The Langlands Classification and Irreducible Characters for Real Reductive Groups

JEFFREY ADAMS

1

DAN BARBASCH

2

DAVID A. VOGAN, JR.

3

1. Introduction.

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(ℝ) of real points of a connected reductive algebraic group G defined over 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

Π(G(ℝ)) ⊃ Π_{unit}(G(ℝ)) ⊃ Π_{temp}(G(ℝ))  (1.1)

to be the set of equivalence classes of irreducible admissible (respectively unitary or tempered) representations of G(ℝ). Now define

Φ(G(ℝ)) ⊃ Φ_{temp}(G(ℝ))  (1.2)

to be the set of Langlands parameters for irreducible admissible (respectively tempered) representations of G(ℝ) (see [Langlands], [Borel], [AV2], section 5, and Definition 22.3). To each ϕ ∈ Φ(G(ℝ)), Langlands attaches a finite set Π_{ϕ} ⊂ Π(G(ℝ)), called an L-packet of representations. The L-packets Π_{ϕ} partition Π(G(ℝ)). If ϕ ∈ Φ_{temp}(G(ℝ)), then the representations in Π_{ϕ} are all tempered, and in this way one gets also a partition of Π_{temp}(G(ℝ)).

Now the classification of the unitary representations of G(ℝ) is one of the most interesting unsolved problems in harmonic analysis. Langlands’ results immediately suggest that one should look for a set between Φ(G(ℝ)) and Φ_{temp}(G(ℝ)) parametrizing exactly the unitary representations. Unfortunately, nothing quite so complete is possible: Knapp has found examples in which some members of the set Π_{ϕ} are unitary and some are not.

1Supported in part by NSF grant DMS-8802586 and a grant from the Sloan Foundation.

2Supported in part by an NSF grant.

3Supported in part by NSF grant DMS-8805665.

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

Ψ(G/ℝ)  (1.3)(a)

of parameters (Definition 22.4). (We write G/ℝ rather than G(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(ℝ) is quasisplit. Then Arthur defines an inclusion

Ψ(G/ℝ) ↪ Φ(G(ℝ)), ψ ↦ ϕ_{ψ}.  (1.3)(b)

Write Φ_{Arthur}(G(ℝ)) for the image of this inclusion. Then

Φ(G(ℝ)) ⊃ Φ_{Arthur}(G(ℝ)) ⊃ Φ_{temp}(G(ℝ)).  (1.3)(c)

Roughly speaking, Arthur proposes that Ψ(G/ℝ) should parametrize all the unitary representations of G(ℝ) that are of interest for global applications. More specifically, he proposed the following problems (still for G(ℝ) quasisplit at first).

Problem A. Associate to each parameter ψ ∈ Ψ(G/ℝ) a finite set Π_{ψ} ⊂ Π(G(ℝ)). This set (which we might call an Arthur packet) should contain the L-packet Π_{ϕ_{ψ}} (cf. (1.3)(b)) and should have other nice properties, some of which are specified below.

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 ψ.

Associated to each Arthur parameter is a certain finite group A_{ψ} (Definition 21.4).

Problem B. Associate to each π ∈ Π_{ψ} a non-zero finite-dimensional representation τ_{π} of A_{ψ}. Problem C. Show that the distribution on G(ℝ)

∑_{π∈Π_{ψ}} (ϵ_{π}dim(τ_{π})) Θ(π)

is a stable distribution in the sense of Langlands and Shelstad ([Langlands2], [Shelstad]). Here ϵ_{π} = ±1 is also to be defined. Problem D. Prove analogues of Shelstad’s theorems on lifting tempered characters (cf. [Shelstad]) in this setting. Problem E. Extend the definition of Π_{ψ} to non-quasisplit G, in a manner consistent with appropriate generalizations of Problems B, C, and D. Problem F. Show that every representation π ∈ Π_{ψ} is unitary. 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(ℝ) in terms of an appropriate geometry on an L-group. So let ^{∨}G be the (complex reductive) dual group of G, and ^{∨}G^{Γ} the (Galois form of the) L-group attached to the real form G(ℝ). The L-group is a complex Lie group, and we have a short exact sequence

1 → ^{∨}G → ^{∨}G^{Γ} → Gal(ℂ/ℝ) → 1  (1.4)(a)

We also need the Weil group W_{ℝ} of ℂ/ℝ; this is a real Lie group, and there is a short exact sequence

1 → ℂ^{×} → W_{ℝ} → Gal(ℂ/ℝ) → 1.  (1.4)(b)

(The Weil group is not a complex Lie group because the action of the Galois group on ℂ^{×} is the non-trivial one, which does not preserve the complex structure.)

Definition 1.5 ([Langlands], [Borel]). A quasiadmissible homomorphism ϕ from W_{ℝ} to ^{∨}G^{Γ} is a continuous group homomorphism satisfying

(a) ϕ respects the homomorphisms to Gal(ℂ/ℝ) defined by (1.4); and

(b) ϕ(ℂ^{×}) consists of semisimple elements of ^{∨}G. (Langlands’ notion of “admissible homomorphism” includes an additional “relevance” hypothesis on ϕ, which will not concern us. This additional hypothesis is empty if G(ℝ) is quasisplit.) Define

P (^{∨}G^{Γ}) = { ϕ : W_{ℝ} → ^{∨}G^{Γ} | ϕ is quasiadmissible }.

Clearly ^{∨}G acts on P (^{∨}G^{Γ}) by conjugation on the range of a homomorphism, and we define

Φ(G/ℝ) = {^{∨}G orbits on P (^{∨}G^{Γ}) }.

(If G(ℝ) is quasisplit, this is precisely the set of parameters in (1.2). In general Langlands omits the “irrelevant” orbits.)

Now a homomorphism ϕ is determined by the value of its differential on a basis of the real Lie algebra of ℂ^{×}, together with its value at a single specified element of the non-identity component of W_{ℝ}; that is, by an element of the complex manifold ∨𝔤 ×^{∨}𝔤 ×^{∨}G^{Γ}. 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

Proposition 1.6. Suppose ^{∨}G^{Γ} is an L-group. The set P (^{∨}G^{Γ}) of quasiadmissible homomorphisms from W_{ℝ} into ^{∨}G^{Γ} may be identified with the set of pairs (y, λ) satisfying the following conditions:

a) y ∈ ^{∨}G^{Γ} −^{∨}G, and λ ∈ ^{∨}𝔤 is a semisimple element;

b) y^{2} = exp(2πiλ); and

c) [λ, Ad(y)λ] = 0.

