that we are in the setting of Definition
10.3, and that $\eta$ is a finite-length canonical projective representation
of type $z$ of a strong real form $\delta$ of $G^\Gamma$.  Thus $\eta$ is in
particular a representation of $G({\Bbb R},\delta)^{can}$.  The {\it character
of $\eta$} is a generalized function $\Theta(\eta)$ on $G({\Bbb
R},\delta)^{can}$, defined as follows.  Suppose $f$ is a compactly supported
smooth density on $G({\Bbb R},\delta)^{can}$.  Then $\eta(f)$ is a
well-defined operator on the space of $\eta$.  (If we write $f$ as a
compactly supported smooth function times a Haar measure, $f(g)dg$, then
$\eta(f)$ is given by the familiar formula
$$\eta(f) = \int_{G({\Bbb R},\delta)^{can}} f(g)\eta(g)dg.)$$
The operator $\eta(f)$ is trace class, and the value of the generalized
function $\Theta(\eta)$ on the test density $f$ is by definition the trace of
this operator:
$$\Theta(\eta)(f) = {\rom tr}(\eta(f)) \eqno(18.1)(a)$$
Of course we can immediately define the trace of any virtual canonical
projective representation (cf.~(15.5)):
$$\Theta: K\Pi^z(G({\Bbb R},\delta)) \rightarrow (\hbox{ generalized
functions on $G({\Bbb R},\delta)^{can}$ }).\eqno(18.1)(b)$$
This map is injective (since characters of inequivalent irreducible
representations are linearly independent).   Recall that an element $g$ of
$G({\Bbb R},\delta)^{can}$ is called {\it strongly regular} if its centralizer
is a Cartan subgroup.  (This condition is slightly stronger than regular
semisimple, which asks only that the centralizer in the Lie algebra be a
Cartan subalgebra.)  Write
$$G({\Bbb R},\delta)^{can}_{SR} \subset G({\Bbb R},\delta)^{can}
\eqno(18.1)(c)$$ 
for the set of strongly regular elements; it is a dense open subset of full
measure.  Harish-Chandra's regularity theorem for invariant
eigendistributions shows that the trace of a virtual representation is
determined by its restriction to this subset, and that this restriction is an
analytic function.  The map of (18.1)(b) therefore becomes
$$\Theta_{SR}: K\Pi^z(G({\Bbb R},\delta)) \hookrightarrow C^\infty(G({\Bbb
R},\delta)^{can}_{SR}).\eqno(18.1)(d)$$
Of course the resulting functions are constant on conjugacy classes of
$G({\Bbb R},\delta)^{can}$.  

\medskip
{\bf Definition 18.2}. ([Langlands2]).  In the setting of (18.1), the virtual
representation $\eta$ is said to be {\it stable}  if whenever $g$ and $g'$ are
strongly regular elements of $G({\Bbb R},\delta)^{can}$ conjugate under the
complex group $G^{can}$, we have
$$\Theta_{SR}(\eta)(g) =  \Theta_{SR}(\eta)(g').$$
\medskip

Now we have in Corollary 1.26 a formal parametrization of all virtual
representations in terms of L-group data, specifically as linear functionals
on a category of perverse sheaves.  It is natural to look for a
characterization of the subset of stable virtual characters.  As we remarked
after the formulation of Theorem 1.29, it is quite difficult to find such a
characterization.  Fortunately the ideas of Langlands and Shelstad about
stable characters admit a slightly different formulation.  To begin, define
$$G({\Bbb R},*)^{can} = \{\,(g,\delta) \mid\delta\in G^\Gamma - G\hbox{ a
strong real form, } g\in G({\Bbb R},\delta)^{can}\,\} \eqno(18.3)(a)$$
This set is analogous to $P({^\vee G^\Gamma})$ (Proposition 5.6), or to
various fiber products appearing in section 6.  The group $G^{can}$ acts on it
by conjugation.  More generally, whenever ${\cal D}$ is a set of strong real
forms of $G$, we can define
$$G({\Bbb R},{\cal D})^{can} = \{\,(g,\delta)\mid \delta\in {\cal D}, g \in
G({\Bbb R},\delta)^{can}\,\} \eqno(18.3)(b)$$
Suppose for example that ${\cal D}_0$ is the equivalence class of the strong
real form  $\delta_0$.  Then 
$$G({\Bbb R},{\cal D}_0)^{can} \simeq G^{can} \times_{G({\Bbb
R},\delta_0)^{can}} G({\Bbb R},\delta_0)^{can}, \eqno(18.3)(c)$$
with $G({\Bbb R},\delta_0)^{can}$ acting by conjugation to define the
induced bundle.  (The base space of the bundle is $G^{can}/G({\Bbb
R},\delta_0)^{can}$, which is just ${\cal D}_0$.)  The singular space
$G({\Bbb R},*)^{can}$ is undoubtedly an interesting place to do analysis, but
we will confine our attention to the smooth open subset
$$G({\Bbb R},*)^{can}_{SR} = \{\,(g,\delta) \in G({\Bbb R},*)^{can} | \hbox{
$g$ is strongly regular}\,\}. \eqno(18.3)(d)$$
This is a disjoint union of open subsets corresponding to equivalence
classes of strong real forms; each open subset is invariant under
conjugation by $G^{can}$, and may be described as a smooth induced bundle
in analogy with (18.3)(c).

\medskip
{\bf Lemma 18.4.} {\it Suppose $G^\Gamma$ is a weak extended group
(Definition 2.13).  In the setting (18.3), to specify a smooth
$G^{can}$-invariant function on $G({\Bbb R},*)^{can}_{SR}$ is equivalent to
specifying (for one representative $\delta_0$ of each equivalence class of
strong real forms of $G$) a smooth $G({\Bbb R},\delta_0)^{can}$-invariant
function on $G({\Bbb R},\delta_0)^{can}_{SR}$.}
\medskip

This is obvious from the definition.  Using this idea, we can reformulate
Langlands' definition of stability.

\medskip
{\bf Lemma 18.5.}  {\it Suppose ${\cal D}_0$ is an equivalence class of
strong real forms of the weak extended group $G^\Gamma$, $\delta_0 \in
{\cal D}_0$, and $\eta$ is a virtual representation of $G({\Bbb
R},\delta_0)^{can}$.  Use the character of $\eta$ to define a smooth
$G^{can}$-invariant function $\Theta_{SR}(\eta,{\cal D}_0)$ on $G({\Bbb
R},{\cal D}_0)^{can}_{SR}$ ((18.3)(b) and Lemma 18.4).  Then $\eta$ is stable
(Definition 18.2) if and only if $\Theta_{SR}(\eta,{\cal D}_0)$ is constant on
the fibers of the first projection
$$p_1:G({\Bbb R},{\cal D}_0)^{can}_{SR} \rightarrow G^{can}, \qquad
p_1(g,\delta) = g.$$}
\medskip

Now Corollary 1.26 suggests considering families of virtual representations
of all strong real forms.  Such families are going to give rise to smooth
functions on $G({\Bbb R},*)^{can}_{SR}$, by means of (18.1)(d) and Lemma
18.4.  Here is the appropriate class of virtual representations.

\medskip
{\bf Definition 18.6} (cf.~Definition 1.27).  Suppose $G^\Gamma$ is a weak
extended group  (Definition 2.13).  As in Corollary 1.26, write
$\overline{K}^z(G/{\Bbb R})$ for the set of formal infinite integral
combinations of irreducible canonical projective representations of strong
real forms of $G$, the group of {\it formal virtual representations}.  Such a
formal virtual representation
$$\eta = \sum_{\xi \in \Xi^z(G/{\Bbb R})} n(\xi)\pi(\xi)$$
is called {\it locally finite} if for each fixed strong real form $\delta$ there
are only finitely many $\xi$ with $n(\xi) \ne 0$ and $\delta(\xi)$ (the
strong real form parametrized by $\xi$) equivalent to $\delta$.  Write 
$$\overline{K}_f\Pi^z(G/{\Bbb R}) \eqno(18.6)(a)$$
for the set of locally finite formal virtual representations.  Evidently the
map of (18.1) extends to 
$$\Theta(\cdot,\delta): \overline{K}_f\Pi^z(G/{\Bbb R}) \rightarrow (\hbox{
generalized functions on $G({\Bbb R},\delta)^{can}$ }). \eqno(18.6)(b)$$
Finally, these maps can be assembled into a single map
$$\Theta_{SR}(\cdot,*): \overline{K}_f\Pi^z(G/{\Bbb R}) \rightarrow
C^\infty(G({\Bbb R},*)^{can}_{SR})) \eqno(18.6)(c)$$
defined by 
$$\Theta_{SR}(\eta,*)(g,\delta) = \Theta(\eta,\delta)(g). \eqno(18.6)(d)$$
That is, given a strong real form $\delta$ and a (strongly regular) element
$g$ of it, we find the (finitely many) irreducible representations of
$G({\Bbb R},\delta)^{can}$ appearing in $\eta$; add their distribution
characters; and evaluate the sum at $g$.

The locally finite formal virtual representation $\eta$ is said to be {\it
strongly stable} if the function $\Theta_{SR}(\eta,*)$ is constant on the
fibers of the first projection
$$p_1:G({\Bbb R},*)^{can}_{SR} \rightarrow G^{can}, \qquad p_1(g,\delta) =
g.$$
\medskip

By Lemma 18.5, the restriction of a strongly stable virtual representation
to a single strong real form must be stable.  The converse is false, however,
since a strongly regular element $g$ of $G^{can}$ may belong to several
inequivalent strong real forms of $G$.  In fact $g$ must belong to a
quasisplit real form (if it belongs to any at all); we have already
encountered this elementary fact in the proof of Proposition 13.6.  It
follows that a strongly stable virtual representation is determined by its
restriction to any quasisplit real form of $G$.  The results of Shelstad in 
[Shelstad1] can be formulated as a converse to this statement, as follows.

\medskip
{\bf Theorem 18.7} ([Shelstad1]).   {\it Suppose $G^\Gamma$ is a weak
extended group for $G$ (Definition 2.13), and $\delta_0$ is a strong real
form of $G$.  Suppose $\eta_0$ is a stable virtual representation of
$G({\Bbb R},\delta_0)^{can}$ (Definition 18.2).  Then there is a strongly
stable locally finite formal virtual representation $\eta$ for $G$
(Definition 18.6) with the property that
$$\Theta_{SR}(\eta,*)(g,\delta_0) = \Theta_{SR}(\eta_0)(g)$$
for every strongly regular element $g$ of $G({\Bbb R},\delta_0)^{can}$
(cf.~(18.6)(c) and (18.1)(d)).  If $\delta_0$ is quasisplit, then $\eta$ is
unique.}
\medskip

We will need a more explicit statement than this; formulating it will also
allow us to explain the relationship between Theorem 18.7 and the
superficially rather different results of [Shelstad1].  We begin with
something in terms of the traditional Langlands classification.  For the rest
of this section we will fix an E-group
$$({^\vee G^\Gamma},{\cal D}) \eqno(18.8)$$
for $G^\Gamma$ with second invariant $z$ (Definition 4.6).

\medskip
{\bf Definition 18.9} ([Shelstad1], section 3).  In the setting of (18.1) and
(18.8), suppose $\phi\in P({^\vee G^\Gamma})$ is a Langlands parameter
(Definition 5.2).  The {\it stable standard (virtual) representation attached
to $\phi$} is by definition the sum of all the (inequivalent) standard final
limit representations of $G({\Bbb R},\delta)^{can}$ parametrized by $\phi$
(Proposition 13.12); to simplify later definitions we insert the sign
$e(G({\Bbb R},\delta))$ attached to the real form by Kottwitz (Definition
15.8).  It is written as $\eta_\phi^{loc}(\delta)$.  (Equivalently,
$\eta_\phi^{loc}(\delta)$ may be defined as the sum of the inequivalent
standard representations induced from discrete series parametrized by
$\phi$.)  In the notation of (1.19),
$$\eta_\phi^{loc}(\delta) = e(G({\Bbb R},\delta))\sum_{\scriptstyle \xi =
(\phi,\tau) \in \Xi^z(G/{\Bbb R}) \atop\scriptstyle \delta(\xi) = \delta}
M(\xi);$$
the sum is over complete Langlands parameters (Definition 5.11), with
$\phi$ fixed.
\medskip

The terminology of the definition is justified by

\medskip
{\bf Lemma 18.10} ([Shelstad1], Lemma 5.2).  {\it In the setting of
Definition 18.9, the virtual representation $\eta_\phi^{loc}(\delta)$ is
stable.}
\medskip

Shelstad proves this under the assumption that $\phi$ is tempered, but the
argument is unchanged in general.  Although she does not state the result
explicitly, her argument also proves

\medskip
{\bf Lemma 18.11.}  {\it In the setting of (18.1), the lattice of stable virtual
representations has as a basis the set of non-zero $\eta_\phi^{loc}(\delta)$
(as $\phi$ varies over $\Phi({^\vee G^\Gamma})$).}
\medskip

To prove Theorem 18.7, we need also a way to compare different real forms.

\medskip
{\bf Theorem 18.12} ([Shelstad1], Theorem 6.3).  {\it In the setting of
Definition 18.9, suppose that $\delta$ and $\delta'$ are strong real forms of
$G$, and that
$$g \in G({\Bbb R},\delta)^{can} \cap G({\Bbb R},\delta')^{can}$$
is strongly regular.  Then
$$\Theta_{SR}(\eta_\phi^{loc}(\delta))(g) =
\Theta_{SR}(\eta_\phi^{loc}(\delta'))(g).$$
Here $\Theta_{SR}$ is as in (18.1).}

\medskip
{\bf Definition 18.13.}  In the setting of Definitions 18.6 and 18.9, suppose
$\phi\in P({^\vee G^\Gamma})$ is a Langlands parameter (Definition 5.2). 
The {\it strongly stable standard formal virtual representation attached to
$\phi$} is by definition the sum of all the (inequivalent) standard final limit
representations parametrized by $\phi$ (Proposition 13.12), normalized by
the signs $e(\xi)$ (Definition 15.8).  It is written as $\eta^{loc}_\phi$.  In
the notation of (1.19),
$$\eta^{loc}_\phi = \sum_{\scriptstyle \xi = (\phi,\tau) \in \Xi^z(G/{\Bbb
R})} e(\xi)M(\xi);$$
the sum is over complete Langlands parameters (Definition 5.11). 
Equivalently,
$$\eta^{loc}_\phi = \sum_\delta \eta_\phi^{loc}(\delta);$$
the sum is over strong real forms.  Finally, we can also write
$$\eta^{loc}_\phi = \sum_{\tau\in
(A^{loc,alg}_\phi)\widehat{}}e(\phi,\tau)M(\phi,\tau),$$
a sum over the (necessarily one-dimensional) irreducible representations of
the universal component group.
\medskip

The terminology is justified by

\medskip
{\bf Theorem 18.14.}  {\it In the setting of Definition 18.12, the formal
virtual representation $\eta^{loc}_\phi$ is strongly stable.  The lattice of
strongly stable formal virtual representations has as a basis the set of
$\eta^{loc}_\phi$ (as $\phi$ varies over $\Phi({^\vee G^\Gamma})$).}
\medskip

Theorems 18.7 and 18.14 follow from Lemmas 18.10 and 18.11, and Theorem
18.12.

To complete the proof of Theorem 1.29, we need to identify the formal
virtual representation $\eta^{loc}_\phi$ in the isomorphism of Corollary
1.26; that is, as a ${\Bbb Z}$-linear functional on the Grothendieck group
$KX({^\vee G^\Gamma})$.  

\medskip
{\bf Lemma 18.15.}  {\it In the setting of Definition 18.13, write $S$ for the
orbit of ${^\vee G}$ on $X({^\vee G^\Gamma})$ corresponding to $\phi$
(Proposition 6.17).  Then
$$\eta^{loc}_\phi = \sum_{\xi = (S,{\cal V}) \in \Xi^z(G/{\Bbb R})}
e(\xi)M(\xi).$$
Here we identify complete geometric parameters as in Definition 7.6; the
sum is therefore over irreducible equivariant local systems ${\cal V}$ on
$S$.  In the isomorphism of Corollary 1.26, this formal virtual
representation corresponds to the map
$$\chi^{loc}_S: {\rom Ob}\,{\cal C}(X({^\vee G^\Gamma}),{^\vee G}^{alg})
\rightarrow {\Bbb N}$$
that assigns to a constructible equivariant sheaf the dimension of its fiber
over $S$ (Definition 1.28).}
\medskip

{\it Proof.}  The first formula is immediate from Definition 18.13 and the
identification of complete Langlands parameters with complete geometric
parameters (Definition 7.6).  For the second, notice first of all that all the
irreducible equivariant local systems on $S$ have fiber dimension 1.  (This
is equivalent to the assertion that the group $A_S^{loc,alg}$ is abelian.  The
proof of Theorem 10.4 exhibits $A_S^{loc,alg}$ as a subquotient of a Cartan
subgroup of the connected reductive complex group ${^\vee G}^{alg}$
(Definition 12.4 and Theorem 12.9), proving the assertion.)  We may
therefore write
$$\chi^{loc}_S(C) = \sum_{\cal V} (\hbox{ multiplicity of ${\cal V}$ in $C$
restricted to $S$ }),\eqno(18.16)$$
the sum extending over irreducible equivariant local systems on $S$. 
Comparing (18.16) with the first formula of the lemma, and using Corollary
1.26(a), we get the second formula.  Q.E.D.

Theorem 1.29 now follows from Lemma 18.15 and Theorem 18.14.

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\centerline{\bf 19. Characteristic cycles, micro-packets, and Corollary
1.32.}
\vskip 1pc

In this section we will repeat in a little more detail the argument given in
the introduction for defining the new strongly stable virtual
representations $\eta_S^{mic}$.  It is convenient to work at first in the
setting of Definition 7.7; that is, that $Y$ is a smooth complex algebraic
variety on which the pro-algebraic group $H$ acts with finitely many orbits. 
At each point $y$ of $Y$, the differential of this action defines a map
$$a_y:{\frak h} \rightarrow T_yY \eqno(19.1)(a)$$
from the Lie algebra ${\frak h}$ of $H$ into the tangent space of $Y$ at $y$. 
If $y$ belongs to the $H$-orbit $S$, then
$$a_y({\frak h}) = T_yS \eqno(19.1)(b)$$
If we regard ${\frak h} \times Y$ as a trivial bundle over $Y$, we get a
bundle map
$$a:{\frak h}\times Y \rightarrow TY \eqno(19.1)(c)$$
Of course the cotangent bundle of $Y$ is dual to the tangent bundle, so we
may define the {\it conormal bundle to the $H$ action} as  the annihilator of
the image of $a$.  Explicitly,
$$T^*_H(Y) = \{\,(\lambda,y) \mid \lambda \in T^*_y(Y), \lambda(a_y({\frak
h})) = 0 \,\} \eqno(19.1)(d)$$
It is clear from this definition that $T^*_H(Y)$ is a closed cone in the
cotangent bundle.   In the situation of (19.1)(b), the fiber of $T^*_H(Y)$ at
$y$ is the conormal bundle to $S$ at $y$ (that is, the annihilator of $T_yS$
in $T^*_y(Y)$):
$$T^*_{H,y}(Y) = T^*_{S,y}(Y). \eqno(19.1)(e)$$
Consequently
$$T^*_H(Y) = \bigcup_{\hbox{$H$-orbits $S$}} T^*_S(Y). \eqno(19.1)(f)$$
That is, the conormal bundle to the action is the union of the conormal
bundles to the orbits.   

\medskip
{\bf Lemma 19.2.}  {\it Suppose the proalgebraic group $H$ acts on the
smooth variety $Y$ with finitely many orbits; use the notation of (19.1).
\smallskip
\item{a)}  The conormal bundle $T^*_H(Y)$ is a closed Lagrangian cone in
$T^*(Y)$.
\item{b)}  The $H$-components (that is, the smallest $H$-invariant unions
of irreducible components) of $T^*_H(Y)$ are the closures
$\overline{T^*_S(Y)}$ of conormal bundles of $H$-orbits $S$ in $Y$.
\medskip}

{\it Proof.}  We have already seen that $T^*_H(Y)$ is a closed cone.  Since
the conormal bundle to a locally closed smooth subvariety is automatically
Lagrangian, part (a) follows from (19.1)(f).  The conormal bundle $T^*_S(Y)$
to an $H$-orbit $S$ has dimension equal to the dimension of $Y$, and is
$H$-irreducible (that is, $H$ permutes the irreducible components
transitively).  Part (b) follows.  Q.E.D.

We turn now to a discussion of the characteristic variety.  We follow
[BorelD], VI,1.9, which the reader may consult for more details.  On any
smooth algebraic variety $Y$, the sheaf ${\cal D}_Y$ of algebraic
differential operators has a natural filtration by order
$${\cal O}_Y = {\cal D}_Y(0) \subset {\cal D}_Y(1) \subset \cdots
\eqno(19.3)(a)$$
The terms here are coherent ${\cal O}_Y$-modules, and the multiplication of
differential operators satisfies
$${\cal D}_Y(p){\cal D}_Y(q) \subset {\cal D}_Y(p+q), \eqno(19.3)(b)$$
Write $p:T^*(Y) \rightarrow Y$ for the natural bundle map.  The symbol
calculus for differential operators is an isomorphism of graded sheaves of
algebras
$${\rom gr}\,{\cal D}_Y \simeq p_*({\cal O}_{T^*(Y)}). \eqno(19.3)(c)$$
Because $p$ is an affine map, $p_*$ defines an equivalence of categories
$$\hbox{( quasicoherent sheaves of ${\cal O}_{T^*(Y)}$-modules )}
\rightarrow \hbox{( quasicoherent sheaves of $p_*{\cal
O}_{T^*(Y)}$-modules )} \eqno(19.3)(d)$$
([Hartshorne], Exercise II.5.17(e)).

Suppose now that ${\cal M}$ is a quasicoherent ${\cal D}_Y$-module.  A {\it
filtration} on ${\cal M}$ is a sequence
$$\cdots \subset {\cal M}(p) \subset {\cal M}(p+1) \subset \cdots
\qquad(p\in {\Bbb Z}) \eqno(19.4)(a)$$
of quasicoherent ${\cal O}_Y$-submodules, satisfying
$${\cal D}(p){\cal M}(q) \subset {\cal M}(p+q), \quad \bigcup_p {\cal M}(p) =
{\cal M},  \quad \bigcap_p {\cal M}(p) = 0 \eqno(19.4)(b)$$
The first condition makes the associated graded sheaf ${\rom gr}\,{\cal M}$
into a sheaf of modules for ${\rom gr}\,{\cal D}_Y$.  By (19.3)(c) and (d), we
may therefore regard ${\rom gr}\,{\cal M}$ as a sheaf of modules on $T^*(Y)$. 
The filtration on ${\cal M}$ is called {\it good} if ${\rom gr}\,{\cal M}$ is a
coherent sheaf (of ${\cal O}_{T^*(Y)}$-modules).   (This implies in particular
that ${\cal M}(p) = 0$ for $p$ sufficiently negative.)

\medskip
{\bf Lemma 19.5} ([BorelD], VI.1.9). {\it Suppose ${\cal M}$ is a coherent
${\cal D}_Y$-module.  Then good filtrations of ${\cal M}$ exist locally on
$Y$.  More precisely, if $U$ is any affine open subset of $Y$, then the
restriction of ${\cal M}$ to $U$ admits a good filtration.}
\medskip

We do not know whether global good filtrations exist in general.  (According
to [KK], Corollary 5.1.11, they exist in the analytic category.)  We will
eventually see that ${^\vee G}^{alg}$-equivariant ${\cal D}$-modules on
$X({\cal O},{^\vee G^\Gamma})$ admit global ${^\vee G}^{alg}$-invariant good
filtrations (Proposition 21.4).

Even when good filtrations exist, they are certainly not unique.  The next
lemma allows us to compare objects defined using different filtrations, and
to deduce that some such objects are independent of the filtration chosen.

\medskip
{\bf Lemma 19.6} ([Ginsburg], Corollary 1.3; cf.~[Assoc], Proposition 2.2).
{\it Suppose ${\cal F}$ and ${\cal G}$ are good filtrations of the coherent
${\cal D}_Y$-module ${\cal M}$.  Then there are finite filtrations
$${\rom gr}({\cal M},{\cal F})(0) \subset {\rom gr}({\cal M},{\cal
F})(1)\subset\cdots\subset {\rom gr}({\cal M},{\cal F})(N) = {\rom gr}({\cal
M},{\cal F})$$
$${\rom gr}({\cal M},{\cal G})(0) \subset {\rom gr}({\cal M},{\cal
G})(1)\subset\cdots\subset {\rom gr}({\cal M},{\cal G})(N) = {\rom gr}({\cal
M},{\cal G})$$ 
by graded coherent sheaves of $p_*{\cal O}_{T^*(Y)}$-modules ), with the
property that the corresponding subquotients are isomorphic:
$${\rom gr}({\cal M},{\cal F})(j)/{\rom gr}({\cal M},{\cal F})(j-1) \simeq {\rom
gr}({\cal M},{\cal G})(j)/{\rom gr}({\cal M},{\cal G})(j-1) \qquad (0 \le j \le
N).$$
(There is a constant $N'$, $0\le N' \le N$, with the property that the $j$th
isomorphism shifts the grading by $j-N'$.)

In particular, the class of ${\rom gr}({\cal M})$ in the Grothendieck group of
coherent sheaves of ${\cal O}_{T^*(Y)}$-modules is independent of the
choice of good filtration.}
\medskip

{\bf Definition 19.7.}  Suppose $V$ is a complex algebraic variety, and ${\cal
N}$ is a coherent sheaf of ${\cal O}_V$-modules.  The {\it associated
variety} (or {\it support}) of ${\cal N}$ is the set of (not necessarily closed)
points $v \in V$ at which the stalk of ${\cal N}$ is not zero:
$${\cal V}({\cal N}) = \{\,v\in V\mid {\cal N}_v \ne 0\,\}. $$
It is a closed subvariety of $V$, defined by the sheaf of ideals ${\rom
Ann}\,{\cal N} \subset {\cal O}_V$.  The set $\{W_i\}$ of irreducible
components of  ${\cal V}({\cal N})$ is the set of maximal points (that is,
points not in the closure of other points) of ${\cal V}({\cal N})$.  (If $V$ is
affine, these correspond to the minimal prime ideals containing the
annihilator of ${\cal N}$.)  To each such point we assign a positive integer
$$m_{W_i}({\cal N}),$$
the {\it multiplicity of ${\cal N}$ along $W_i$}.   One way to define it is as
the length of ${\cal N}_{W_i}$ as a module for the local ring ${\cal
O}_{V,W_i}$.  The {\it associated cycle of ${\cal N}$} is the formal sum
$${\rom Ch}({\cal N}) = \sum_i m_{W_i}({\cal N}) W_i.$$
This is a finite formal integer combination of irreducible subvarieties of
$V$, with no containments allowed among the subvarieties.  Notice that the
support of ${\cal N}$ is the union of the subvarieties appearing in the
associated cycle.
\medskip

The theory of associated cycles is elementary, and well-known to experts;
but it is not easy to give references with enough details for the novice.  The
theory is essentially local, so it suffices to discuss it for affine varieties. 
In that case the necessary commutative algebra is summarized in section 2
of [Assoc].

\medskip
{\bf Lemma 19.8.} {\it Suppose ${\cal N}$ and ${\cal N}'$ are coherent
sheaves of modules on the complex algebraic variety $V$.  Assume that
${\cal N}$ and ${\cal N}$ admit finite filtrations (as ${\cal O}_V$-modules)
with the property that ${\rom gr}\,{\cal N} \simeq {\rom gr}\,{\cal N}'$.  Then 
${\rom Ch}\,{\cal N} = {\rom Ch}\,{\cal N}'$.
\medskip}

Again this is well-known, and again we refer to section 2 of [Assoc] for a
proof.

\medskip
{\bf Definition 19.9} Suppose ${\cal M}$ is a coherent ${\cal D}_Y$-module
on the smooth algebraic variety $Y$.  Assume first that ${\cal M}$ admits a
good filtration.  The {\it characteristic variety of ${\cal M}$} (or {\it
singular support}) is the support of the ${\cal O}_{T^*(Y)}$-module ${\rom
gr}\,{\cal M}$ (cf.~(19.4)):
$${\rom SS}({\cal M}) = {\cal V}({\rom gr}\,{\cal M}) \subset T^*Y.$$
By Lemmas 19.6 and 19.8, this is independent of the choice of good
filtration.  Since ${\rom gr}\,{\cal M}$ is graded, the characteristic variety
is a closed cone in $T^*(Y)$. The {\it characteristic cycle of ${\cal M}$} is
the associated cycle of ${\rom gr}\,{\cal M}$:
$${\rom Ch}({\cal M}) = {\rom Ch}({\rom gr}\,{\cal M}).$$
It is a formal sum of irreducible subvarieties of $T^*(Y)$, with positive
integer coefficients; the union of these subvarieties is the characteristic
variety of ${\cal M}$.  Lemmas 19.6 and 19.8 guarantee that it is
independent of the choice of good filtration.

In general (that is, even if ${\cal M}$ does not admit a good filtration) we
can define the characteristic variety and characteristic cycle over each
affine open set in $Y$ using Lemma 19.5; the resulting invariants agree on
overlapping affines, and so make sense on all of $Y$ (cf.~[BorelD], VI.1.9).
\medskip

Obviously the characteristic variety of ${\cal M}$ is contained in the
inverse image under $p$ (cf.~(19.3)) of the support of ${\cal M}$.  In fact
$${\rom SS}({\cal M}) \cap T^*_y(Y) \ne \emptyset \Leftrightarrow y\in {\rom
Supp}({\cal M}). \eqno(19.10)$$
In this sense the characteristic variety refines the notion of support for
${\cal D}$-modules (or, via the Riemann-Hilbert correspondence of Theorem
7.9, for perverse sheaves).

Recall that the dimension of the characteristic variety of ${\cal M}$ is
(everywhere locally on ${\rom SS}({\cal M})$) greater than or equal to the
dimension of $Y$ ([BorelD], Theorem VI.1.10) and that (by definition) ${\cal
M}$ is holonomic exactly when equality holds.

\medskip
{\bf Examples 19.11.}  If ${\cal M}$ is equal to ${\cal D}_Y$, then the usual
filtration by degree is good.  The associated graded module is ${\cal
O}_{T^*(Y)}$, so ${\rom Ch}({\cal D}_Y) = 1\cdot T^*(Y)$.  

Suppose ${\cal M}$ is equal to ${\cal O}_Y$.  Then we can define
$${\cal M}(p) = {\cal O}_Y\qquad (p = 0,1,2,\ldots).$$
(This module is regular holonomic.)  The associated graded module is ${\cal
O}_Y$; that is, it is ${\cal O}_{T^*(Y)}$ modulo the ideal defining the zero
section of the cotangent bundle.  Consequently ${\rom Ch}({\cal O}_Y) =
1\cdot Y$

Suppose ${\cal M}$ is the ${\cal D}_Y$ module generated by a delta function
supported at the point $y$; that is,
$${\cal M} = {\cal D}_Y/{\cal D}_Y{\cal I}(y).$$
(Here we write ${\cal I}(y)$ for the ideal in ${\cal O}_Y$ defining the point
$y$.  Again the module is regular holonomic.)  Then ${\cal M}$ inherits a good
quotient filtration from ${\cal D}_Y$.  The associated graded module is the
sheaf of functions on the cotangent space $T^*_y(Y)$, so ${\rom Ch}({\cal M})
= 1\cdot T^*_y(Y)$.
\medskip

We return now to the equivariant setting.

\medskip
{\bf Proposition 19.12.}  {\it Suppose we are in the setting of Lemma 19.2,
and that ${\cal M}$ is an $H$-equivariant coherent ${\cal D}_Y$-module.  
\smallskip
\item{a)} For each $X\in {\frak h}$, let $a(X)$ be the vector field on $Y$
induced by the action of $H$, a first-order differential operator on $Y$. 
Then the action of $a(X)$ on ${\cal M}$ induced by the ${\cal D}_Y$-module
structure agrees with the action of $X$ defined by differentiating the action
of $H$ on ${\cal M}$.
\item{b)} The characteristic variety of ${\cal M}$ is contained in the
conormal bundle to the $H$ action (cf.~(19.1)).
\item{c)} The $H$-components (Lemma 19.2) of ${\rom SS}(\cal M)$ are
closures of conormal bundles of $H$-orbits on $Y$.  Consequently
$${\rom Ch}({\cal M}) = \sum_{\hbox{$H$-orbits $S$}} \chi_S^{mic}({\cal
M})\overline{T^*_S(Y)},$$
for some non-negative integers $\chi_S^{mic}({\cal M})$.
\item{d)} The support of ${\cal M}$ is given by
$${\rom Supp}({\cal M}) = \bigcup_{\chi_S^{mic}({\cal M}) \ne 0}
\overline{S}.$$
\item{e)}  The functions $\chi_S^{mic}$ are additive for short exact
sequences (of $H$-equivariant coherent ${\cal D}_Y$-modules). 
Consequently they define ${\Bbb Z}$-linear functionals
$$\chi_S^{mic}:K(Y,H) \rightarrow {\Bbb Z}$$
(notation (7.10)).
\medskip}

{\it Proof.}  Part (a) is a trivial exercise in understanding the definition of
equivariant ${\cal D}$-modules ([BorelD], VII.12.10). (We couldn't do it
ourselves, but never mind.)  For (b), assume for simplicity that there is a
global $H$-invariant good filtration.  Fix $X \in {\frak h}$.  We may regard
the vector field $a(X)$ as a first-order differential operator, and therefore 
$$a(X):{\cal M}(p) \rightarrow {\cal M}(p+1).$$
On the other hand, the action of $H$ preserves degree in the filtration, so
the differentiated action of ${\frak h}$ does as well.  By (a), it follows that
$$a(X):{\cal M}(p) \rightarrow {\cal M}(p).$$
It follows that the principal symbol of $a(X)$ annihilates ${\rom gr}({\cal
M})$.  This symbol is nothing but $a(X)$, regarded as a function on $T^*(Y)$. 
It follows that
$${\cal V}({\rom gr}\,{\cal M}) \subset \{\,(\lambda,y)\in T^*(Y)\mid
\lambda(a_y(X)) = 0\,\}.$$
Comparing this with (19.1)(d) gives (b).  Part (c) follows from (b) and
Lemma 19.2.  Part (d) follows from (c) and (19.10).  Part (e) is clear from
the definitions.  Q.E.D.

This completes our exposition of Definition 1.30.  Corollary 1.32 is an
immediate consequence of Theorem 1.31 (Kashiwara's local index theorem)
and Theorem 1.29.  It will be convenient to introduce a bit more notation in
that context, however.

\medskip
{\bf Definition 19.13.}  In the setting of (19.1), fix $S\in \Phi(Y,H)$
(Definition 7.1); that is, an orbit of $H$ on $Y$.  To every complete
geometric parameter $\xi' \in \Xi(Y,H)$ is associated an irreducible
perverse sheaf $P(\xi')$ (cf.~(7.10)(d)).  The conormal bundle $T^*_S(Y)$ has
a non-negative integral multiplicity $\chi_S^{mic}(\xi')$ in the
characteristic cycle ${\rom Ch}\,P(\xi')$ (Proposition 19.12).  We define the
{\it micro-packet of geometric parameters attached to $S$} to be the set of
$\xi$ for which this multiplicity is non-zero:
$$\Xi(Y,H)^{mic}_S = \{\,\xi\mid \chi_S^{mic}(\xi')\ne 0\,\}.$$
\medskip

{\bf Lemma 19.14.}  {\it In the setting of Definition 19.13, suppose
$\xi'=(S',{\cal V}')$ is a geometric parameter (Definition 7.1).  
\smallskip
\item{a)} If $\xi' \in \Xi(Y,H)^{mic}_S$, then $S\subset \overline{S'}$.
\item{b)} if $S' = S$, then $\xi' \in \Xi(Y,H)^{mic}_S$.  In this case
$\chi_S^{mic}(\xi')$ is equal to the rank of the local system ${\cal V}'$; that
is, to the dimension of the corresponding representation of $A_S^{loc}$
(Definition 7.1).
\medskip}

{\it Proof.}  Part (a) is clear from (19.12)(d), as is the first assertion in (b). 
The rest of (b) (which is ``well-known'') can be deduced from the
construction of the ${\cal D}_Y$-module corresponding to the perverse sheaf
$P(\xi')$ ([BorelD]; some of the main points are contained in the proof of
Kashiwara's theorem VI.7.11 in [BorelD]).  Q.E.D.