The Langlands classification described after (1.2) is thus already geometric: Lpackets 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 Π_{ϕ} may be parametrized using ^{∨}G-equivariant local systems on the ^{∨}G orbit of ϕ.

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 ^{∨}G-orbits on P (^{∨}G^{Γ}). 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 ^{∨}G action, having the same set of orbits as P (^{∨}G^{Γ}), 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 ^{∨}G.)

Definition 1.7. Suppose H is a complex reductive group, with Lie algebra 𝔥, and λ ∈ 𝔥 is a semisimple element. Set

𝔥(λ)_{n} = { μ ∈ 𝔥 | [λ, μ] = nμ } (n ∈ ℤ)  (1.7)(a)
𝔫(λ) = ∑_{n=1,2,...} 𝔥(λ)_{n}  (1.7)(b)
e(λ) = exp(2πiλ) ∈ H.  (1.7)(c)

The canonical flat through λ is the affine subspace

F(λ) = λ + 𝔫(λ) ⊂ 𝔥.  (1.7)(d)

We will see in section 6 that the canonical flats partition the semisimple elements of 𝔥 — in fact they partition each conjugacy class — and that the map e is constant on each canonical flat. If Λ is a canonical flat, we may therefore write

e(Λ) = exp(2πiλ)  (any λ ∈ Λ).  (1.7)(e)

Finally, write F(𝔥) for the set of all canonical flats in 𝔥.

Definition 1.8 Suppose ^{∨}G^{Γ} is the L-group of a real reductive group (cf. (1.4)). The geometric parameter space for ^{∨}G^{Γ} is the set

X = X(^{∨}G^{Γ}) = { (y, Λ) | y ∈ ^{∨}G^{Γ} −^{∨}G, Λ ∈ F(^{∨}𝔤), y^{2} = e(Λ) }.

This is our proposed substitute for Langlands’ space P (^{∨}G^{Γ}). The set F(^{∨}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.

Lemma 1.9 (cf. Proposition 6.16 below). In the setting of Definition 1.8, fix a single orbit O of ^{∨}G on the semisimple elements of ^{∨}𝔤, and set

X(O,^{∨}G^{Γ}) = { (y, Λ) ∈ X | Λ ⊂ O }.

Then X(O,^{∨}G^{Γ}) has in a natural way the structure of a smooth complex algebraic variety, on which ^{∨}G acts with a finite number of orbits.

The variety X(O,^{∨}G^{Γ}) need not be connected or equidimensional, but this will cause no difficulties. We topologize X by making the subsets X(O,^{∨}G^{Γ}) 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.

Proposition 1.10. (cf. Proposition 6.17 below). Suppose ^{∨}G^{Γ} is an L-group. Then there is a natural ^{∨}G-equivariant map

p : P (^{∨}G^{Γ}) → X(^{∨}G^{Γ}), p(y, λ) = (y, F(λ))

inducing a bijection on the level of ^{∨}G-orbits. The fibers of p are principal homogeneous spaces for unipotent algebraic groups. More precisely, suppose x = p(ϕ). Then the isotropy group ^{∨}G_{ϕ} is a Levi subgroup of ^{∨}G_{x}.

This proposition shows that (always locally over ℝ!) the geometric parameter space X shares all the formal properties of P (^{∨}G^{Γ}) needed for the Langlands classification. In particular, if G(ℝ) is quasisplit, L-packets in Π(G(ℝ)) are parametrized precisely by ^{∨}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.

Proposition 1.11. Suppose G(ℝ) is quasisplit. Let ϕ, ϕ′ ∈ Φ(G/ℝ) be two Langlands parameters, and S, S′ ⊂ X the corresponding ^{∨}G-orbits. Then the following conditions are equivalent:

i) S is contained in the closure of S′. ii) there are irreducible representations π ∈ Π_{ϕ} and π′ ∈ Π_{ϕ′} with the property that

π′ is a composition factor of the standard representation of which π is the unique quotient.

If ϕ is a tempered parameter, then the orbit S is open in the variety X(O,^{∨}G^{Γ}) containing it (cf. Lemma 1.9).

(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(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 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 ^{∨}G is trivial.

Definition 1.12. Suppose G is a complex connected reductive algebraic group. An extended group for G/ℝ is a pair (G^{Γ}, W), subject to the following conditions.

(a) G^{Γ} is a real Lie group containing G as a subgroup of index two, and every element of G^{Γ} − G acts on G (by conjugation) by antiholomorphic automorphisms.

(b) W is a G-conjugacy class of triples (δ, N, χ), with

(1) The element δ belongs to G^{Γ} − G, and δ^{2} ∈ Z(G) has finite order. (Write σ = σ(δ) for the conjugation action of δ on G, and G(ℝ) or G(ℝ, δ) for the fixed points of σ; this is a real form of G.)

(2) N ⊂ G is a maximal unipotent subgroup, and δ normalizes N. (Then N is defined over ℝ; write N(ℝ) = N(ℝ, δ) for the subgroup of real points.)

(3) The element χ is a one-dimensional non-degenerate unitary character of N(ℝ). (Here “non-degenerate” means non-trivial on each simple restricted root subgroup of N.)

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(ℝ) appearing in the definition are quasisplit (because of (b)(2)) and the pair (N(ℝ), χ) is the set of data needed to define a Whittaker model for G(ℝ).

Definition 1.13. Suppose (G^{Γ}, W) is an extended group. A strong real form of (G^{Γ}, W) (briefly, of G) is an element δ ∈ G^{Γ} − G such that δ^{2} ∈ Z(G) has finite order. Given such a δ, we write σ = σ(δ) for its conjugation action on G, and

G(ℝ) = G(ℝ, δ)

for the fixed points of σ. Two strong real forms δ and δ′ are called equivalent if they are conjugate by G; we write δ ∼ δ′. (The elements δ of Definition 1.12 constitute a single equivalence class of strong real forms, but in general there will be many others.)

The usual notion of a real form can be described as an antiholomorphic involution σ 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(ℝ, δ) (for δ a strong real form of (G^{Γ}, W)) constitute exactly one inner class of real forms of G.

Definition 1.14. Suppose (G^{Γ}, W) is an extended group. A representation of a strong real form of (G^{Γ}, W) (briefly, of G) is a pair (π, δ), subject to

(a) δ is a strong real form of (G^{Γ}, W) (Definition 1.13); and

(b) π is an admissible representation of G(ℝ, δ). Two such representations (π, δ) and (π′, δ′) are said to be (infinitesimally) equiv-

alent if there is an element g ∈ G such that gδg^{−1} = δ′, and π ∘ Ad(g^{−1}) is (infinitesimally) equivalent to π′. (In particular, this is possible only if the strong real forms are equivalent.) Finally, define

Π(G^{Γ}, W) = Π(G/ℝ)

to be the set of (infinitesimal) equivalence classes of irreducible representations of strong real forms of G.

Lemma 1.15. Suppose (G^{Γ}, W) is an extended group for G (Definition 1.12). Choose representatives { δ_{s} | s ∈ Σ } for the equivalence classes of strong real forms of G (Definition 1.13). Then the natural map from left to right induces a bijection

∐_{s∈Σ} Π(G(ℝ, δ_{s})) ≃ Π(G/ℝ)

(Definition 1.14; the set on the left is a disjoint union).

This lemma is an immediate consequence of the definitions; we will give the argument in section 2.

The set Π(G/ℝ) is the set of representations we wish to parametrize. To do so requires one more definition on the geometric side.

Definition 1.16. Suppose ^{∨}G^{Γ} is the L-group of the inner class of real forms represented by the extended group G^{Γ} (cf. (1.4) and Definition 1.13). The algebraic universal covering^{∨}G^{alg} is the projective limit of all the finite coverings of^{∨}G. This is a pro-algebraic group, of which each finite-dimensional representation factors to some finite cover of ^{∨}G.

With the algebraic universal covering in hand, we can define a complete set of geometric parameters for representations.

Definition 1.17. Suppose G is a connected reductive algebraic group endowed with an inner class of real forms, and ^{∨}G^{Γ} is a corresponding L-group for G. A complete geometric parameter for G is a pair

ξ = (S, V),

where

(a) S is an orbit of ^{∨}G on X(^{∨}G^{Γ}) (Definition 1.8); and

(b) V is an irreducible ^{∨}G̃-equivariant local system on S, for some finite covering ∨G̃ of ^{∨}G.

We may write (S_{ξ}, V_{ξ}) to emphasize the dependence on ξ. In (b), it is equivalent to require V to be ^{∨}G^{alg}-equivariant. Write Ξ(G/ℝ) for the set of all complete geometric parameters.

A slightly different formulation of this definition is sometimes helpful. Fix a ∨G-orbit S on X, and a point x ∈ S. Write ^{∨}G_{x}^{alg} for the stabilizer of x in ^{∨}G^{alg}, and define

A_{S}^{loc,alg} = ^{∨}G_{x}^{alg}/ (^{∨}G_{x}^{alg})_{0}

for its (pro-finite) component group. We call A_{S}^{loc,alg} the 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

ξ = (S, τ),

where

(a) S is an orbit of ^{∨}G on X(^{∨}G^{Γ}); and

(b) τ is an irreducible representation of A_{S}^{loc,alg}. Again we may write (S_{ξ}, τ_{ξ}).

Theorem 1.18. Suppose (G^{Γ}, W) is an extended group for G (Definition 1.12), and ^{∨}G^{Γ} is an L-group for the corresponding inner class of real forms. Then there is a natural bijection between the set Π(G/ℝ) of equivalence classes of irreducible representations of strong real forms of G (Definition 1.14), and the set Ξ(G/ℝ) of complete geometric parameters for G (Definition 1.17). In this parametrization, the set of representations of a fixed real form G(ℝ) corresponding to complete geometric parameters supported on a single orbit is precisely the L-packet for G(ℝ) attached to that orbit (Proposition 1.10).

As we remarked after Proposition 1.6, one can find results of this nature in [Shelstad].

For each complete geometric parameter ξ, we define (using Theorem 1.18 and Definition 1.14)

(π(ξ), δ(ξ)) = some irreducible representation parametrized by ξ (1.19)(a)
M(ξ) = standard representation with Langlands quotient π(ξ).  (1.19)(b)

As a natural setting in which to study character theory, we will also use

KΠ(G/ℝ) = free ℤ-module with basis Π(G/ℝ).  (1.19)(c)

We will sometimes call this the 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 ρ has a well-defined image

[ρ] ∈ KΠ(G/ℝ).

By abuse of notation, we will usually drop the brackets, writing for example M(ξ) ∈ KΠ(G/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 δ of G, and a complete geometric parameter ξ. By the proof of Lemma 1.15, there is at most one irreducible representation π of G(ℝ, δ) so that (π, δ) is equivalent to (π(ξ), δ(ξ)). We define

π(ξ, δ) = π.  (1.19(d)

If no such representation π exists, then we define

π(ξ, δ) = 0.  (1.19)(e)

Similarly we can define M(ξ, δ).

Lemma 1.20 (Langlands — see [Green], [IC4]). The (image in KΠ(G/ℝ) of the) set

{ M(ξ) | ξ ∈ Ξ(G/ℝ) }

is a basis for KΠ(G/ℝ).

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 {π(ξ)} and {M(ξ)} of KΠ(G/ℝ):

M(ξ) = ∑_{γ∈Ξ} m_{r}(γ, ξ)π(γ).  (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 multiplicity matrix m_{r}(γ, ξ) 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.)