\medskip
{\bf Definition 19.15.}  In the setting of Theorem 10.4, fix an equivalence
class $\phi \in \Phi^z(G/{\Bbb R})$ of Langlands parameters (Definition 5.3),
and write $S = S_\phi$ for the corresponding orbit of ${^\vee G}$ on
$X({^\vee G^\Gamma})$ (Definition 7.6).  Then we can define the {\it
micro-packet of geometric parameters attached to $\phi$} as
$$\Xi^z(G/{\Bbb R})_\phi^{mic} = \Xi(X({^\vee G^\Gamma}),{^\vee
G}^{alg})^{mic}_{S_\phi}$$
(notation as in Definition 19.13).  The {\it micro-packet of $\phi$} is the
corresponding set of irreducible representations of strong real forms of
$G$:
$$\Pi^z(G/{\Bbb R})^{mic}_\phi = \{\,\pi(\xi')\mid \xi'\in \Xi^z(G/{\Bbb
R})_\phi^{mic}\,\}.$$
\medskip

{\bf Corollary 19.16.}  {\it In the setting of Theorem 10.4, fix an equivalence
class $\phi \in \Phi^z(G/{\Bbb R})$ of Langlands parameters.  Then the
micro-packet $\Pi^z(G/{\Bbb R})^{mic}_\phi$ contains the L-packet
$\Pi^z(G/{\Bbb R})_\phi$.  If $\pi(\xi')$ is any irreducible representation
occurring in the micro-packet, then the corresponding orbit $S'$ on $X({^\vee
G^\Gamma})$ contains $S_\phi$ in its closure.  There is a strongly stable
formal virtual character
$$\eta_\phi^{mic} = \sum_{\pi'\in \Pi^z(G/{\Bbb R})^{mic}_\phi}
e(\pi')(-1)^{d(\pi')-d(\phi)}\chi_\phi^{mic}(\pi')\pi'.$$
Here $e(\pi') = \pm 1$ is Kottwitz' sign attached to the real form of which
$\pi'$ is a representation; $d(\pi')$ is the dimension of the orbit $S_{\pi'}$
on $X({^\vee G^\Gamma})$;  and the positive integer $\chi_\phi^{mic}(\pi')$
is the multiplicity of the conormal bundle to $S_\phi$ in the characteristic
variety of the irreducible perverse sheaf corresponding to $\pi'$.  This is
therefore a sum over the full L-packet $\Pi^z(G/{\Bbb R})^{mic}_\phi$,
together with ``correction terms'' taken from L-packets corresponding to
larger orbits on $X({^\vee G^\Gamma})$.

The set $\{\,\eta^{mic}_\phi\,\}$ (as $\phi$ varies over $\Phi^z(G/{\Bbb
R})$) is a basis of the lattice of strongly stable formal virtual
representations.}

\medskip

This is just a reformulation of Corollary 1.32 (taking into account the
information in Lemma 19.14).  The point of the result is that under favorable
circumstances the characteristic variety of an equivariant ${\cal
D}$-module is relatively small; at best it may just be the conormal bundle
of a single orbit.  The sum in Corollary 19.16 therefore does not extend over
too many representations.  By contrast, the sum in Lemma 18.15 (when
expressed in terms of irreducible representations) tends to include most of
the representations for which the corresponding irreducible perverse sheaf
has $S_\phi$ in its support.  

\medskip
{\bf Example 19.17.}  Suppose that $G$ is adjoint.  In this case strong real
forms are the same as real forms, and ${^\vee G}$ is simply connected (so
there are no coverings to consider).  Take $({^\vee G^\Gamma},{\cal S})$ to
be an L-group for $G$ in the sense of Definition 4.14.  Let ${\cal O}\subset
{^\vee {\frak g}}$ be the orbit of half sums of positive coroots; this
corresponds to representations of $G$ of infinitesimal character equal to
that of the trivial representation (Lemma 15.4).  The conjugacy class ${\cal
C}({\cal O})$ of (6.10) consists of the single (central) element $z(\rho)$. 
Since $\rho$ is regular and integral, each canonical flat $\Lambda \subset
{\cal F}({\cal O})$ is preserved by a unique Borel subgroup ${^dB}(\Lambda)$
(notation (6.6)).  We may therefore identify the geometric parameter space
$X({\cal O},{^\vee G^\Gamma})$ with the set of pairs $(y,{^dB})$, with $y\in
{^\vee G^\Gamma}-{^\vee G}$, $y^2 = z(\rho)$, and $^dB$ a Borel subgroup of
${^\vee G}$ (Definition 6.9). 

Now by definition the L-group structure provides a distinguished orbit $S_0
= {\cal S}$ of ${^\vee G}$ on  $X({\cal O},{^\vee G^\Gamma})$.  We write
$\phi_0$ for the corresponding Langlands parameter.   Fix a point
$(y,{^dB})\in S_0$.  Since ${^\vee G}$ is simply connected, the fixed point
group $K(y)$ of the involution $\theta_y$ is connected.  Since $\theta_y$
preserves $^dB$, the intersection ${^dB}\cap K(y)$ is a Borel subgroup of
$K(y)$, so it is connected as well.  Evidently ${^dB}\cap K(y)$ is just the
isotropy group of the action of ${^\vee G}$; so it follows that the canonical
component group $A_{\phi_0}^{loc,alg}$ is trivial.  The L-packet
$\Pi_{\phi_0}$ therefore consists of a single representation.  Inspecting the
constructions in the proof of Theorem 10.4, we find that $M(\phi_0)$ is a
spherical principal series representation of the quasisplit real form of $G$
(in our fixed inner class), and that $\pi(\phi_0)$ is the trivial
representation of this group.  The strongly stable standard formal virtual
representation $\eta^{loc}_{\phi_0}$ of Definition 18.13 is $M(\phi)$ alone. 
This representation has a great many irreducible composition factors,
however, occurring with relatively high multiplicity; so as a ``stabilization''
of the trivial representation $\pi(\phi_0)$, $M(\phi_0)$ is not very useful. 
(Another way to say this is that the relationship between the characters of
$\pi(\phi_0)$ and $M(\phi_0)$ is quite weak.)

We consider therefore the virtual representation $\eta_{\phi_0}^{mic}$.  To
calculate it, we need to know the multiplicity of the conormal bundle
$T^*_{S_0}(X({\cal O},{^\vee G}))$ in the characteristic variety of any
irreducible perverse sheaf.  We will consider some techniques for making
such calculations in sections 21 and 27 (see for example Theorem 27.18). 
For the moment we simply state the result: the micro-packet of $\phi_0$
consists of one trivial representation of each real form of $G$ in our inner
class.  The multiplicities $\chi_{S_0}^{mic}$ are equal to 1, and the
differences in orbit dimensions have the same parity as Kottwitz' invariant. 
The formula in Corollary 19.16 therefore reduces to
$$\eta_{\phi_0}^{mic} = \sum_{\rom real\ forms\ \delta} (\hbox{ trivial
representation of $G({\Bbb R},\delta)$ }).$$
This is a charming (if rather obvious) stabilization of the trivial
representation.  Notice that all real forms (in the inner class) appear, even
though $\phi_0$ is relevant only for the quasisplit one.  
\medskip

%\end
%\input mssymb.tex
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\vskip 3pc
\centerline{\bf 20.  Characteristic cycles and Harish-Chandra modules.}
\vskip 1pc

Recall that Theorem 8.5 related the category of equivariant perverse
sheaves (or, equivalently, equivariant ${\cal D}$-modules) on the geometric
parameter space to certain categories of Harish-Chandra modules.  Our goal
in this section is to see what this relationship has to say about the
characteristic cycles.  Theorem 8.5 implies that the characteristic cycles
must somehow be encoded in the Harish-Chandra modules.  We are not able
to break that code, but we get some useful information about it (Theorem
20.18).  This will later be the key to relating our definition of Arthur's
unipotent representations to the original one of Barbasch and Vogan. 
Roughly speaking, Arthur's representations will be characterized on the
E-group side by the occurrence of certain ``regular'' components in a
characteristic cycle (Definition 20.7).  We therefore seek to understand that
occurrence on the level of Harish-Chandra modules.

We begin with analogues of Propositions 7.14 and 7.15.

\medskip
{\bf Proposition 20.1.} {\it Suppose $X$ and $Y$ are smooth algebraic
varieties, and
$$f:X\rightarrow Y$$
is a smooth surjective morphism having connected geometric fibers of
dimension $d$.  Suppose ${\cal M}$ is a ${\cal D}_Y$-module.  
\item{a)} There is a natural inclusion
$$f^*(T^*Y) \hookrightarrow T^*X$$
identifying the pullback of the cotangent bundle of $Y$ with a subbundle of
$T^*X$.  This induces a bundle map
$$\tilde{f}:f^*(T^*Y) \rightarrow T^*Y$$
which  is again a smooth surjective morphism having connected geometric
fibers of dimension $d$.
\item{b)} The ${\cal O}_X$-module $f^*{\cal M}$ carries a natural ${\cal
D}_X$-module structure.
\item{c)}  The functor $f^*$ is a fully faithful exact functor from coherent
${\cal D}_Y$ modules to coherent ${\cal D}_X$ modules.
\item{d)} Suppose ${\cal M}(p)$ is a good filtration of ${\cal M}$. Then
$$(f^*{\cal M})(p) = f^*({\cal M}(p))$$
is a good filtration of $f^*{\cal M}$.  The associated graded module ${\rom
gr}(f^*{\cal M})$ is defined on the subvariety $f^*(T^*Y)$ of $T^*X$, where it
may be computed by
$${\rom gr}(f^*{\cal M}) \simeq \tilde{f}^*{\rom gr}({\cal M}).$$
The characteristic cycle of $f^*{\cal M}$ is therefore
$${\rom Ch}(f^*{\cal M}) =  \tilde{f}^{-1}({\rom Ch}({\cal M})).$$
\item{e)} Suppose we are in the setting of Proposition 7.15 (that is, that a
pro-algebraic group $H$ acts on everything).  Suppose ${\cal M}$ is an
$H$-equivariant coherent ${\cal D}_Y$-module, and $S$ is an $H$-orbit on
$Y$.  Define $f^*S$ to be the unique dense $H$-orbit in $f^{-1}S$ (cf.~(7.16)). 
Then the multiplicities of Proposition 19.12 are given by
$$\chi_{f^*S}^{mic}(f^*{\cal M}) = \chi_S^{mic}({\cal M}).$$
\medskip}

{\it Proof.}  Part (a) is formal.  Part (b) may be found in [BorelD], VI.4.1. 
Part (c) is [BorelD], Proposition VI.4.8.  For (d), we know that $f^*$ is an
exact functor on ${\cal O}_Y$-modules (since $f$ is smooth), and that it
preserves coherence; the only thing to check is the degree condition in
(19.4)(b).  This is immediate from the definition of the ${\cal D}_X$-module
structure.  The rest of (d) is a straightforward consequence.  (Notice that
the inverse image under a smooth map with connected fibers of an
irreducible variety is irreducible, so the last assertion makes sense.)  The
formula of (e) just interprets the last relation in (d).  Q.E.D.

For an analogue of Proposition 7.14, we will need to recall (in that setting)
that the induction functor $G\times_H$ is an equivalence of categories from
$H$-equivariant (quasi)-coherent ${\cal O}_Y$-modules to $G$-equivariant
(quasi)-coherent ${\cal O}_X$-modules.  We have already used this for local
systems, but we will need it now more generally.  (The definitions of the
functors may be found for example in the appendix to [Dix].)

\medskip
{\bf Proposition 20.2.}  {\it Suppose we are in the setting of Proposition
7.14.  Regard ${\frak g}\times TY$ as a vector bundle over $Y$, containing
the bundle ${\frak h}\times Y$ of (19.1).  We make $H$ act on the bundle by
the adjoint action on ${\frak g}$.  Write $i:{\frak h}\times Y \rightarrow
{\frak g}\times Y$ for the inclusion, and consider the bundle map
$$i\times a:{\frak h}\times Y \rightarrow {\frak g}\times TY.$$
Write $Q$ for the quotient bundle: the fiber at $y$ is
$$Q_y = ({\frak g}\times T_yY)/\{(X,a_y(X)\mid X\in {\frak h}\}.$$
\smallskip
\item{a)} The tangent bundle $TX$ of $X= G\times_H Y$ is naturally
isomorphic to the bundle on $X$ induced by $Q$:
$$TX \simeq G\times_H Q.$$
\item{b)} The action mapping $a_x:{\frak g}\rightarrow T_xX$ (cf.~(19.1))
may be computed as follows.  Fix a representative $(g,y)$ for the point $x$
of $G\times_H Y$, and an element $Z\in {\frak g}$.  Then
$$a_x(Z) = \hbox{ class of } (g,({\rom Ad}(g^{-1})Z,(y,0))).$$
Here $(y,0)$ is the zero element of $T_yY$, so the term paired with $g$ on
the right side represents a class in $Q_y$.
\item{c)} The conormal bundle to the $G$ action on $G\times_H Y$ is
naturally induced by the conormal bundle to the $H$ action on $Y$:
$$T^*_G(G\times_H Y) \simeq G\times_H T^*_H(Y).$$ 
\smallskip
Suppose ${\cal M}$ is an $H$-equivariant coherent ${\cal D}_Y$-module
admitting an $H$-invariant good filtration ${\cal M}(p)$. Let ${\cal N}$ be
the corresponding $G$-equivariant ${\cal D}_X$-module (Proposition
7.14(d)).
\smallskip
\item{d)} As a $G$-equivariant ${\cal O}_X$-module, ${\cal N}$ is
isomorphic to $G\times_H {\cal M}$.
\item{e)} The filtration
$${\cal N}(p) = G\times_H {\cal M}(p)$$
is a $G$-equivariant good filtration of ${\cal N}$.   In particular,
$${\rom gr}({\cal N}) \simeq G\times_H {\rom gr}({\cal M}).$$
\item{f)} The characteristic cycle of ${\cal N}$ may be identified as a cycle
in $T^*_GX$ (see (c) above) as
$${\rom Ch}({\cal N}) = G\times_H {\rom Ch}({\cal M}).$$
In particular, the multiplicities are given by
$$\chi_{G\times_H S}^{mic}({\cal N}) = \chi_S^{mic}({\cal M}).$$

\medskip}
{\it Proof.}  Parts (a) and (b) are elementary.  Part (c) is a consequence of
(b).  Parts (d) and (e) of course involve the explicit equivalence of
categories in Proposition 7.14(d), which we have not written down.  The
idea, as in Proposition 7.14, is to apply Proposition 20.1 to the two smooth
morphisms $f_G:G\times Y \rightarrow Y$ (projection on the second factor)
and $f_H:G \times Y \rightarrow G\times_H Y$ (dividing by the $H$ action). 
Then $f^*_H{\cal M} \simeq f^*_G{\cal N}$; this is more or less the
definition of ${\cal N}$.  Part (f) follows from (e).  Q.E.D.

We work now in the setting of Theorem 8.3.  Fix a complex connected
reductive algebraic group $G$, and a complete homogeneous space $Y$ for
$G$.  If $y$ is any point of $Y$, then the isotropy group $G_y$ is a parabolic
subgroup of $G$; in this way $Y$ may be identified with a conjugacy class
${\cal P}$ of parabolic subgroups in $G$ (or parabolic subalgebras in ${\frak
g}$).  As in (19.1) or (8.1) there is a bundle map
$$a:{\frak g} \times Y \twoheadrightarrow TY. \eqno(20.3)(a)$$
It is surjective because $Y$ is a homogeneous space.  The transpose of $a$
is a bundle map
$$a^*:T^*Y \hookrightarrow {\frak g}^* \times Y. \eqno(20.3)(b)$$
It is a closed immersion (and therefore projective) since $a$ is surjective. 
Composing $a^*$ with projection on the first factor gives the {\it moment
map}
$$\mu:T^*Y \rightarrow {\frak g}^*\eqno(20.3)(c)$$
([BorBI], 2.3).  The first projection is a projective morphism because $Y$ is
a projective variety; so $\mu$ is also projective.  Clearly
$$\mu(T^*_y(Y)) = ({\frak g}/{\frak g}_y)^*, \eqno(20.3)(d)$$
so that the image of $\mu$ consists of all linear functionals annihilating
some parabolic subalgebra in the class ${\cal P}$ ([BorBI], Proposition 2.4). 
We write 
$${\cal N}^*_Y = \mu(T^*Y), \eqno(20.3)(e)$$
an irreducible closed cone in ${\frak g}^*$.

\medskip
{\bf Proposition 20.4} (Richardson --- see [Rich] or [BorBI], 2.6). {\it In the
setting of (20.3), the cone ${\cal N}^*_Y$ is contained in the cone of
nilpotent elements of ${\frak g}^*$.  It is the union of finitely many orbits
under $G$.  In particular there is a unique open orbit
$${\cal Z}_Y \subset {\cal N}^*_Y,$$
the Richardson orbit attached to $Y$.  The dimension of ${\cal Z}_Y$ is equal
to the dimension of $T^*Y$; so $\mu$ is generically finite.  The preimage
$\mu^{-1}({\cal Z}_Y)$ is an open orbit of $G$ on $T^*Y$.

The degree of $\mu$ may be computed as follows.  Fix $z\in {\cal Z}_Y$, and
a point $\zeta \in T^*_y(Y)$ with $\mu(\zeta) = z$.  Write $G_z$ for the
isotropy group of $z$, $G_y$ for the isotropy group of $y$ (a parabolic
subgroup in ${\cal P}$), and $G_\zeta$ for the isotropy group of $\zeta$. 
Then
$$G_\zeta = G_y \cap G_z, \quad G_z \supset G_\zeta \supset (G_z)_0.$$
The degree of $\mu$ is equal to the index of $G_\zeta$ in $G_z$. 
Equivalently, it is the number of distinct parabolic subalgebras in ${\cal P}$
annihilated by the linear functional $z\in {\frak g}^*$.}
\medskip

We should say a word about the meaning of ``nilpotent elements'' of ${\frak
g}^*$.  The Lie algebra ${\frak g}$ is reductive, and so may be identified
with its dual using an invariant bilinear form.  Using this identification, we
may therefore speak of nilpotent or semisimple elements in ${\frak g}^*$.  A
more intrinsic discussion may be found in section 5 of [Assoc].

Finally, suppose $M$ is a finitely generated $U({\frak g})$-module.  By an
argument parallel to (but technically much simpler than) Definition 19.9, we
can define the {\it associated variety of $M$},
$${\cal V}(M) \subset {\frak g}^* \eqno(20.5)$$
(see [Assoc] or [BorBI], 4.1.)  Again the irreducible components of the
associated variety have well-defined multiplicities, so that we can define
the {\it associated cycle} ${\rom Ch}(M)$ ([Assoc], Definition 2.4 or [BorBIII],
1.2).  

\medskip
{\bf Theorem 20.6} (Borho-Brylinski --- see [BorBIII], Theorem 1.9).  {\it In
the setting (20.3), suppose $M$ is a finitely generated module for $U({\frak
g})/I_Y$ (cf.~(8.1)(c)) and ${\cal M}$ is the corresponding ${\cal
D}_Y$-module (Theorem 8.3).  Then
$${\cal V}(M) = \mu({\rom SS}({\cal M})).$$}
\medskip
 
Keeping track of multiplicities is a more delicate task, because the fibers
of $\mu$ can be quite complicated. 

\medskip
{\bf Theorem 20.7} (Borho-Brylinski).  {\it In the setting of Theorem 20.6,
fix good filtrations of ${\cal M}$ and $M$.  Consider the higher direct images
$$R^i\mu_*({\rom gr}\,{\cal M}) = N^i,$$
which we regard as finitely generated graded $S({\frak g})$-modules.  Then
$N^i$ is supported on
$$\{\,z\in {\cal N}^*_Y\mid \dim(\mu^{-1}(z) \cap {\rom SS}\,{\cal M}) \ge
i\,\}\subset {\cal V}(M).$$
In the Grothendieck group of coherent modules supported on ${\cal V}(M)$,
we have
$${\rom gr}\,M = \sum_i (-1)^i N^i.$$}
\medskip

{\it Proof.}  That $N^i$ is coherent follows from the fact that $\mu$ is
projective ([Hartshorne], Theorem III.8.8).  The statement about support
follows from [Hartshorne], Theorem III.11.1.  A version of the main assertion
is proved in [BorBIII], Corollary A.6 after twisting by a sufficiently positive
line bundle on $Y$.  (This has the effect of killing the higher direct images.) 
Their argument proves this result in general.  Q.E.D.

There are several ways to describe the modules $N^i$ of Theorem 20.6.  One
is
$$N^i \simeq H^i(T^*Y,{\rom gr}\,{\cal M}) \eqno(20.8)(a)$$
([Hartshorne, Proposition III.8.5).  Here we are using (19.3) to regard ${\rom
gr}\,{\cal M}$ as a sheaf on $T^*Y$.  Elements of $S({\frak g})$ define global
functions on $T^*Y$, and so act on this cohomology.  Another is
$$N^i \simeq H^i(Y,{\rom gr}\,{\cal M}) \eqno(20.8)(b)$$
([Hartshorne, Exercise III.8.2).  Here we are regarding ${\rom gr}\,{\cal M}$ as
a quasicoherent sheaf on $Y$ that happens to be also a module for $p_*{\cal
O}_{T^*Y}$.  The extra structure is needed to make $N^i$ an $S({\frak g})$
module, but not to compute it as a vector space or as a module for some
group.

\medskip
{\bf Definition 20.9.}  In the setting of Proposition 20.4, we say that an
element $\zeta\in T^*Y$ is {\it regular} if it belongs to the inverse image of
the $G$-orbit ${\cal Z}_Y$.  Similarly, an element $z \in {\cal N}^*_Y$ is {\it
$Y$-regular} (or simply {\it regular}) if it belongs to ${\cal Z}_Y$.  An
irreducible subvariety $W$ of $T^*Y$ or ${\cal N}^*_Y$ is {\it regular} if it
contains a regular element; equivalently, if its subset of regular elements
is open and dense.
\medskip

{\bf Corollary 20.10.}  {\it In the setting of Proposition 20.4, suppose that
$W \subset T^*Y$ is a regular irreducible subvariety (Definition 20.9).  Then
$\mu(W)$ is a $Y$-regular irreducible subvariety of ${\cal N}^*_Y$, and
every $Y$-regular irreducible subvariety arises in this way.  If in addition
$\mu$ has degree 1, then this correspondence of regular irreducible
subvarieties is bijective.

In the setting of Theorem 20.7, assume that $\mu$ has degree 1.  Then
$\mu$ defines a multiplicity-preserving bijection from the regular
irreducible components of ${\rom SS}({\cal M})$ onto the regular irreducible
components of ${\cal V}(M)$.}
\medskip

{\bf Definition 20.11.}  A {\it symmetric pair} consists of a complex
reductive Lie algebra ${\frak g}$, an involutive automorphism $\theta$ of
${\frak g}$, and a reductive pro-algebraic group $K$, with the following
properties:
\smallskip
\item{(a)}  $({\frak g},K)$ is a compatible pair (Definition 8.2);
\item{(b)}  ${\frak k} = {\frak g}^\theta$; and
\item{(c)}  $\theta$ commutes with ${\rom Ad}(K)$.
\smallskip
We will sometimes write ${\frak s}$ for the $-1$-eigenspace of $\theta$ on
${\frak g}$, so that
$${\frak g} = {\frak k} + {\frak s},$$
the {\it Cartan decomposition}.  
\medskip

In the setting of (20.3), we assume in addition now that
$$\hbox{$({\frak g},K)$ is a symmetric pair, and $K$ acts compatibly on $Y$}
\eqno(20.12)(a)$$
Since $Y$ may be identified with a conjugacy class ${\cal P}$ of parabolic
subalgebras, the action of $K$ may be identified with
$$k\cdot {\frak p} = {\rom Ad}(k)({\frak p}). \eqno(20.12)(b)$$
(The only reason we need to assume in (a) that the action exists is that if
$K$ is disconnected, it could fail to preserve ${\cal P}$.)  Recall from (19.1)
the conormal bundle $T^*_K(Y)$ to the $K$ action on $Y$ (the union of the
conormal bundles of the orbits).  The fiber of this bundle at the point $y$
may be identified with
$$T^*_{K,y} = ({\frak g}/({\frak g}_y + {\frak k}))^*. \eqno(20.12)(c)$$
Set
$${\cal N}^*_{Y,K} = \mu(T^*_K(Y)) = \bigcup_{{\frak p}\in {\cal P}} ({\frak
g}/({\frak p} + {\frak k}))^*; \eqno(20.12)(d)$$
the second description is immediate from (20.12)(c).  This is a closed cone
in ${\cal N}^*_Y$.  
\medskip
{\bf Lemma 20.13.}  {\it In the setting (20.12), suppose $M \in {\cal
F}({\frak g},K,I_Y)$ (Definition 8.2); that is, that $M$ is a finite-length
$({\frak g},K)$-module annihilated by the kernel of the operator
representation on $Y$.  Let ${\cal M}$ be the associated $K$-equivariant
${\cal D}_Y$-module (Theorem 8.3)(c)), so that $M$ may be identified with
the space of global sections of ${\cal M}$.
\smallskip
\item{a)}  The associated variety ${\cal V}(M)$ is a closed $K$-invariant
cone in ${\cal N}^*_Y$.
\item{b)}  The module ${\cal M}$ admits a $K$-invariant good filtration.
\medskip}

{\it Proof.}  Part (a) follows from Theorem 20.6, Proposition 19.12(c), and
(20.12)(d).  For (b), choose a finite-dimensional $K$-invariant generating
subspace $M(0)$ of $M \simeq \Gamma{\cal M}$, and let ${\cal M}(0)$ be the
(coherent) ${\cal O}_Y$-submodule of ${\cal M}$ generated by $M(0)$.  Then
${\cal M}(0)$ generates ${\cal M}$ as a ${\cal D}_Y$ module (Theorem 8.3). 
It follows that
$${\cal M}(p) = {\cal D}_Y(p){\cal M}(0)$$
is a $K$-invariant good filtration of ${\cal M}$.  Q.E.D.

\medskip
{\bf Lemma 20.14} (Kostant-Rallis; see [KR] or [Assoc], Corollary 5.20). {\it
In the setting of (20.12), the group $K$ has a finite number of orbits on
${\cal N}^*_{Y,K}$.  If $z \in {\cal N}^*_{Y,K}$, then
$$\dim K\cdot z = 1/2 \dim G\cdot z.$$
In particular, $z$ is regular (Definition 20.9) if and only if $\dim K\cdot z =
\dim Y$.}
\medskip

{\bf Theorem 20.15} (Borho-Brylinski)  {\it In the setting (20.12), suppose
$M \in {\cal F}({\frak g},K,I_Y)$ (Definition 8.2); that is, that $M$ is a
finite-length $({\frak g},K)$-module annihilated by the kernel of the
operator representation on $Y$.  Let ${\cal M}$ be the associated
$K$-equivariant ${\cal D}_Y$-module (Theorem 8.3)(c)), so that $M$ may be
identified with the space of global sections of ${\cal M}$.  Then the
following four conditions are equivalent:
\smallskip
\item{1)} the Gelfand-Kirillov dimension ${\rom Dim}\,M$ is equal to the
dimension of $Y$;
\item{2)} the annihilator ${\rom Ann}\,M$ is equal to $I_Y$;
\item{3)} the associated variety ${\cal V}(M)$ contains $Y$-regular
elements (Definition 20.9); and
\item{4)} the singular support ${\rom SS}\,{\cal M}$ contains regular
elements (Definition 20.9).
\medskip}

{\it Proof.}  We know that the Gelfand-Kirillov dimension of $M$ is half that
of $U({\frak g})/{\rom Ann}\,M$ ([GKDim], Theorem 1.1); so condition (1) is
equivalent to
$${\rom Dim}\, U({\frak g})/{\rom Ann}\,M = 2 \dim Y.$$
It is well-known that the right side is equal to the Gelfand-Kirillov
dimension of $U({\frak g})/I_Y$.  (This follows from Theorem 20.6, for
example.)  On the other hand, $I_Y$ is a prime ideal (since it is the kernel of
a map into a ring without zero divisors), and ${\rom Ann}\,M \supset I_Y$.  It
follows that 
$${\rom Dim}\, U({\frak g})/{\rom Ann}\,M \ge {\rom Dim}\, U({\frak g})/I_Y,$$
with equality only if ${\rom Ann}\,M = I_Y$.  This gives the equivalence of (1)
and (2).  Since the Gelfand-Kirillov dimension of $M$ is the dimension of the
largest component of ${\cal V}(M)$, the equivalence of (1) and (3) follows
from Lemma 20.14.  That of (3) and (4) follows from Corollary 20.10.  Q.E.D.

In light of this result and Proposition 19.12, we now seek to understand the
$K$-orbits $S$ on $Y$ with the property that $T^*_S Y$ is regular.

\medskip
{\bf Lemma 20.16.}  {\it In the setting (20.12), suppose $S$ is an orbit of
$K$ on $Y$.  Then $\mu(\overline{T^*_S Y})$ is the closure of a single orbit
${\cal Z}_S$ of $K$ on ${\cal N}^*_{Y,K}$.  We have
$$\dim {\cal Z}_S = \dim Y -\hbox{\rom fiber dimension of $\mu|_{T^*_S
Y}$}.$$
\medskip}

{\it Proof.}  Since $T^*_S Y$ is a vector bundle over a homogeneous space for
$K$, it is ``$K$-irreducible;'' that is, $K$ permutes its irreducible
components transitively.  Consequently $\mu(T^*_S Y)$ is also
$K$-irreducible.  By Lemma 20.14, it must be contained in the closure of a
single orbit.  Since the dimension of $T^*_S Y$ is equal to the dimension of
$Y$, the last statement follows.  Q.E.D.

\medskip
{\bf Definition 20.17.}  In the setting (20.12), an orbit $S$ of $K$ on $Y$ is
called {\it regular} if any of the following equivalent conditions is
satisfied:
\smallskip
\item{(a)} $T^*_S Y$ is regular;
\item{(b)} $\mu(T^*_S Y)$ is $Y$-regular;
\item{(c)} the restriction of $\mu$ to $T^*_S Y$ has finite degree; or
\item{(d)} $\dim \mu(T^*_S Y) = \dim Y$.
\medskip

The equivalence of the conditions follows from Corollary 20.10, Lemma
20.14, and Lemma 20.16.

\medskip
{\bf Theorem 20.18.}  {\it In the setting (20.12), suppose that the moment
map $\mu$ has degree 1 on $T^* Y$.  Then the correspondence of Lemma
20.16 defines a bijection from the set of regular orbits $S$ of $K$ on $Y$
onto the set of regular orbits ${\cal Z}_S$ of $K$ on ${\cal N}^*_{Y,K}$.  This
bijection is multiplicity-preserving in the following sense.  Suppose the
$K$-equivariant ${\cal D}_Y$-module ${\cal M}$ corresponds to the
finite-length $({\frak g},K)$-module $M$ (Theorem 8.3).  Then for every
regular $K$-orbit $S$ on $Y$, the multiplicity $\chi_S^{mic}({\cal M})$
(Proposition 19.12) is equal to the multiplicity of $\overline{{\cal Z}_S}$ in
the associated cycle ${\rom Ch}(M)$.}
\medskip

This is a special case of Corollary 20.10.  It is not very difficult to analyze
the case when $\mu$ has degree greater than one.  For example, if we fix a
regular orbit ${\cal Z}_0$ of $K$ on ${\cal N}^*_{Y,K}$, then
$$\sum_{{\cal Z}_S = {\cal Z}_0} \hbox{\rom  degree of $\mu$ on $T^*_S Y$}
= \hbox{\rom   degree of $\mu$ on $T^* Y$}. \eqno(20.19)$$

We conclude this section with some examples illustrating the notion of
regular.
\medskip

{\bf Example 20.20.}  Suppose that $Y$ is the variety of Borel subgroups of
$G$.  Then ${\cal N}^*_Y$ is the nilpotent cone in ${\frak g}^*$, and ${\cal
Z}_Y$ is the open orbit of principal nilpotent elements.  Suppose $S$ is a
regular orbit.  We claim that $S$ consists of $\theta$-stable Borel
subalgebras having no compact simple roots.  (These are the Borel
subalgebras of large type for $\theta$ in the sense of Definition 4.9.)  To
see this, suppose $z \in \mu(T^*_{S,y} Y)$ is a regular element.  If $y$
corresponds to the Borel subalgebra ${\frak b}$, this means (by (20.12)(c))
that $z \in ({\frak g}/({\frak b} + {\frak k}))^*$; that is, that $z$ annihilates
${\frak b}$.  Since $\theta{\frak b} \subset {\frak b} + {\frak k}$, $z$ also
annihilates $\theta{\frak b}$.  The linear span of any two distinct Borel
subalgebras must contain a larger parabolic subalgebra; so if ${\frak b} \ne
\theta{\frak b}$, then there is a parabolic ${\frak p}$ properly containing
${\frak b}$ and annihilated by $z$.  It follows from Proposition 20.4 that the
$G$ orbit of $z$ has dimension at most twice the dimension of $G/P$; so $z$
is not principal, a contradiction.  Consequently ${\frak b} = \theta{\frak b}$. 
If some simple root $\alpha$ is compact, we find that $z$ must also
annihilate $s_\alpha{\frak b}$, and reach a contradiction again.  Conversely,
if $(\theta,{\frak b})$ is large, then it is easy to find a principal nilpotent
element $z$ annihilating ${\frak b}$ with $\theta z = -z$ ([AV2], Proposition
A.7).  Then $z$ annihilates ${\frak k}$, and so is a regular element of
$\mu(T^*_{S,y} Y)$.  We have therefore completely characterized the regular
orbits in this case.  (The moment map has degree one, since the stabilizer of
a principal nilpotent element is contained in a Borel subgroup.)
\smallskip

{\bf Example 20.21.}  Suppose $G=Sp(4)$, and $Y = {\Bbb P}^3$, the variety of
(isotropic) lines in ${\Bbb C}^4$; the Levi subgroup for the corresponding
parabolic is $GL(1)\times Sp(2)$.  It turns out that ${\cal Z}_Y$ consists of
all short root vectors in the Lie algebra, and that the moment map has
degree 2.  We take $K=GL(2)$, acting on ${\Bbb C}^4={\Bbb C}^2 + {\Bbb C}^2$
by block diagonal matrices with blocks $g$ and $(g^t)^{-1}$.  The two copies
of ${\Bbb C}^2$ are in natural duality by the symplectic form.  We therefore
think of ${\Bbb C}^4$ as consisting of pairs $(v,\eta)$, with $v\in {\Bbb
C}^2$ and $\eta$ a linear functional on ${\Bbb C}^2$.  Obviously there are
four orbits of $K$ on $Y$:
$$S^+_1 = \hbox{ lines with $v=0$},\qquad S^-_1 = \hbox{ lines with
$\eta=0$} \eqno(20.21)(a)$$
$$S_2 = \hbox{ lines with $\eta(v) = 0$ not in $S^\pm_1$},\qquad S_3 =
\hbox{ lines with $\eta(v) \ne 0$}. \eqno(20.21)(b)$$
The subscripts indicate the dimensions of the orbits.  The orbit $S_3$ is
obviously not regular, since its conormal bundle is zero.  It turns out that
the other three orbits are all regular, and that the corresponding orbits on
${\cal N}^*_{K,Y}$ are precisely the three regular orbits there.  Consequently
(cf.~(20.19)) the moment map has degree 2 on the conormal bundle of each
regular orbit.  
\smallskip

{\bf Example 20.22.}  Take $G=Sp(2n)$ and $Y$ the variety of Lagrangian
subspaces of ${\Bbb C}^{2n}$; the corresponding parabolic has Levi factor
$GL(n)$, and the moment map has degree 1.  We take $K= GL(n)$, acting on
${\Bbb C}^n + ({\Bbb C}^n)^*$ by diagonal blocks $g$ and $(g^t)^{-1}$.  Using a
little linear algebra, we can identify $Y$ with the set of triples
$(V_0,V_1,Q)$.  Here $V_0\subset V_1\subset {\Bbb C}^n$ are subspaces,
and $Q$ is a non-degenerate quadratic form on $V_1/V_0$.  We will not
recall in detail why this is so, but here are the definitions.  If $L$ is a
Lagrangian subspace, define
$$V_0 = L\cap {\Bbb C}^n, \qquad V_1 = \{\,v\in {\Bbb C}^n\mid
(v,\lambda)\in L \hbox{ for some }\lambda \in ({\Bbb C}^n)^*\,\}.
\eqno(20.22)(a)$$
Suppose then that $(v,\lambda)$ and $(v',\lambda')$ belong to $L$.  The
quadratic form is defined by
$$Q(v,v') = \lambda(v') = \lambda'(v); \eqno(20.22)(b)$$
these are equal because their difference is the symplectic pairing of
$(v,\lambda)$ and $(v',\lambda')$.
>From this description of $Y$, it is clear that the orbits of $K$ are
parametrized by integers $p$ and $q$ with $p+q\le n$: we can take $p = \dim
V_0$, $q = \dim({\Bbb C}^n/V_1).$  The orbit $S_{p,q}$ is open if and only if
$p=q=0$, and closed if and only if $p+q=n$.  The $n+1$ closed orbits are also
precisely the regular orbits; they correspond by Theorem 20.18 to the $n+1$
regular orbits of $K$ on ${\cal N}^*_{Y,K}$.
\medskip
%\end

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\vskip 3pc
\centerline{\bf 21. The classification theorem and Harish-Chandra modules
for the dual group.}
\vskip 1pc

We assemble here the results we have established linking the
representation theory of real forms of $G$ to Harish-Chandra modules for
certain subgroups of ${^\vee G}$.  This is for the most part a reformulation
of the results of [IC4].  We work therefore in the setting of Proposition 6.24. 
Specifically, we fix always a weak E-group ${^\vee G^\Gamma}$, and a
semisimple orbit
$${\cal O} \subset {^\vee {\frak g}} \eqno(21.1)(a)$$
We fix a canonical flat $\Lambda \subset {\cal O}$, or sometimes just a
point $\lambda \in {\cal O}$.  In either case we get as in (6.6) a
well-defined semisimple element
$$e(\Lambda) = \exp(2\pi i \lambda) \in {^\vee G} \qquad(\lambda \in
\Lambda) \eqno(21.1)(b)$$
and subgroups
$$P(\Lambda) \subset {^\vee G}(\Lambda)_0 \eqno(21.1)(c)$$
The reductive group ${^\vee G}(\Lambda)_0$ is the identity component of the
centralizer of $e(\Lambda)$.  Its parabolic subgroup $P(\Lambda)$ is the
stabilizer of $\Lambda$ in the action of ${^\vee G}$ on ${\cal F}({\cal O})$
(cf.~(6.10)).  Define ${\cal P}(\Lambda)^0$ as in Proposition 6.24 to be the
${^\vee G}(\Lambda)_0$-orbit of $\Lambda$.  Then
$$ {\cal P}(\Lambda)^0 \simeq {^\vee G}(\Lambda)_0/P(\Lambda),
\eqno(21.1)(d)$$
a flag manifold for ${^\vee G}(\Lambda)_0$.  (This space will play the r\^ole
of $Y$ in section 20.)  Recall from (8.1) the ideal 
$$I_{{\cal P}(\Lambda)^0} \subset U({^\vee {\frak g}}(\Lambda)),
\eqno(21.1)(e)$$
and from Proposition 20.4 the nilpotent orbit
$${\cal Z}_{{\cal P}(\Lambda)^0} \subset {^\vee {\frak g}}(\Lambda)^*.
\eqno(21.1)(f)$$
Recall from (6.10)(f) the set ${\cal I}(\Lambda)$, on which ${^\vee
G}(\Lambda)_0$ acts with finitely many orbits ${\cal
I}^0_1(\Lambda),\ldots,{\cal I}^0_s(\Lambda)$; choose representatives
$$y_j \in {\cal I}^0_j(\Lambda). \eqno(21.1)(g)$$
Write $\theta_j$ for the involutive automorphism of ${^\vee
G}(\Lambda)_0$ defined by $y_j$, $K^0_j$ for its group of fixed points, and
$${^\vee {\frak g}}(\Lambda) = {\frak k}_j + {\frak s}_j \eqno(21.1)(h)$$
for the corresponding Cartan decomposition.  Finally, recall from
Proposition 6.24 that the geometric parameter space $X({\cal O},{^\vee
G^\Gamma})$ is the disjoint union of $s$ smooth connected subvarieties
$X_j({\cal O},{^\vee G^\Gamma})$, with
$$X_j({\cal O},{^\vee G^\Gamma}) \simeq {^\vee G} \times_{K_j^0} ({^\vee
G}(\Lambda)_0/P(\Lambda)) \eqno(21.1)(i)$$
In particular, the orbits of ${^\vee G}$ on $X_j({\cal O},{^\vee G^\Gamma})$
are naturally in one-to-one correspondence with the orbits of $K^0_j$ on
${^\vee G}(\Lambda)_0/P(\Lambda)$.