Definition 1.22. Suppose Y is a complex algebraic variety on which the proalgebraic group H acts with finitely many orbits. Define

C(Y, H) = category of H-equivariant constructible sheaves on Y. (1.22)(a)
P(Y, H) = category of H-equivariant perverse sheaves on Y.  (1.22)(b)

(For the definition of perverse sheaves we refer to [BBD]. The definition of Hequivariant 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

ξ = (S_{ξ}, V_{ξ}) = (S, V)  (1.22)(c)

with S an orbit of H on Y, and V an irreducible H-equivariant local system on S. The set of all such pairs will be written Ξ(Y, H), the set of 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} (y ∈ S)

and its representations. We write μ(ξ) for the irreducible constructible sheaf corresponding to ξ (the extension of ξ by zero), and P (ξ) for the irreducible perverse sheaf (the “intermediate extension” of ξ — cf. [BBD], Definition 1.4.22).

The Grothendieck groups of the two categories P(Y, H) and 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 (ξ) | ξ ∈ Ξ} and {μ(ξ) | ξ ∈ Ξ} are obviously bases of their respective Grothendieck groups, but they are not identified by the isomorphism. Write d(ξ) for the dimension of the underlying orbit S_{ξ}. We can write in K(Y, H)

μ(ξ) = (−1)^{d(ξ)} ∑_{γ∈Ξ(Y,H)} m_{g}(γ, ξ)P (γ)  (1.22)(d)

with m_{g}(γ, ξ) an integer. (The subscript g stands for “geometric.”) In this formula, it follows easily from the definitions that

m_{g}(ξ, ξ) = 1,  m_{g}(γ, ξ) ≠ 0 only if S_{γ} ⊂ (\overline{S_{ξ}} − S_{ξ}) (γ ≠ ξ).  (1.22)(e)

The matrix m_{g}(γ, ξ) 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.

Definition 1.23. In the setting of Definition 1.8, define

C(X(^{∨}G^{Γ}),^{∨}G^{alg})

to be the direct sum over semisimple orbits O ⊂ ^{∨}𝔤 of the categories C(X(O,^{∨}G^{Γ}),^{∨}G^{alg}) of Definition 1.22. The objects of this category are called (by a slight abuse of terminology) ^{∨}G^{alg}-equivariant constructible sheaves on X. Similarly we define

P(X(^{∨}G^{Γ}),^{∨}G^{alg}),

the ^{∨}G^{alg}-equivariant perverse sheaves on X. The irreducible objects in either category are parametrized by Ξ(G/ℝ) (cf. Definition 1.22), and we write

KX(^{∨}G^{alg})

for their common Grothendieck group. We write μ(ξ) and P (ξ) for the irreducible objects constructed in Definition 1.22, or their images in KX(^{∨}G^{alg}). These satisfy (1.22)(d) and (e).

Since Theorem 1.18 tells us that the two Grothendieck groups KX(^{∨}G^{alg}) and KΠ(G/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.

Theorem 1.24 Suppose (G^{Γ}, W) is an extended group for G (Definition 1.12), and ^{∨}G^{Γ} is an L-group for the corresponding inner class of real forms. Then there is a natural perfect pairing

<, >: KΠ(G/ℝ) × KX(^{∨}G^{alg}) → ℤ

between the Grothendieck group of the category of finite length representations of strong real forms of G, and that of ^{∨}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(ξ), μ(ξ′) >= e(G(ℝ, δ(ξ)))δ_{ξ,ξ′}.

Here we use the notation of (1.19) and Definition 1.22. The group G(ℝ, δ(ξ)) is the real form represented by M(ξ); the constant e(G(ℝ)) = ±1 is the one defined in [Kott1] (see also Definition 15.8), and the last δ is a Kronecker delta. In terms of the other bases of (1.19) and Definition 1.23, we have

< π(ξ), P (ξ′) >= e(G(ℝ, δ(ξ)))(−1)^{d(ξ)}δ_{ξ,ξ′}.

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.

Corollary 1.25.

a) The matrices m_{r} and m_{g} of (1.21) and Definition 1.22(d) are essentially inverse transposes of each other:

∑_{γ}(−1)^{d(γ)}m_{r}(γ, ξ)m_{g}(γ, ξ′) = (−1)^{d(ξ)}δ_{ξ,ξ′}.

b) The multiplicity of the irreducible representation π(γ) in the standard representation M(ξ) is up to a sign the multiplicity of the local system V_{ξ} in the restriction to S_{ξ} of the Euler characteristic of the perverse sheaf P (γ):

m_{r}(γ, ξ) = (−1)^{d(γ)−d(ξ)} ∑_{i}(−1)^{i}(multiplicity of V_{ξ} in H^{i}P (γ) |_{S_{ξ}}).

c) The coefficient of the standard representation M(γ) in the expression of the irreducible representation π(ξ) is equal to (−1)^{d(γ)−d(ξ)} times the multiplicity of the perverse sheaf P (ξ) in the expression of μ(γ)[−d(γ)].

Here part (c) refers to the expansion of π(ξ) in the Grothendieck group as a linear combination of standard representations (cf. Lemma 1.20); and similarly for μ(γ).

Another way to think of Theorem 1.24 is this.

Corollary 1.26. In the setting of Theorem 1.24, write

\overline{K} = \overline{K}Π(G/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 ℤ-linear functionals on the Grothendieck group KX(^{∨}G^{alg}):

\overline{K}Π(G/ℝ) ≃ Hom_{ℤ}(KX(^{∨}G^{alg}), ℤ).

In this identification,

a) the standard representation M(ξ) of G(ℝ, δ(ξ)) corresponds to e(G(ℝ, δ(ξ))) times the linear functional “multiplicity of V_{ξ} in the restriction to S_{ξ} of the constructible sheaf C;” and

b) the irreducible representation π(ξ) of G(ℝ, δ(ξ)) corresponds to e(G(ℝ, δ(ξ)))(−1)^{d(ξ)} times the linear functional “multiplicity of P (ξ) as a composition factor of the perverse sheaf Q.”

Here in (a) we are interpreting KX(^{∨}G^{alg}) as the Grothendieck group of constructible sheaves, and in (b) as the Grothendieck group of perverse sheaves.

We call elements of \overline{K}Π(G/ℝ) 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.

Definition 1.27. In the setting of Definition 1.14 and (1.19), suppose

η = ∑_{ξ∈Ξ} n(ξ)(π(ξ), δ(ξ))

is a formal virtual character. We say that η is locally finite if for each strong real form δ there are only finitely many ξ with n(ξ) ≠ 0 and δ(ξ) equivalent to δ. Suppose that η is locally finite, and that δ is a strong real form of G. There is a finite set π_{1}, …, π_{r} of inequivalent irreducible representations of G(ℝ, δ) so that each (π_{j}, δ) is equivalent to some (π(ξ_{j}), δ(ξ_{j})) with n(ξ_{j}) ≠ 0. Each of these representations has a character Θ(π_{j}), a generalized function on G(ℝ, δ); and we define

Θ(η, δ) = ∑_{j} n(ξ_{j})Θ(π_{j}),

a generalized function on G(ℝ, δ). This generalized function has well-defined values at the regular semisimple elements of G(ℝ, δ), and these values determine Θ(η, δ). In the notation of (1.19)(d,e), we can write

Θ(η, δ) = ∑_{ξ} n(ξ)Θ(π(ξ, δ)).

We say that η is strongly stable if it is locally finite, and the following condition is satisfied. Suppose δ and δ′ are strong real forms of G, and g ∈ G(ℝ, δ) ∩ G(ℝ, δ′) is a strongly regular semsimple element. Then

Θ(η, δ)(g) = Θ(η, δ′)(g).

A necessary condition for η to be strongly stable is that each Θ(η, δ) should be stable in Langlands’ sense. Conversely, Shelstad’s results in [Shelstad] imply that if Θ is a stable finite integer combination of characters on a real form G(ℝ, δ), then there is a strongly stable η with Θ = Θ(η, δ).

Corollary 1.26 gives a geometric interpretation of formal virtual characters. We can now give a geometric interpretation of the notion of stability.

Definition 1.28. In the setting of Definition 1.22, fix an H-orbit S ⊂ Y, and a point y ∈ 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

χ_{S}^{loc} : Ob C(Y, H) → ℕ, χ_{S}^{loc}(C) = dim(C_{y})

is independent of the choice of y in S. It is additive for short exact sequences, and so defines a ℤ-linear map

χ_{S}^{loc} : K(Y, H) → ℤ,

the local multiplicity along S. If we regard K(Y, H) as a Grothendieck group of perverse sheaves, then the formula for χ_{S}^{loc} on a perverse sheaf P is

χ_{S}^{loc}(P) = ∑(−1)^{i}dim(H^{i}P)_{y}.

Any ℤ-linear functional η on K(Y, H) is called geometrically stable if it is in the ℤ-span of the various χ_{S}^{loc}.

In the setting of Definition 1.23, a ℤ-linear functional η on K(Y, H) is called geometrically stable if its restriction to each summand K(X(O),^{∨}G^{alg}) is geometrically stable, and vanishes for all but finitely many O.

Theorem 1.29. In the identification of Corollary 1.26, the strongly stable formal virtual characters correspond precisely to the geometrically stable linear functionals.

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 χ_{S}^{loc}. 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).

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) ⊂ 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 characteristic cycle

Ch(P) = ∑_{S} χ_{S}^{mic}(P)\overline{T_{S}^{∗}(Y)}.

Here the coefficients χ_{S}^{mic}(P) are non-negative integers, equal to zero unless S is contained in the support of P. One way to construct 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 Dmodules on Y, and the characteristic cycle of a D-module is fairly easy to define (see for example [Ginsburg] or [Kashiwara]). The functions χ_{S}^{mic} are additive for short exact sequences, and so define ℤ-linear functionals

χ_{S}^{mic} : K(Y, H) → ℤ,

the microlocal multiplicity along S.

Theorem 1.31 (Kashiwara — see [Kold], [Kashiwara], Theorem 6.3.1, or [Ginsburg], Theorem 8.2.) The linear functionals χ_{S}^{mic} of Definition 1.30 are geometrically stable. More precisely, for every H-orbit S′ such that \overline{S′} ⊃ S there is an integer c(S, S′) so that for every H-equivariant perverse sheaf on Y,

χ_{S}^{mic}(P) = ∑_{S′} c(S, S′)χ_{S′}^{loc}(P).

Here χ_{S′}^{loc} is defined in Definition 1.28.

In fact Kashiwara’s interest was in an inverted form of this relationship, expressing χ_{S}^{loc} in terms of the various χ_{S′}^{mic}. (The invertibility of the matrix c(S, S′) is an immediate consequence of the facts that c(S, S) = (−1)^{dim S}, and that c(S, S′) ≠ 0 only if \overline{S′} ⊃ S.) We could therefore have defined geometrically stable in terms of the linear functionals χ_{S}^{mic}.

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 χ_{S}^{mic} 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 ≠ 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.

Corollary 1.32. Suppose (^{∨}G, W) is an extended group for G (Definition 1.12), and ^{∨}G^{Γ} is an L-group for the corresponding inner class of real forms. Fix an orbit S of ^{∨}G on X(^{∨}G^{Γ}) (Definition 1.8) (or, equivalently, an L-packet for the quasisplit form of G (Proposition 1.10)). Then the linear functional χ_{S}^{mic} on KX(^{∨}G^{alg}) (Definition 1.30) corresponds via Corollary 1.26 to a strongly stable formal virtual representation η_{S}^{mic}. The irreducible representations of strong real forms occurring in η_{S}^{mic} are those for which the corresponding perverse sheaf P has χ_{S}^{mic}(P) ≠ 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(ℝ, δ) is

Θ(η_{S}^{mic}, δ) = e(G(ℝ, δ)) ∑_{ξ′∈Ξ}(−1)^{d(ξ′)−dim S}χ_{S}^{mic}(P (ξ′))Θ(π(ξ′, δ)).

In terms of standard representations, this distribution may be expressed as

Θ(η_{S}^{mic}, δ) = e(G(ℝ, δ))(−1)^{dim S} ∑_{ξ′∈Ξ} c(S, S_{ξ′})Θ(M(ξ′, δ)).

The set { η_{S}^{mic} } (as S varies) is a basis of the lattice of strongly stable formal virtual representations.