We will sometimes abuse this notation in the following way.  The element
$e(\Lambda)$, the group ${^\vee G}(\Lambda)_0$, and the sets ${\cal
I}^0_j(\Lambda)$ depend only on the ${^\vee G}(\Lambda)_0$ orbit ${\cal
P}(\Lambda)^0$, and not on $\Lambda$ itself.  We may therefore allow
$\Lambda$ to vary over ${\cal P}(\Lambda)^0$, rather than keep it fixed.

Here is a reformulation of Theorem 8.5.

\medskip
{\bf Theorem 21.2.}  {\it Suppose we are in the setting (21.1).  Then there
are natural bijections among the following sets:
\smallskip
\item{1)} irreducible $({^\vee {\frak g}}(\Lambda),K^{alg}_j)$-modules
annihilated by $I_{{\cal P}(\Lambda)^0}$;
\item{2)} irreducible $K_j^{alg}$-equivariant ${\cal D}_{{\cal
P}(\Lambda)^0}$-modules; and
\item{3)} irreducible ${^\vee G}^{alg}$-equivariant ${\cal D}$-modules on
$X_j({\cal O},{^\vee G^\Gamma})$.
\smallskip
These bijections arise from natural equivalences among the three
corresponding categories.  If in addition we are in the setting of Theorem
10.4, then the union over $j$ of these sets is in natural bijection with
\smallskip
\item{4)} equivalence classes of irreducible canonical projective
representations of type $z$ and infinitesimal character ${\cal O}$ of strong
real forms of $G$.
\medskip}

{\it Proof.}  The bijection between (1) and (2) is the Beilinson-Bernstein
localization theorem 8.3.  That between (2) and (3) is the smooth base
change of Proposition 7.14 (for which the hypotheses are established in
Proposition 6.24).  That between (3) and (4) is Theorem 10.4.  Q.E.D.

We want to apply the results of section 20 on characteristic varieties to
these bijections.

\medskip
{\bf Proposition 21.3.}  {\it Suppose we are in the setting (21.1).  Regard $x
= (y_j,\Lambda)$ as a point of $X({\cal O},{^\vee G^\Gamma})$ (Definition
6.9), so that
$$S = {^\vee G}\cdot x \subset X({\cal O},{^\vee G^\Gamma}),\qquad S_K =
K_j^0\cdot \Lambda \subset {\cal P}(\Lambda)^0$$
are corresponding orbits.
\smallskip
\item{a)}  The isotropy group of the action of ${^\vee G}$ at $x$ (or of the
action of $K_j^0$ at $\Lambda$) is $P(\Lambda)\cap K_j^0$.
\item{b)}  The conormal space to the orbit $S$ at $x$ (or to the orbit $S_K$
at $\Lambda$) may be identified naturally (as a module for $P(\Lambda)\cap
K_j^0$) with
$$T^*_{S,x}(X({\cal O},{^\vee G^\Gamma})) = T^*_{{^\vee G},x}(X({\cal
O},{^\vee G^\Gamma})) = ({^\vee{\frak g}}(\Lambda)/({\frak p}(\Lambda) +
{\frak k}_j))^*.$$
\item{c)}  Fix a non-degenerate ${^\vee G^\Gamma}$-invariant symmetric
bilinear form on $^\vee{\frak g}$, and use it to identify ${^\vee{\frak
g}}(\Lambda)^*$ with ${^\vee{\frak g}}(\Lambda)$.  Then the subspace
appearing in (b) corresponds to 
$$({^\vee{\frak g}}(\Lambda)/({\frak p}(\Lambda) + {\frak k}_j))^* \simeq
{\frak n}(\Lambda)\cap {\frak s}_j.$$
\medskip}

{\it Proof.}  Part (a) is obvious (and contained in Proposition 6.16).  For (b),
combine the last description of $X({\cal O},{^\vee G^\Gamma})$ in
Proposition 6.24 with Proposition 20.2(a) and (20.12)(d).  For (c), the left
side is clearly identified with ${\frak p}(\Lambda)^\perp \cap {\frak
k}_j^\perp$; here the orthogonal complements are to be taken with respect
to the bilinear form.  It is well known (and easy to check) that the
orthogonal complement of a parabolic subalgebra is its nil radical;  and the
Cartan decomposition of (20.11) is orthogonal since (by assumption) ${\rom
Ad}(y_j)$ respects the form.  Q.E.D.

\medskip
{\bf Proposition 21.4.} {\it Suppose we are in the setting (21.1).
\smallskip 
\item{a)} Every ${^\vee G}^{alg}$-equivariant ${\cal D}$-module on $X({\cal
O},{^\vee G^\Gamma})$ admits a ${^\vee G}^{alg}$-invariant good filtration.
\item{b)}  Suppose that the ${^\vee G}^{alg}$-equivariant ${\cal D}$-module
${\cal N}$ corresponds to the $K^{0,alg}_j$-equivariant ${\cal D}$-module
${\cal M}$ in the equivalence of Theorem 21.2.  Suppose $S$ is any orbit of
${^\vee G}$ on $X({\cal O},{^\vee G^\Gamma})$; write $S_K$ for the
corresponding orbit of $K^{0,alg}_j$ on ${\cal P}^0(\Lambda)$.  Then the
multiplicities in the characteristic cycle are related by
$$\chi^{mic}_{S}({\cal N}) = \chi^{mic}_{S_K}({\cal M}).$$
\medskip}

{\it Proof.}  Part (a) follows from Proposition 20.2(e) and Lemma 20.13. 
Part (b) is Proposition 20.2(f).  Q.E.D.

Recall now from (20.3) and (20.12) the moment map
$$\mu:T^*({\cal P}(\Lambda)^0) \rightarrow {\cal N}^*_{{\cal P}(\Lambda)^0}
\subset {^\vee {\frak g}}(\Lambda)^* \eqno(21.5)(a)$$
and its restriction
$$\mu_{K_j^0}:T^*_{K_j^0}({\cal P}(\Lambda)^0) \rightarrow {\cal
N}^*_{{\cal P}(\Lambda)^0,K_j^0} \subset ({^\vee {\frak g}}(\Lambda)/{\frak
k}_j)^*. \eqno(21.5)(b)$$
Write
$${\cal Z}_{{\cal P}(\Lambda)^0,K_j^0}  = {\cal Z}_{{\cal P}(\Lambda)^0} \cap
({^\vee {\frak g}}(\Lambda)/{\frak k}_j)^*, \eqno(21.5)(c)$$
a finite union of orbits of $K_j^0$ of dimension equal to $\dim {\cal
P}(\Lambda)^0$ (Lemma 20.14).  Recall (Definition 20.17) that an orbit
$S_K$ of $K_j^0$ on ${\cal P}(\Lambda)^0$ is called regular if and only if
$\mu(T^*_{S_K}({\cal P}(\Lambda)^0))$  meets ${\cal Z}_{{\cal
P}(\Lambda)^0}$.  We say that an orbit $S$ of ${^\vee G}$ on $X_j({\cal
O},{^\vee G^\Gamma})$ is {\it regular} if the corresponding orbit of $K_j^0$
is regular (cf.~(21.1)(i)).

\medskip
{\bf Theorem 21.6.}  {\it Suppose we are in the setting (21.1); use the
notation (21.5).  Suppose that the irreducible $({^\vee {\frak
g}}(\Lambda),K^{alg}_j)$-module $M$, the irreducible
$K_j^{alg}$-equivariant ${\cal D}_{{\cal P}(\Lambda)^0}$-module ${\cal M}$,
and the irreducible ${^\vee G}^{alg}$-equivariant ${\cal D}$-module ${\cal
N}$ correspond in the bijections of Theorem 21.2.  Then the following
conditions are equivalent.
\smallskip
\item{a)}  The singular support $SS({\cal N})$ contains the conormal bundle
of a regular orbit ${\cal S}$ (Definition 19.9 and (21.5)).
\item{b)}  The singular support $SS({\cal M})$ contains the conormal bundle
of a regular orbit ${\cal S_K}$.
\item{c)}  The associated variety ${\cal V}(M)$ meets ${\cal Z}_{{\cal
P}(\Lambda)^0,K_j^0}$ ((20.5) and (21.5)).
\item{d)}  The annihilator ${\rom Ann}\,M$ is equal to $I_{{\cal
P}(\Lambda)^0}$ ((21.1)(e)).
\item{e)}  The Gelfand-Kirillov dimension ${\rom Dim}\,M$ is equal to $\dim
{\cal P}(\Lambda)^0$.
\medskip}

{\it Proof.}  The equivalence of (a) and (b) follows from the definition of
regular and Proposition 21.4(b).  The rest of the result is Theorem 20.15. 
Q.E.D.

\medskip
{\bf Theorem 21.7.} {\it In the setting of (21.1), assume that the moment
map $\mu$ of (21.5)(a) has degree one.  Then there are natural bijections
among the following three sets:
\smallskip
\item{1)} regular orbits $S$ of ${^\vee G}$ on $X_j({\cal O},{^\vee
G^\Gamma})$;
\item{2)} regular orbits $S_K$ of $K_j^0$ on ${\cal P}(\Lambda)^0$; and 
\item{3)} orbits ${\cal Z}_S$ of $K_j^0$ on ${\cal Z}_{{\cal
P}(\Lambda)^0,K_j^0}$ ((21.5)(c)).
\smallskip
This bijection is multiplicity-preserving in the following sense.  Suppose
$S$, $S_K$, and ${\cal Z}_S$ correspond as above, and ${\cal N}$, ${\cal M}$,
and $M$ correspond as in Theorem 21.2.  Then $\chi_S^{mic}({\cal N}) =
\chi_{S_K}^{mic}({\cal M})$, and these are equal to the multiplicity of
$\overline{{\cal Z}_S}$ in the associated cycle ${\rom Ch}(M)$ (cf.~(20.5)).}
\medskip

{\it Proof.}  The first bijection is a definition, and the equality
$\chi_S^{mic}({\cal N}) = \chi_{S_K}^{mic}({\cal M})$ is Proposition 21.4(b). 
The rest of the result is Theorem 20.18.  Q.E.D.


\medskip
{\bf Theorem 21.8} (cf.~[BVunip], Proposition 3.24). {\it Suppose we are in
the setting (21.1).  Then there is a natural order-reversing bijection
$$ I \rightarrow {^\vee I}$$
from the set of primitive ideals in $U({\frak g})$ of infinitesimal character
${\lambda}$, onto the set of primitive ideals in $U({^\vee {\frak
g}}(\Lambda))$ containing $I_{{\cal P}(\Lambda)^0}$.  In particular, the
maximal primitive ideal $J_{max}(\lambda)$ of infinitesimal character
$\lambda$ corresponds to $I_{{\cal P}(\Lambda)^0}$:
$$ {^\vee J}_{max}(\lambda) = I_{{\cal P}(\Lambda)^0}.$$

In the setting of Theorem 21.2 and Theorem 10.4, suppose that the
irreducible $({^\vee {\frak g}}(\Lambda),K_j^0)$-module $M$ corresponds to
the irreducible canonical projective representation $\pi$ of $G({\Bbb
R},\delta)^{can}$.  Then the annihilators correspond:
$$^\vee({\rom Ann}\,\pi) = {\rom Ann}\,M.$$
In particular, the annihilator of $\pi$ is a maximal primitive ideal if and
only if the annihilator of $M$ is equal to $I_{{\cal P}(\Lambda)^0}$.}
\medskip

Recall that Theorem 21.6 provides several characterizations of the modules
$M$ with annihilator equal to $I_{{\cal P}(\Lambda)^0}$.
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\centerline{\bf 22.  Arthur parameters.}
\vskip 1pc

In this section we recall from [Arthur2] the parameters appearing in
Arthur's conjectures.  We begin with some simple facts about conjugacy
classes in complex algebraic groups, which will be applied to E-groups.

\medskip
{\bf Definition 22.1.}  Suppose $H$ is a complex algebraic group.  An element
$h\in H$ is called {\it elliptic} if the closure in the analytic topology of the
group $\langle h \rangle$ generated by $h$ is compact.  Similarly, an
element $X\in {\frak h}$ is called {\it elliptic} if the closure in the analytic
topology of the one-parameter subgroup $\{\,\exp(tX)\mid t\in {\Bbb R}\,\}$
is compact.
\medskip
{\bf Lemma 22.2.} {\it Suppose $H$ is a complex algebraic group, $h\in H$,
and $X\in {\frak h}$.
\smallskip
\item{a)}  If $h$ is elliptic, then $h$ is semisimple; and if $X$ is elliptic,
then $X$ is semisimple.
\item{b)}  The element $h$ is elliptic if and only if $h^n$ is elliptic for
every non-zero integer $n$.
\smallskip
\noindent Suppose $T\subset H$ is an algebraic torus.  Define
$${\frak t}_{\Bbb R} = X_*(T) \otimes_{\Bbb Z} {\Bbb R} \subset {\frak t} $$
(cf.~(9.1)(d) and Lemma 9.9), so that $X_*(T)$ is a lattice in the real vector
space ${\frak t}_{\Bbb R}$.  Finally, define
$$T_c = e({\frak t}_{\Bbb R}) \simeq {\frak t}_{\Bbb R}/X_*(T),$$
a compact analytic torus in $T$ (cf.~Lemma 9.9).
\smallskip
\item{c)}  Suppose $h \in T$.  Then $h$ is elliptic if and only if $h\in T_c$.
\item{d)}  Suppose $X\in {\frak t}$.  Then $X$ is elliptic if and only if $X\in
i{\frak t}_{\Bbb R}$.
\medskip}

We leave the elementary argument to the reader.  The point of (a) and (b) is
that they reduce the problem of testing for ellipticity (even if $H$ is
disconnected) to the cases treated in (c) and (d)).

\medskip
{\bf Definition 22.3.}  Suppose ${^\vee G^\Gamma}$ is a weak E-group.  A
Langlands parameter $\phi\in P({^\vee G^\Gamma})$ (Definition 5.2) is said
to be {\it tempered} if it satisfies any of the following equivalent
conditions.
\smallskip
\item{(a)} The closure of $\phi(W_{\Bbb R})$ in the analytic topology is
compact.
\item{(b)} In the notation of Proposition 5.6, the Lie algebra element
$\lambda + {\rom Ad}(y)\lambda$ is elliptic.
\item{(c)} Suppose ${^d{T}^\Gamma}$ is a Cartan subgroup of ${^\vee
G^\Gamma}$, and that $\phi(W_{\Bbb R}) \subset {^d{T}^\Gamma}$.  Fix an
E-group structure on ${^d{T}^\Gamma}$, and let $T$ be an algebraic torus
defined over ${\Bbb R}$ admitting ${^d{T}^\Gamma}$ as an E-group.  Then the
canonical projective character $\pi(\phi)$ of $T({\Bbb R})^{can}$ associated
to $\phi$ (Proposition 10.6) is unitary.
\item{(d)} Fix an E-group structure on ${^\vee G^\Gamma}$, and let
$(G^\Gamma,{\cal W})$ be an extended group admitting ${^\vee G^\Gamma}$
as an E-group.  Then at least one of the irreducible canonical projective
representations of strong real forms of $G$ parametrized by $\phi$
(Theorem 10.4) is tempered.
\item{e)} In the setting of (d), all the representations parametrized by
$\phi$ are tempered.
\smallskip
We write $P_{temp}({^\vee G^\Gamma})$ for the set of tempered Langlands
parameters, and $\Phi_{temp}({^\vee G^\Gamma})$ for the corresponding set
of equivalence classes.
\medskip
\noindent To see that the conditions in the definition are equivalent, we can
argue as follows.  First, the Weil group is the direct product of the positive
real numbers ${\Bbb R}^+$ and a compact group; so (a) is equivalent to
ellipticity of the generator of $\phi({\Bbb R}^+)$.  This generator is
precisely $\lambda + {\rom Ad}(y)\lambda$ (cf.~(5.5)(a)).  Hence (a) and (b)
are equivalent.  In the setting of (c), $T({\Bbb R})$ is the direct product of a
compact group and a vector group ([AV2], Corollary 3.16), so $\pi(\phi)$ is
unitary if and only if its restriction to the vector group is.  Now $\lambda +
{\rom Ad}(y)\lambda$ is identified with the differential of that restriction,
so the equivalence of (b) and (c) follows ([AV2], Proposition 4.11).  From the
definitions underlying Theorem 10.4 (Proposition 13.12), it follows that (c)
is equivalent to versions of (d) and (e) in which ``tempered representation''
is replaced by ``unitary limit character'' (Definitions 11.2 and 12.1).  To
finish, we need to observe that in the classification of Theorem 11.4,
tempered representations correspond precisely to unitary final limit
characters.  This is a fairly serious result, but it is well-known (see
[Langlands]). 

\medskip
{\bf Definition 22.4} (cf.~[Arthur2], section 6).  Suppose ${^\vee G^\Gamma}$
is a weak E-group (Definition 4.3).  An {\it Arthur parameter} is a
homomorphism
$$\psi:W_{\Bbb R} \times SL(2,{\Bbb C}) \rightarrow {^\vee G^\Gamma}$$
(see Definition 5.2) satisfying
\smallskip
\item{(a)}  the restriction of $\psi$ to $W_{\Bbb R}$ is a tempered
Langlands parameter; and
\item{(b)}  the restriction of $\psi$ to $SL(2,{\Bbb C})$ is holomorphic.
\smallskip
\noindent Two such parameters are called {\it equivalent} if they are
conjugate by the action of ${^\vee G}$.  The set of Arthur parameters is
written $Q({^\vee G^\Gamma})$; the set of equivalence classes is written
$\Psi({^\vee G^\Gamma})$.  In the setting of Definition 5.3, we may also
write $\Psi(G/{\Bbb R})$.

Suppose $\psi$ is an Arthur parameter.  In analogy with Definition 5.11, we
define
$${^\vee G}_\psi = \hbox{ centralizer in ${^\vee G}$ of $\psi(W_{\Bbb
R}\times SL(2,{\Bbb C}))$},$$
and
$$A_\psi = {^\vee G}_\psi/({^\vee G}_\psi)_0,$$
the {\it Arthur component group for $\psi$}.  Similarly,
$$A_\psi^{alg} = {^\vee G}^{alg}_\psi/({^\vee G}^{alg}_\psi)_0,$$
the {\it universal component group for $\psi$}.

The {\it associated Langlands parameter for $\psi$} is the homomorphism
$$\phi_\psi:W_{\Bbb R} \rightarrow {^\vee G}, \qquad \phi_\psi(w) =
\psi(w,\pmatrix|w|^{1/2}&0\cr 0&|w|^{-1/2}\cr\endpmatrix)\quad (w\in W_{\Bbb R}).$$
Attached to $\phi_\psi$ is an orbit $S_{\phi_\psi}$ of ${^\vee G}$ on
$X({^\vee G^\Gamma})$ (Proposition 6.17); define
$$S_\psi = S_{\phi_\psi}.$$
Write
$$P_{Arthur}({^\vee G^\Gamma}) = \{\,\phi_\psi\mid \psi \in Q({^\vee
G^\Gamma})\,\} \subset P({^\vee G^\Gamma})$$
for the image of the map on parameters, and 
$$\Phi_{Arthur}({^\vee G^\Gamma}) \subset \Phi({^\vee G^\Gamma})$$
for the corresponding set of equivalence classes.
\medskip

{\bf Proposition 22.5} (Arthur [Arthur1], p.10).  {\it Suppose ${^\vee
G^\Gamma}$ is a weak E-group.  The map $\psi \mapsto \phi_\psi$ from
Arthur parameters to Langlands parameters defines a bijection on the level
of equivalence classes
$$\Psi({^\vee G^\Gamma}) \simeq \Phi_{Arthur}({^\vee G^\Gamma}).$$
We have
$$\Phi_{temp}({^\vee G^\Gamma}) \subset \Phi_{Arthur}({^\vee G^\Gamma})
\subset \Phi({^\vee G^\Gamma}).$$
If $\psi$ is an Arthur parameter, there is an inclusion
$${^\vee G}^\Gamma_\psi \subset {^\vee G}^\Gamma_{\phi_\psi}.$$
The induced maps on component groups
$$A_\psi \rightarrow A_{\phi_\psi}^{loc}, \qquad A^{alg}_\psi \rightarrow
A_{\phi_\psi}^{loc,alg}$$
are surjective.}
\medskip

Here is the definition of the ``Arthur packets'' discussed in the introduction.

\medskip
{\bf Definition 22.6.}  In the setting of Theorem 10.4, suppose $\psi\in
\Psi({^\vee G^\Gamma})$ is an Arthur parameter (Definition 22.4).  The {\it
Arthur packet $\Pi^z_\psi$ of $\psi$} is by definition the micro-packet of
the Langlands parameter $\phi_\psi$ attached to $\psi$ (Definition 19.15,
Definition 22.4):
$$\Pi^z(G/{\Bbb R})_\psi = \Pi^z(G/{\Bbb R})^{mic}_{\phi_\psi}.$$

A little more explicitly , write $S = S_\psi$ as in Definition 22.4.  Then
$\Pi^z_\psi$ consists of those irreducible representations with the
property that the corresponding irreducible perverse sheaf (Theorem 1.24 or
Theorem 15.12) contains the conormal bundle $T^*_S(X({^\vee G^\Gamma}))$
in its characteristic cycle (Proposition 19.12).
\medskip

{\bf Theorem 22.7.}  {\it Suppose $(G^\Gamma,{\cal W})$ is an extended
group for $G$ (Definition 1.12), and $({^\vee G^\Gamma},{\cal D})$ is an
E-group for the corresponding inner class of real forms, with second
invariant $z$ (Definition 4.6). Fix an Arthur parameter $\psi\in
\Psi^z(G/{\Bbb R})$ (Definition 22.4).  Define the Arthur packet $\Pi_\psi$
as in Definition 22.6.
\smallskip
\item{a)}  $\Pi_\psi$ contains the L-packet $\Pi_{\phi_\psi}$, and at most
finitely many additional representations of each strong real form of $G$.
\item{b)}  There is a strongly stable formal virtual character
$$\eta_\psi = \sum_{\pi'\in \Pi_\psi} e(\pi')(-1)^{d(\pi')-d(\phi_\psi)}
\chi_\psi(\pi') \pi'.$$
Here the terms are as explained in Corollary 19.16.  In particular,
$e(\pi')=\pm 1$ is Kottwitz' sign attached to the real form of which $\pi'$
is a representation, and $\chi_\psi(\pi')$ is a positive integer.
\medskip}

This is immediate from Corollary 19.16 and the definitions.  Essentially it
solves Problems A, C, and E from the introduction.  For Problem B, we must
attach to each $\pi' \in \Pi_\psi$ a representation of $A_\psi^{alg}$, of
dimension equal to $\chi_\psi(\pi')$.  This will turn out to be standard
micro-local geometry.  To see that, however, we need a micro-local
interpretation of $A_\psi^{alg}$.

We turn therefore to an analysis of the geometry of an Arthur parameter
$\psi$.  Write $\psi_0$ for the restriction of $\psi$ to $W_{\Bbb R}$, a
tempered Langlands parameter.  According to Proposition 5.6, $\psi_0$ is
determined by a pair
$$(y_0,\lambda_0),\qquad y_0\in {^\vee G^\Gamma}-{^\vee G},\qquad
\lambda_0 \in {^\vee {\frak g}}. \eqno(22.8)(a)$$
Write $\psi_1$ for the restriction of $\psi$ to $SL(2,{\Bbb C})$, an
algebraic homomorphism into ${^\vee G}$.  We can define
$$y_1 = \psi_1\pmatrix i&0\cr 0&-i\cr\endpmatrix \in {^\vee G}, \qquad \lambda_1 =
d\psi_1\pmatrix 1/2&0\cr0&-1/2\cr\endpmatrix \in {^\vee {\frak g}}. \eqno(22.8)(b)$$
An elementary calculation from Definition 22.4 and Proposition 5.6 shows
that the Langlands parameter $\phi_\psi$ is associated to the pair
$(y,\lambda)$, with
$$y = y_0y_1,\qquad  \lambda = \lambda_0 + \lambda_1. \eqno(22.8)(c)$$
(Here and elsewhere in the following discussion, one should keep in mind
that elements of $\psi_0(W_{\Bbb R})$ commute with elements of
$\psi_1(SL(2,{\Bbb C})$.)  Using $\lambda$ and $y$, we can introduce all of
the structure considered in section 6 (cf.~(6.1), (6.10)).  We also write
$\theta_y$ for the conjugation action of $y$ on ${^\vee G}(\lambda)$, and
$${^\vee {\frak g}}(\lambda) = {\frak k}(y) + {\frak s}(y) \eqno(22.8)(c)$$
for the corresponding Cartan decomposition.  Define
$$E_\psi = d\psi_1\pmatrix 0&1\cr0&0\cr\endpmatrix \in {^\vee {\frak g}}, \qquad
F_\psi = d\psi_1\pmatrix 0&0\cr1&0\cr\endpmatrix \in {^\vee {\frak g}};
\eqno(22.8)(d)$$
these are nilpotent elements.  By calculation in ${\frak s}{\frak l}(2)$,
$E_\psi$ lies in the $+1$-eigenspace of ${\rom ad}(\lambda_1)$.  Since
$\lambda_0$ and $E_\psi$ commute, it follows that
$$E_\psi \in {^\vee {\frak g}}(\lambda)_1 \subset {\frak n}(\lambda).
\eqno(22.8)(e)$$
Similarly, we check that ${\rom Ad}(y_1)(E_\psi) = -E_\psi$, and deduce
$$E_\psi \in {\frak s}(y).  \eqno(22.8)(f)$$

Here is a micro-local description of the Arthur component group. The result
is a slight generalization of work of Kostant and Rallis in [KR]; the proof
follows their ideas very closely.

\medskip
{\bf Proposition 22.9.}  {\it Suppose ${^\vee G^\Gamma}$ is an E-group, and
$\psi \in Q({^\vee G^\Gamma})$ is an Arthur parameter.  Let $x =
p(\phi_\psi) \in X({^\vee G^\Gamma})$ (Proposition 6.17) be the
corresponding point, and $S = {^\vee G}\cdot x$ the corresponding orbit.  Fix
a non-degenerate ${^\vee G^\Gamma}$-invariant symmetric bilinear form on
${^\vee {\frak g}}$, and use it as in Proposition 21.3 to identify the conormal
space $T^*_{S,x}(X({^\vee G^\Gamma}))$ with ${\frak n}(\lambda)\cap {\frak
s}(y)$.  Accordingly regard $E_\psi$ as an element of this conormal space
(cf.~(22.8)(e),(f)):
$$E_\psi \in {\frak n}(\lambda)\cap {\frak s}(y) \simeq T^*_{S,x}(X({^\vee
G^\Gamma})).$$
\smallskip
\item{a)}  We have
$$[{\frak p}(\lambda)\cap {\frak k}(y),E_\psi] = {\frak n}(\lambda)\cap
{\frak s}(y).$$
\item{b)} The orbit $(P(\lambda)\cap K(y))\cdot E_\psi$ (with the natural
action of the isotropy group on the conormal space) is Zariski dense in
$T^*_{S,x}(X({^\vee G^\Gamma}))$.
\item{c)}  The orbit ${^\vee G}\cdot E_\psi$ is Zariski dense in the full
conormal bundle $T^*_S(X({^\vee G^\Gamma}))$.
\item{d)}  The isotropy groups of $E_\psi$ in the actions of (b) and (c)
coincide; write ${^\vee G}_{E_\psi}$ for this subgroup of ${^\vee G}_x$. 
Then ${^\vee G}_\psi$ is a Levi subgroup of ${^\vee G}_{E_\psi}$.
\item{e)}  The inclusion in (d) identifies the Arthur component group
(Definition 22.4) naturally with the component group of ${^\vee
G}_{E_\psi}$:
$$A_\psi \simeq {^\vee G}_{E_\psi}/({^\vee G}_{E_\psi})_0.$$
Similarly,
$$A^{alg}_\psi \simeq {^\vee G}^{alg}_{E_\psi}/({^\vee G}^{alg}_{E_\psi})_0.
$$
\medskip}

{\it Proof.}  By the representation theory of ${\frak s}{\frak l}(2)$, ${\rom
ad}(\lambda_1)$ has half-integral eigenvalues.  (For this discussion, we call
any element of $1/2{\Bbb Z}$ a half-integer.)  By definition of ${\frak
g}(\lambda)$, ${\rom ad}(\lambda)$ has integral eigenvalues on ${\frak
g}(\lambda)$.  It follows that $\lambda_0 = \lambda - \lambda_1$ has
half-integral eigenvalues on ${\frak g}(\lambda)$.  In analogy with (6.1)(b),
we define for half-integers $r$ and $s$
$${^\vee{\frak g}}_{r,s} = \{\,\mu \in {^\vee{\frak g}}\mid [\lambda_0,\mu] =
r\mu,[\lambda_1,\mu] = s\mu\,\}. \eqno(22.10)(a)$$
Then
$${^\vee{\frak g}}(\lambda)_n = \sum_{r+s = n} {^\vee{\frak g}}_{r,s}. 
\eqno(22.10)(b)$$
Consulting the definitions (6.1), we deduce that
$${\frak n}(\lambda) = \sum_{r+s \in {\Bbb N}-\{0\}}{^\vee{\frak
g}}_{r,s},\qquad {\frak p}(\lambda) = \sum_{r+s \in {\Bbb N}}{^\vee{\frak
g}}_{r,s}.\eqno(22.10)(c)$$

Now $\theta_y(\lambda_0) = \theta_{y_0}(\lambda_0)$ commutes with
$\lambda_0$ and with $\lambda_1$ (Proposition 5.6(c)) and so acts
diagonalizably on each ${^\vee{\frak g}}_{r,s}$.  Since $\psi_0$ is tempered,
$\lambda_0 + \theta_{y_0}(\lambda_0)$ has purely imaginary eigenvalues
(Definition 22.3(b)).  It follows that $\theta_y(\lambda_0)$ acts by $-r$ on
${^\vee{\frak g}}_{r,s}$.  On the other hand, 
$$\theta_y(\lambda_1) = {\rom Ad}(y_1)(\lambda_1) = \lambda_1;$$
so $\theta_y(\lambda_1)$ acts by $s$ on ${^\vee{\frak g}}_{r,s}$.  Combining 
these two facts, we get
$$\theta_y: {^\vee{\frak g}}_{r,s} \rightarrow {^\vee{\frak g}}_{-r,s}.
\eqno(22.10)(d)$$

Combining the descriptions of $\theta_y$ and ${\frak p}(\lambda)$ in terms
of the eigenspaces ${^\vee{\frak g}}_{r,s}$, we get
$${\frak k}(y) \cap {\frak p}(\lambda) = \{\,X+\theta_y X\mid X\in
{^\vee{\frak g}}_{r,s}, s\pm r \in {\Bbb N}\,\} \eqno(22.10)(e)$$
$${\frak s}(y) \cap {\frak n}(\lambda) = \{\,X-\theta_y X\mid X\in
{^\vee{\frak g}}_{r,s}, s\pm r \in {\Bbb N}-\{0\}\,\} \eqno(22.10)(f).$$
To study the adjoint action of $E_\psi$, we use the representation theory of
${\frak s}{\frak l}(2)$.  We deduce that
$${\rom ad}(E_\psi):{^\vee{\frak g}}_{r,s-1} \rightarrow {^\vee{\frak
g}}_{r,s}; \eqno(22.10)(g)$$
this map is surjective for $s\ge 1/2$ and injective for $s\le 1/2$.

We can now prove (a).  Since both sides are vector spaces, it is enough by
(22.10)(f) to show that every element of the form $X-\theta_y X$ belongs to
the left side, with $X\in {^\vee{\frak g}}_{r,s}$ and $s\pm r$ positive
integers.  By (22.10)(g), we can find $Y \in {^\vee{\frak g}}_{r,s-1}$ with 
$[Y,E_\psi] = X$.  Since $\theta_y E_\psi = -E_\psi$, it follows that
$$[Y + \theta_y Y,E_\psi] = X - \theta_y X.$$
Furthermore $(s-1)\pm r$ must be non-negative integers, so (by (22.10)(e))
$Y + \theta_y Y \in {\frak k}(y)\cap {\frak p}(\lambda)$.  This is (a).