(Recall that the tempered representations correspond to open orbits; in that case χ_{S}^{mic} = (−1)^{dim S}χ_{S}^{loc}, and we get nothing new.)

As the second formula of Corollary 1.32 shows, obtaining explicit character formulas for η_{S}^{mic} 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 ψ a Langlands parameter ϕ_{ψ}, and thus (by Proposition 1.10) an orbit S_{ψ}. We define Π_{ψ} to consist of all those representations appearing in η_{S_{ψ}}^{mic}; that is, representations for which the corresponding perverse sheaf has the conormal bundle of S_{ψ} 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_{ψ} turns out (for S = S_{ψ}) to be a quotient of a certain geometrically defined group A_{S}^{mic,alg}; the difference arises only from our use of the algebraic universal covering of ^{∨}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 τ_{S}^{mic}(P) of A_{S}^{mic,alg}, of dimension equal to χ_{S}^{mic}(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(^{∨}G^{Γ}) obviously depends covariantly on ^{∨}G^{Γ}. 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, 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.

Index of notation.

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.

A_{ϕ}^{loc,can}: group of connected components of the centralizer in ^{∨}G^{can} of a Langlands parameter ϕ (Definition 5.11). A Langlands parameter ϕ gives a point x of the geoemetric parameter space X(^{∨}G^{Γ}) (Proposition 6.17), which generates an orbit S under ^{∨}G. The ^{∨}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_{ϕ}^{loc,can} (Lemma 7.5).

A_{ϕ}^{mic,can}: micro-component group for a Langlands parameter ϕ (Definition 23.13). Write S for the orbit on X(^{∨}G^{Γ}) corresponding to ϕ, and (x, ν) for a sufficiently generic conormal vector to S at x ∈ S. Then A_{ϕ}^{mic,can} is the group of connected components of the stablizer of (x, ν) in ^{∨}G^{can}.

A_{ψ}^{can}: group of connected components of the centralizer in ^{∨}G^{can} of an Arthur parameter ψ (Definition 21.4). Naturally isomorphic to the micro-component group A_{ϕ_{ψ}}^{mic,can} for the corresponding Langlands parameter (Proposition 21.9 and Definition 23.7).

d(ξ): dimension of the underlying orbit for a geometric parameter ξ ((1.22),(7.10)). e(G_{ℝ}), e(ξ): Kottwitz’ sign attached to a complete geometric parameter or to

the corresponding real form of G (Definition 15.8, Lemma 15.9).

e(·): normalized exponential map exp(2πi·); usually from semisimple elements or canonical flats in ^{∨}𝔤 ((6.2), Lemma 6.11).

G: connected complex reductive algebraic group.

G^{Γ}: weak extended group for G (Definition 2.13).

(G^{Γ}, W): extended group for G (Definition 1.12, Proposition 3.6).

G(ℝ, δ): real points of G with real form defined by δ ∈ G^{Γ} − G (Definition 2.13). ∨G: dual group for G (Definition 4.2). ∨G^{Γ}: weak L-group or E-group for G; an algebraic extension of the Galois group

of ℂ/ℝ by a dual group (Definition 4.3).

(^{∨}G^{Γ}, D): L-group or E-group for G, essentially as defined by Langlands (in Galois form); D is a class of distinguished splittings.

(^{∨}G^{Γ}, 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 S is different from D.

H(O): Hecke algebra attached to a regular semisimple conjugacy class O in ^{∨}𝔤 ((16.10)).

KX(O,^{∨}G^{Γ}): Grothendieck group of ^{∨}G^{can}-equivariant perverse sheaves (or constructible sheaves, or holonomic D-modules) on X(O,^{∨}G^{Γ}) (Definition 7.13, (7.10)).

KΠ^{z}(O, G_{ℝ}): Grothendieck group of finite-length canonical projective representations of G_{ℝ} of type z and infinitesimal character O ((15.5)). Without the O, no assumption on infinitesimal character. With G/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}_{ℂ}, complex-linear combinations of irreducible representations are allowed (Proposition 25.1).

KΠ^{z}(O, G/ℝ): mixed Grothendieck group of finite-length canonical projective representations of strong real forms of G of type z and infinitesimal character O ((16.12)). (Formally, K denotes extension of scalars to ℤ[u^{1/2}, u^{−1/2}].)

l^{I}(Λ), l^{I}(ξ): integral length for a standard limit character of regular infinitesimal character, or for the corresponding complete geometric parameter (Definition 16.16).

L^{z}(G_{ℝ}): equivalence classes of final standard limit characters of Cartan subgroups of G_{ℝ} (Definition 11.13).

L^{z}(G/ℝ): equivalence classes of final standard limit characters of Cartan subgroups of G^{Γ} (Definition 12.1).

M^{z}(G_{ℝ}), M^{z}(G/ℝ): category of (canonical projective of type z) representations of a real reductive group G_{ℝ}, or of strong real forms of G^{Γ} ((15.2) and (15.7)).

M(Λ), M(ξ): standard limit representation attached to a limit character Λ, or to the corresponding complete geometric parameter ξ (Definition 11.2, (15.7)).

O: semisimple ^{∨}G-conjugacy class in ^{∨}𝔤; parametrizes an infinitesimal character for 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).

T^{Γ}: Cartan subgroup of G^{Γ} (Definition 12.1).

Λ: 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).

χ_{ϕ}^{mic}(π)(σ): trace at σ ∈ A_{ϕ}^{mic,can} of the representation τ_{ϕ}^{mic,can} (Definition 23.13). Without the argument σ, the dimension of this representation, which is equal to the multiplicity of the conormal bundle to the orbit S_{ϕ} in the characteristic cycle of the perverse sheaf attached to π (Definition 19.13, Theorem 23.8).

η_{ϕ}^{loc}: strongly stable standard formal virtual representation of strong real forms of G attached to a Langlands parameter ϕ (Definition 18.13). We may replace ϕ by the corresponding ^{∨}G-orbit S on the geometric parameter space X(^{∨}G^{Γ}).

η_{ϕ}^{mic}, η_{S}^{mic}: strongly stable formal virtual representation of strong real forms of G attached to a Langlands parameter ϕ or to the corresponding orbit S on the geometric parameter space X(^{∨}G^{Γ}) (Corollary 19.16, Corollary 1.32).

ξ: complete geometric parameter (S_{ξ}, V_{ξ}). Here S_{ξ} is an orbit of ^{∨}G on the geometric parameter space X(^{∨}G^{Γ}), and V_{ξ} is an irreducible equivariant local system on S_{ξ} (Definition 1.17, Definition 7.6).

π(ξ): irreducible representation corresponding to the complete geometric parameter ξ (Theorem 10.4, Definition 10.8, and (1.19)). This is a representation of G(ℝ, δ(ξ))^{can}.

π(Λ): Langlands quotient of the standard representation M(Λ) (Definition 11.2(e), Theorem 11.14).

Π^{z}(G/ℝ): equivalence classes of pairs (π, δ), with δ a strong real form (for a fixed extended group G^{Γ}) and π an irreducible representation of G(ℝ, δ)^{can} of type z (Definition 10.3, Definition 1.14). This set is in one-to-one correspondence with Ξ^{z}(G/ℝ).

Π^{z}(G/ℝ)_{ϕ}: the L-packet attached to the Langlands parameter ϕ; that is, the subset of Π^{z}(G/ℝ) parametrized by complete Langlands parameters of the form (ϕ, τ), or by complete geometric parameters supported on the orbit attached to ϕ (Definition 10.8). Includes at most finitely many representations of each strong real form.

Π^{z}(G/ℝ)_{ϕ}^{mic}: the micro-packet attached to the Langlands parameter ϕ (Definition 19.15); the set of irreducible constituents of the stable virtual representation η_{ϕ}^{mic}. Contains the L-packet.

Ξ(^{∨}G^{Γ}): complete geometric parameters; that is, pairs ξ = (S, V) with S an orbit of^{∨}G on the geometric parameter space X(^{∨}G^{Γ}), and V an irreducible^{∨}G^{can}- equivariant local system on S (Definition 7.6). Equivalently, complete Langlands parameters; that is, equivalence classes of pairs (ϕ, τ), with ϕ a Langlands parameter and τ an irreducible representation of A_{ϕ}^{loc,can} (Definition 5.11). When ^{∨}G^{Γ} has the structure of an E-group with second invariant z, we may write instead Ξ^{z}(G/ℝ).

Φ(^{∨}G^{Γ}): equivalence classes of Langlands parameters; that is, ^{∨}G-conjugacy classes of quasiadmissible homomorphisms ϕ from W_{ℝ} to ^{∨}G^{Γ} (Definition 5.2). Equivalently, equivalence classes of geometric parameters; that is, ^{∨}G-orbits on X(^{∨}G^{Γ}) (Definition 7.6). When ^{∨}G^{Γ} has the structure of an E-group with second invariant z, we write instead Φ^{z}(G/ℝ).

Ψ(^{∨}G^{Γ}): equivalence classes of Arthur parameters; that is, ^{∨}G-conjugacy classes of homomorphisms from W_{ℝ} × SL(2, ℂ) satisfying some additional conditions (Definition 21.4). We sometimes write Ψ^{z}(G/ℝ) as for Langlands parameters.

τ_{ϕ}^{mic}(π): representation (possibly reducible) of A_{ϕ}^{mic,can} attached to an irreducible representation π of a strong real form of G (Definition 23.13). Non-zero if and only if π ∈ Π^{z}(G/ℝ)_{ϕ}^{mic}.

Θ(η): distribution character of the virtual representation η ((18.1)). When the argument includes a strong real form δ, η is a (locally finite) virtual representation of all strong real forms, and Θ(η, δ) is the character of those defined on the group G(ℝ, δ)^{can}.

Θ_{SR}(η): smooth function on the strongly regular elements obtained by restricting the character of η.

2. Structure theory: real forms.

In this section we review the basic facts about real forms of reductive groups. Since our concern is entirely with local problems over 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 real form of G is an antiholomorphic involutive automorphism

σ : G → G.  (2.1)(a)

Here “antiholomorphic” means that if f is any algebraic function on G, then the function

g ↦ \overline{f(σg)}  (2.1)(b)

is also algebraic. Equivalently, the differential of σ (an automorphism of 𝔤 = Lie(G)) satisfies

dσ(ix) = −idσ(x).  (2.1)(b′)

“Involutive” means that σ^{2} is the identity. The group of real points of σ is

G(ℝ, σ) = G(ℝ) = { g ∈ G | σg = g }.  (2.1)(c)

Condition (2.1)(b′) shows that the Lie algebra 𝔤(ℝ) determines dσ, and hence determines σ. We may therefore speak of G(ℝ) as the real form of G without danger of confusion. However, not every real form of the Lie algebra 𝔤 exponentiates to a real form of G. The multiplicative group G = ℂ^{×} has exactly two real forms, given by the automorphisms

σ_{s}(z) = \overline{z},  σ_{c}(z) = \overline{z}^{−1}.  (2.2)(a)

The corresponding groups are

G(ℝ, σ_{s}) = ℝ^{×},  G(ℝ, σ_{c}) = S^{1}.  (2.2)(b)

The real forms of the Lie algebra ℂ, on the other hand, are parametrized by real lines in ℂ. The corresponding subgroups {e^{tz} | t ∈ 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 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 is a group of real points.)

Two real forms σ and σ′ are called equivalent if there is an element g ∈ G such that

σ′ = Ad(g) ∘ σ ∘ Ad(g^{−1}).  (2.3)(a)

We can write this as

σ′ = Ad(gσ(g^{−1})) ∘ σ.  (2.3)(a′)

In terms of the groups of real points, it is equivalent to

G(ℝ, σ′) = gG(ℝ, σ)g^{−1}.  (2.3)(a′′)

The set of real forms equivalent to a fixed real form σ is therefore a homogeneous space G/H; the isotropy group is

H = { g ∈ G | gσ(g^{−1}) ∈ Z(G) }.  (2.3)(b)

This isotropy group contains G(ℝ, σ)Z(G), but may be larger. (For example, if G(ℝ) = SL(2, 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(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, ℂ) has two isomorphic but inequivalent real forms (the isomorphism class is represented by SO^{∗}(4n)) if n ≥ 1. The involutions are conjugate by the non-identity component of O(4n, ℂ). (There is a unique equivalence class of real forms of SO(4n + 2, ℂ) with G(ℝ) isomorphic to SO^{∗}(4n + 2).) A less subtle example is provided by the inequivalent real forms ℝ^{×} ×S^{1} and S^{1} ×ℝ^{×} of ℂ^{×} × ℂ^{×}.