For (b), recall that the conormal space $T^*_{S,x}(X({^\vee G^\Gamma}))$ has
been identified with ${\frak s}(y)\cap {\frak n}(\lambda)$.  This
identification sends the isotropy action of ${^\vee G}_x = K(y) \cap
P(\lambda)$ (Proposition 6.16) to the restriction of the adjoint action.  It
follows that the tangent space to the orbit $(P(\lambda)\cap K(y))\cdot
E_\psi$ is just $[{\frak k}(y)\cap {\frak p}(\lambda),E_\psi]$.  (Here we
identify the tangent space to the vector space ${\frak s}(y)\cap {\frak
n}(\lambda)$ at the point $E_\psi$ with the vector space.)  By (a), this
tangent space is the full tangent space at $E_\psi$; so the orbit is open, as
we wished to show.

For (c), the conormal bundle $T^*_S(X({^\vee G^\Gamma})$ is an equivariant
bundle over the homogeneous space $S = {^\vee G}/{^\vee G}_x$.  The orbits
of such an action are in one-to-one correspondence with those of ${^\vee
G}_x$ on the fiber over $x$ (cf.~Lemma 6.15); this bijection preserves
isotropy groups and codimension of orbits.  Since $(P(\lambda)\cap
K(y))\cdot E_\psi$ has codimension 0 in $T^*_{S,x}(X({^\vee G^\Gamma}))$, it
follows that ${^\vee G}\cdot E_\psi$ has codimension 0 in $T^*_S(X({^\vee
G^\Gamma}))$, as we wished to show.

We have already explained the first claim of (d).  For the rest, we begin with
the Levi decomposition
$$P(\lambda)\cap K(y) = (L(\lambda)\cap K(y))(N_{\theta_y} \cap K(y))
\eqno(22.11)(a)$$
of Proposition 6.17 and Lemma 6.18.  We claim that this decomposition is
inherited by the subgroup ${^\vee G}_{E_\psi}$:
$${^\vee G}_{E_\psi} = (L(\lambda)\cap K(y))_{E_\psi}(N_{\theta_y} \cap
K(y))_{E_\psi}. \eqno(22.11)(b)$$
So suppose that $ln$ is the decomposition of an element of ${^\vee
G}_{E_\psi}$; it is enough to show that $l\in {^\vee G}_{E_\psi}$.  Using
(22.10), one calculates easily that
$${\frak n}_{\theta_y} = \sum_{s\ge |r|,s>0,s+r\in {\Bbb Z}} {^\vee{\frak
g}}_{r,s}. \eqno(22.11)(c)$$
It follows that
$${\rom Ad}(n)(E_\psi) = E_\psi +\hbox{ (terms in ${^\vee{\frak g}}_{r,s}$
with $s > 1$)}. \eqno(22.11)(d)$$
(Recall that $E_\psi \in {^\vee{\frak g}}_{0,1}$.)  Applying ${\rom Ad}(l)$ to
this, we get
$$E_\psi = {\rom Ad}(ln)(E_\psi) = {\rom Ad}(l)(E_\psi) + \hbox{ (terms in
${^\vee{\frak g}}_{r,s}$ with $s > 1$)}. \eqno(22.11)(e)$$
Projecting on the $1$-weight space of $\lambda_1$, we find that ${\rom
Ad}(l)(E_\psi) = E_\psi$, proving (22.11)(b).

To complete the proof of (d), it suffices to show that 
$$(L(\lambda)\cap K(y))_{E_\psi} = {^\vee G}_\psi. \eqno(22.12)(a)$$
That the left side contains the right is trivial from the definitions; so
suppose $l \in (L(\lambda)\cap K(y))_{E_\psi}$.  We must show that $l$
centralizes the image of $\psi$.  It is enough to show that it fixes the
images under $d\psi_1$ of the basis $\{E_\psi,2\lambda_1,F_\psi\}$ of
${\frak s}{\frak l}(2)$ (see (22.8)). Since $l$ commutes with $\lambda$ and
$y$, it commutes also with $\theta_y \lambda$.  Now it follows from
(22.10)(c) that the only eigenvalue of $\lambda_0+\theta_y\lambda_0$ on
${^\vee {\frak g}}(\lambda)$ is 0; so it is central in ${^\vee {\frak
g}}(\lambda)$, and in particular is fixed by $l$.  Hence
$$2\lambda_1 = (\lambda + \theta_y\lambda) -
(\lambda_0+\theta_y\lambda_0) \eqno(22.12)(b)$$
is also fixed by $l$. By the injectivity statement in (22.10)(g), there is at
most one element $F\in {^\vee {\frak g}}_{0,-1}$ with the property that
$$[E_\psi,F] = 2\lambda_1. \eqno(22.12)(c)$$
By calculation in ${\frak s}{\frak l}(2)$, $F_\psi$ satisfies (22.12)(c). 
Since $l$ fixes $E_\psi$ and $2\lambda_1$, ${\rom Ad}(l)(F_\psi)$ is a
second solution of (22.12)(d).  By the uniqueness, $l$ fixes $F_\psi$. 
Consequently $l$ commutes with $E_\psi$, $2\lambda_1$, and $F_\psi$, as
we wished to show.

Part (e) follows from (d) exactly as in Lemma 7.5.  Q.E.D.

%\end

%\input mssymb.tex
%\font\Bbb=cmr10
%\font\frak=cmr10
%\hsize=27pc
%\vsize=42pc
%\hoffset=3.5pc
%\voffset=5.5pc
%\pageno=142
\vskip 3pc
\centerline{\bf 23. Local geometry of constructible sheaves.}
\vskip 1pc
As preparation for the serious geometry of the next section, we present
here some trivial reformulations of material from section 7.  The goal is to
find local results analogous to the  microlocal ones we want.  Constructible
sheaves have a local behavior analogous to the microlocal behavior of
perverse sheaves, so we will concentrate on them.

We work in the setting of Definition 7.7, so that the pro-algebraic group $H$
acts on the smooth algebraic variety $Y$ with finitely many orbits:
$$ Y = \bigcup_{\hbox{$H$-orbits $S$}} S. \eqno(23.1)(a)$$
Each orbit $S$ is smooth and locally closed.  If $y\in S$, then
$$S \simeq H/H_y \eqno(23.1)(b)$$

\medskip
{\bf Proposition 23.2.} {\it In the setting (23.1), suppose $C$ is an
$H$-equivariant constructible sheaf on $Y$.  Then the restriction of $C$ to
an $H$-orbit $S$ is an $H$-equivariant local system $Q_S^{loc}(C)$ on $S$.
\smallskip
\item{a)} The rank of $Q_S^{loc}(C)$ is equal to the dimension of the stalks
of $C$ along $S$.
\item{b)} Suppose that $\xi = (S,{\cal V})$ is a complete geometric
parameter for $H$ acting on $Y$ (Definition 7.1), and that $\mu(\xi)$ (the
extension of $\xi$ by zero --- see (7.10(c)) is the corresponding irreducible
$H$-equivariant constructible sheaf.  Then $Q_S^{loc}(\mu(\xi)) = \xi$, and
$Q_{S'}^{loc}(\mu(\xi)) = 0$ for $S' \ne S$.
\item{c)} $Q_S^{loc}$ is an exact functor from ${\cal C}(Y,H)$ to
$H$-equivariant local systems on $S$.
\medskip}

This is entirely trivial.  Using the description of equivariant local systems
in Lemma 7.3, we deduce

\medskip
{\bf Corollary 23.3.}  {\it In the setting (23.1), suppose $C$ is an
$H$-equivariant constructible sheaf on $Y$, and $S\subset Y$ is an
$H$-orbit.  Attached to $C$ there is a representation $\tau_S^{loc}(C)$ of
the equivariant fundamental group $A_S^{loc}$ (Definition 7.1).
\smallskip
\item{a)}  The dimension of $\tau_S^{loc}(C)$ is equal to the dimension of
the stalks of $C$ on $S$.
\item{b)}  Suppose that $\xi = (S,\tau)$ is a local complete geometric
parameter for $H$ acting on $Y$ (with $\tau$ an irreducible representation
of $A_S^{loc}$), and $\mu(\xi)$ is the corresponding irreducible
constructible sheaf (see (7.10)(c)).  Then $\tau_S^{loc}(\mu(\xi)) = \tau$,
and $\tau_{S'}^{loc}(\mu(\xi)) = 0$ for $S\ne S'$.
\item{c)}  The functor $\tau_S^{loc}$ from ${\cal C}(Y,H)$ to
representations of $A_S^{loc}$ is exact.  In particular, it gives a
well-defined map (also denoted $\tau_S^{loc}$) from the Grothendieck group
$K(Y,H)$ (cf.~(7.10)) to virtual representations of $A_S^{loc}$.
\medskip}

{\bf Definition 23.4.}  Suppose $P \in {\cal P}(Y,H)$ is an $H$-equivariant
perverse sheaf on $Y$.  (In fact we could take for $P$ any $H$-equivariant
constructible complex on $Y$.)  Then the cohomology sheaves $H^iP$ are
$H$-equivariant constructible sheaves on $Y$ (Lemma 7.8), so we can attach
to each orbit $S$ an $H$-equivariant local system
$$(Q_S^{loc})^i(P) = Q_S^{loc}(H^iP)$$
on $S$ (Proposition 23.2), and a representation
$$(\tau_S^{loc})^i(P) = \tau_S^{loc}(H^iP)$$
of the equivariant fundamental group $A_S^{loc}$ (Corollary 23.3).   
We write
$$\tau_S^{loc}(P) = \sum (-1)^i (\tau_S^{loc})^i(P),$$
a virtual representation of $A_S^{loc}$.
\medskip

Because of Lemma 7.8, this notation is consistent with that of Corollary
23.3(c).  We have already seen the virtual representation $\tau_S^{loc}(P)$,
at least for $P$ irreducible, as the geometric character matrix.  More
precisely, suppose $\gamma = (S_\gamma,\tau_\gamma)$ and $\xi =
(S_\xi,\tau_\xi)$ are complete geometric parameters (Definition 7.1), with
$\tau_\gamma$ an irreducible representation of $A_{S_\gamma}^{loc}$ and
$\tau_\xi$ an irreducible representation of $A_{S_\xi}^{loc}$.  Write
$P(\gamma)$ for the irreducible perverse sheaf parametrized by $\gamma$
(cf.~(7.10)).  Then
$$\hbox{\rom multiplicity of $\tau_\xi$ in $\tau_{S_\xi}^{loc}(P(\gamma))$}
= (-1)^{\dim S_\xi} c_g(\xi,\gamma) \eqno(23.5)$$
(notation (7.11)).

\medskip
{\bf Definition 23.6.}  Suppose $F \in K(Y,H)$ (notation (7.10)), $S$ is an
$H$-orbit on $Y$, and $\sigma \in A_S^{loc}$.  The {\it local trace of
$\sigma$ on $F$} is
$$\chi_S^{loc}(F)(\sigma) = {\rom tr}(\tau_S^{loc}(F)(\sigma)).
\eqno(23.6)(a)$$
(The trace of an element in a finite-dimensional virtual representation of a
group is well-defined.)  If $C$ is an $H$-equivariant constructible sheaf, or
perverse sheaf, or constructible complex representing $F$ (cf.~(7.10)(a)),
then we write $\chi_S^{loc}(C)(\sigma)$ instead of
$\chi_S^{loc}(F)(\sigma)$.  Explicitly, fix $y\in S$, so that $A_S^{loc}$ is
the group of connected components of $H_y$, and choose a representative
$s\in H_y$ for $\sigma$.  Then
$$\chi_S^{loc}(C)(\sigma) = \sum (-1)^i {\rom tr}(s\ {\rom on}\ (H^iC)_y),
\eqno(23.6)(b)$$
the alternating sum of the traces of $s$ on the stalks of the cohomology
sheaves of $C$.

The {\it local multiplicity of $F$ along $S$} is
$$\chi_S^{loc}(F) = \chi_S^{loc}(F)(1) \eqno(23.6)(c)$$
In the setting of (23.6)(b), this is the alternating sum of the dimensions of
the stalks of the cohomology sheaves.  (In particular, it is non-negative for
constructible sheaves.)
\medskip

We can use this notation to reformulate the result of Proposition 7.18.

\medskip
{\bf Proposition 23.7.} {\it Suppose we are in the setting (7.17).
\smallskip
\item{a)}  Suppose $C \in {\rom Ob}\,{\cal C}(X,G)$ is  a $G$-equivariant
constructible sheaf on $X$, so that $\epsilon^*C$ is an $H$-equivariant
constructible sheaf on $Y$ (Proposition 7.18).  Fix an orbit $S$ of $H$ on $Y$. 
Then the representation $\tau_S^{loc}(\epsilon^*C)$ (Corollary 23.3) is
given by
$$\tau_S^{loc}(\epsilon^*C) = A^{loc}(\epsilon) \circ
\tau^{loc}_{\Phi(\epsilon)S}(C)$$
(notation (7.17)).
\item{b)}  Suppose $\xi' = (S',\tau') \in \Xi(X,G)$ is a local complete
geometric parameter for $G$ acting on $X$, with $\tau'$ an irreducible
representation of $A^{loc}_{S'}$ (Definition 7.1). Then
$$\epsilon^*\mu(\xi') = \sum_{\scriptstyle \xi = (S,\tau) \in \Xi(Y,H)\atop
\scriptstyle S' = \Phi(\epsilon)S} \hbox{(multiplicity of $\tau$ in
$A^{loc}(\epsilon)\circ \tau'$)} \mu(\xi).$$
\item{c)} Suppose $F \in K(X,G)$ (cf.~(7.10)(a)).  Fix an $H$-orbit $S\subset
Y$, and an element $\sigma\in A_S^{loc}$ (Definition 7.1).  Define
$$S' = \Phi(\epsilon)S \subset X,\qquad \sigma' = A^{loc}(\epsilon)(\sigma)
\in A_{S'}^{loc}.$$
Then the local trace of $\sigma$ on $\epsilon^*F$ (Definition 23.6) is
$$\chi_S^{loc}(\epsilon^*F)(\sigma) = \chi_{S'}^{loc}(F)(\sigma').$$
In particular, the local multiplicity of $\epsilon^*F$ along $S$ is equal to
the local multiplicity of $F$ along $S'$.
\medskip}

{\it Proof.}  Part (a) is a reformulation of Proposition 7.18.  Part (b) follows
from (a) and Corollary 23.3(b).  If $F$ is represented by a constructible
sheaf $C$, then (c) follows from (a) by taking the trace of the action of
$\sigma$.  The general case follows from the fact that both sides are
homomorphisms from $K(X,G)$ to ${\Bbb C}$.  Q.E.D.

%\end\input mssymb.tex
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\centerline{\bf 24.  Microlocal geometry of perverse sheaves.}
\vskip 1pc
Up until now we have described the integers that are to be the dimensions of
the representations whose existence is predicted by Arthur's conjectures
(Theorem 22.7).  Here the key idea was that of the multiplicity of a
component of a characteristic cycle.  To produce the representations
themselves, we must exhibit such a multiplicity as the dimension of a
natural vector space, on which something like the Arthur component group
can act.  Now it is relatively easy to find a vector space whose dimension is
the multiplicity, and in this way to construct (in the notation of Definition
22.4) a representation of something like ${^\vee G}_\psi^{alg}$.  (Something
very similar is done in [Assoc], Theorem 2.13.)  The difficulty arises in
showing that this representation is trivial on the identity component.  (The
analogous result in [Assoc] is Theorem 8.7.)  Before going into further
detail, we introduce some notation.

We work in the setting of Definition 7.7, so that the pro-algebraic group $H$
acts on the smooth algebraic variety $Y$ with finitely many orbits.  Recall
from (19.1) the conormal bundle
$$T^*_H(Y) = \bigcup_{\hbox{$H$-orbits $S$}} T^*_S(Y). \eqno(24.1)(a)$$
A covector $(\lambda,y)\in T^*_S(Y)$ is called {\it degenerate} if it belongs
to the closure in $T^*Y$ of some other conormal bundle $T^*_{S'}(Y)$
(cf.~[GM], Definition I.1.8).  (In this case necessarily $S\subset
\overline{S'}$.)  We indicate non-degenerate conormal covectors with the
subscript $reg$, so that
$$T^*_H(Y)_{reg} = \bigcup_{\hbox{$H$-orbits $S$}} T^*_S(Y)_{reg}.
\eqno(24.1)(b)$$
The set of degenerate covectors at $y$ is evidently a closed $H_y$-invariant
cone in the conormal space, so the complementary set is an open cone.  It
follows that
$$T^*_S(Y)_{reg} \simeq H \times_{H_y} T^*_{S,y}(Y)_{reg}, \eqno(24.1)(c)$$
a smooth cone bundle over $S \simeq H/H_y$.

At this point things become much simpler if $H$ has an open orbit on
$T^*_S(Y)$ (or, equivalently, if $H_y$ has an open orbit on the vector space
$T^*_{S,y}(Y)$.  This is automatic in the case of orbits attached to Arthur
parameters (Proposition 22.9) but not in general.  Here are two examples.

\medskip
{\bf Example 24.2.}  Suppose $Y$ is the projective line ${\Bbb C}{\Bbb P}^1$,
and $H$ is the additive group ${\Bbb C}$ (acting by translation on the affine
line, and fixing the point at infinity).  Then the conormal bundle to the fixed
point $y$ is just the cotangent space at $y$:  a one-dimensional vector
space.  The only degenerate covector is $0$.  The action of $H = H_y$ on this
cotangent space is unipotent algebraic, and therefore trivial; so there are
infinitely many orbits of $H_y$, and none is open.  (This example cannot be
realized as a geometric parameter space.)

For a more interesting example, take $Y$ to be the complete flag variety of
Borel subgroups in $SL(4)$, and $H = B$ the standard Borel subgroup of upper
triangular matrices.  (This example --- or rather the essentially equivalent
(Proposition 20.2) $SL(4) \times_H Y$ --- does occur as a geometric
parameter space, in connection with representations of $PGL(4,{\Bbb C})$.) 
According to the Bruhat decomposition, $B$ has 24 orbits on $Y$,
corresponding to the various $B$-conjugacy classes of Borel subgroups. 
Suppose $B'$ is such a subgroup, corresponding to a point $y \in Y$ and an
orbit $S\subset Y$.  Write $N$ and $N'$ for the corresponding  unipotent
radicals, and $T$ for a maximal torus in $B\cap B'$.  Then
$$B_y = B\cap B'= T(N\cap N'),\qquad T^*_{S,y}(Y) \simeq {\frak n}\cap
{\frak n}' \eqno(24.2)(a)$$
(compare Proposition 20.14).  Suppose now $B'$ is chosen so that $N\cap N'$
is abelian.   Then the unipotent radical $N\cap N'$ of $B\cap B'$ acts
trivially on ${\frak n}\cap {\frak n}'$, so 
$$\hbox{ orbits of $B_y$ on $T^*_{S,y}(Y)$ } \simeq \hbox{ orbits of $T$ on
${\frak n}\cap {\frak n}'$. }\eqno(24.2)(b)$$
Now let $B'$ be obtained from $B$ by permuting the first two and the last
two coordinates.  Then
$${\frak n}\cap {\frak n}' = \pmatrix 0&A\cr 0&0\cr\endpmatrix; \eqno(24.2)(c)$$
here we use two by two block matrices.  Consequently ${\frak n}\cap {\frak
n}'$ is abelian and four-dimensional.  Since $T$ is three-dimensional, there
can be no open orbits of $T$ on ${\frak n}\cap {\frak n}'$.  By (24.2)(b), there
are no open orbits of $B_y$ on $T^*_{S,y}(Y)$.
\medskip
{\bf Lemma 24.3.}  {\it Suppose the complex algebraic group $G$ acts on the
smooth irreducible variety $X$.  Write $G_x$ for the stabilizer in $G$ of the
point $x\in X$.
\smallskip
\item{a)}  There is a variety $G_X$ and a morphism $f:G_X\rightarrow X$ (a
group scheme over $X$) with the property that the fiber over $x$ is $G_x$.  
\item{b)}  Put $m = \dim G_X - \dim X$.  There is an open $G$-invariant
subvariety $U_0$ of $X$ with the property that $\dim G_x = m$ for all $x\in
U_0$.  
\item{c)}  There is an open $G$-invariant subvariety $U_1\subset U_0$ with
the property that $G_{U_1} = f^{-1}(U_1)$ is smooth, and the restriction of
$f$ to $G_{U_1}$ is a smooth morphism .
\item{d)}  There is an open $G$-invariant subvariety $U_2 \subset U_1$
with the following property.  Put $G_{U_2} = f^{-1}(U_2)$, and let
$(G_{U_2})_0$ be the union of the identity components $(G_x)_0$ (for $x\in
U_2$).  Then $(G_{U_2})_0$ is (the set of closed points of) an open and
closed subgroup scheme of $G_{U_2}$.
\item{e)}  There is a an \'etale group scheme $A_{U_2}$ over $U_2$, with
the property that the fiber $A_x$ over $x\in U_2$ is isomorphic to the
component group $G_x/(G_x)_0$. 
\item{f)}  There is an open $G$-invariant subvariety $U_3 \subset U_2$ over
which the \'etale group scheme of (e) is finite.  The family of component
groups $A_x = G_x/(G_x)_0$ is in a natural way a local system of finite
groups over $U_3$.  
\item{g)}  For each $x\in U_3$, the (algebraic) fundamental group
$\pi_1(U_3;x)$ acts by automorphisms on $A_x$.  If $y$ is any other point of
$U_3$, then $A_y$ is isomorphic to $A_x$ by an isomorphism that is
canonical up to an automorphism coming from $\pi_1(U_3;x)$.
\medskip}

{\it Proof.}  The definition of the action provides a morphism $a:G\times X
\rightarrow X$.  We define $G_X$ to be the subvariety of $G\times X$
$$G_X = \{\,(g,x)\mid a(g,x) = x\,\}.$$
Obviously this is a closed group subscheme  of $G\times X$.  The morphism
$f$ is just the restriction to $G_X$ of projection on the second factor.  Part
(b) is a general property of morphisms of varieties ([Hartshorne], Exercise
II.3.23).  (More precisely, we choose $U'$ as in the general property, then let
$U_0 = G\cdot U'$.)

To prove (c), we work in the analytic topology; this does not affect the
notion of smooth points of a variety or a morphism.  The strategy we use for
showing that $f$ is smooth near some point $(g_0,x_0)$ in $G_X$ is this. 
We look for a neighborhood $X_0$ of $x_0$, a neighborhood  $Z_0$ of $0$ in
${\Bbb C}^m$, and a holomorphic immersion $\gamma:Z_0 \times
X_0\rightarrow G \times X$, with the following properties: $\gamma(z,x)\in
G_X$, $f\circ\gamma(z,x) = x$, and $\gamma(z_0,x_0) = (g_0,x_0)$.  If such
a map exists, then $f$ is smooth of relative dimension $m$ at $(g_0,x_0)$,
and the image of $\gamma$ is a neighborhood of $(g_0,x_0)$ in $G_X$.  (We
will call $\gamma$ a {\it local trivialization}, even though it is not really
trivializing the whole bundle $G_X$ over $X_0$.)  Conversely, if $G_X$ and
$f$ are smooth of relative dimension $m$ at $(g_0,x_0)$, then the implicit
function theorem guarantees the existence of a local trivialization.

For (c), we may as well assume $X = U_0$, so that $G_x$ has dimension $m$
for all $x$.  We will first show that $f$ is smooth near ${e}\times X \subset
G_X$.  The family of Lie algebras ${\frak g}_x$ may be regarded as a map
from the smooth variety $X$ to the Grassmanian variety of $m$-dimensional
subspaces of ${\frak g}$.  In an analytic neighborhood $X_0$ of any point
$x_0$ on $X$, we can find $m$ holomorphic functions
$$s_i:X_0 \rightarrow {\frak g}\qquad(i=1,\ldots,m)$$
so that $\{s_i(x)\}$ is a basis of ${\frak g}_x$.  For a sufficiently small
neighborhood of zero $Z_0$ in ${\Bbb C}^m$, the map
$$\gamma_e:X_0\times Z_0 \rightarrow G_X, \qquad \gamma(x,z) =
(\exp(\sum z_is_i(x)),x)\in G\times X$$
is a local trivialization at $(e,x_0)$; so $f$ is smooth near ${e}\times X$.
 
Next, fix an irreducible component $C$ of $G_X$, and suppose that the
restriction of $f$ to $C$ is dominant (that is, that $f(C)$ contains an open
set in $X$).  Fix a smooth point $(g_0,x_0)$ of $C$ at which the restriction
of $f$ to $C$ is smooth; this is possible by [Hartshorne], Lemma III.10.5. 
Then on an appropriate analytic neighborhood $X_0$ of $x_0$, we can choose
a holomorphic section
$$\sigma:X_0\rightarrow C,\qquad \sigma(x_0) = (g_0,x_0).$$
Now the group scheme structure $\mu$ (that is, the multiplication in $G$ in
the first factor) gives a holomorphic map
$$\gamma:X_0\times Z_0 \rightarrow V,\qquad \gamma(x,z) = 
\mu(\gamma_e(x,z),\sigma(x)).$$
The map $\gamma$ is a local trivialization at $(g_0,x_0)$, and it follows
that the restriction of $f$ to $C$ is smooth of relative dimension $m$. 

If $C$ is an irreducible component of $V$ on which $f$ is not dominant, then
$f(C)$ is contained in a Zariski-closed $G$-invariant proper subvariety of
$X$.  After removing such subvarieties, we may as well assume that no such
components exist.  Write $V$ for the (Zariski open and dense) set of points
of $G_X$ at which $f$ is smooth of relative dimension $m$.  The argument of
the preceding paragraph now shows that $V$ is a subgroup scheme in $G_X$. 
(We multiply a local section at one smooth point $(g_0,x_0)$ by a local
trivialization at another $(g_0',x_0)$ to get a local trivialization at the
product point $(g_0g_0',x_0)$.)  We also know that $V$ contains the identity
component of $G_x$ for every $x$.  Write $Z$ for the complement of $V_0$. 
Every fiber $Z_x$ is a union of cosets of $G_x$, so has dimension equal to
$m$ (if it is non-empty).  Since $Z$ has dimension less than the dimension
of $G_X$, $f(Z)$ must be contained in a proper $G$-invariant subvariety of
$X$.  After removing this subvariety, we are left with $f$ smooth
everywhere.

For (d), we replace $X$ by $U_1$ as in (c).  Corollary VI-B 4.4 (page 349) of
[SGA 3] guarantees the existence of an open subgroup scheme $(G_X)_0$
with the property that the fibers of $f$ on $(G_X)_0$ are the identity
components $(G_x)_0$.  Let $Z$ be the complement of $(G_X)_0$ in its
closure.  Then $\dim Z < m + \dim X$.  On the other hand, the fibers of $f$ on
$Z$ are unions of cosets of $(G_x)_0$, and therefore have dimension $m$.  It
follows that $f(Z)$ is contained in a proper $G$-invariant closed subvariety
of $X$.  After removing such a subvariety, we find that $(G_X)_0$ is closed,
as we wished to show.  

We will not prove (e) in detail (but see [Artin], Lemma 1.17, where a related
result is also not proved in detail).  Of course $A_X$ is the quotient of
$G_X$ by $(G_X)_0$.  If we had not arranged for $(G_X)_0$ to be closed as
well as open, this quotient would exist only as an algebraic space, and not
as a scheme.

For (f), an \'etale morphism must be finite over an open set ([Hartshorne],
exercise II.3.7).  Part (g) is a general statement about local systems of
finite groups.  Q.E.D.  

\medskip
{\bf Example 24.4.}  Let $G$ be the group of upper triangular two-by-two
matrices of determinant one, and $X$ the Lie algebra of $G$ (with the
adjoint action).  Then the stabilizer of the origin is all of $G$, and so has
dimension 2.  Every other point has a one-dimensional stabilizer.  The
stabilizers of the semisimple points (those with non-zero diagonal entries)
are the (connected) maximal tori of $G$.  The stabilizer of a non-zero
nilpotent element is the subgroup of upper triangular matrices with
diagonal entries $\pm 1$; it has two connected components.  Explicitly,
$$G_X = \left\{\,\left(\pmatrix a&b\cr 0&a^{-1}\cr\endpmatrix,\pmatrix
    x&y\cr
0&-x\endpmatrix\right)\mid 2xab -y(a^2-1) = 0\,\right\}, \eqno(24.4)(a)$$
a three-dimensional irreducible variety.  The morphism $f$ is projection on
the second factor.  By the Jacobian criterion (that is, by inspection of the
differential of the defining equation) $G_X$ is smooth except at the two
points $(\pm I,0)$.  At smooth points of $G_X$, we can test for smoothness
of $f$ by determining whether $df$ is surjective.  The conclusion is that $f$
is smooth except at $x=y=0$.  In the notation of Lemma 24.3, we can
therefore take $U_0 = U_1 = X-{0}$.  The open subgroup scheme discussed in
the proof of (d) is $G_{U_1} - \{x=0,a=-1\}$.  This is dense in $G_{U_1}$, and
the complementary closed set is the two-dimensional variety
$$Z = \{\,a=-1,x=0,y\ne 0\,\}.$$
Its image in $X$ is the one-dimensional cone of non-zero nilpotent
elements; so we take
$$U_2 = \left\{\,\pmatrix x&y\cr 0&-x\endpmatrix\mid x\ne 0\,\right\}.$$
The group scheme $A_{U_2}$ is trivial, so we can take $U_3= U_2$.
\medskip

{\bf Definition 24.5.}  In the setting of Lemma 24.3, the finite group $A_x$
attached to any $x\in U_3$ is called the {\it generic component group} for
the action of $G$ on $X$.  If $G$ is only assumed to be pro-algebraic, then
$A_x$ is pro-finite.  
\medskip

The hypotheses of Lemma 24.3 may be weakened slightly.  For example, we
need not assume that $X$ is smooth; there will in any case be a
$G$-invariant open smooth subvariety.  We can allow $X$ to be reducible, as
long as $G$ permutes the irreducible components transitively.  It would be
nice to have a well-defined ``generic isotropy group,'' but this is not
possible.  The difficulty is that the unipotent radical $U_x$ of $G_x$ may
vary continuously over an open set.  Nevertheless the dimension of $U_x$
and the isomorphism class of the (reductive) quotient $L_x = G_x/U_x$ will
be constant on an open set. We will make no use of these facts, however.

\medskip
{\bf Lemma 24.6.}  {\it Suppose the complex pro-algebraic group $G$ acts on
the smooth irreducible variety $X$.  Write $A = G/G_0$ for the pro-finite
component group of $G$.  Choose an open $G$-invariant subvariety $U\subset
X$ as in Lemma 24.3(f), so that the pro-finite component groups $A_x =
G_x/(G_x)_0$ form a local system over $U$.
\smallskip
\item{a)}  There is a locally constant family of natural homomorphisms
$$i_x:A_x \rightarrow A\qquad (x\in U).$$
The image of these homomorphisms is independent of $x$.
\item{b)}  If $G$ has an open orbit on $X$, then the homomorphisms of (a)
are surjective.
\item{c)}  Suppose ${\cal V}$ is a $G$-equivariant local system of complex
$r$-dimensional vector spaces on $X$.  Then ${\cal V}$ defines a local
system of $r$-dimensional representations of $A_x$. 
\medskip}

{\it Proof.}  The maps in (a) are just
$$i_x:G_x/(G_x)_0 \rightarrow G/G_0.$$
That they are locally constant is clear from the construction of the local
system $A_U$ in Lemma 24.3.  Since $U$ is connected, the image must be
constant.  Clearly the image of $i_x$ consists of the classes of elements in
$G$ that preserve the orbit $G_0\cdot x$.  If $X$ has an open orbit $S$, then
we may take $x$ in $S$.  Since $X$ is irreducible, $S$ must be connected; so
$S = G\cdot x = G_0\cdot x$, and $i_x$ is surjective.  For (c), we apply
Lemma 7.3 to the restriction of ${\cal V}$ to the orbit $G\cdot x$.  That is,
we take the isotropy action of $G_x$ on the fiber ${\cal V}_x$.  Q.E.D.

\medskip
{\bf Definition 24.7.}  Suppose $Y$ is a smooth complex algebraic variety on
which the pro-algebraic group $H$ acts with finitely many orbits.  Fix an
orbit $S$ of $H$ on $Y$, and a point $y\in S$.  Consider the action of $H_y$
on the regular part $T^*_{S,y}(Y)_{reg}$ of the conormal space (cf.~(24.1)),
and choose an open set $U_{S,y} \subset T^*_{S,y}(Y)_{reg}$ as in Lemma
24.3(f).  Notice that 
$$U_S = H\times_{H_y} U_{S,y}$$
is then an open set in $T^*_S(Y)_{reg}$.  Fix a point $\nu\in U_{S,y}$.  The
{\it (local) equivariant micro-fundamental group at $(y,\nu)$} is the
pro-finite group
$$A^{mic}_{y,\nu} = H_{y,\nu}/(H_{y,\nu})_0.$$
Notice that this is just the generic component group for the action of $H_y$
on $T^*_{S,y}(Y)$, or for the action of $H$ on $T^*_S(Y)$ (Definition 24.5). 
By Definition 7.1 and Lemma 24.6, it comes equipped with a natural
homomorphism
$$i_{y,\nu}:A^{mic}_{y,\nu} \rightarrow A^{loc}_y,$$
which is surjective if $H_y$ has an open orbit on $T^*_{S,y}(Y)$ (or,
equivalently, if $H$ has an open orbit on $T^*_S(Y)$).

The {\it equivariant micro-fundamental group for $S$} is
$$A^{mic}_S = A^{mic}_{y,\nu} \qquad ((y,\nu) \in U_S\subset T^*_S(Y)).$$
By Lemma 24.3, it is independent of the choice of $(y,\nu)$ up to
automorphism; if $H$ has an open orbit on $T^*_S(Y)$, then it is independent
of choices up to {\it inner} automorphism.  There is a natural homomorphism
$$i_S:A^{mic}_S \rightarrow A^{loc}_S,$$
which is surjective if $H$ has an open orbit on $T^*_S(Y)$.
\medskip

{\bf Theorem 24.8} ([KK], [GM]).  {\it In the setting (24.1), suppose $P$ is an
$H$-equivariant perverse sheaf on $Y$.  Attached to $P$ there is an
$H$-equivariant local system $Q^{mic}(P)$ of complex vector spaces on
$T^*_H(Y)_{reg}$ (notation (24.1)(b)).  
\smallskip
\item{a)}  The rank of $Q^{mic}(P)$ at any point $(y,\nu)$ of
$T^*_S(Y)_{reg}$ is equal to the multiplicity $\chi_S^{mic}(P)$ of $P$ in the
characteristic cycle of $P$ (Proposition 19.12).
\item{b)}  Suppose that $P$ is supported on the closure of the $H$-orbit $S$. 
Then the restriction of $Q^{mic}(P)$ to $T^*_S(Y)_{reg}$ is the pullback of
$(Q_S^{loc})^{-\dim S}(P)$ (Definition 23.4) by the projection
$T^*_S(Y)_{reg} \rightarrow S$.
\item{c)} $Q^{mic}$ is an exact functor from ${\cal P}(Y,H)$ to
$H$-equivariant local systems on $T^*_H(Y)_{reg}$.
\medskip}
{\bf Corollary 24.9.}  {\it In the setting (24.1), suppose $P$ is an
$H$-equivariant perverse sheaf on $Y$, and $S\subset Y$ is an $H$-orbit. 
Attached to $P$ there is a representation $\tau_S^{mic}(P)$ of the
equivariant micro-fundamental group $A^{mic}_S$ (Definition 24.7).
\smallskip
\item{a)}  The dimension of $\tau_S^{mic}(P)$ is equal to the multiplicity
$\chi_S^{mic}(P)$.
\item{b)}  Suppose that $\xi = (S,\tau)$ is a local complete geometric
parameter for $H$ acting on $Y$ (with $\tau$ an irreducible representation
of $A^{loc}_S$), and $P(\xi)$ is the corresponding irreducible perverse
sheaf.  Then $\tau_S^{mic}(P(\xi)) = \tau\circ i_S$ (Definition 24.7).
\item{c)}  The functor $\tau_S^{mic}$ from $H$-equivariant perverse
sheaves on $Y$ to representations of $A^{mic}_S$ is exact.  In particular, it
gives rise to a map (also denoted $\tau_S^{mic}$) from the Grothendieck
group $K(Y,H)$ (cf.~(7.10)) to virtual representations of $A^{mic}_S$.
\medskip}

This theorem is absolutely fundamental to ``microlocal geometry,'' and it is
well-known to all mathematicians working in the field.  Nevertheless it
appears to be very difficult to find a complete proof in the literature.  We
are certainly not qualified to fill this gap, but we will try to indicate what
is involved.

There are several ways to define the local system $Q^{mic}(P)$.  Kashiwara
and Kawai begin with the  $H$-equivariant  ${\cal D}$-module ${\cal M}$
corresponding to $P$ under Theorem 7.9.  By extension of scalars, they get a
module ${\cal M}^{mic}$ for the sheaf ${\cal E}$ of micro-differential
operators.  The restriction of ${\cal M}^{mic}$ to $T^*_S(Y)_{reg}$ can be
understood completely: it gives rise to the local system that we want ([KK],
Theorem 1.3.1, and [Kashiwara], Theorem 3.2.1).  Unfortunately, even to
formulate their definition of the local system requires not only ${\cal E}$,
but also the much larger sheaf of algebras ${\cal E}^{\Bbb R}$.  Some of the
necessary proofs appear in [Kashiwara], but several key steps (like the
flatness results that make the extensions of scalars exact) may apparently
be found only in [SKK]. 

A closely related possibility is to work with the ${\cal O}_{T^*Y}$-module
${\rom gr}({\cal M})$ (cf.~(19.4)).  We first consider what can be done easily. 
If the filtration of ${\cal M}$ is chosen to be $H$-invariant (as is possible
on the geometric parameter space by Proposition 20.15) then it follows
easily that ${\rom gr}({\cal M})$ is annihilated by the defining ideal of
$T^*_H(Y)$ (see the proof of Propostion 19.12(b)).  We may therefore regard
${\rom gr}({\cal M})$ as a sheaf of modules on $T^*_H(Y)$, which we may
restrict to a single conormal space $T^*_S(Y)$.  Because it is
$H$-equivariant, this restriction is of the form $H\times_{H_y} {\cal N}$
for some $H_y$-equivariant coherent ${\cal O}_{T^*_{S,y}(Y)}$-module
${\cal N}$ (cf.~(24.1)(c)).  There is an open $H_y$-invariant subvariety $U$
of $T^*_{S,y}(Y)$ over which ${\cal N}$ is (the sheaf of sections of) an
$H_y$-equivariant vector bundle ${\cal V}$  (compare [Shaf], Proposition
VI.3.1).  Essentially by definition, the rank of ${\cal V}$ is the multiplicity
of $T^*_S(Y)$ in the characteristic cycle of ${\cal M}$.  If $\nu\in U$ is
arbitrary, we get an isotropy representation $\tau_{y,\nu}$ of $H_{y,\nu}$
on the stalk of ${\cal V}$ at $\nu$.  It is a consequence of Lemma 19.6 that  
the class of $\tau_{y,\nu}$ in the Grothendieck group of virtual
representations of $H_{y,\nu}$ is independent of the choice of good
filtration on ${\cal M}$.  In order to prove at least Corollary 24.9 (which is
all we will use) the main difficulty is to prove that $\tau_{y,\nu}$ must be
trivial on the identity component of $H_{y,\nu}$.  It seems likely that this
can be done in a direct and elementary way, as in the proof of Theorem 8.7
of [Assoc].  We have not done this, however.  It turns out that the
``canonical'' good filtration constructed in [KK], Corollary 5.1.11 has the
property that ${\rom gr}({\cal M})$ is actually a local system on
$T^*_H(Y)_{reg}$; but this uses again the machinery of [SKK].

Another possibility is to use the vanishing cycles functor to construct
$Q^{mic}$.  One basic result is then [Ginsburg], Proposition 7.7.1.  (The proof
given there appears to use analytic microlocal methods extensively.)  This
statement falls short of Theorem 24.8, however, and it is not easy for a
non-expert to evaluate the difficulty of extending it. 

The approach we will adopt is the Morse-theoretic method of [GM].  The
results actually proved in [GM] are not quite as general as we need, but at
least one can find there most of the statements and techniques needed in
general (see in particular sections II.6.3 and II.6.A).  We outline the
construction.  Fix $(y,\nu) \in T^*_S(Y)_{reg}$.  First, we choose a (smooth)
complex analytic submanifold $N$ of $Y$ that meets $S$ transversally at
$y$.  Second, we choose a (real-valued smooth) function $f$ on $Y$ with
$f(y) = 0$ and $df(y) = \nu$.  (The assumptions on $f$ and $\nu$ guarantee
that $f$ is Morse on $N$ near $y$, with respect to the stratification by
$H$-orbits intersected with $N$ ([GM], I.2.1.) Finally, we choose a
Riemannian metric on $Y$, and sufficiently small positive numbers
$\epsilon$ and $\delta$ ([GM], I.3.6). Write $B_\delta(y)$ for the closed ball
of radius $\delta$ about $y$ in $Y$, and define a pair of compact spaces
$$J = N\cap B_\delta(y) \cap f^{-1}[-\epsilon,\epsilon]\supset K = N\cap
B_\delta(y) \cap f^{-1}(-\epsilon) \eqno(24.10)(a)$$
Then the stalk of the local system $Q^{mic}(P)$ at the point $(y,\nu)$ is by
definition
$$Q^{mic}(P)_{y,\nu} = H^{-\dim S}(J,K;P),\eqno(24.10)(b)$$
the hypercohomology of the pair $(J,K)$ with coefficients in the
constructible complex $P$.  That this is a local system on $T^*_H(Y)_{reg}$
is [GM], Proposition II.6.A.1.

We have not been able to find a good reference for Theorem 24.8(a) (which is
stated for example in [GM], II.6.A.4).  One reasonable approach is to drop our
old definition of $\chi_S^{mic}(P)$ completely, replacing it by Theorem
24.8(a) as a definition.  We then need a proof of Theorem 1.31 using the new
definition of $\chi_S^{mic}$.  We will outline such a proof at the end of this
section.

Let us verify (b).  By the transversality of the intersection of $S$ and $N$,
$N\cap B_\delta(y)$ meets $\overline{S}$ (the support of $P$) only in the
point $y$.  In particular, $K$ does not meet the support of $P$ at all. 
Consequently
$$Q^{mic}(P)_{y,\nu} = H^{-\dim S}(\{y\},\emptyset;P) = \hbox{\rom stalk at
$y$ of } H^{-\dim S}(P).$$
By Definition 23.4, this is the stalk of $(Q^{loc}_S)^{-\dim S}(P)$ at $y$, as
we wished to show.

The exactness of $Q^{mic}(P)$ in $P$ (part (c)) is an immediate consequence
of the ``purity''  theorem of Kashiwara-Schapira and Goresky-MacPherson
([KS], section 7.2 and Theorem 9.5.2, or [GM], section II.6.A and Theorem
II.6.4), which says simply that
$$H^i(J,K;P) = 0\qquad (\hbox{\rom $P$ perverse and $i\ne -\dim S$}).
\eqno(24.10)(c)$$
(The proof in [KS] uses [SKK], and the definition of pure is formulated a
little differently.)

This concludes our discussion of Theorem 24.8.  The construction allows us
to formulate an analogue of Definition 23.4.

\medskip
{\bf Definition 24.11.}  Suppose $C$ is any $H$-equivariant constructible
sheaf on $Y$.  (In fact we could take for $C$ any $H$-equivariant
constructible complex on $Y$.)  Then there is attached to $C$ a family
$(Q^{mic})^i(C)$ of $H$-equivariant local systems on $T^*_H(Y)_{reg}$, as
follows.  Fix a point $(y,\nu) \in T^*_S(Y)_{reg}$) as in (24.10), and define
$J\supset K$ as in (24.10)(a) .  Then the stalk of $(Q^{mic})^i(C)$ at $(y,\nu)$
is
$$H^{i-\dim S}(J,K;C).$$
That this is a local system is [GM], Proposition II.6.A.1.  By Lemma 24.6, it
carries a representation $(\tau_S^{mic})^i(C)$ of $A^{mic}_S$.  (When we
wish to emphasize the base point, we will write $(\tau_{y,\nu}^{mic})^i(C)$.) 
The virtual representation of $A^{mic}_S$ in Corollary 24.9(c) is
$$\tau_S^{mic}(C) = \sum_i (-1)^i (\tau_S^{mic})^i(C).$$
\medskip
{\bf Definition 24.12.} Suppose $F \in K(Y,H)$ (notation (7.10)), $S$ is an
$H$-orbit on $Y$, and $\sigma\in A_S^{mic}$.  The {\it microlocal trace of
$\sigma$ on $F$} is 
$$\chi_S^{mic}(F)(\sigma) = {\rom tr}(\tau_S^{mic}(F)(\sigma)).
\eqno(24.12)(a)$$
If $C$ is an $H$-equivariant constructible sheaf, or perverse sheaf, or
constructible complex representing $F$ (cf.~(7.10)(a)), then we write
$\chi_S^{mic}(C)(\sigma)$ instead of $\chi_S^{mic}(F)(\sigma)$.  Explicitly,
fix $(y,\nu)\in T^*_S(Y)_{reg}$, so that $A_S^{mic}$ is the group of
connected components of $H_{y,\nu}$, and choose a representative $s\in
H_{y,\nu}$ for $\sigma$.  Then
$$\chi_S^{mic}(C)(\sigma) = (-1)^{\dim S}\sum (-1)^i {\rom tr}(s\ {\rom on}\
H^i(J,K;C)).$$
(If $s$ preserves the pair $(J,K)$, then its action on the cohomology is the
natural one.  In general it will carry $(J,K)$ to another pair $(J',K')$ of the
same type, and we need to use the canonical isomorphism $H^i(J,K;C)\simeq
H^i(J',K';C)$ explained in [GM] to define the action of $s$.)

\medskip
{\bf Definition 24.13.}  In the setting of Theorem 10.4, fix an equivalence
class $\phi \in \Phi^z(G/{\Bbb R})$ of Langlands parameters (Definition 5.3),
and write $S = S_\phi$ for the corresponding orbit of ${^\vee G}$ on
$X({^\vee G^\Gamma})$ (Definition 7.6).  The {\it micro-component group for
$\phi$} is by definition the ${^\vee G}^{alg}$-equivariant micro-fundamental
group $A_S^{mic}$ (Definition 24.7):
$$A_\phi^{mic,alg} = A_S^{mic}.$$
By Definition 24.7, there is a natural homomorphism
$$i_\phi:A_\phi^{mic,alg} \rightarrow A_\phi^{loc,alg},$$
which is surjective if ${^\vee G}$ has an open orbit on $T^*_S(X({^\vee
G^\Gamma}))$.  To each irreducible representation $\pi \in \Pi^z(G/{\Bbb
R})^{mic}_\phi$ (Definition 19.15), we associate a (possibly reducible)
representation $\tau_\phi^{mic}(\pi)$, as follows.  Let $P(\pi)$ be the
irreducible perverse sheaf corresponding to $\pi$.  Then
$$\tau_\phi^{mic}(\pi) = \tau_S^{mic}(P(\pi))$$
(Corollary 24.9).  This definition makes sense for any irreducible
representation $\pi$ (not necessarily in the micro-packet of $\psi$). 
Corollary 24.9(a) guarantees that $\tau_\phi^{mic}(\pi) \ne 0$ if and only if
$\pi \in \Pi^z(G/{\Bbb R})^{mic}_\phi$.  Following Definition 24.12, we
write for $\sigma\in A_\phi^{mic,alg}$
$$\chi_\phi^{mic}(\pi)(\sigma) = {\rom tr}(\tau_\phi^{mic}(\pi)(\sigma)).$$

\medskip

\medskip
{\bf Theorem 24.14.}  {\it Suppose $(G^\Gamma,{\cal W})$ is an extended
group for $G$, and $({^\vee G^\Gamma},{\cal D})$ is an E-group for the
corresponding inner class of real forms, with second invariant $z$
(Definition 4.6).  Fix an equivalence class $\phi\in \Phi^z(G/{\Bbb R})$ of
Langlands parameters, and define the map $\tau_\psi$ (from irreducible
representations in the micro-packet $\Pi^z(G/{\Bbb R})^{mic}_\phi$ to
representations of $A_\phi^{mic,alg}$) as in Definition 24.13.
\smallskip
\item{a)}  Suppose $\pi$ belongs to the L-packet $\Pi^z_\phi$, and
$\tau^{loc}_\phi(\pi)$ is the corresponding irreducible representation of
$A^{loc,alg}_\phi$ (Definition 5.11 and Theorem 10.4).   Then
$$\tau^{mic}_\phi(\pi) = \tau^{loc}_\phi(\pi) \circ i_\phi.$$
\item{b)} The dimension of $\tau^{mic}_\phi(\pi)$ is the multiplicity
$\chi_\phi^{mic}(\pi)$ (Corollary 19.16).
\medskip}

This is immediate from Corollary 24.9 and the definitions.  In the special
case of Arthur parameters, we get a solution to Problem B of the
introduction.  Here it is.

\medskip
{\bf Definition 24.15.}  In the setting of Theorem 10.4, suppose $\psi \in
\Psi({^\vee G^\Gamma})$ is an Arthur parameter (Definition 22.4), and
$\pi\in \Pi^z_\psi$ is an irreducible representation in the corresponding
Arthur packet (Definition 22.6).  We define a (possibly reducible)
representation $\tau_\psi(\pi)$ of the universal component group
$A_\psi^{alg}$ as follows.  Let $x = p(\phi_\psi) \in X({^\vee G^\Gamma})$ 
be the geometric parameter corresponding to the Langlands parameter
$\phi_\psi$, and $S \subset X({^\vee G^\Gamma})$ the corresponding orbit. 
Define $E_\psi \in T^*_{S,x}({^\vee G^\Gamma})$ as in Proposition 22.9, so
that the orbit of $E_\psi$ is open and dense in $T^*_S({^\vee G^\Gamma})$. 
By Proposition 22.9 and Definition 24.7, $A_\psi^{alg}$ is naturally
identified with the equivariant micro-fundamental group $A_S^{mic}$ for
the action of ${^\vee G}^{alg}$ on $X({^\vee G^\Gamma})$; that is, with
$A_{\phi_\psi}^{mic,alg}$.  We can therefore define
$$\tau_\psi(\pi) = \tau_{\phi_\psi}^{mic,alg}(\pi),\qquad
\chi_\psi(\pi)(\sigma) = {\rom tr}\,\tau_\psi(\pi)(\sigma)$$
(Definition 24.13).
\medskip

We leave to the reader the formulation of a special case of Theorem 24.14
for Arthur packets, in analogy with the deduction of Theorem 22.7 from
Corollary 19.16.

We conclude this section with a proof of Theorem 1.31 using Theorem
24.8(a) as the definition of $\chi_S^{mic}$.  The main point is the following
observation of MacPherson taken from [Ch].

\medskip
{\bf Proposition 24.16.} {\it Suppose $X$ is a compact space with a finite
Whitney stratification ${\cal S} = \{S_j\}$ having connected strata, and $C$
is a complex of sheaves on $X$ constructible with respect to ${\cal S}$.
Define $F_{ij}$ to be the dimension of the stalks of the $i$th cohomology
sheaf $H^iC$ at points of $S_j$, and set
$$F_j = \sum_i (-1)^i F_{ij}.$$
Then the Euler characteristic of the hypercohomology
$$\chi(X;C) = \sum (-1)^i \dim H^i(X;C)$$
depends only on the integers $F_j$.  More precisely, define
$$c_j = \chi_c(S_j) = \sum_i (-1)^i \dim H^i_c(S_j;{\Bbb C}),$$
the Euler characteristic with compact supports of the stratum $S_j$.  Then
$$\chi(X;C) = \sum_j c_jF_j.$$}
\medskip
\noindent The elementary proof is sketched in [GMFP], 11.3.  (Proofs that all
the cohomology groups are finite-dimensional may be found in [BorelIC],
Lemma V.10.13.)  Connectedness of the strata is needed only to make
$F_{ij}$ well-defined; we may replace the $S_j$ by arbitrary unions of
strata, as long as the stalks of $H^iC$ have constant dimension on each
$S_j$.

To prove Theorem 1.31, fix $S$ and any point $(y,\nu)\in T^*_S(Y)_{reg}$. 
Define $J \supset K$ as in (24.10).  These compact sets have Whitney
stratifications so that the various $J\cap S'$  and $K\cap S'$ (with $S'$ an
$H$-orbit containing $S$ in its closure) are unions of strata.  The
cohomology sheaves $H^i(P)$ are locally constant on these sets, of constant
rank $\dim (\tau^{loc}_{S'})^i(P)$ (Definition 23.4).  The alternating sum of
these ranks is $\chi_{S'}^{loc}(P)$ (Definition 23.6):
$$\sum_i (-1)^i \dim H^i(P)_x = \chi_{S'}^{loc}(P)\qquad(x \in J\cap S').
\eqno(24.17)(a)$$
Define
$$c(S,S') = (-1)^{\dim S}\chi_c(J\cap S',K\cap S') = (-1)^{\dim S}\sum_i
(-1)^i\dim H^i_c(J\cap S',K\cap S';{\Bbb C}). \eqno(24.17)(b)$$
Finally, recall that we are defining
$$\chi_S^{mic}(P) = (-1)^{\dim S}\sum_i \dim H^i(J,K;P). \eqno(24.17)(c)$$
Then Theorem 1.31 is immediate from Proposition 24.16 and (24.17).  Q.E.D. 

%\end


%\input mssymb.tex
%\font\Bbb=cmr10
%\font\frak=cmr10
%\hsize=27pc
%\vsize=42pc
%\hoffset=3.5pc
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%\pageno=152
\vskip 3pc
\centerline{\bf 25. A fixed point formula.}
\vskip 1pc
 
We alluded in the introduction to a fixed point formula relevant to the
theory of endoscopic lifting.  In this section we will explain the formula;
the connection with endoscopy will appear in the next section.

We work in the setting of Definition 7.7, so that the pro-algebraic group $G$
acts on the smooth variety $X$ with a finite number of orbits.  We wish to
study the action of an automorphism of finite order on this situation.  That
is, we assume that we are given compatible automorphisms of finite order
$$\sigma:G\rightarrow G,\quad \sigma:X\rightarrow X \eqno(25.1)(a)$$
(A little more formally, we could consider an action of some finite cyclic
group ${\Bbb Z}/n{\Bbb Z}$ on everything, with $\sigma$ the action of the
distinguished generator $1 + n{\Bbb Z}$.)  The compatibility means for
example that
$$(\sigma \cdot g)\cdot (\sigma \cdot x) = \sigma \cdot(g\cdot
x).\eqno(25.1)(b)$$
We fix a subgroup $H$ of $G$, with the property that
$$(G^\sigma)_0 \subset H \subset G^\sigma. \eqno(25.1)(c)$$
We fix also a subvariety $Y$ of $X^\sigma$, with the property that 
$$\hbox{\rom $H\cdot Y  = Y$, and $Y$ is open and closed in $X^\sigma$.}
\eqno(25.1)(d)$$
We write
$$\epsilon:Y \rightarrow X,\qquad \epsilon:H \rightarrow G \eqno(25.1)(e)$$
for the inclusions.  Typically we will also have an $G$-equivariant
constructible complex $C$ on $X$.   We will assume that $C$ is also endowed
with an automorphism $\sigma$ of finite order, compatible with those of
(25.1)(a):
$$ \sigma:C\rightarrow C\eqno(25.1)(f)$$
According to Definition 24.11, we can attach to $C$ a family of
$G$-equivariant local systems $(Q^{mic})^i(C)$ on $T^*_G(X)_{reg}$; the
stalks at a point $(x,\nu)$ are given by
$$(Q^{mic})^i(C)_{x,\nu} = H^{i-\dim G\cdot x}(J,K;C),\eqno(25.1)(g)$$
with $J\supset K$ as in (24.10).  Accordingly we get an action of $\sigma$
on $(Q^{mic})^i(C)$:
$$\sigma:(Q^{mic})^i(C)_{x,\nu} \rightarrow
(Q^{mic})^i(C)_{\sigma\cdot(x,\nu)}. \eqno(25.1)(h)$$
Assume finally that we are given a point $x_0\in Y \subset X$, and a
conormal covector
$$(x_0,\nu_0)\in T^*_G(X)_{reg},\qquad \sigma\cdot(x_0,\nu_0) =
(x_0,\nu_0), \eqno(25.1)(i)$$
so that $\sigma$ defines an automorphism (denoted
$(\tau_{x_0,\nu_0}^{mic})^i(C)(\sigma)$, in analogy with Definition 24.11)
of finite order of each of the finite-dimensional vector spaces
$(Q^{mic})^i(C)_{x_0,\nu_0}$.  In analogy with Definition 24.12, we write
$$\chi_{x_0,\nu_0}^{mic}(C)(\sigma) = \sum (-1)^i {\rom tr}
(\tau_{x_0,\nu_0}^{mic})^i(C)(\sigma), \eqno(25.1)(j)$$
the {\it microlocal trace of $\sigma$ on $C$ at $(x_0,\nu_0)$}.  The problem
we consider is the calculation of this microlocal trace.  Because of
(25.1)(g), it may be regarded as a Lefschetz number --- that is, as an
alternating sum of traces on cohomology groups --- so we may hope to
compute it on the fixed points of $\sigma$.  That is what we will do in
Theorem 25.8.

For our purposes the most important examples of (25.1) arise in the
following way.  Fix an element $s \in G$ of finite order, and put
$$\sigma \cdot g = sgs^{-1} \qquad (g\in G). \eqno(25.2)(a)$$
We make $\sigma$ act on $X$ and $C$ as $s$ does.  In this case
$(\tau_{x_0,\nu_0}^{mic})^i(C)(\sigma)$ is just the action of $s$ on the
stalk of the local system $(Q^{mic})^i(C)$ at the point $(x_0,\nu_0)$. 
Suppose in addition that $(x_0,\nu_0)$ is sufficiently generic that
$$G_{x_0,\nu_0}/(G_{x_0,\nu_0})_0 = A_{x_0,\nu_0}^{mic} \eqno(25.2)(b)$$
(Definition 24.7).  Then the element $s$ (which by assumption (25.1)(i)
belongs to $G_{x_0,\nu_0}$) defines a coset
$$\overline{s} = s(G_{x_0,\nu_0})_0 \in A_{x_0,\nu_0}^{mic}
\eqno(25.2)(c)$$
(Definition 24.7).  It follows that 
$$(\tau_{x_0,\nu_0}^{mic})^i(C)(\sigma) = (\tau_{x_0,\nu_0}^{mic})^i
(C)(\overline{s}) \eqno(25.2)(d)$$
(Definition 24.11).  Taking the alternating sum of traces, we get
$$\chi_{x_0,\nu_0}^{mic}(C)(\sigma) = \chi_{x_0,\nu_0}^{mic}(C)
(\overline{s}). \eqno(25.2)(e)$$
Here the term on the left is defined in (25.1)(j), and that on the right by
Definition 24.12.

In order to formulate the fixed point theorem, we need to know that the pair
$(Y,H)$ of (25.1) satisfies the requirements of Definition 7.7.

\medskip
{\bf Lemma 25.3.} {\it Suppose $Z$ is a smooth algebraic variety, and
$\sigma$ is an algebraic automorphism of $Z$ of finite order.  Write
$Z^\sigma$ for the subvariety of fixed points.  
\smallskip
\item{a)} $Z^\sigma$ is a closed smooth subvariety of $Z$.
\item{b)} The tangent space to $Z^\sigma$ at a fixed point of $\sigma$ may
be identified with the fixed points of the differential of $\sigma$ on the
tangent space to $Z$:
$$T_z(Z^\sigma) = T_z(Z)^{d\sigma}.$$
\medskip}

{\it Proof.}  (This is well-known, and we have omitted many much more
difficult arguments; the reader may take it as an opportunity to relax a
moment.)  Every point $z$ of $Z^\sigma$ belongs to an open affine set $U$,
and therefore to a $\sigma$-invariant affine open set $\bigcap_{n \in {\Bbb
Z}} \sigma^n \cdot U$.  (The intersection is finite since $\sigma$ has finite
order.)  We may therefore assume $Z = {\rom Spec}\,A$ is affine.  Then
$\sigma$ arises from an automorphism (also called $\sigma$) of $A$, also
of finite order.  The subvariety $Z^\sigma$ is defined by the ideal $I$
generated by elements $f-\sigma \cdot f$, with $f\in A$.   Since $\sigma$
acts trivially on $A/I$ and has finite order, we have
$$A/I \simeq A^\sigma/(A^\sigma \cap I).\eqno(25.4)(a)$$
Finally, write ${\frak m}$ for the maximal ideal of the point $z$;
necessarily $I\subset {\frak m}$.  The Zariski cotangent spaces to $Z$ and
$Z^\sigma$ at $z$ are 
$$ T^*_z(Z) = {\frak m}/{\frak m}^2,\qquad T^*_z(Z^\sigma) = {\frak
m}/({\frak m}^2 + I)\simeq {\frak m}^\sigma/({\frak m}^\sigma \cap ({\frak
m}^2 + I)).\eqno(25.4)(b)$$
(The last isomorphism uses (25.4)(a).)  The (equivalent) cotangent version of
(b) in the lemma therefore amounts to
$${\frak m}^\sigma / ({\frak m}^\sigma \cap {\frak m}^2) \simeq {\frak
m}^\sigma/({\frak m}^\sigma \cap ({\frak m}^2 + I)).\eqno(25.4)(c)$$
Here the first space evidently maps surjectively to the second, so the
isomorphism is equivalent to
$$({\frak m}^2 + I)^\sigma \subset {\frak m}^2.$$
Since $\sigma$ preserves both ${\frak m}^2$ and $I$, this in turn is
equivalent to
$$I^\sigma \subset {\frak m}^2 \eqno(25.4)(d)$$
(This says that a $\sigma$-invariant function vanishing on $Z^\sigma$ must
actually vanish to first order there.)  To prove (25.4)(d), suppose $f\in
I^\sigma$.   By the definition of $I$, we can write $f = \sum f_i(g_i -
\sigma \cdot g_i)$.  We can expand each $f_i$ and $g_i$ in eigenfunctions of
$\sigma$.  This gives
$$f = \sum f_j(g_j - \sigma \cdot g_j),\qquad \sigma \cdot f_j =
\lambda_jf_j, \quad \sigma \cdot g_j = \mu_j g_j$$
for some roots of unity $\lambda_j$ and $\mu_j$.  Consequently
$$f = \sum (1-\mu_j)f_j g_j,\qquad \sigma \cdot f = \sum
(1-\mu_j)\lambda_j\mu_j (f_j g_j).$$
It follows that the terms in the expansion of $f$ in eigenfunctions of
$\sigma$ may be found by restricting to the summands with
$\lambda_j\mu_j$ fixed.  Since $f$ is fixed by $\sigma$, we may throw
away all the terms with $\lambda_j \ne (\mu_j)^{-1}$.  Of course we may
also throw away the terms with $\mu_j = 1$.  This  leaves
$$ f = \sum (1-\mu_j)f_jg_j, \qquad \sigma \cdot f_j = (\mu_j)^{-1}f_j,
\quad \sigma \cdot g_j = \mu_j g_j, \qquad \mu_j \ne 1.$$
Therefore
$$f = \sum (1-\mu_j^{-1})^{-1}(f_j - \sigma \cdot f_j)(g_j - \sigma \cdot
g_j),$$
which exhibits $f$ as an element of $I^2 \subset {\frak m}^2$.  This proves
(25.4)(d), and hence (b) of the lemma.  

For (a), we must show that the dimension of the Zariski cotangent space is
locally constant on $Z^\sigma$; for the dimension must jump up at a
singular point.  If $z\in Z^\sigma$, choose functions $f_1,\cdots,f_n$ on $Z$
so that $(df_1,\cdots,df_n)$ is a basis of $T^*_zZ$.  Clearly we may assume
that the $f_i$ are eigenfunctions for  $\sigma$; say $\sigma \cdot f_i =
\lambda_i f_i$.  There is a Zariski open set $U \subset Z$ with the property
that the differentials of the $f_i$ are a basis of the tangent spaces at each
point of $U$.  By (b), it follows that
$$\dim T^*_y(Z^\sigma) = \hbox{\rom number of $i$ such that $\lambda_i =
1$}$$
for all $y \in U\cap Z^\sigma$.  Thus this dimension is constant on a
neighborhood of $z$ in $Z^\sigma$, as we wished to show.  Q.E.D.

This argument can be extended to the case of a reductive group action on a
smooth variety without essential change.

\medskip
{\bf Lemma 25.5.} {\it Suppose $S$ is a homogeneous space for an algebraic
group $G$, and $\sigma$ is an automorphism of finite order of the pair
$(S,G)$.  Then the orbits of the group of fixed points $G^\sigma$ on
$S^\sigma$ are both open and closed; they are finite in number.}
\medskip

{\it Proof.}  The tangent space to $S$ at a point $x$ may be identified (using
the action mapping of (19.1)) with the quotient of Lie algebras
$$T_xS \simeq {\frak g}/{\frak g}_x.$$
Here $G_x$ is the isotropy group of the action at $x$.  If $\sigma \cdot x =
x$, then $\sigma$ must preserve $G_x$, and so act on ${\frak g}_x$.  By
Lemma 25.3(b), 
$$T_x(S^\sigma) \simeq {\frak g}^\sigma/{\frak g}_x^\sigma.$$
This is also the tangent space to the orbit $G^\sigma \cdot x$.  We conclude
that the (smooth) orbit $G^\sigma \cdot x$ has the same dimension at $x$ as
the (smooth) fixed point set $S^\sigma$, and therefore that the orbit
contains a neighborhood of $x$ in $S^\sigma$.  It follows that every orbit of
$G^\sigma$ on  $S^\sigma$ is open.  As $S^\sigma$ is the disjoint union of
orbits of $G^\sigma$, every orbit must also be closed.  Since an algebraic
variety has only finitely many connected components, the lemma follows. 
Q.E.D.

\medskip
{\bf Theorem 25.6.}  {\it Suppose we are in the setting (25.1)(a)--(e).  Then
$Y$ is a smooth algebraic variety on which $H$ acts with finitely many
orbits.  In particular, the map $\epsilon:(Y,H) \rightarrow (X,G)$ of (25.1)
has the properties considered in (7.17).  Fix a point $y\in Y$.
\smallskip
\item{a)}  The conormal bundle to the $H$ action on $Y$ at $y$ (cf.~(19.1))
may be identified naturally as
$$T^*_{H,y}(Y) = \left( T^*_{G,y}(X) \right)^\sigma.$$
\item{b)}  In the identification of (a), we have
$$\left( T^*_{G,y}(X)_{reg} \right)^\sigma \subset T^*_{H,y}(Y)_{reg}.$$
\item{c)}  Suppose $y\in Y$.  Then there is a natural homomorphism of
equivariant fundamental groups (Definition 7.1)
$$A^{loc}(\epsilon):A_y^{loc}(Y,H) \rightarrow A_y^{loc}(X,G).$$
\item{d)}  Suppose $(y,\nu) \in T^*_{H,y}(Y)$; use (a) to identify $(y,\nu)$
with a point of $T^*_{G,y}(X)$.  Assume that these points belong to the open
sets $U_{H\cdot y}$ and $U_{G\cdot y}$ of Definition 24.7.  Then there is a
natural homomorphism of equivariant micro-fundamental groups
$$A^{mic}(\epsilon):A_{y,\nu}^{mic}(Y,H) \rightarrow A_{y,\nu}^{mic}(X,G).$$
\medskip}

{\it Proof.}  Through (a), this is a straightforward consequence of Lemmas
25.3 and 25.5 and the definitions.  For (b), suppose that the normal covector
$(y,\nu)$ to $H\cdot y$ fails to be regular.  By (24.1), $(y,\nu)$ is a limit of
normal covectors to orbits in $Y$ distinct from $H\cdot y$.  By (a), these
may be identified with normal covectors to orbits in $X$.  By Lemma 25.5,
the corresponding points of $X$ must (except for finitely many) belong to
orbits of $G$ distinct from $G\cdot y$.  Consequently $(y,\nu)$ fails to be
regular for $X$, as we wished to show.  The homomorphisms in (c) and (d)
arise from the corresponding inclusions of isotropy groups; part (c) is just
(7.17)(d), and part (d) is similar.  (The regularity hypothesis on $\nu$ is
included only so that we can call the component group a micro-fundamental
group; it is not needed for the proof.)  Q.E.D.

In the setting (25.1), we can introduce categories
$${\cal C}(X,G;\sigma),\qquad {\cal P}(X,G;\sigma),\qquad {\cal
D}(X,G;\sigma) \eqno(25.7)(a)$$
in analogy with Definition 7.7; we include in the objects an automorphism
$\sigma$ of finite order, compatible with $\sigma$ on $(X,G)$.  Thus for
example an object of ${\cal C}(X,G;\sigma)$ is an $G$-equivariant
constructible sheaf on $X$, endowed with an automorphism $\sigma$ of
finite order, and satisfying some obvious compatibility conditions.  In the
special case (25.2), we simply use the categories of Definition 7.7.  There
are obvious analogues of Lemma 7.8 and Theorem 7.9 in this setting.  In
particular, the three categories of (25.7)(a) have a common Grothendieck
group
$$K(X,G;\sigma). \eqno(25.7)(b)$$
By Theorem 25.6 and Proposition 7.18, there are corresponding categories
$${\cal C}(Y,H;\sigma), \qquad {\cal P}(Y,H;\sigma), \qquad {\cal
D}(Y,H;\sigma) \eqno(25.7)(c)$$
and a homomorphism
$$\epsilon^*:K(X,G;\sigma) \rightarrow K(Y,H;\sigma). \eqno(25.7)(d)$$
Now the action of $\sigma$ on $(Y, H)$ is trivial.  It follows that any object
in one of these categories is a direct sum of ``eigenobjects'' on which
$\sigma$ acts by scalar multiplication (necessarily by a root of unity).  For
example, a constructible sheaf $C \in {\rom Ob}\,{\cal C}(Y,H;\sigma)$ can be
written
$$C = \sum_{\lambda\in {\Bbb C}^\times} C_\lambda \eqno(25.7)(e)$$
using the eigenspace decomposition of the action of $\sigma$ on each stalk.
This decomposition is inherited by $K(Y,H;\sigma)$.  (An unnecessarily fancy
way to say this is that $K(Y,H;\sigma)$ is the tensor product of $K(Y,H)$
with the group algebra of the group of roots of unity.)

\medskip
{\bf Theorem 25.8.} {\it Suppose we are in the setting (25.1).  (Thus
$\sigma$ is an automorphism of finite order of a triple $(X,G,C)$; here $G$
has finitely many orbits on the smooth variety $X$, and $C$ is an
$G$-equivariant constructible complex on $X$.)  Then the microlocal trace of
$\sigma$ at the $\sigma$-fixed point $(x_0,\nu_0) \in T^*_G(X)_{reg}$
(cf.~(25.1)(j)) may be computed along the fixed points of $\sigma$.  More
precisely, use Theorem 25.6(b) to identify $(x_0,\nu_0)$ as a point of
$T^*_H(Y)_{reg}$.  Write $\epsilon^* C$ for the restriction to $Y$ of $C$,
with its inherited action of $\sigma$ (Proposition 7.18).  Then
$$\chi_{x_0,\nu_0}^{mic}(C)(\sigma) = (-1)^{\dim G\cdot x_0 - \dim H\cdot
x_0 } \chi_{x_0,\nu_0}^{mic}(\epsilon^* C)(\sigma) \eqno(25.9)(a)$$
(notation (25.1)(j)).  (The trace on the left is for $X$, and on the right for
$Y$.)

Suppose in particular that we are in the setting (25.2).  Then 
$$\chi_{x_0,\nu_0}^{mic}(C)(\overline{s}) = (-1)^{\dim G\cdot x_0 - \dim
H\cdot x_0} \chi_{x_0,\nu_0}^{mic}(\epsilon^* C)
(A^{mic}(\epsilon)(\overline{s})) \eqno(25.9)(b)$$
(notation as in Theorem 25.6(d)).}
\medskip
\noindent We will be most interested in the last case, with $C$ an
irreducible perverse sheaf.  In that case the left side of (25.9)(b) is a value
of a character of a representation.  The complex $\epsilon^* C$ need not be
perverse, however, so the right side is only the value of a character of a
virtual representation.   
\smallskip

{\it Proof.}  Recall from (24.10) the construction of the pair of spaces
$J\supset K$ attached to the regular conormal covector $(x_0,\nu_0)$.  The
normal slice $N$ may be taken (near $x_0$) to be the set of common zeros of
any set of holomorphic functions vanishing at $x_0$, whose differentials
form a basis of the conormal space $T^*_{G,x_0}(X)$.  Obviously we may
choose these functions to be eigenfunctions of $\sigma$; and in this case
$N$ will be preserved by $\sigma$.  Of course we can choose the Riemannian
metric to be preserved by $\sigma$.  Since $(x_0,\nu_0)$ is assumed to be
fixed by $\sigma$, we may replace the smooth function $f$ by the average
of its (finitely many) translates by $\sigma$ without affecting the
requirements $f(x_0)= 0$ and $df(x_0) = \nu_0$.  That is, we may assume
that $\sigma\cdot f = f$.  When the choices are made in this way, the
automorphism $\sigma$ preserves $J\supset K$, and so we may interpret
the microlocal trace directly as a Lefschetz number:
$$\chi_{x_0,\nu_0}^{mic}(C)(\sigma) = (-1)^{\dim G\cdot x_0} \sum
(-1)^i{\rom tr}\,(\sigma\ {\rom on}\ H^i(J,K;C)). \eqno(25.9)(a)$$
By the long exact sequence for the pair, this is also the difference of the
Lefschetz numbers of $\sigma$ on $J$ and $K$ with coefficients in $C$.

By the analysis of tangent spaces in Lemma 25.3, and the choice of $N$ made
above, we see that $N^\sigma$ is a smooth submanifold meeting $H\cdot
x_0$ transversally in the single point $x_0$.  The restriction of $f$ to $Y$ 
still has differential $\nu_0$; so we see that the pair of spaces $J^\sigma
\supset K^\sigma$ may be taken as the ones constructed in $Y$ using
$(x_0,\nu_0)$ (cf.~(24.10)).  Consequently
$$\chi_{x_0,\nu_0}^{mic}(\epsilon^*C)(\sigma) = (-1)^{\dim H\cdot x_0}
\sum (-1)^i{\rom tr}\,(\sigma\ {\rom on}\ H^i(J^\sigma,K^\sigma;\epsilon^*C)).
\eqno(25.9)(b)$$
Again this is a difference of Lefschetz numbers for $J^\sigma$ and
$K^\sigma$ separately.

We now apply to each of $J$ and $K$ the Lefschetz fixed point formula in the
form established in [GMFP].  We take for the ``indicator map'' needed in
[GMFP] the function
$$t(y) = (0,\ \hbox{\rom distance to $Y$ });$$
this is $\sigma$-invariant (since the Riemannian metric is) and therefore
satisfies Goresky and MacPherson's requirements
$$t^{-1}(0,0) = Y,\qquad t_1(\sigma \cdot y) \ge t_1(y),\qquad t_2(\sigma
\cdot y) \le t_1(y).$$
Their local group $A_3^i$ ([GMFP], Definition 4.4) for $J$ is precisely
$$H^i(J^\sigma;\epsilon^* C),$$
and similarly for $K$.  Their Theorem 1 ([GMFP], section 4.7) now says 
$$ \sum (-1)^i{\rom tr}\,(\sigma\ {\rom on}\ H^i(J;C)) = \sum (-1)^i{\rom
tr}\,(\sigma\ {\rom on}\ H^i(J^\sigma;\epsilon^* C)),\eqno(25.9)(c)$$
and similarly for $K$.  Combining (25.9)(a), (b), and (c) gives the formula we
want.  Q.E.D.

Most fixed point theorems (including the one in [GMFP]) are concerned with
the possibility of pathological behavior of the map whose fixed points are
considered, and this is what makes them difficult.  For automorphisms of
finite order things are much simpler, and one should expect the most
na{\"\i}ve results --- for example, that the Lefschetz number of the
automorphism is the Euler characteristic of the fixed point set --- to hold
under very mild hypotheses.  To prove such a statement for finite polyhedra
is an easy exercise.  Our topological skills were insufficient to produce a
direct elementary proof of Theorem 25.8, but we still believe that one
exists.   

%\end\input mssymb.tex
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\centerline{\bf 26. Endoscopic lifting.}
\vskip 1pc
The theory of endoscopic lifting created by Langlands and Shelstad concerns
correspondences from stable characters of various smaller reductive groups
into the (unstable virtual) characters of real forms of $G$.  We begin with a
fairly general setting for Langlands functoriality.  Suppose
$(G^\Gamma,{\cal W})$ and $(H^\Gamma,{\cal W_H})$ are extended groups
(Definition 1.12).  (Recall that this essentially means that $G$ and $H$ are
complex connected reductive algebraic groups endowed with inner classes
of real forms.)  Suppose $({^\vee G^\Gamma},{\cal D})$ and $({^\vee
H^\Gamma},{\cal D}_H)$ are E-groups for these extended groups (Definition
4.6), say with second invariants $z \in Z({^\vee G})^{\theta_Z}$ and $z_H \in
Z({^\vee H})^{\theta_Z}$ respectively.  Suppose we are given an
L-homomorphism 
$$\epsilon:{^\vee H^\Gamma} \rightarrow {^\vee G^\Gamma} \eqno(26.1)(a)$$
(Definition 5.1).  As in Definition 10.10, fix a quotient $Q$ of $\pi_1({^\vee
G})^{alg}$, and form the corresponding quotient
$$1\rightarrow Q \rightarrow {^\vee G}^Q \rightarrow {^\vee G}\rightarrow
1 \eqno(26.1)(b)$$
of ${^\vee G}^{alg}$.  As in (5.13)(b), we can pull this extension back by
$\epsilon$ to
$$1\rightarrow Q \rightarrow {^\vee H}^Q \rightarrow {^\vee H} \rightarrow
1 $$
Define $Q_H$ to be the intersection of $Q$ with the identity component
$({^\vee H}^Q)_0$, so that
$$1\rightarrow Q_H \rightarrow {^\vee H}^{Q_H} \rightarrow {^\vee
H}\rightarrow 1 \eqno(26.1)(c)$$
is a connected pro-finite covering of ${^\vee H}$.  It is easy to see that this
is a quotient of the canonical cover of ${^\vee H}$, so $Q_H$ is naturally a
quotient of $\pi_1({^\vee H})^{alg}$.  Proposition 7.18 provides a natural
homomorphism
$$\epsilon^*:KX({^\vee G^\Gamma})^Q \rightarrow KX({^\vee
H^\Gamma})^{Q_H} \eqno(26.1)(d)$$
(cf.~(7.19)(d)).

As in Theorem 10.11, we now define a subgroup $J \subset
Z(G)^{\sigma_Z,fin}$ so that
$$\widehat{Q} \simeq J, \eqno(26.1)(e)$$
and similarly for $J_H$.  Since $Q_H$ is by its definition a subgroup of $Q$,
$J_H$ may be regarded as a quotient of $J$.  The Langlands classification
theorem provides bijections
$$\Pi^z(G/{\Bbb R})_J \leftrightarrow \Xi^z(G/{\Bbb R})^Q,  \qquad
\Pi^z(H/{\Bbb R})_{J_H} \leftrightarrow \Xi^z(H/{\Bbb R})^{Q_H}.
\eqno(26.1)(f)$$

To formulate Langlands functoriality in this setting, we need to consider
formal complex-linear combinations of representations.  Here is an
appropriate modification of Corollary 1.26.

\medskip
{\bf Theorem 26.2.} {\it Suppose we are in the setting of Theorem 10.11. 
Write
$$\overline{K}_{\Bbb C}\Pi^z(G/{\Bbb R})_J$$
for the set of (possibly infinite) formal complex combinations of irreducible
canonical projective representations of type $z$ (Definition 10.3) of strong
real forms of $G$ of type $J$ (Definition 10.10).  Then $\overline{K}_{\Bbb
C}\Pi^z(G/{\Bbb R})_J$ may be identified with the space of complex-valued
linear functionals on the Grothendieck group $KX({^\vee G^\Gamma})^Q$:
$$\overline{K}_{\Bbb C}\Pi^z(G/{\Bbb R})_J \simeq {\rom Hom}_{\Bbb
Z}(KX({^\vee G^\Gamma})^Q,{\Bbb C}).$$
\medskip}

We will call elements of $\overline{K}_{\Bbb C}\Pi^z(G/{\Bbb R})_J$ {\it
formal complex virtual representations}, or simply {\it virtual
representations} if no confusion can arise.  As in Definition 18.6, we write
$\overline{K}_{{\Bbb C},f}\Pi^z(G/{\Bbb R})_J$ for the virtual
representations involving only finitely many irreducible representations of
each strong real form; these will have well-defined characters (Definition
18.6).

\medskip
{\bf Definition 26.3.}  Suppose we are in the setting (26.1).  {\it Langlands
functoriality} is a linear map from formal complex virtual (canonical
projective of type $z_H$) representations of (strong real forms of type
$J_H$ of) $H$ to formal complex virtual representations of $G$,
$$\epsilon_*:\overline{K}_{\Bbb C}\Pi^{z_H}(H/{\Bbb R})^{Q_H} \rightarrow
\overline{K}_{\Bbb C}\Pi^z(G/{\Bbb R})^Q. \eqno(26.3)(a)$$
It is by definition the transpose of $\epsilon^*$ (cf.~(26.1)(d)) with respect
to the isomorphisms of Proposition 26.2 for $H$ and $G$.  In terms of the
pairing of Definition 15.11, this means that the defining property is
$$\langle \epsilon_* \eta_H,F_G\rangle_G = \langle \eta_H, \epsilon^*F_G
\rangle_H. \eqno(26.3)(b)$$
Here $\eta_H$ is a formal complex virtual representation for $H$, and $F_G$ 
belongs to the Grothendieck group of equivariant constructible sheaves on
$X({^\vee G^\Gamma})$. 
\medskip
 
We can calculate the Langlands functoriality map $\epsilon_*$ using
Definition 15.11 and Theorem 15.12.  To see what it does to standard
representations for $H$, we need to see what the restriction map
$\epsilon^*$ does to irreducible constructible sheaves on $X({^\vee G})$. 
This is easy (Proposition 23.7).  To see what it does to irreducible
representations for $H$, we need to see what $\epsilon^*$ does to
irreducible perverse sheaves.  This is much harder.  Here is a precise
statement.

\medskip
{\bf Proposition 26.4.}  {\it Suppose we are in the setting (26.1).  Fix
complete geometric parameters $\xi_H = (S_H,\tau_H) \in \Xi^{z_H}(H/{\Bbb
R})^{Q_H}$ and $\xi_G = (S_G,\tau_G) \in \Xi^z(G/{\Bbb R})^Q$.  (Here  $S_H
\in \Phi(H/{\Bbb R})$ is an orbit of ${^\vee H}$ on $X({^\vee H^\Gamma})$,
$\tau_H$ is an irreducible representation of the corresponding
$Q_H$-component group $A^{loc,Q_H}_{S_H}$, and similarly for $G$.)
\smallskip
\item{a)} If $\Phi(\epsilon)(S_H) \ne S_G$ (cf.~(7.19)(b)) then the standard
representation $M(\xi_G)$ does not occur in the expression of
$\epsilon_*M(\xi_H)$ in terms of standard representations.
\item{b)}  Suppose $\Phi(\epsilon)(S_H) = S_G$; recall from (7.19)(c) the
induced map $A^{loc}(\epsilon):A^{loc,Q_H}_{S_H} \rightarrow
A^{loc,Q}_{S_G}$.  Then the multiplicity of $M(\xi_G)$ in the expression of
$\epsilon_*M(\xi_H)$ in terms of standard representations is equal to the
multiplicity of $\tau_H$ in the representation $A^{loc}(\epsilon)\circ
\tau_G$ of $A^{loc,Q_H}_{S_H}$, multiplied by the quotient
$e(\xi_G)/e(\xi_H)$ of the Kottwitz invariants (Definition 15.8).
\item{c)} If $\Phi(\epsilon)(S_H)$ is not contained in the closure of $S_G$,
then the irreducible representation $\pi(\xi_G)$ does not occur in the
expression of $\epsilon_*\pi(\xi_H)$ in terms of irreducible
representations.
\item{d)}  Suppose $\Phi(\epsilon)(S_H) \subset \overline{S_G}$.   Recall
from (7.10) the irreducible equivariant perverse sheaf $P(\xi_G)$.  Its
restriction $\epsilon^*P(\xi_G)$ is a ${^\vee H}^{Q_H}$-equivariant
constructible complex on $X({^\vee H^\Gamma})$, and may therefore be
written (in the Grothendieck group $KX({^\vee H^\Gamma})$) as a sum of
irreducible equivariant perverse sheaves, with integer coefficents.  Write
$m$ for the coefficient of $P(\xi_H)$ in this sum.  Then the multiplicity of
$\pi(\xi_G)$ in the expression of $\epsilon_*\pi(\xi_H)$ in terms of
irreducible representations is equal to 
$$m(e(\xi_G)(-1)^{\dim S_G})/(e(\xi_H)(-1)^{\dim S_H}).$$
\medskip}
{\it Proof.}  According to Definition 15.11, the multiplicity of $M(\xi_G)$ in
any formal virtual representation $\eta_G$ is equal to $e(\xi_G)\langle
\eta_G,\mu(\xi_G)\rangle_G$.  Applying Definition 26.3(b), we find that the
multiplicity of $M(\xi_G)$ in $\epsilon_*M(\xi_H)$ is equal to 
$$e(\xi_G)\langle \epsilon_*M(\xi_H),\mu(\xi_G)\rangle_G =
e(\xi_G)\langle M(\xi_H),\epsilon^*\mu(\xi_G)\rangle_H.$$
Applying Definition 15.11 for $H$, we find that the right side is
$e(\xi_G)/e(\xi_H)$ times the multiplicity of $\mu(\xi_H)$ in
$\epsilon^*\mu(\xi_G)$.  This last number is computed by Proposition
23.7(b), which gives (a) and (b).  Parts (c) and (d) are proved in exactly the
same way, using Theorem 15.12 in place of Definition 15.11.  Q.E.D.

In part (d) of the proposition, we could use the alternating sum of the
perverse cohomology groups ${^pH}^i(\epsilon^*P(\xi_G))$ (see [BBD]) to
represent the image of $\epsilon^*P(\xi_G)$ in the Grothendieck group of
equivariant perverse sheaves.  These cohomology groups are equivariant
perverse sheaves, so $m$ is the alternating sum of the non-negative
integers
$$m_i = \hbox{\rom multiplicity of $P(\xi_H)$ in
${^pH}^i(\epsilon^*P(\xi_G))$}.$$
This perhaps sounds a little more concrete, but seems to add nothing in the
way of computability to the formulation in the proposition.

The reader may wonder why we define Langlands functoriality in such
generality, when our applications will be to the very special case of
endoscopy.  One reason is that it seems to us to be easier to understand the
definition in a setting including only what is needed.  Another is that we
expect to find applications more general than endoscopy.  For example, the
cohomological induction functors used to construct discrete series
representations from characters of compact Cartan subgroups implement
some non-endoscopic examples of functoriality; this was critical to the
proof of the Langlands classification theorem.  In any case, the reader is of
course welcome to think of $H$ as an endoscopic group.

We now introduce some virtual representations that behave particularly
well under functoriality. 

\medskip
{\bf Definition 26.5.}  In the setting of Theorem 10.11, suppose $\phi \in
P({^\vee G^\Gamma})$ is a Langlands parameter, and $x \in X({^\vee
G^\Gamma})$ is the corresponding point of the geometric parameter space. 
Write $S$ for the orbit of $x$, so that the ${^\vee G}^Q$-equivariant
fundamental group of $S$ is isomorphic to the $Q$-component group for
$\phi$ (Definition 7.6):
$$A_S^{loc,Q} = A_x^{loc,Q} \simeq A_\phi^{loc,Q}$$
(Definition 7.6).  Fix $\sigma\in A_\phi^{loc,Q}$.  The {\it standard formal
complex virtual representation attached to $\phi$ and $\sigma$} is a sum of
the standard final limit representations in the L-packet $\Pi^z(G/{\Bbb
R})^Q_\phi$, with coefficients given by character values for the
corresponding representation of $A_\phi^{loc,Q}$.  Specifically,
$$\eta^{loc,Q}_\phi(\sigma) = \sum_{\xi = (\phi,\tau) \in \Xi^z(G/{\Bbb
R})^Q} e(\xi){\rom tr}\,\tau(\sigma) M(\xi);$$
the sum is over complete Langlands parameters of type $Q$ with $\phi$
fixed (Definition 5.11 and (5.13)).  We may also write this as
$\eta^{loc,Q}_S(\sigma)$ or $\eta^{loc,Q}_x(\sigma)$.  When we wish to
emphasize the group $G$, we will write $\eta^{loc,Q}_{\phi,G}(\sigma)$. 
Incorporating the definition of $e(\xi)$ (Definition 15.8), we may rewrite
the definition as
$$\eta^{loc,Q}_\phi(\sigma) = \sum_{\tau\in
(A^{loc,Q}_\phi)\widehat{}}{\rom tr} \,\tau(\sigma \overline{z(\rho)})
M(\phi,\tau),$$
a sum over the irreducible representations of the $Q$-component group.

This virtual representation is locally finite (Definition 18.6).  If $\delta$ is
a strong real form of $G$ of type $J$ (Definition 10.10) we can write
$$\eta^{loc,Q}_\phi(\sigma)(\delta) = e(G({\Bbb
R},\delta))\sum_{\scriptstyle \xi = (\phi,\tau) \in \Xi^z(G/{\Bbb R})
\atop\scriptstyle \delta(\xi) = \delta} {\rom tr}\,\tau(\sigma) M(\xi)$$ 
for the part of $\eta^{loc}_\phi(\sigma)$ living on $G({\Bbb R},\delta)$. 
This is a finite combination of irreducible representations with complex
coefficients, so it has a well-defined character
$$\Theta(\eta^{loc,Q}_\phi(\sigma),\delta),$$
a generalized function on $G({\Bbb R},\delta)^{can}$.
\medskip

The term $\overline{z(\rho)}$ appears unnatural here; it is justified by
Shelstad's Theorem 18.12. 

\medskip
{\bf Lemma 26.6.}  {\it In the setting of Definition 26.5, the isomorphism of
Theorem 26.2 identifies the formal complex virtual representation
$\eta^{loc,Q}_\phi(\sigma)$ with the map $\chi_S^{loc,Q}(\cdot)(\sigma)$
(Definition 23.6) that assigns to a constructible equivariant sheaf $C$ the
trace of $\sigma$ acting on the stalk $C_x$.  In terms of irreducible
representations, we therefore have
$$\eta^{loc,Q}_\phi(\sigma) = \sum_{\xi \in \Xi^z(G/{\Bbb R})^Q} e(\xi)
(-1)^{d(\xi)} \chi_S^{loc,Q}(P(\xi))(\sigma)\pi(\xi).$$
Here as usual $P(\xi)$ is the irreducible perverse sheaf and $\pi(\xi)$ the
irreducible representation corresponding to the complete geometric
parameter $\xi$.}
\medskip
\noindent This is proved in exactly the same way as Lemma 18.15; the
$\overline{z(\rho)}$ has disappeared because of its occurrence in Definition
15.11.

\medskip
{\bf Proposition 26.7.}  {\it Suppose we are in the setting (26.1).  Fix a
Langlands parameter $\phi_H \in P({^\vee H^\Gamma})$, so that $\phi_G =
\epsilon\circ\phi_H$ is a Langlands parameter for ${^\vee G^\Gamma}$.  Fix
an element $\sigma_H \in A^{loc,Q_H}_{\phi_H}$, and write 
$$\sigma_G = A^{loc}(\epsilon)(\sigma_H) \in A^{loc,Q}_{\phi_G}$$
(cf. (5.14)(d)). Then Langlands functoriality sends the virtual representation
attached to $\phi_H$ and $\sigma_H$ to the one attached to $\phi_G$ and
$\sigma_G$:
$$\epsilon_*\eta^{loc,Q_H}_{\phi_H}(\sigma_H) =
\eta^{loc,Q}_{\phi_G}(\sigma_G).$$
\medskip}

{\it Proof.}  This is a reformulation of Proposition 23.7(c) in a special case. 
It may also be deduced from Proposition 26.4(a) and (b), using the formulas
in Definition 26.5 to express the standard virtual representations in terms
of standard representations.  Q.E.D. 

The reason this result is so easy is that it has so little content.  We have
shown that Langlands functoriality is computable, but not that it has any
nice properties.

Using the results of section 23, we can now give an unstable generalization
of the microlocal stable characters of section 19.

\medskip
{\bf Definition 26.8.}  In the setting of Theorem 10.11, suppose $\phi \in
P({^\vee G^\Gamma})$ is a Langlands parameter, and $x \in X({^\vee
G^\Gamma})$ is the corresponding point of the geometric parameter space. 
Write $S$ for the orbit of $x$, so that the equivariant micro-fundamental
group of $S$ is isomorphic to the micro-component group for $\phi$
(Definition 24.13):
$$A_S^{mic,Q}  \simeq A_\phi^{mic,Q}.$$
Fix $\sigma\in A_\phi^{mic,Q}$.  The {\it formal complex virtual
representation attached to $\phi$ and $\sigma$} is a sum of the irreducible
representations in the micro-packet $\Pi^z(G/{\Bbb R})^{mic}_{J,\phi}$,
with coefficients given by character values for the corresponding
representation of $A_\phi^{mic,Q}$.  Specifically,
$$\eta^{mic,Q}_\phi(\sigma) = \sum_{(\pi,\delta) \in \Pi^z(G/{\Bbb
R})^{mic}_{J,\phi}}
e(\delta)(-1)^{d(\pi)-d(\phi)}\left(\chi_\phi^{mic}(\pi)(\sigma) \right)\pi$$
(notation 24.13).  As in Definition 26.5, we may replace the Kottwitz
invariant $e(\delta)$ with an extra factor $\overline{z(\rho)}$ next to
$\sigma$.  We may also separate the terms
$\eta^{mic,Q}_\phi(\sigma)(\delta)$ corresponding to each strong real form,
and attach a generalized function to each of these.

Suppose $\psi \in Q({^\vee G^\Gamma})$ is an Arthur parameter (Definition
22.4).  We identify the $Q$-component group $A^Q_\psi$ with
$A_{\phi_\psi}^{mic,Q}$ (Definition 24.15).  This allows us to define
$$\eta_\psi^Q(\sigma) = \eta_{\phi_\psi}^{mic,Q}(\sigma)$$
for any $\sigma \in A^Q_\psi$.  Using the notation of Definitions 22.6 and
24.15, this amounts to
$$ \eta_\psi^Q(\sigma) = \sum_{(\pi,\delta) \in \Pi^z(G/{\Bbb R})_{J,\psi}}
e(\delta)(-1)^{d(\pi)-d(\psi)}\left(\chi_\psi(\pi)(\sigma) \right)\pi.$$

\medskip
{\bf Lemma 26.9.}  {\it In the setting of Definition 26.8, fix a sufficiently
generic point $(x,\nu) \in T^*_S(X({^\vee G^\Gamma}))_{reg}$ (Definition
24.7).  The formal complex virtual representation
$\eta^{mic,Q}_\phi(\sigma)$ corresponds in the isomorphism of Theorem
26.2 to the map $(-1)^{d(\phi)}\chi_{(x,\nu)}^{mic,Q}(\cdot)(\sigma)$
(Definition 24.12) that assigns to a ${^\vee G}^Q$-equivariant perverse sheaf
$P$ the trace of $\sigma$ acting on the stalk of $Q^{mic}(P)$ at $(x,\nu)$,
multiplied by $(-1)^{d(\phi)}$.  In terms of standard representations, we
therefore have
$$\eta^{mic,Q}_\phi(\sigma) = (-1)^{d(\phi)}\sum_{\xi \in \Xi^z(G/{\Bbb R})}
e(\xi)  \chi_S^{mic}(\mu(\xi))(\sigma)M(\xi).$$
\medskip}

{\it Proof.}  We use Theorem 15.12 to compute the pairing of
$\eta^{mic,Q}_\phi(\sigma)$ with the irreducible perverse sheaf $P(\pi)$. 
The result is
$$e(\delta)(-1)^{d(\pi)-d(\phi)}\left(\chi_\phi^{mic}(\pi)(\sigma)
\right)\langle \pi,P(\pi)\rangle = (-1)^{d(\phi)}
\chi_\phi^{mic}(\pi)(\sigma).$$
The first assertion follows.  The second is a consequence, because of
Definition 15.11.  Q.E.D

We should make here a few remarks about the computability of these virtual
characters.  The pa\-ram\-e\-triza\-tion of standard final limit
representations in Theorem 10.11 is fairly concrete; there is no difficulty
in listing all the induced from limits of discrete series representations
corresponding to a given Langlands parameter $\phi$, or in describing the
associated characters of $A^{loc,Q}_\phi$.  In this sense the virtual
representations $\eta_\phi^{loc,Q}(\sigma)$ are very well understood.  The
characters of standard limit representations are computable as well,
although it is not quite so easy to get answers in closed form.  
Nevertheless the generalized functions $\Theta(\eta_\phi^{loc,Q}(\sigma),
\delta)$ may be regarded as known.  To describe $\eta_\phi^{loc,Q}(\sigma)$
in terms of irreducible representations is not very different from
describing the composition series of the standard representations.  This is
accomplished in principle by the Kazhdan-Lusztig algorithms (see sections
15 -- 17), although one can get useful answers in closed form only under
very special circumstances.

The situation for $\eta_\phi^{mic,Q}(\sigma)$ is less satisfactory.  We do
not know even in principle how to compute characteristic cycles of perverse
sheaves, so we cannot compute even the size of the micro-packets of
Definition 19.15 in general.  Nevertheless, we expect them to be moderately
small --- larger than L-packets, but not too much.  It should actually be
simpler to compute characteristic cycles for irreducible perverse sheaves
than for irreducible constructible sheaves (or at least the answer should be
simpler).  The expression of $\eta_\phi^{mic,Q}(\sigma)$ in terms of
standard representations (which would provide explicit formulas for its
character) therefore promises to be very complicated.

We would like a version of Proposition 26.7 for the virtual representations
$\eta_\phi^{mic,Q}(\sigma)$.  It is immediately clear that some additional
hypotheses are required: there is no homomorphism of equivariant
micro-fundamental groups $A^{mic}(\epsilon)$ attached to a general
L-homomorphism, so we cannot even formulate an analogous statement.  But
we found in section 24 geometric conditions under which such a
homomorphism does exist, and even leads to a result like Proposition 23.7
(which was the geometric part of Proposition 26.7).  In the present context,
these conditions lead directly to the Langlands-Shelstad theory of
endoscopic groups.  Before we discuss that theory, it is convenient to
develop a small extension of the results of section 18.  

\medskip
{\bf Definition 26.10.}  Suppose we are in the setting (5.14).  Recall from
Proposition 4.4 the group $Z({^\vee G})^{\theta_Z}$; write $Z({^\vee
G})^{\theta_Z,Q}$ for its preimage in ${^\vee G}^Q$, so that we have a short
exact sequence
$$1 \rightarrow Q \rightarrow Z({^\vee G})^{\theta_Z,Q} \rightarrow Z({^\vee
G})^{\theta_Z} \rightarrow 1.$$
If $\phi$ is any Langlands parameter for ${^\vee G^\Gamma}$, then this
group centralizes the image of $\phi$:
$$Z({^\vee G})^{\theta_Z,Q} \subset {^\vee G}^Q_\phi \eqno(26.10)(a)$$
(notation as in (5.14)).  The {\it universal $Q$-component group for ${^\vee
G^\Gamma}$} is
$$A^Q_{univ}({^\vee G^\Gamma}) = Z({^\vee G})^{\theta_Z,Q}/(Z({^\vee
G})^{\theta_Z,Q})_0. \eqno(26.10)(b)$$
The map of (26.10)(a) provides a natural homomorphism
$$i_{univ}^{loc}:A^Q_{univ}({^\vee G^\Gamma}) \rightarrow A^{loc,Q}_\phi
\eqno(26.10)(c)$$
for every Langlands parameter $\phi$; similarly we get maps into the
various geometrically defined groups of section 7, into the $Q$-component
group for an Arthur parameter, and into the micro-component groups of
Definition 24.13.

Suppose now that we are in the setting of Theorem 10.10, and that
$(\pi,\delta)$ is an irreducible canonical projective representation of type
$z$ of a strong real form $\delta$ of type $J$.  Theorem 10.10 associates to
$(\pi,\delta)$ a complete Langlands parameter
$(\phi(\pi,\delta),\tau(\pi,\delta))$, with $\tau(\pi,\delta)$ an irreducible
representation of $A^{loc,Q}_\phi$.  Define 
$$\tau_{univ}(\pi,\delta) = \tau(\pi,\delta)\circ i_{univ}^{loc}: A^Q_{univ}
({^\vee G^\Gamma}) \rightarrow {\Bbb C}^\times, \eqno(26.10)(d)$$
a one-dimensional character of $A^Q_{univ}$.  In terms of geometric
parameters, $\tau_{univ}(\pi,\delta)$ is the character by which $Z({^\vee
G})^{\theta_Z,Q}$ acts on the stalks of the equivariant local system (or the
equivariant perverse sheaf) corresponding to $(\pi,\delta)$.
\medskip

We have already considered several cases of this definition.  The element
$z(\rho)$ belongs to $Z({^\vee G})^{\theta_Z,Q}$, and Definition 15.8 says
that
$$\tau_{univ}(\pi,\delta)(\overline{z(\rho)}) = e(\delta), \eqno(26.11)(a)$$
the Kottwitz invariant of the real form.  As another example, let $z_0\in
Z(G)^{\sigma_Z,fin}$ be the second invariant of the extended group
$(G^\Gamma,{\cal W})$ (cf.~(3.5)).  Then Lemma 10.9(b) says (in the notation
defined there)
$$\tau_{univ}(\pi,\delta)(\overline{z^Q}) = \chi_{z_0\delta^2}(z^Q)
\qquad(z^Q \in Q). \eqno(26.11)(b)$$
These examples are special cases of the following general fact.

\medskip
{\bf Lemma 26.12.}  {\it In the setting of Definition 26.10, the character
$\tau_{univ}(\pi,\delta)$ of $Z({^\vee G})^{\theta_Z,Q}$ depends only on the
$G$-conjugacy class of $\delta$, and not on the representation $\pi$.  We
may therefore write it as $\tau_{univ}(\delta)$.}
\medskip

We have not found a particularly compelling proof of this result, but one can
proceed along the following lines.  The group $Z({^\vee G})^{\theta_Z,Q}$
must act by scalars on an irreducible ${^\vee G}^Q$-equivariant perverse
sheaf on the geometric parameter space.  By Theorem 15.12, it follows that
$\tau_{univ}(\pi,\delta)$ is unchanged if $\pi$ is replaced by any
composition factor of the standard limit representation having $\pi$ as a
quotient.  In this way (using the subquotient theorem) we can reduce to the
case of ordinary principal series representations of $G({\Bbb
R},\delta)^{can}$.  Since all of these may be regarded as associated to the
same Cartan subgroup and set of positive imaginary roots, it is fairly easy
to apply the definition of $\tau(\pi,\delta)$ directly (see section 13) and
verify the lemma.  We leave the details to the reader.

\medskip
{\bf Definition 26.13.}  Suppose we are in the setting of Definition 26.10,
and $\sigma \in A_{univ}^Q({^\vee G^\Gamma})$.  Using the notation of
(18.3), we now define a function $\zeta_{\sigma}$ on $G({\Bbb
R},*)^{can}_{SR}$ by
$$\zeta_{\sigma}(g,\delta) = \tau_{univ}(\delta)(\sigma) \eqno(26.13)(a)$$
The locally finite formal complex virtual representation $\eta \in
\overline{K}_{{\Bbb C},f}\Pi^z(G/{\Bbb R})_J$ is said to be {\it strongly
$\sigma$-stable} if the function
$$\Theta_{SR}(\eta,*)(\zeta_{\sigma})^{-1} \eqno(26.13)(b)$$
is constant on the fibers of the first projection $p_1:G({\Bbb R},*)^Q_{SR}
\rightarrow G^Q$  (Definition 18.6).  Define
$$\overline{K}_{{\Bbb C},f}\Pi^z(G/{\Bbb R})_J^{\sigma-st}
\eqno(26.13)(c)$$
to be the vector space of these virtual representations.  
\medskip

{\bf Theorem 26.14.}  {\it Suppose we are in the setting of Theorem 10.11,
and $\sigma \in A_{univ}^Q({^\vee G^\Gamma})$ (Definition 26.10). 
\smallskip
\item{a)} The restriction of a strongly $\sigma$-stable formal complex
virtual representation to single strong real form is stable (Definition 18.2).
\item{b)} Suppose $\delta_0$ is a strong real form of $G$ of type $J$, and
$\eta_0$ is a stable virtual representation of $G({\Bbb R},\delta_0)^{can}$
(Definition 18.2).  Then there is a strongly $\sigma$-stable locally finite
formal complex virtual  representation $\eta$ of $G$ with the property that 
$$\Theta_{SR}(\eta,*)(g,\delta_0) = \Theta_{SR}(\eta_0)(g)$$
for every strongly regular element $g \in G({\Bbb R},\delta_0)^{can}$.  If
$\delta_0$ is quasisplit, then $\eta$ is unique.
\item{c)}  The vector space $\overline{K}_{{\Bbb C},f}\Pi^z(G/{\Bbb
R})_J^{\sigma-st}$ has as a basis the set
$\{\,\eta_\phi^{loc,Q}(i_{univ}^{loc}(\sigma))\,\}$ (Definitions 26.5 and
26.10), as $\phi$ varies over $\Phi({^\vee G^\Gamma})$.
\item{d)}  The vector space $\overline{K}_{{\Bbb C},f}\Pi^z(G/{\Bbb
R})_J^{\sigma-st}$ has as a basis the set
$\{\,\eta_\phi^{mic,Q}(i^{mic}_{univ}(\sigma))\,\}$ (Definitions 26.8 and
26.10), as $\phi$ varies over $\Phi({^\vee G^\Gamma})$.
\medskip}

{\it Proof.}  Part(a) follows from Lemma 18.5.  The function
$\zeta_{\sigma}$ is constant on strong real forms and nowhere zero, so (b)
is a formal consequence of Theorem 18.7.  Similarly (c) follows from
Theorem 18.14.  For (d), Theorem 1.31 shows that this set is related to the
basis of (c) by an upper triangular matrix with $\pm 1$ in each diagonal
entry.  Q.E.D.
 
In the setting of Example 19.17, it is not difficult to see that the function
$\zeta_\sigma$ is nothing but the character of the complex formal virtual
representation $\eta^{mic,Q}_{\phi_0}(\sigma)$.  
 
\medskip
{\bf Definition 26.15.}  Suppose ${^\vee G^\Gamma}$ is a weak E-group
(Definition 4.3), and $Q$ is any quotient of $\pi_1({^\vee G})^{alg}$
(Definition 10.10).  A {\it weak endoscopic datum for ${^\vee G^\Gamma}$} is
a pair $(s^Q,{^\vee H^\Gamma})$, subject to the following conditions.
\smallskip
\item{i)} $s^Q \in {^\vee G}^Q$ is a semisimple element, and ${^\vee
H^\Gamma} \subset {^\vee G^\Gamma}$ is a weak E-group.  Write $s$ for the
image of $s^Q$ in ${^\vee G}$.
\item{ii)} ${^\vee H^\Gamma}$ is open in the centralizer of $s$ in ${^\vee
G^\Gamma}$.
\smallskip
In (i), it is understood that the weak E-group structure ${^\vee H^\Gamma}
\rightarrow \Gamma$ is the restriction of that for $G$; that is, that the
inclusion
$$\epsilon:{^\vee H^\Gamma} \rightarrow {^\vee G^\Gamma}$$
is an L-homomorphism (Definition 5.1).  An {\it endoscopic datum} is a weak
endoscopic datum endowed with an E-group structure on ${^\vee H^\Gamma}$
(Definitions 4.6, 4.14).  That is, it is a triple $(s^Q,{^\vee H^\Gamma},{\cal
D}_H)$, subject to the conditions above and
\smallskip
\item{iii)}  ${\cal D}_H$ is a ${^\vee H}$-conjugacy class of elements of
finite order in ${^\vee H^\Gamma} - {^\vee H}$, each acting by conjugation on
${^\vee H}$ as a distinguished involution.
\smallskip
Endoscopic data $(s^Q,{^\vee H^\Gamma},{\cal D}_H)$ and $((s^Q)', ({^\vee
H^\Gamma})', ({\cal D}_H)')$ are {\it equivalent} if there is an element $g^Q
\in {^\vee G}^Q$ with the property that
$$g({^\vee H^\Gamma})' g^{-1} = {^\vee H^\Gamma},\qquad g({\cal D}_H)'
g^{-1} = {\cal D}_H, \qquad g^Q(s^Q)'(g^Q)^{-1} \in s^Q(Z({^\vee
H})^{\theta_Z,Q})_0.$$
Here $g$ is the image of $g^Q$ in ${^\vee G}$.  The last condition says that 
$s^Q$ and $g^Q(s^Q)'(g^Q)^{-1}$ should have the same image in
$A^{Q_H}_{univ}({^\vee H^\Gamma})$.
\medskip

This definition of endoscopic data is compatible with the one given by
Langlands and Shelstad in [LS], 1.2 (although of course they do not use a
covering group); their element $s$ satisfies slightly weaker conditions, but
can obviously be modified within their notion of equivalence to satisfy the
conditions here.  What is more serious is that the notion of equivalence in
[LS] is substantially less stringent: they replace our coset $s^QZ({^\vee
H^Q})_0$ by $sZ({^\vee H^\Gamma})_0Z({^\vee G})$.  This is part of the reason
that the endoscopic lifting of [Shelstad] is defined only up to a sign.  In any
case, we find a natural surjective correspondence from equivalence classes
of endoscopic data in our sense to those of [LS]. 

Notice first of all that every element $s^Q \in Z({^\vee G})^{\theta_Z,Q}$
(notation as in Proposition 4.4) defines an endoscopic datum $(s^Q,{^\vee
G^\Gamma},{\cal D})$.  Two of these are equivalent if and only if $s^Q$ and
$(s^Q)'$ have the same image in $A^Q_{univ}$ (Definition 26.10).  In general,
suppose $s^Q$ is any semisimple element of ${^\vee G}^Q$; write $s$ for its
image in ${^\vee G}$.  Define
$$S= \hbox{\rom \ centralizer of $s$ in ${^\vee G^\Gamma}$,}\qquad S_1 =
\hbox{\rom \ centralizer of $s$ in ${^\vee G}$,}$$
and write $S_0$ for the identity component of $S$.  Then the number of
weak endoscopic data corresponding to $s^Q$ is the number of elements of
order 2 in the finite group $S/S_0$ not belonging to $S_1/S_0$.  

Here is an important construction for weak endoscopic data.  Suppose
$\phi\in P({^\vee G^\Gamma})$ is a Langlands parameter, and $s^Q \in {^\vee
G}^Q_\phi$ is a semisimple element (Definition 5.11).  (One should think of
$s^Q$ as representing an element $\sigma \in A^{loc,Q}_\phi$.)  Let ${^\vee
H}$ be the identity component of the centralizer of $s$ in ${^\vee G}$.  Then
$${^\vee H^\Gamma} = ({^\vee H})\phi(W_{\Bbb R}) \eqno(26.16)$$
is a weak E-group, and the pair $(s^Q, {^\vee H^\Gamma})$ is a set of weak
endoscopic data.

We can now describe the setting for endoscopic lifting.  We begin in the
setting of Theorem 10.11, with an extended group $(G^\Gamma,{\cal W})$ a
corresponding E-group $({^\vee G^\Gamma},{\cal D})$ with second invariant
$z$, and a quotient $Q$ of $\pi_1({^\vee G})^{alg}$.  Fix a set of endoscopic
data
$$(s^Q,{^\vee H^\Gamma},{\cal D}_H) \eqno(26.17)(a)$$
for ${^\vee G^\Gamma}$ as in Definition 26.15, say with second invariant
$z_H$.  (We may also call this endoscopic data for $G$.)  An {\it extended
endoscopic group} attached to this set of data is an extended group
$$(H^\Gamma,{\cal W}_H) \eqno(26.17)(b)$$
with E-group ${^\vee H^\Gamma}$ (Definition 4.3) and second invariant $1$
(Proposition 3.6).  Recall from Definition 1.12 that the extended group
structure ${\cal W_H}$  includes an equivalence class of quasisplit strong
real forms of $H$.  Fix one of these $\delta_0$.  In the terminology of [LS],
$H({\Bbb R},\delta_0)$ is an endoscopic group attached to the endoscopic
data.  The L-homomorphism
$$\epsilon:{^\vee H^\Gamma}\hookrightarrow {^\vee G^\Gamma}
\eqno(26.17)(c)$$
now places us in the setting (26.1), so we can make use of all the
constructions introduced there.  (It is easy to check that in the present
situation the group $Q_H$ of (26.1)(c) is actually equal to $Q$; but this is
something of an accident that disappears in the case of twisted endoscopy,
so we will make no use of it.)  In particular, we have a restriction
homomorphism
$$\epsilon^*:KX({^\vee G^\Gamma})^Q \rightarrow KX({^\vee
H^\Gamma})^{Q_H} \eqno(26.17)(d)$$
and Langlands functoriality
$$\epsilon_*:\overline{K}_{\Bbb C}\Pi^{z_H}(H/{\Bbb R})^{Q_H} \rightarrow
\overline{K}_{\Bbb C}\Pi^z(G/{\Bbb R})^Q. \eqno(26.17)(e)$$
With these definitions, it is easy to check that the element $s^Q$ from the
endoscopic datum belongs to $Z({^\vee H})^{\theta_Z,Q_H}$, and therefore
defines a class
$$\sigma_H \in A^{Q_H}_{univ}({^\vee H^\Gamma}) \eqno(26.17)(f)$$
(Definition 26.10).  

\medskip
{\bf Definition 26.18.}  Suppose we are in the setting (26.17).  {\it
Endoscopic lifting} is the restriction of Langlands functoriality to
$\sigma_H$-stable virtual representations.  That is,
$${\rom Lift}:\overline{K}_{{\Bbb C},f}\Pi^{z_H}(H/{\Bbb
R})_{J_H}^{\sigma_H-st} \rightarrow \overline{K}_{{\Bbb C},f}\Pi^z(G/{\Bbb
R})_J.$$
Fix a quasisplit strong real form $\delta_0$ of $H$ from the distinguished
class given by ${\cal W}_H$, and write $$\overline{K}_{{\Bbb
C},f}\Pi^{z_H}(H({\Bbb R},\delta_0))^{st}$$
for the Grothendieck group of stable complex virtual representations
(projective of type $z_H$) of the group $H({\Bbb R},\delta_0)$.  By Theorem
26.14 (which relies on Shelstad's Theorem 18.7), restriction to $H({\Bbb
R},\delta_0)$ defines an isomorphism
$$\overline{K}_{{\Bbb C},f}\Pi^{z_H}(H/{\Bbb R})_{J_H}^{\sigma_H-st}
\simeq K_{\Bbb C}\Pi^{z_H}(H({\Bbb R},\delta_0))^{st}.$$
Using this identification, we may regard lifting as defined on stable virtual
(canonical projective) representations of the endoscopic group $H({\Bbb
R},\delta_0)$:
$${\rom Lift}_0:K_{\Bbb C}\Pi^{z_H}(H({\Bbb R},\delta_0))^{st} \rightarrow
\overline{K}_{{\Bbb C},f}\Pi^z(G/{\Bbb R})_J.$$
If we fix a strong real form $\delta$ of $G$ of type $J$ (that is, satisfying
$\delta^2 \in Jz_0$,  with $z_0$ the second invariant of the extended group
$G^\Gamma$), then we may project the image of lift on the complex virtual
(projective of type $z$) representations of $G({\Bbb R},\delta)$.  This gives
a map
$${\rom Lift}_0(\delta):K_{\Bbb C}\Pi^{z_H}(H({\Bbb R},\delta_0))^{st}
\rightarrow K_{\Bbb C}\Pi^z(G({\Bbb R},\delta)).$$
It is essentially this restricted map that Langlands and Shelstad use.
\medskip

{\bf Lemma 26.19.}  {\it Endoscopic lifting depends only on the equivalence
class of endoscopic data.  More precisely, suppose $(s^Q,{^\vee
H^\Gamma},{\cal D}_H)$ and $((s^Q)', ({^\vee H^\Gamma})', ({\cal D}_H)')$ are 
endoscopic data for $G$ (Definition 26.15), say with second invariants
$z_H$ and $z_H'$.  Fix associated extended endoscopic groups
$(H^\Gamma,{\cal W}_H)$ and $((H^\Gamma)',{\cal W}'_H)$ (cf.~(26.17)).  Fix
an equivalence $^\vee j$ from $(s^Q,{^\vee H^\Gamma},{\cal D}_H)$ to
$((s^Q)', ({^\vee H^\Gamma})', ({\cal D}_H)')$, implemented by conjugation by
an element $g^Q \in {^\vee G}^Q$ (Definition 26.15).  Finally, choose
distinguished (quasisplit) strong real forms $\delta_0$ and $\delta'_0$ for
$H$ and $H'$.
\smallskip
\item{a)}  The isomorphism $^\vee j$ carries $\sigma_H$ to $\sigma_{H'}$
(cf.~(26.17)(f)).
\item{b)} The isomorphism $^\vee j$ induces an isomorphism
$$j:(H^\Gamma,{\cal W}_H) \buildrel\sim\over \rightarrow
((H^\Gamma)',{\cal W}'_H)$$
unique up to an inner automorphism from $H$.  
\item{c)}  The isomorphism $j$ allows us to identify equivalence classes of
irreducible canonical projective representations of strong real forms of $H$
and $H'$:
$$j:\Pi^{z_H}(H/{\Bbb R})_{J_H}  \buildrel\sim\over \rightarrow
\Pi^{z_{H'}}(H'/{\Bbb R})_{J_{H'}}.$$ 
\item{d)} The isomorphism $j$ induces an isomorphism
$$j_0:H({\Bbb R},\delta_0) \buildrel\sim\over \rightarrow H'({\Bbb
R},\delta'_0)$$
unique up to an inner automorphism from the normalizer of $H'({\Bbb
R},\delta_0)$ in $H$.
\item{e)} The isomorphism $j_0$ allows us to identify stable virtual
representations of $H({\Bbb R},\delta_0)$ and $H'({\Bbb R},\delta'_0)$:
$$j_0:K_{\Bbb C}\Pi^{z_H}(H({\Bbb R},\delta_0))^{st} \buildrel\sim\over
\rightarrow K_{\Bbb C}\Pi^{z_{H'}}(H'({\Bbb R},\delta'_0))^{st}.$$
This isomorphism depends only on $^\vee j$.
\item{f)}  If $\eta_H$ is a $\sigma_H$-stable formal complex virtual
(canonical projective of type $z_H$) representation of strong real forms of
$H$ of type $J_H$, then
$${\rom Lift}(\eta_H) = {\rom Lift}(j(\eta_H)).$$
\item{g)}  If $(\eta_0)_H$ is a complex virtual (canonical projective of type
$z_H$) representation of $H({\Bbb R},\delta_0)$, then
$${\rom Lift}_0((\eta_0)_H) = {\rom Lift}_0(j_0((\eta_0)_H)).$$
\medskip}

{\it Proof.}  Part (a) is immediate from the definition of equivalence and the
definition of $A^{Q_H}_{univ}$ (Definition 26.10).  Part (b) is Proposition
3.6, and then (c) is immediate.  For (d), $j(\delta_0)$ must be a
distinguished strong real form coming from ${\cal W}_{H'}$.  Since these
strong real forms constitute a single conjugacy class, conjugation by an
element of $H'$ carries $j(\delta_0)$ to $\delta'_0$.  The composition of
this inner automorphism with $j$ gives $j_0$.   For (e), we need only know
that the normalizer of $H'({\Bbb R},\delta'_0)$ in $H'$ acts trivially on
stable virtual representations.  This is obvious from the definition of
stability (Definition 18.2).  Parts (f) and (g) are clear from the definitions. 
Q.E.D.

The first serious problem we face is showing that Definition 26.18 agrees
with the definition of [Shelstad], up to a multiplicative constant (which can
be chosen more or less arbitrarily in Shelstad's formulation, and which
depends on the choice of strong real form $\delta$ in ours).    To do this,
notice that both kinds of lifting can be computed explicitly on the basis
elements $\eta_{\phi_H}^{loc,Q_H}(\sigma_H)(\delta_0)$: this is
Proposition 26.7 in our case and Theorem 4.1.1 of [Shelstad] in hers.  In each
case the answer is a sum over the final standard limit representations in
the L-packet $\Pi^z(G({\Bbb R},\delta)_{\phi_G}$, with coefficients given in
an explicit combinatorial way in terms of a single Cartan subgroup. 
Comparing the definitions is a tedious but straightforward exercise (to
which Proposition 13.12 is particularly relevant).  We omit the details.  

Our main theorem on endoscopic lifting concerns the virtual representations
of Definition 26.8.  In order to apply the results of section 25, we need an
element of finite order.

\medskip
{\bf Lemma 26.20.}  {\it Suppose $G$ is a (possibly disconnected) complex
algebraic group, and $a_0\in G$ is a semisimple element.  Define $H =
G_{a_0}$ to be the centralizer of $a_0$ in $G$, $Z(H)$ the center of $H$, and
$Z(H)_s$ the group of semisimple elements in $Z(H)$.  
\smallskip
\item{a)}  The set of elements of finite order is Zariski dense in $Z(H)_s$.
\item{b)}  The set
$$U =\{\,a\in Z(H)_s \mid \dim G_a = \dim H \,\}$$
is Zariski open in $Z(H)_s$ and contains $a_0$.
\item{c)}  There is an element $a' \in a_0(Z(H)_s)_0$ of finite order such
that $H$ is open in the centralizer of $a'$ in $G$.
\medskip}

{\it Proof.}  The identity component $(Z(H)_s)_0$ is a product of copies of
${\Bbb C}^\times$.  The elements of finite order are therefore dense in it. 
We need only show that they meet every other component.  Suppose $b\in
Z(H)_s$.  Since the group of connected components of $Z(H)$ is finite, $b^n
\in (Z(H)_s)_0$ for some positive integer $n$.  Since ${\Bbb C}^\times$ is a
divisible group, there is an element $b_0 \in (Z(H)_s)_0$ with $b_0^n = b^n$. 
Then $bb_0^{-1}$ is an element of order $n$ in the component $b(Z(H)_s)_0$.

For (b), we have $G_a \supset H$ for every $a\in Z(H)_s$.  Consequently 
$$U = \{\,a\in Z(H)_s \mid \det({\rom Ad}(a)|_{{\frak g}/{\frak h}}) \ne
0\,\}.$$
This is obviously an open set.  Part (c) follows from (a) and (b).  Q.E.D.

Suppose we are in the setting (26.17).  By Lemma 26.20, we can find an
element $s' \in sZ({^\vee H^\Gamma})_0$ so that $s'$ has finite order, and 
$$\hbox{\rom ${^\vee H^\Gamma}$ is open in the centralizer of $s'$ in ${^\vee
G^\Gamma}$.} \eqno(26.21)(a)$$
Now $Z({^\vee H^\Gamma})_0$ is just $Z({^\vee H})^{\theta_Z}_0$.  We may
therefore find a preimage
$$(s^Q)' \in s^Q(Z({^\vee H})^{\theta_Z,Q_H})_0 \subset {^\vee G}^Q
\eqno(26.21)(b)$$
for $s'$.  When the pro-finite group $Q$ is infinite, this preimage may not
itself have finite order; but it will have finite order in every algebraic
quotient of ${^\vee H}^Q$, and this is all we need to apply the results from
section 25.  (For example, any particular ${^\vee G}^Q$-equivariant
constructible sheaf on $X({^\vee G^\Gamma})$ will actually be equivariant
for an algebraic quotient of ${^\vee G}^Q$.)  By Definition 26.15,
$$((s^Q)',{^\vee H^\Gamma},{\cal D}_H) \eqno(26.21)(c)$$
is again an endoscopic datum, equivalent to $(s^Q,{^\vee H^\Gamma},{\cal
D}_H)$ by the identity map on ${^\vee H^\Gamma}$.  In this setting, we will
write $\sigma$ for the action of $s'$ on $X({^\vee G^\Gamma})$ or on ${^\vee
G}$ (by conjugation); these are compatible automorphisms of finite order. 
By the choice of $s'$,
$$({^\vee G}^\sigma)_0 \subset {^\vee H} \subset {^\vee G}^\sigma.
\eqno(26.21)(d)$$

\medskip
{\bf Lemma 26.22}  {\it In the setting (26.21), $X({^\vee H^\Gamma})$ is
open and closed in $X({^\vee G^\Gamma})^\sigma$.}
\medskip

{\it Proof.}  It is immediate from the definition of the geometric parameter
space that 
$$X({^\vee G^\Gamma})^\sigma = \{\,(y,\Lambda)\mid y \in {^\vee G}^\sigma,
{\rom Ad}(s')\Lambda = \Lambda\,\}.$$
Because a canonical flat $\Lambda$ is an affine space, and therefore convex,
it contains a fixed point of each linear automorphism of finite order
preserving it.  (Take the center of mass of an orbit.)  Hence
$$X({^\vee G^\Gamma})^\sigma = \{\,(y,\Lambda)\mid y \in {^\vee G}^\sigma,
\Lambda \cap {\frak h} \ne 0\,\}.$$
On the other hand, the image of the closed immersion $X(\epsilon)$
(Corollary 6.21) is
$$\{\,(y,\Lambda)\mid y \in {^\vee H}, \Lambda \cap {\frak h} \ne 0\,\}.$$
Since ${^\vee H}$ is open and closed in ${^\vee G}^\sigma$, the lemma
follows.  Q.E.D.

\medskip
{\bf Proposition 26.23.}  {\it In the setting (26.17), suppose $x \in X({^\vee
H^\Gamma})$; write $\epsilon(x)$ for the corresponding point of $X({^\vee
G^\Gamma})$.
\smallskip
\item{a)}  The element $s\in {^\vee G}$ from the endoscopic datum defines
an automorphism $\sigma$ of the conormal space $T^*_{{^\vee
G},\epsilon(x)} (X({^\vee G^\Gamma}))$ to the ${^\vee G}$-orbit of
$\epsilon(x)$.
\item{b)}  There is a natural isomorphism
$$\epsilon:T^*_{{^\vee H},x}(X({^\vee H^\Gamma})) \buildrel \sim \over
\rightarrow  \left( T^*_{{^\vee G},\epsilon(x)} (X({^\vee G^\Gamma}))
\right)^\sigma.$$ 
\item{c)}  In the isomorphism of (b),
$$\left( T^*_{{^\vee G},\epsilon(x)} (X({^\vee G^\Gamma}))_{reg}
\right)^\sigma \subset \epsilon\left(T^*_{{^\vee H},x}(X({^\vee
H^\Gamma}))_{reg} \right).$$ 
\item{d)}  Suppose $(x,\nu) \in  T^*_{{^\vee H},x}(X({^\vee H^\Gamma}))$, so
that $(\epsilon(x),\epsilon(\nu)) \in  T^*_{{^\vee G},\epsilon(x)} (X({^\vee
G^\Gamma}))$. With notation as in Definition 24.7, assume that
$$(x,\nu) \in U_{{^\vee H}\cdot x},\qquad (\epsilon(x),\epsilon(\nu)) \in
U_{{^\vee G}\cdot \epsilon(x)}.$$
(This is automatic if the orbit ${^\vee G}\cdot (\epsilon(x), \epsilon(\nu))$
is open in the conormal bundle.)  Then there is a natural homomorphism of
micro-component groups
$$A^{mic}(\epsilon):A^{mic,Q_H}_{x,\nu} \rightarrow A^{mic,Q}_{\epsilon(x),
\epsilon(\nu)}.$$
 \medskip}

{\it Proof.}  This result is true, and not difficult to prove, in exactly the
form stated.  For our applications, however, it is enough to know that it is
true after the endoscopic datum is modified in accordance with (26.21). 
(This will change the automorphism $\sigma$, but not its subspace of fixed
points.)  In that case it follows from Theorem 25.6, which we are allowed to
apply by Lemma 26.22.  Q.E.D.

\medskip
{\bf Theorem 26.24.}  {\it Suppose we are in the setting (26.17). Fix a
Langlands parameter $\phi_H \in P({^\vee H^\Gamma})$, so that $\phi_G =
\epsilon\circ\phi_H$ (that is, $\phi_H$ regarded as a map into the larger
group ${^\vee G^\Gamma}$) is a Langlands parameter for $G$.  Write $x_0 \in
X({^\vee H^\Gamma})$ for the point corresponding to $\phi_H$, so that
$\epsilon(x_0)\in  X({^\vee G^\Gamma})$ corresponds to $\phi_G$.  Fix a
generic conormal covector $\nu_0 \in U_{{^\vee H}\cdot x_0}$ (Definition
24.7), so that the micro-component group for $\phi_H$ may be computed at
$(x_0,\nu_0)$:
$$A_{\phi_H}^{mic,Q_H} = A_{(x_0,\nu_0)}^{mic,Q_H}$$
(Definition 24.13).  Using Proposition 26.23, regard
$(\epsilon(x_0),\epsilon(\nu_0))$ as a conormal covector to ${^\vee G}\cdot
\epsilon(x_0)$.  Assume that this covector is also generic, so that
$$A_{\phi_G}^{mic,Q} = A_(\epsilon(x_0),\epsilon(\nu_0))^{mic,Q},$$
and Proposition 26.23 provides a homomorphism
$$A^{mic}(\epsilon):A_{\phi_H}^{mic,Q_H}\rightarrow A_{\phi_G}^{mic,Q}.$$

Define $\sigma_H \in A^{Q_H}_{univ}({^\vee H^\Gamma})$ as in (26.17)(f). 
We write also
$$\sigma^{mic}_{\phi_H} = i^{mic}_{univ}(\sigma_H) \in
A_{\phi_H}^{mic,Q_H},\qquad \sigma^{mic}_{\phi_G} =
A^{mic}(\epsilon)(\sigma^{mic}_{\phi_H}) \in A_{\phi_G}^{mic,Q}.$$
Then endoscopic lifting (Definition 26.18) sends the $\sigma_H$-stable
formal complex virtual representation attached to $\phi_H$ and
$\sigma^{mic}_{\phi_H}$ to the one attached to $\phi_G$ and
$\sigma^{mic}_{\phi_G}$:
$${\rom Lift}(\eta_{\phi_H}^{mic,Q_H}(\sigma^{mic}_{\phi_H})) =
\eta_{\phi_G}^{mic,Q}(\sigma^{mic}_{\phi_G}).$$
In terms of the restricted map of Langlands and Shelstad, we have
$${\rom Lift}_0(\delta)(\eta^{mic,Q_H}_{\phi_H}(\delta_0))  =   e(\delta)
\sum_{\pi \in \Pi^z(G({\Bbb R},\delta))^{mic}_{\phi_G}} (-1)^{d(\pi)-
d(\phi_G)}(\chi_{\phi_G}^{mic,Q}(\pi)(\sigma^{mic}_{\phi_G})) \pi
$$
for each strong real form $\delta$ of type $J$.
\medskip}

{\it Proof.}  By Lemma 26.19, we may replace the endoscopic datum by an
equivalent one without changing ${\rom Lift}$.  By (26.21), we may therefore
assume that $s$ has finite order.  Now regard ${\rom
Lift}(\eta_{\phi_H}^{mic,Q_H} (\sigma^{mic}_{\phi_H}))$ as a linear
functional on constructible ${^\vee G}^Q$-equivariant sheaves (Theorem
26.2), and evaluate it on such a sheaf $C$.  By Lemma 26.9, the result is
$(-1)^{d(\phi_H)}$ times the alternating sum of the traces of
$\sigma^{mic}_{\phi_H}$ on the on the stalks at $(x_0,\nu_0)$ of
$(Q^{mic})^i(\epsilon^*C)$.  By Theorem 25.8, this is equal to
$(-1)^{d(\phi_G)}$ times the alternating sum of the traces of
$\sigma^{mic}_{\phi_G}$ on the stalks at $(\epsilon(x_0),\epsilon(\nu_0))$
of $(Q^{mic})^i(C)$.  By Lemma 26.9 again, this alternating sum is
$\eta_{\phi_G}^{mic,Q}(\sigma^{mic}_{\phi_G})$ evaluated at $C$, as we
wished to show.  Q.E.D.

Here is the solution to Problem D from the introduction.

\medskip
{\bf Theorem 26.25.}  {\it Suppose we are in the setting (26.17). Fix an
Arthur parameter $\psi_H \in Q({^\vee H^\Gamma})$ (Definition 22.4), so
that $\psi_G = \epsilon\circ \psi_H$ (that is, $\psi_H$ regarded as a map
into the larger group ${^\vee G^\Gamma}$) is an Arthur parameter for $G$. 
Write
$$A^{Arthur}(\epsilon):A_{\psi_H}^{Q_H} \rightarrow A_{\psi_G}^Q$$
for the induced map on component groups.  Define
$$\sigma_{\psi_H} = i^{Arthur}_{univ}(\sigma_H) \in
A_{\psi_H}^{Q_H},\qquad \sigma_{\psi_G} =
A^{Arthur}(\epsilon)(\sigma_{\psi_H}) \in A_{\psi_G}^Q.$$
Then endoscopic lifting satisfies
$${\rom Lift}(\eta_{\psi_H}(\sigma_{\psi_H})) = \eta_{\psi_G}
(\sigma_{\psi_G}).$$
In terms of the restricted lifting map of Langlands and Shelstad, 
$${\rom Lift}_0(\delta)(\eta_{\psi_H}(\sigma_{\psi_H})(\delta_0)) =
e(\delta) \sum_{\pi\in \Pi^z(G({\Bbb R},\delta))_{\psi_G}} (-1)^{d(\pi) -
d(\psi_G)} (\chi_{\psi_G}(\pi)(\sigma_{\psi_G}))\pi$$
for each strong real form $\delta$ of $G$ of type $J$.}
\medskip
\noindent In light of the discussion at Definition 24.15, this is just a
special case of Theorem 26.24.

%\end
%\input mssymb.tex
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\centerline{\bf 27. Special unipotent representations.}
\vskip 1pc

Arthur's parameters are in many respects most interesting when they are as
far as possible from being tempered; that is, when the tempered part of the
parameter is as trivial as possible.  The corresponding representations are
the {\it special unipotent representations}.  These were defined already in
[BVunip].  (The theorems of that paper are proved only in the complex case,
but the basic definition works in general.)  In this section we will consider
the unipotent case in more detail, and prove that our new definition of
Arthur's representations agrees with the old one (Corollary 27.13).

The reader may wonder about the qualifying adjective ``special'' in the
terminology.  Arthur's representations should include all the unitary
representations of real forms of $G$ that can appear in spaces of
square-integrable automorphic forms (with respect to congruence
subgroups).  They do not, however, include all the interesting unitary
representations; perhaps the simplest example of one which is omitted is
(either irreducible component of) the metaplectic representation of the
complex rank two symplectic group.  This representation should be an
example of a (still undefined) larger class of interesting unitary
representations, for certain of which we are optimistically reserving the
term ``unipotent.''  A longer discussion (if not a more illuminating one) may
be found in [Orange].

\medskip
{\bf Definition 27.1.}  Suppose ${^\vee G^\Gamma}$ is a weak E-group.  An
Arthur parameter $\psi$ (Definition 22.4) is said to be {\it unipotent} if its
restriction to the identity component ${\Bbb C}^\times$ of $W_{\Bbb R}$ is
trivial.  Write $Q_{unip}({^\vee G^\Gamma})$ for the set of unipotent Arthur
parameters, and $\Psi_{unip}({^\vee G^\Gamma})$ for the corresponding set
of equivalence classes.  Following Definition 22.4, write
$$P_{unip}({^\vee G^\Gamma}) \subset P_{Arthur}({^\vee G^\Gamma}) \subset
P({^\vee G^\Gamma})$$
for the corresponding Langlands parameters $\phi_\psi$, and 
$$\Phi_{unip}({^\vee G^\Gamma}) \subset\Phi_{Arthur}({^\vee G^\Gamma})
\subset \Phi({^\vee G^\Gamma})$$
for their equivalence classes.
\medskip

{\bf Proposition 27.2.}  {\it Suppose ${^\vee G^\Gamma}$ is a weak E-group. 
The set of unipotent Arthur parameters for ${^\vee G^\Gamma}$ (Definition
27.1) may be identified naturally with the set of L-homomorphisms
(Definition 5.1) from the L-group of $PGL(2)$ to ${^\vee G^\Gamma}$. 
Suppose that $\psi$ corresponds to $\epsilon$.  Then the Langlands
parameter associated to $\psi$ (Definition 22.4) is the lift by $\epsilon$ of
the Langlands parameter of the trivial representation of $PGL(2,{\Bbb R})$
(Proposition 5.4).}
\medskip

{\it Proof.}  The dual group of $PGL(2)$ is $SL(2)$.  Since $PGL(2,{\Bbb R})$
is split, the corresponding automorphism $a$ of the based root datum
(Corollary 2.16) is trivial.  The L-group is therefore the direct product of
the dual group and the Galois group $\Gamma$ (section 4):
$$\hbox{\rom L-group of $PGL(2)$ } = SL(2) \times \Gamma.$$
The L-homomorphisms in question are therefore certain homomorphisms
from $SL(2) \times \Gamma$ to ${^\vee G^\Gamma}$.  On the other hand, the
Weil group $W_{\Bbb R}$ modulo its identity component is just $\Gamma$,
so it is immediate from Definitions 22.4 and 27.1 that unipotent parameters
are certain homomorphisms from $\Gamma \times SL(2)$ to ${^\vee
G^\Gamma}$.  One checks immediately that the additional conditions
imposed on these homomorphisms by Definition 22.4 correspond precisely to
those imposed on L-homomorphisms by Definition 5.1.  

Now write $\gamma:W_{\Bbb R} \rightarrow \Gamma$ for the natural
quotient map (Definition 5.2).  The Langlands parameter of the trivial
representation of $PGL(2,{\Bbb R})$ is represented by the homomorphism
$$\phi:W_{\Bbb R} \rightarrow SL(2) \times \Gamma, \qquad \phi(w) =
\left( \pmatrix |w|^{1/2}&0\cr 0&|w|^{-1/2}\cr\endpmatrix,\gamma(w)\right).$$
(This is well-known: in addition to the definitions of sections 11--14, one
must understand exactly which standard representation of $PGL(2,{\Bbb R})$
contains the trivial representation.)  Comparing this with the definition of
$\phi_\psi$ in Definition 22.4 gives the last assertion.  Q.E.D.

\medskip
{\bf Corollary 27.3.}  {\it Suppose ${^\vee G^\Gamma}$ is a weak E-group. 
Fix an algebraic homomorphism
$$\psi_1:SL(2,{\Bbb C}) \rightarrow {^\vee G}, $$
and define
$$S = \hbox{\rom centralizer of $\psi_1(SL(2,{\Bbb C}))$ in ${^\vee
G^\Gamma}$}$$
$$S_0 = S\cap {^\vee G},\qquad S_1 = S - S_0,\qquad S_1^{(2)} = \{\,y_0 \in
S_1 \mid y_0^2 = 1\,\}.$$
Then the set of unipotent Arthur parameters $\psi$ restricting to $\psi_1$
on $SL(2,{\Bbb C})$ may be naturally identified with $S_1^{(2)}$.  Two such
parameters are equivalent if and only if the corresponding elements of
$S_1^{(2)}$ are conjugate under $S_0$.  If the Arthur parameter $\psi$
corresponds to $y_0 \in S_1^{(2)}$, then the Arthur component group for
$\psi$ is the component group of the centralizer $S_0(y_0)$ of $y_0$ in
$S_0$:
$$A_\psi = S_0(y_0)/(S_0(y_0))_0,\qquad A^{alg}_\psi =
S_0(y_0)^{alg}/(S_0(y_0)^{alg})_0$$
(Definition 22.4).}
\medskip
\noindent This is immediate from the definitions.
\medskip
{\bf Corollary 27.4.}  {\it Suppose ${^\vee G^\Gamma}$ is a weak E-group. 
Then the set $\Psi_{unip}({^\vee G^\Gamma})$ of equivalence classes of
unipotent Arthur parameters is finite.}
\medskip

{\bf Proof.}  According to a classical result of Dynkin, the number of
$H$-conjugacy classes of algebraic homomorphisms of $SL(2,{\Bbb C})$ into
a complex reductive algebraic group $H$ is finite.  Applying this to ${^\vee
G}$, we find that there are only finitely many possible $\psi_1$ up to
equivalence.  Now Corollary 27.3 completes the proof. Q.E.D.

\medskip
{\bf Example 27.5.}  Suppose $G=GL(n)$, endowed with the inner class of real
forms including $GL(n,{\Bbb R})$.  Then the L-group is
$${^\vee G^\Gamma} = GL(n,{\Bbb C}) \times \Gamma. \eqno(27.5)(a)$$
A homomorphism $\psi_1$ of $SL(2)$ into ${^\vee G}$ is therefore an
$n$-dimensional representation of $SL(2)$.  Such a representation may be
decomposed as a direct sum of irreducible representations of various
dimensions $p_1 > p_2 > \cdots > p_r$; say the representation of dimension
$p_i$ occurs $m_i$ times, so that $n = \sum m_ip_i$.  It follows from
Schur's lemma that the centralizer in $GL(n)$ of the image of $\psi_1$ is
isomorphic to the product of the various $GL(m_i)$; so in the notation of
Corollary 27.3,
$$S_0 = GL(m_1) \times \cdots \times GL(m_r),\qquad S = S_0\times
\Gamma, \qquad S_1^{(2)} = (\hbox{\rom elements of order $2$}) \times
\{\sigma\}; \eqno(27.5)(b)$$
here $\sigma$ is the non-trivial element of $\Gamma$.  The equivalence
classes of $\psi$ associated to $\psi_1$ therefore correspond to conjugacy
classes of elements of order $2$ in $S_0$.  Now an element of $GL(m)$ has
order $2$ if and only if it is diagonalizable with all eigenvalues $1$ or $-1$. 
There are exactly $m+1$ such conjugacy classes (indexed by the multiplicity
of the eigenvalue $-1$); so we find finally $(m_1+1)\cdots (m_r+1)$
different unipotent Arthur parameters associated to $\psi_1$.

It turns out that the Arthur packet of each of these parameters contains
exactly one representation of $GL(n,{\Bbb R})$.  We now describe these
representations.  Let $P = MN$ be the standard parabolic subgroup of $GL(n)$
with Levi factor
$$ M = GL(p_1)^{m_1} \times \cdots \times GL(p_r)^{m_r}. \eqno(27.5)(c)$$
For each $i$, fix an integer $k_i$ between $0$ and $m_i$.  Consider the
one-dimensional character $\pi_i$ of $GL(p_i)^{m_i}$ defined by
$$\pi_i(g_1,\ldots,g_{m_i}) = {\rom sgn}(\det(g_1g_2\ldots g_{k_i})). 
\eqno(27.5)(d)$$
Set 
$$\pi = {\rom Ind}_{P({\Bbb R})}^{GL(n,{\Bbb R})} (\pi_1\otimes \cdots
\otimes \pi_r) \eqno(27.5)(e)$$
Then $\pi$ is a unipotent representation attached to the Arthur parameter
specified by $\psi_1$ and the various $k_i$.  (This can be verified by
computing $\phi_\psi$; what is also true, but not so immediately clear, is
that there are no other representations of $GL(n,{\Bbb R})$ in the Arthur
packet of $\psi$.)
\medskip
{\bf Example 27.6.}  Suppose $G = G_1 \times G_1$, endowed with the inner
class of real forms including $G_1$ (regarded as a real group).  The L-group
of $G$ is
$${^\vee G^\Gamma} = ({^\vee G}_1 \times {^\vee G}_1)\rtimes \{1,\delta \},
\eqno(27.6)(a)$$
with $\delta$ acting on the first factor by interchanging the two factors
${^\vee G}_1$.   A homomorphism $\psi_1$ of $SL(2)$ into ${^\vee G}$ is
specified by a pair $\psi_1^L$, $\psi_1^R$ of such homomorphisms into
${^\vee G}_1$; the centralizer $S_0$ of its image is a product $S_0^L \times
S_0^R$.  An element $(g^L,g^R)\delta \in {^\vee G^\Gamma}-{^\vee G}$
conjugates $(\psi_1^L,\psi_1^R)$ to $({\rom Ad}(g^L)\psi_1^R,{\rom
Ad}(g^R)\psi_1^L)$.  It follows at once that the set $S_1^{(2)}$ of Corollary
27.3 is empty (and so there are no unipotent Arthur parameters) unless
$\psi_1^L$ is conjugate to $\psi_1^R$.  So we assume that they are
conjugate; after replacing $\psi_1$ by a conjugate, we may even assume
$$\psi_1^L = \psi_1^R,\qquad S_0^L  = S_0^R. \eqno(27.6)(b)$$
An elementary calculation now shows that 
$$S_1^{(2)} = \{\,(x,x^{-1})\delta \mid x \in S_0^L\,\}, \eqno(27.6)(c)$$
and that this set is precisely the $S_0$ conjugacy class of $\delta$.  There
is therefore exactly one equivalence class of unipotent Arthur parameters
associated to $\psi_1$.  The corresponding representations are (as we will
see) the ones considered in [BVunip].
\medskip

Fix now an algebraic homomorphism
$$\psi_1:SL(2,{\Bbb C}) \rightarrow {^\vee G}. \eqno(27.7)(a)$$
Define $\lambda = \lambda_1$ as in (22.8).  Write $\Lambda$ for the
canonical flat through $\lambda$ (Definition 1.7 and (6.4)), and ${\cal O}$
for the orbit of $\lambda$ in ${^\vee {\frak g}}$.  Representations in Arthur
packets attached to unipotent Arthur parameters extending $\psi_1$ must
have infinitesimal character ${\cal O}$, and we wish to apply to them the
results of section 21.  We therefore use the notation of (21.1) and (21.5).  In
particular, 
$$e(\Lambda) = \psi_1(-I) . \eqno(27.7)(b)$$
The automorphism ${\rom Ad}\,e(\lambda)$ acts by $+1$ on the integer
eigenspaces of ${\rom ad}(\lambda)$, and by $-1$ on the half-integer
eigenspaces.  The map $\psi_1$ is called {\it even} if $e(\lambda)$ is
central; that is, if ${\rom ad}(\lambda)$ has no half-integral eigenspaces. 
Define 
$$E = d\psi_1\pmatrix 0&1\cr 0&0\cr\endpmatrix \in {\frak n}(\Lambda) \subset
{^\vee{\frak g}}(\Lambda) \eqno(27.7)(c)$$
as in (22.8).  As in Proposition 22.9, we can use an invariant bilinear form to
identify $({^\vee {\frak g}}(\Lambda)/{\frak p}(\Lambda))^*$ with ${\frak
n}(\Lambda)$, and so regard $E$ as an element of $T^*_{eP(\Lambda)}({\cal
P}(\Lambda)^0)$ or of ${\cal N}^*_{{\cal P}(\Lambda)^0}$ (cf.~(21.5)).

\medskip
{\bf Lemma 27.8.} {\it In the setting (27.7), the element $E$ is ${\cal
P}(\Lambda)^0$-regular (Definition 20.9); that is, it belongs to the
Richardson class ${\cal Z}_{{\cal P}(\Lambda)^0}$ in ${^\vee {\frak
g}}(\Lambda)$ associated to $Y$ (Proposition 20.4).  The moment mapping
$\mu$ has degree 1; equivalently, the only conjugate of ${\frak
p}(\Lambda)$ to which $E$ belongs is ${\frak p}(\Lambda)$ itself.}
\medskip

{\it Proof.}  This is very well-known, but we sketch an argument.  We begin
with a direct construction of the parabolic subalgebra ${\frak p}(\Lambda)$
of ${^\vee {\frak g}}(\Lambda)$ from $E$.  Now ${\frak p}(\Lambda)$ is the
sum of the non-negative eigenspaces of $\lambda$ on ${^\vee {\frak
g}}(\Lambda)$.  Suppose $(\pi,W)$ is any finite-dimensional representation
of $SL(2)$, with $E = d\pi\pmatrix 0&1\cr 0&0\cr\endpmatrix$.  Then the sum of the
non-negative weight spaces of $W$ is
$$W^E + W^{E^2}\cap E\cdot W +  W^{E^3}\cap E^2\cdot W + \cdots 
\eqno(27.9)(a)$$
(It suffices to check this in an irreducible representation $W$, where it is
immediate.)  Consequently
$${\frak p}(\Lambda) = {^\vee {\frak g}}(\Lambda)^{{\rom Ad}\,E} + {^\vee
{\frak g}}(\Lambda)^{({\rom Ad}\,E)^2} \cap ({\rom Ad}\,E){^\vee {\frak
g}}(\Lambda) + \cdots \eqno(27.9)(b)$$
The first consequence of this description is that the centralizer of $E$ in
${^\vee G}(\Lambda)$ must normalize ${\frak p}(\Lambda)$, and
consequently must belong to $P(\Lambda)$:
$${^\vee G}(\Lambda)^E \subset P(\Lambda). \eqno(27.9)(c)$$
In the setting of (27.9)(a), the operator $E$ must carry the sum of the
non-negative weight spaces onto the sum of the positive weight spaces.  In
our case the sum of the positive weight spaces is just the nil radical
${\frak n}(\Lambda)$ of ${\frak p}(\Lambda)$, so 
$${\rom ad}(E){\frak p}(\Lambda) = {\frak n}(\Lambda). \eqno(27.9)(d)$$
This says that the $P(\Lambda)$ orbit of $E$ is open in ${\frak
n}(\Lambda)$, which certainly implies that $E$ is in the Richardson class
for $P(\Lambda)$.  Proposition 20.4 now says that the containment (27.9)(c)
is equivalent to the assertion about the degree of $\mu$.  Q.E.D.

We can now reorganize the classification of unipotent parameters in
Corollary 27.3 to be consistent with Proposition 6.24 and Theorem 21.2. 
Expressed in terms of the corresponding Langlands parameters $\phi_\psi =
\phi(y,\lambda)$ (cf.~(5.8) and (22.8)) the difference is this.  In Corollary
27.3, we fix $\psi_1$ (and therefore $\lambda_1 = \lambda$ and $y_1$) and
look for possible elements $y_0 = yy_1^{-1}$ to go with it.  In Theorem 21.2,
we fix representatives for the possible $y$, and then look for $\lambda$. 

Now the equivalence classes of unipotent Arthur parameters $\psi$ of
infinitesimal character ${\cal O}$ are in one-to-one correspondence with
the equivalence classes of associated Langlands parameters $\phi_\psi$
(Definition 22.4).  These in turn correspond to certain ${^\vee G}$ orbits on
$X({\cal O},{^\vee G^\Gamma})$ (Proposition 1.10) and therefore to certain
$K_j$ orbits on ${\cal P}(\Lambda)^0$ (Proposition 6.24).  It is these latter
orbits that we wish to identify.

\medskip
{\bf Theorem 27.10.}  {\it Suppose we are in the setting (21.1), and that the
orbit ${\cal O}$ arises from a homomorphism of $SL(2,{\Bbb C})$ as in
(27.7).  The following sets are in one-to-one correspondence.
\smallskip
\item{1)} Equivalence classes of unipotent Arthur parameters supported on
$X_j({\cal O},{^\vee G^\Gamma})$.
\item{2)} $K_j$-conjugacy classes of parabolic subgroups $P' \in {\cal
P}(\Lambda)^0$ with the following two properties:
\itemitem{a)} $\theta_j P' = P'$; and
\itemitem{b)} ${\frak n}' \cap {\frak s}_j \cap {\cal Z}_{{\cal
P}(\Lambda)^0} \ne \emptyset$.
Here ${\frak n}'$ is the nil radical of the Lie algebra of $P'$.
\item{3)} $K_j$ orbits on ${\frak s}_j \cap {\cal Z}_{{\cal P}(\Lambda)^0}$.
\smallskip
The Arthur (respectively canonical) component group for a parameter may be
identified with the component group for the stabilizer of a point in the
corresponding orbit of $K_j$ (respectively $K_j^{alg}$) on ${\frak s}_j \cap
{\cal Z}_{{\cal P}(\Lambda)^0}$.}
\medskip
\noindent Notice that the $K_j$-conjugacy classes in (2) are precisely the
``regular orbits'' of $K_j$ on ${\cal P}(\Lambda)^0$ in the sense of Definition
20.17; the bijection between (2) and (3) is a consequence of Theorem 20.18,
although we will prove it again here.

{\it Proof.}  Suppose $S$ is an orbit on $X_j({\cal O},{^\vee G^\Gamma})$
corresponding to a unipotent Arthur parameter.  According to Proposition
6.24, we may choose a point $(y_j,\Lambda')$ in $S$, with $y_j$ one of the
representatives chosen in (21.1)(g) (uniquely determined) and $P' =
P(\Lambda') \in {\cal P}(\Lambda)^0$ (determined up to conjugation by
$K_j$).  We want to show that $P'$ satisfies the conditions in (2).  By the
assumption on $S$, we can find a unipotent Arthur parameter $\psi'$ for
which the corresponding elements $y'$, $\lambda'$, and $E' = E_{\psi'}$ 
(cf.~(22.8)(c)) satisfy
$$y' = y_j, \qquad \lambda' \in \Lambda'. \eqno(27.11)(a)$$
Since $E'$ is in the $+1$ eigenspace of ${\rom ad}\,\lambda'$, it follows that
$E' \in {\frak n}(\lambda')$.  Since ${\rom Ad}(y')E' = -E'$, $E'$ must also be in
the $-1$ eigenspace ${\frak s}_j$ of $\theta_j$.  By Lemma 27.8, $E' \in
{\cal Z}_{{\cal P}(\Lambda)^0}$.  This gives (2)(b).  Since $y'$ and $\lambda'$
commute, $\theta_j$ fixes $P(\lambda') = P'$.  This is (2)(a).

Next, suppose we are given a parabolic $P'$ as in (2); we associate to it the
unique $K_j$ orbit on ${\frak s}_j \cap {\cal Z}_{\cal P}(\Lambda)^0$ that is
dense in ${\frak s}_j \cap {\cal Z}_{\cal P}(\Lambda)^0$.  

Finally, suppose we are given an orbit of $K_j$ on ${\frak s}_j \cap {\cal
Z}_{{\cal P}(\Lambda)^0}$.  Fix a point $E'$ of this orbit, and construct a
homomorphism $\psi_1'$ from $SL(2)$ to ${^\vee G}(\lambda)$ satisfying
$$d\psi_1'\pmatrix 0&1\cr 0&0\cr\endpmatrix = E',\qquad \psi_1'(\theta_{SL(2)}x) =
\theta_j\psi_1'(x). \eqno(27.11)(b)$$
(Here $\theta_{SL(2)}$ is the involutive automorphism conjugation by
$\pmatrix i&0\cr 0&-i\cr\endpmatrix$.)  That this is possible is proved in [KR].  From
[KR] one can also deduce that $\psi_1'$ is unique up to conjugation by $K_j$
(more precisely, by the unipotent radical of the stabilizer of $E'$ in $K_j$). 
Since $E'$ is conjugate by ${^\vee G}(\lambda)$ to our original $E$, it
follows that the element $\lambda'$ attached to $\psi_1'$ must be
conjugate to $\lambda$.  Because $e(\lambda)$ is central in ${^\vee
G}(\lambda)$, it follows that 
$$\psi_1'(-I) = e(\lambda). \eqno(27.11)(c)$$
Define
$$y_0 = y_j \psi_1'\pmatrix -i&0\cr 0&i\cr\endpmatrix. \eqno(27.11)(d)$$
Because of (27.11)(b), $y_0$ commutes with the image of $\psi_1'$. 
Because of (27.11)(c), $y_0^2 = 1$.  By Corollary 27.3, the pair
$(y_0,\psi_1')$ gives rise to a unipotent Arthur parameter; by construction
it depends (up to conjugacy by $K_j$) only on the $K_j$ orbit from which we
began.  

We leave to the reader the easy verification that performing any three of
these correspondences in succession recovers the equivalence class from
which one started.  Q.E.D.

\medskip
{\bf Theorem 27.12.} {\it Suppose we are in the setting (21.1), and that the
orbit ${\cal O}$ arises from a homomorphism of $SL(2,{\Bbb C})$ as in
(27.7).  Fix an equivalence class of unipotent Arthur parameters supported
on $X_j({\cal O},{^\vee G^\Gamma})$, say corresponding to a ${^\vee G}$ orbit
$S$ on $X_j({\cal O},{^\vee G^\Gamma})$.  Write $S_K$ for the corresponding
orbit of $K_j$ on $Y$, and ${\cal Z}_S$ for the corresponding orbit of $K_j$
on ${\frak s}_j \cap {\cal Z}_{{\cal P}(\Lambda)^0}$ (Theorem 27.10).  Then
the bijections of Theorem 21.2 identify the following sets:
\smallskip
\item{1)} irreducible $({^\vee {\frak g}}(\Lambda),K^{alg}_j)$-modules $M$
annihilated by $I_{{\cal P}(\Lambda)^0}$, with the property that ${\cal Z}_S
\subset {\rom Ch}(M)$ (notation (20.5));
\item{2)} irreducible $K^{alg}_j$-equivariant ${\cal D}_{{\cal
P}(\Lambda)^0}$-modules ${\cal M}$, with the property that
$T^*_{S_K}({\cal P}(\Lambda)^0) \subset {\rom Ch}({\cal M})$ (Definition
19.9);  and 
\item{3)} irreducible ${^\vee G}^{alg}$-equivariant ${\cal D}$-modules
${\cal N}$ on $X_j({\cal O},{^\vee G^\Gamma})$, with the property that
$T^*_S(X_j) \subset {\rom Ch}({\cal N})$.
\smallskip
Suppose in addition we are in the setting of Theorem 10.4.  Then each of
these sets is in bijection with
\smallskip
\item{4)} $\Pi^z(G/{\Bbb R})_\psi$; that is, equivalence classes of
irreducible canonical projective representations of type $z$ of strong real
forms of $G$, belonging to the Arthur packet for $\psi$.
\smallskip
In cases (1)--(3), the bijections identify the corresponding multiplicities in
characteristic cycles.  Their common value (a positive integer) is equal to
the weight $\chi_\psi(\pi)$ of Theorem 22.7 for the representation in (4).}

\medskip
{\bf Proof.}  The equivalence between (1) and (2) is given by Theorem 20.18;
between (2) and (3) by Proposition 21.4; and between (3) and (4) by
Definition 22.6 (see Definitions 19.13 and 19.15).  Q.E.D.

One can extend Theorem 27.12 from multiplicities to the component group
representations discussed in Definition 24.15.  For the relation between (3)
and (4) this is a matter of definition again, and for (2) and (3) it is
straightforward.  For the relation between (1) and (2) there is a problem,
however.  In the setting of (1) we have referred to [Assoc] for a definition of
the component group representation.  That definition is most easily related
to the microdifferential operator definition in the setting of (2), which we
elected in section 24 not to discuss.  For this reason we will not formulate
(or use) such an extension.  

The bijection between (1) and (4) in Theorem 27.12 can be formulated using
just the results of [IC4].  In this form it was proposed several years ago by
Barbasch and Vogan (in various lectures) as the definition of the Arthur
packet attached to a unipotent parameter.  They also observed the following
corollary, which appeared in [BVunip], Definition 1.17 as the definition of
special unipotent representations in the complex case.

\medskip
{\bf Corollary 27.13.}  {\it Suppose that the infinitesimal character ${\cal
O}$ (cf.~Lemma 15.4) arises from a homomorphism of $SL(2,{\Bbb C})$ into
the dual group as in (27.7).  Write $J_{max}({\cal O})$ for the maximal
primitive ideal in $U({\frak g})$ of infinitesimal character ${\cal O}$. 
Suppose $\pi$ is an irreducible canonical projective representation of type
$z$ of a strong real form of $G$, having infinitesimal character ${\cal O}$. 
Then $\pi$ is a special unipotent representation --- that is,  $\pi$ belongs
to an Arthur packet $\Pi^z(G/{\Bbb R})_{\psi}$ for some unipotent parameter
$\psi$ --- if and only if $\pi$ is annihilated by $J_{max}({\cal O})$.}
\medskip

{\it Proof.}  This follows from Theorem 27.12, Theorem 21.6, and Theorem
21.8.  Q.E.D.

To say that a representation has a large annihilator amounts to saying that
the representation is small (see [GKDim]).  The condition in Corollary 27.13
is therefore equivalent to requiring the Gelfand-Kirillov dimension of $\pi$
to be minimal, or to requiring the associated variety of the Harish-Chandra
module of $\pi$ to be contained in the closure of a certain complex nilpotent
orbit in ${\frak g}^*$ (namely the associated variety of $J_{max}({\cal O})$). 
The techniques of [BVPC] make it possible to compute exactly how many
such representations exist; that is, to find the order of the union of the
various unipotent Arthur packets for a single real form and infinitesimal
character.  Different Arthur packets may overlap, however, and we do not
know in general how to calculate the cardinality of an individual one.

\medskip
{\bf Example 27.14.}  Suppose $G = Sp(4)$ (by which we mean a group of rank
2), endowed with the inner class of real forms including $Sp(4,{\Bbb R})$. 
The L-group is
$${^\vee G^\Gamma} = SO(5,{\Bbb C}) \times \Gamma. \eqno(27.14)(a)$$
Henceforth we will drop the ${\Bbb C}$; since we consider only complex
orthogonal groups, no confusion should arise.  The group $SL(2)$ is a double
cover of $SO(3)$; composing the covering map with the inclusion of $SO(3)$
in $SO(5)$ gives a homomorphism
$$\psi_1:SL(2) \rightarrow SO(5) \simeq {^\vee G}. \eqno(27.14)(b)$$
The centralizer in $SO(5)$ of the image of $\psi_1$ is evidently
$$S_0 = \left\{\pmatrix \epsilon I_3&0\cr 0&A\cr\endpmatrix\mid A\in O(2),\det A =
\epsilon \right\} \simeq O(2). \eqno(27.14)(c)$$
Here $I_3$ is the 3 by 3 identity matrix, and $\epsilon = \pm 1$.  Just as in
Example 27.5, we find that the equivalence classes of unipotent Arthur
parameters $\psi$ associated to $\psi_1$ correspond to conjugacy classes
of elements of order 2 in $S_0$.  There are exactly three such classes,
represented by the elements
$$s_{++} = \pmatrix 1&0\cr 0& 1\cr\endpmatrix,\qquad s_{+-} = \pmatrix
1&0\cr
0&-1\cr\endpmatrix,\qquad s_{--} = \pmatrix -1&0\cr 0& -1\cr\endpmatrix \eqno(27.14)(d)$$
in $O(2) \simeq S_0$.  Write $\psi_{++}$, etc., for the corresponding Arthur
parameters.  The corresponding centralizers are just the centralizers in
$O(2)$ of these elements, namely
$$S_{\psi_{++}} = O(2),\qquad S_{\psi_{+-}} = O(1)\times O(1),\qquad
S_{\psi_{--}} = O(2).  \eqno(27.14)(e)$$
The component groups are therefore
$$A_{\psi_{++}} = {\Bbb Z}/2{\Bbb Z},\qquad A_{\psi_{+-}} = {\Bbb Z}/2{\Bbb
Z} \times {\Bbb Z}/2{\Bbb Z}, \qquad A_{\psi_{--}} = {\Bbb Z}/2{\Bbb Z}.
\eqno(27.14)(f)$$

We also want to calculate the various $A_\psi^{alg}$.  The algebraic
universal cover ${^\vee G}^{alg}$ is the double cover $Spin(5)$, which is also
isomorphic to $Sp(4)$.  Perhaps the easiest way to calculate $S_0^{alg}$ is
to identify the lift $\psi_1^{alg}:SL(2) \rightarrow Sp(4)$.  This turns out to
be the diagonal map of $Sp(2)$ (which is isomorphic to $SL(2)$) into
$Sp(2)\times Sp(2)$, followed by the obvious embedding of $Sp(2)\times
Sp(2)$ into $Sp(4)$.  From this one can deduce that $S_0^{alg} \simeq O(2)$;
the covering map
$$O(2)\simeq S_0^{alg} \rightarrow S_0 \simeq O(2) \eqno(27.15)(a)$$
is the double cover of $SO(2)$ on the identity component.  It follows that 
$$S^{alg}_{\psi_{++}} = O(2),\qquad S^{alg}_{\psi_{+-}} = \hbox{\rom
quaternion group of order 8,}\qquad S^{alg}_{\psi_{--}} = O(2) 
\eqno(27.15)(b)$$
$$A^{alg}_{\psi_{++}} = A_{\psi_{++}}, \qquad A^{alg}_{\psi_{+-}} =
\hbox{\rom quaternion group of order 8,}\qquad A^{alg}_{\psi_{--}} =
A_{\psi_{--}}. \eqno(27.15)(c)$$

We do not know any entirely elementary way to calculate the Arthur
packets, but (using Theorem 27.12) this is not too difficult once one knows
something about Harish-Chandra modules for $SO(5)$.  We will simply
record some of the results.  It turns out that there are exactly four
equivalence classes of strong real forms of $G$, having representatives that
we denote $\delta_s$, $\delta_{(1,1)}$, $\delta_{(2,0)}$, and
$\delta_{(0,2)}$.  The last two of these represent the compact real form,
which can have no representations of the (singular) infinitesimal character
associated to $\psi_1$.  The other two represent the inequivalent real
forms $Sp(4,{\Bbb R})$ and $Sp(1,1)$.  It turns out that $\delta_{(1,1)}^2 =
-I$, so representations of $Sp(1,1)$ correspond to ${^\vee
G}^{alg}$-equivariant perverse sheaves on which $\pi_1({^\vee G})^{alg}$
acts non-trivially (Lemma 10.9(b)). It follows from (27.15)(c) that the
Arthur packets for $\psi_{++}$ and $\psi_{--}$ can contain no
representations of $Sp(1,1)$.  Furthermore, the representation
$\tau_{\psi_{+-}}(\pi)$ of $A^{alg}_{\psi_{+-}}$ attached to any irreducible
representation $\pi \in \Pi(G/{\Bbb R})_{\psi_{+-}}$ (Definition 24.15) must
satisfy
$$ \tau_{\psi_{+-}}(\pi)(-I) = \cases 1, &\text{ if } \pi \text{ is a
    representation of } Sp(4,{\Bbb R});\\
-1,& \text{ if } \pi \text{ is a representation of } Sp(1,1).\cr\endcases \eqno(27.16)$$

We mentioned after Corollary 27.13 that one can count unipotent
representations using the ideas of [BVPC].  In this way (or any of several
others) one finds that $Sp(4,{\Bbb R})$ has exactly eight unipotent
representations attached to $\psi_1$.  They are the three irreducible
constituents of ${\rom Ind}_{GL(2,{\Bbb R})}^{Sp(4,{\Bbb R})} (|\det|)$; the three irreducible constituents of ${\rom Ind}_{GL(2,{\Bbb R})}^{Sp(4,{\Bbb R})} (\det)$; and the two irreducible constituents of ${\rom Ind}_{GL(1,{\Bbb R})\times Sp(2,{\Bbb R})}^{Sp(4,{\Bbb R})} ({\rom sgn}(\det)\otimes 1)$.  These representations may be distinguished by their lowest $K$-types; the maximal compact subgroup is $U(2)$, so its representations are parametrized by decreasing pairs of integers.  The lowest $K$-types of the three sets of representations described above are $(0,0)$, $(2,2)$, and $(-2,-2)$; $(1,1)$, $(-1,-1)$, and $(1,-1)$; and $(1,0)$ and $(0,-1)$.  We denote the representation of lowest $K$-type $(m,n)$ by $\pi_s(m,n)$.

An even simpler calculation shows that $Sp(1,1)$ has a unique irreducible unipotent representation attached to $\psi_1$, namely the unique spherical representation of the correct infinitesimal character.  (It lies one-third of the way along the complementary series.)  We denote this representation $\pi_{(1,1)}(0)$. 

Here are the Arthur packets.
$$\Pi(G/{\Bbb R})_{\psi_{++}} = \{ \pi_s(0,0),\pi_s(1,-1) \} \eqno(27.17)(a)$$
$$\Pi(G/{\Bbb R})_{\psi_{--}} = \{ \pi_s(1,0),\pi_s(0,-1) \} \eqno(27.17)(b)$$
$$\Pi(G/{\Bbb R})_{\psi_{+-}} = \{ \pi_s(1,1),\pi_s(-1,-1) ,\pi_s(2,2),\pi_s(-2,-2), \pi_{(1,1)}(0)\} \eqno(27.17)(c)$$
In each case the map $\pi \mapsto \tau_\psi(\pi)$ is a bijection from the packet onto the set of equivalence classes of irreducible representations of $A_\psi^{alg}$; we have listed the representations so that the one with $\tau_\psi(\pi)$ trivial is first.   In particular, $\tau_{\psi_{+-}}(\pi_{(1,1)}(0))$ is the irreducible two-dimensional representation of the quaternion group (cf.~(27.16)).  
\medskip

We conclude with a careful study of the case of one-dimensional representations.  One of the points of this is to understand the failure of injectivity of the map $\pi \mapsto \tau_\psi(\pi)$.  (Failure of surjectivity appears already in the complex case; it corresponds to the fact that we had to use Lusztig's quotient of the component group in [BVunip], rather than just a component group.)  All of the examples that we know of this failure seem to arise finally from this very special case.  (Part (e) of the following theorem is taken from [Langlands], Lemma 2.11.)

\medskip
{\bf Theorem 27.18.}  {\it Suppose $(G^\Gamma,{\cal W})$ is an extended group, and (${^\vee G^\Gamma},{\cal D})$ is an L-group for $G^\Gamma$.  Fix a principal three-dimensional subgroup
$$\psi_1:SL(2) \rightarrow {^\vee G}.$$
\smallskip
\item{a)} The centralizer $S_0$ of $\psi_1$ in ${^\vee G}$ is $Z({^\vee G})$.  \item{b)} The set of equivalence classes of unipotent Arthur parameters attached to $\psi_1$ may be identified with 
$$\{\,z\in Z({^\vee G})\mid z\theta_Z(z) = 1\,\}/\{\,w\theta_Z^{-1}(w)\mid w\in Z({^\vee G})\,\} = H^1(\Gamma,Z({^\vee G})).$$
\item{c)} If $\psi$ is any unipotent Arthur parameter attached to $\psi_1$, then
$$A_\psi^{alg} = Z({^\vee G})^{\theta_Z,alg}/(Z({^\vee G})^{\theta_Z, alg})_0.$$
This is precisely the universal component group $A^{alg}_{univ}({^\vee G^\Gamma})$ of Definition 26.10.
\item{d)} The unipotent representations (of some real form $G({\Bbb R},\delta)$) attached to $\psi_1$ are precisely the representations trivial on the identity component $G({\Bbb R},\delta)_0$.
\item{e)}  Suppose $\delta$ is any strong real form of $G^\Gamma$.  Then there is a natural surjection
$$H^1(\Gamma,Z({^\vee G})) \twoheadrightarrow {\rom Hom}(G({\Bbb R},\delta)/G({\Bbb R},\delta)_0,{\Bbb C}^\times).$$
If $G({\Bbb R},\delta)$ is quasisplit, this is an isomorphism.
\smallskip
Suppose $\psi$ is a unipotent Arthur parameter attached to $\psi_1$, and $\delta$ is a strong real form of $G^\Gamma$.  Write $\pi(\psi,\delta)$ for the character of $G({\Bbb R},\delta)/G({\Bbb R},\delta)_0$ attached to $\psi$ by composing the bijection of (b) with the surjection of (e).
\smallskip
\item{f)} Suppose $\pi$ is an irreducible unipotent representation (attached to $\psi_1$) of a strong real form $G({\Bbb R},\delta)$.  Then the integer $\chi_\psi(\pi)$ of Theorem 22.7 is given by
$$\chi_\psi(\pi) = \cases 1 & \text{ if } \pi \simeq  \pi(\psi,\delta) \\
0 & \text{ otherwise.} \cr\endcases$$
The character $\tau_\psi(\pi)$ of $A_\psi^{alg} \simeq A_{univ}^{alg}({^\vee G^\Gamma})$ (see (c) above) is
$$\tau_\psi(\pi,\delta) = \chi_\psi(\pi)\tau_{univ}(\delta). $$
\medskip}

\noindent If we fix the unipotent parameter $\psi$ attached to $\psi_1$, we therefore find exactly one unipotent representation for each strong real form.  The correspondence $\pi \mapsto \tau_\psi(\pi)$ therefore fails to be injective as soon as the number of strong real forms exceeds the cardinality of $A_\psi^{alg} = Z({^\vee G})^{\theta_Z,alg}/(Z({^\vee G})^{\theta_Z,alg})_0$.  The simplest example of this is perhaps for $PGL(n)$ endowed with the class of inner forms including the projective unitary groups $PU(p,q)$.  There are $[(n+2)/2]$ real forms.  The dual group is $SL(n)$, with $\theta_Z$ acting on the center by inversion.  Consequently $A_\psi^{alg}$ consists of the elements of order 2 in the cyclic group of order $n$; injectivity fails for $n \ge 3$.  For simple groups not of type $A$ the situation is even clearer: typically the number of real forms is on the order of the rank, but $A_\psi^{alg}$ has order at most four.

{\it Proof of Theorem 27.18.}  Define $E$, $F$, and $\lambda$ for $\psi_1$ as in (22.8).  Because $\psi_1$ is principal, $\lambda$ is a regular element with integral eigenvalues.  It follows that $P(\lambda)$ is a Borel subgroup of ${^\vee G}$; write it as $^d B$, and write $^d T$ for the unique maximal torus containing $\lambda$ in its Lie algebra.  Then the centralizer of $\lambda$ in ${^\vee G}$ is $^d T$, so $S_0 \subset {^d T}$.  The element $E$ is the sum of certain root vectors $\{X_\alpha\}$ for the simple roots, so the centralizer of $E$ in $^d T$ is the intersection of the kernels of the simple roots, which is $Z({^\vee G})$.  This proves (a).  For (b), write $S$ for the centralizer of $\psi_1$ in ${^\vee G^\Gamma}$.  By Definition 4.6, we can find an element ${^\vee \delta} \in {\cal D}$ so that
$${^\vee \delta} \in {\rom Aut}({^\vee G},{^d B},{^d T},\{X_\alpha\}) \eqno(27.19)(a)$$
(see Proposition 2.11).  In particular, $^\vee \delta$ centralizes $E$.  Now $\lambda$ belongs to $[{^\vee {\frak g}},{^\vee {\frak g}}]$, and $\alpha(\lambda) = 1$ for each simple root $\alpha$.  These properties characterize $\lambda$, and they are preserved by $^\vee \delta$.  Consequently $^\vee \delta$ also centralizes $\lambda$.  Finally, it follows from the representation theory of $SL(2)$ that $F$ is characterized by the property $[E,F] = 2\lambda$, so $^\vee \delta$ also centralizes $F$.  Therefore
$${^\vee \delta} \in S ,\qquad S = Z({^\vee G}) \cup Z({^\vee G}){^\vee \delta} \eqno(27.19)(b)$$
Since ${^\vee G^\Gamma}$ is an L-group, ${^\vee \delta}^2 = 1$.   Part (b) therefore follows from Corollary 27.3 and (27.19)(b).  Part (c) follows from (b) and Corollary 27.3.

For (d), we have already observed that $\lambda$ belongs to the derived algebra and takes the value 1 on each simple root.  The corresponding infinitesimal character is therefore $\rho$, half the sum of a set of positive roots for $G$.  This is the infinitesimal character of the trivial representation, so the corresponding maximal ideal in $U({\frak g})$ is the augmentation ideal, the annihilator of the trivial representation.  Now (d) follows from Corollary 27.13.

For (e), we begin with the isomorphism
$$Z({^\vee G}) \simeq {\rom Hom}(\pi_1(G),{\Bbb C}^\times) \eqno(27.20)(a)$$
of Lemma 10.2(a).  Along the lines of Lemma 10.2(d), we deduce that
$$H^1(\Gamma,Z({^\vee G})) \simeq {\rom Hom}(H^1(\Gamma,\pi_1(G)),{\Bbb C}^\times) \eqno(27.20)(b)$$
To prove (e), it is therefore enough to exhibit an injection
$$G({\Bbb R},\delta)/G({\Bbb R},\delta)_0 \hookrightarrow H^1(\Gamma,\pi_1(G)). \eqno(27.20)(c)$$
For this, we begin with the short exact sequence of groups with $\Gamma$ actions
$$1 \rightarrow \pi_1(G) \rightarrow G^{sc} \rightarrow G \rightarrow 1. \eqno(27.20)(d)$$
(Here we make complex conjugation act by $\delta$ on $G$, and pull this automorphism back to $G^{sc}$.)  The corresponding exact sequence in cohomology includes
$$ H^0(\Gamma,G^{sc}) \rightarrow H^0(\Gamma,G) \rightarrow H^1(\Gamma,\pi_1(G)). \eqno(27.20)(e)$$
Now $H^0$ is just the $\Gamma$-invariant subgroup; that is (for $G$ and $G^{sc}$) the group of real points.  Hence
$$G^{sc}({\Bbb R},\delta) \rightarrow G({\Bbb R},\delta) \rightarrow H^1(\Gamma,\pi_1(G)). \eqno(27.20)(f)$$
Since $G^{sc}$ is simply connected, its group of real points is connected.  The image of the first map is therefore $G({\Bbb R},\delta)_0$.  The second map provides (by the exactness of (27.20)(f)) the inclusion in (27.20)(c).

To see that the map of (e) is surjective when $G({\Bbb R},\delta)$ is quasisplit, it suffices to find $|H^1(\Gamma,Z({^\vee G}))|$ linearly independent functions on $G({\Bbb R},\delta)/G({\Bbb R},\delta)_0$.  The functions we use are the stable characters $\Theta(\eta_\psi(\delta))$ (notation as in Theorem 22.7 and (18.1)), with $\psi$ a unipotent Arthur parameter attached to $\psi_1$.  The strongly stable characters $\eta_\psi$ are part of the basis $\{\,\eta_\phi^{mic}\,\}$ of strongly stable virtual characters (Corollary 19.16), and are therefore linearly independent.  By Shelstad's theorem 18.7, the stable characters $\Theta(\eta_\psi(\delta))$ are also linearly independent.  By (d), these characters may be regarded as functions on $G({\Bbb R},\delta)/G({\Bbb R},\delta)_0$.  By (b), there are $|H^1(\Gamma,Z({^\vee G}))|$ such functions, as we wished to show.

For (f), suppose first that $G({\Bbb R},\delta)$ is quasisplit.  The L-packets attached by the Langlands classification to the parameters $\phi_\psi$ (with $\psi$ unipotent attached to $\psi_1$) are disjoint, and each is a non-empty set of representations of the component group of $G({\Bbb R},\delta)$.  It follows from (e) that the  L-packet $\Pi_{\phi_\psi}$ contains exactly one representation, and that these exhaust the representations of the component group.  We leave to the reader the verification from the definitions that this single representation is actually $\pi(\psi,\delta)$.  Because the integer $d(\pi)$ (the dimension of the corresponding orbit on the geometric parameter space) is the same for all finite-dimensional irreducible representations of a fixed group, it follows from Corollary 19.16 that the only representation of $G({\Bbb R},\delta)$ in the Arthur packet $\Pi_\psi$ is $\pi(\psi,\delta)$.  Now the two assertions of (f) are immediate.

Finally we consider the general case of (f).  Because of the naturality of the surjection in (e), one can check easily that the virtual representation
$$\eta'_\psi = \sum_\delta \pi(\psi,\delta)$$
is strongly stable.  By the preceding paragraph, it agrees with the Arthur character $\eta_\psi$ on each quasisplit real form.  By Theorem 18.7, $\eta'_\psi = \eta_\psi$.  This is the first claim of (f).  The second follows immediately.  Q.E.D.

We leave to the reader the formulation of an analogous result in the
case of a general E-group.  What happens there is that some real forms
(possibly all) will have no projective representations of type $z$
trivial on the identity component.  The set of equivalence classes of
unipotent Arthur parameters attached to $\psi_1$ may be empty; if not,
then it is a principal homogeneous space for $H^1(\Gamma,Z({^\vee
  G}))$.

%\end

%\input mssymb.tex
%\pageno=181
\vskip 1pc
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\vskip 3pc
{\obeylines
Jeffrey Adams
Department of Mathematics
University of Maryland
College Park, MD 20742
\vskip 3pc
Dan Barbasch
Department of Mathematics
Cornell University
Ithaca, NY 14853
\vskip 3pc
David A. Vogan, Jr.
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA 02139
}
\end