Two real forms σ and σ′ are said to be inner to each other if there is an element g ∈ G such that

σ′ = Ad(g) ∘ σ.  (2.4)

This is an equivalence relation. (If σ is a real form, the automorphism σ′ defined by (2.4) will certainly 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, 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.

Example 2.5. Suppose G = GL(n, ℂ), and σ_{c} is the compact real form

σ_{c}(g) =^{t} \overline{g}^{−1}, G(ℝ, σ_{c}) = U(n).

For 0 ≤ p ≤ n, let g_{p} be the diagonal matrix with p entries equal to 1 and n − p equal to −1. Set

σ_{p} = Ad(g_{p}) ∘ σ_{c}.

Then

G(ℝ, σ_{p}) = U(p, q),

so all these forms are inner to each other. On the other hand, the split real form

σ_{s}(g) = \overline{g}, G(ℝ, σ_{s}) = GL(n, ℝ)

is not inner to σ_{c} (as one can see — rather unfairly — by examining their restrictions to the center of G).

We recall that a real form σ is called quasisplit if there is a Borel subgroup B ⊂ G such that

σ(B) = B.  (2.6)

Proposition 2.7. Suppose σ′ 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 σ′. Specifically, there is a quasisplit real form σ of G and an element g of G so that

σ′ = Ad(g) ∘ σ.

Before giving the proof, we recall the basic structural fact on which it is based.

Proposition 2.8. Suppose G is a connected reductive complex algebraic group. Fix a Borel subgroup B of G, a maximal torus T ⊂ B, and a set of basis vectors {X_{α}} for the simple root spaces of T in the Lie algebra 𝔟. Let B′, T′, {X_{α}′} be another set of choices of these objects. Then there is an element g ∈ G such that

gBg^{−1} = B′,  gT g^{−1} = T′,  Ad(g)({X_{α}}) = {X_{α}′}.

The inner automorphism 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.

To get a formulation valid over any algebraically closed field k, one need only replace the Lie algebra elements X_{α} by one-parameter subgroups x_{α} : k → B.

Proof of Proposition 2.7. Fix B, T ⊂ B, and {X_{α}} as in Proposition 2.8. Then the automorphism σ′ carries these objects to others of the same kind:

σ′(B) = B′,  σ′(T) = T′ ⊂ B′,  dσ(X_{α}) = X_{α}′.  (2.9)(a)

By Proposition 2.8, we can therefore find an element g ∈ G such that

Ad(g^{−1})(B′) = B,  Ad(g^{−1})(T′) = T,  Ad(g^{−1})({X_{α}′}) = {X_{α}}. (2.9)(b)

Define σ = Ad(g^{−1}) ∘ σ′; then σ is an antiholomorphic automorphism of G, and

σ(B) = B,  σ(T) = T,  σ({X_{α}}) = {X_{α}}.  (2.9)(c)

Furthermore σ^{2} is an inner automorphism of G (namely Ad(g^{−1}) ∘ Ad(σ′(g^{−1}))) with the properties in (2.9)(c). By Proposition 2.8, such an inner automorphism is necessarily trivial, so σ is an involution. It is therefore a real form. Since σ preserves B, it is quasisplit. By construction it is inner to σ′.

It remains to prove the uniqueness of the equivalence class of σ. So suppose σ′′ is another quasisplit real form in the inner class of σ. By definition of quasisplit, this means that there is a Borel subgroup B′′ fixed by σ′′. The first problem is to find a σ′′-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{σ′′}, 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{σ′′} on the character lattice X^{∗}(\overline{T′′}). Let \overline{Z} be any element of the Lie algebra \overline{𝔱′′} on which all the positive roots take positive values; then \overline{Z} + \overline{σ′′}(\overline{Z}) has the same property. It follows that any preimage of \overline{Z} + \overline{σ′′}(\overline{Z}) in 𝔟′′ is a regular semisimple element. Let Z ∈ 𝔟 be any preimage of \overline{Z}; then Z +σ′′(Z) is a preimage of \overline{Z} + \overline{σ}(\overline{Z}), so it is a σ′′-fixed regular semisimple element of 𝔟′′. Its centralizer in B′′ is therefore a σ′′-stable maximal torus.

Finally, we want to find a σ′′-stable set of basis vectors for the simple root spaces of T′′ in B′′. We know that σ′′ permutes these root spaces by a permutation of order 2. We consider first a pair {α, β} of distinct simple roots interchanged by σ′′. Choose any basis vector X_{α} for the α root space, and define X_{β} = σ′′(X_{α}). Then the relation (σ′′)^{2} = 1 forces X_{α} = σ′′(X_{β}). Next, suppose the root α is fixed by σ′′. Let Y_{α} be any basis vector for the root space. Then there is a complex number c_{α} with σ′′(Y_{α}) = c_{α}Y_{α}. Applying σ′′ to this relation, and using the fact that σ′′ is an antiholomorphic involution, we get c_{α}\overline{c_{α}} = 1. Let z_{α} be a square root of c_{α}; then it follows easily that X_{α} = z_{α}Y_{α} is fixed by σ′′.

We have shown how to construct B′′, T′′ ⊂ B′′, and {X_{α}′′} as in Proposition 2.8 preserved by σ′′. By Proposition 2.8, we can find a g ∈ G carrying B to B′′, etc. It follows that the antiholomorphic involution

σ̃′′ = Ad(g) ∘ σ ∘ Ad(g^{−1})

preserves B′′, etc. Therefore (σ′′)^{−1}σ̃′′ preserves them as well. But the assumption that σ and σ′′ are in the same inner class means that this last automorphism is inner. By Proposition 2.8, it is trivial. So σ′′ = σ̃′′. The right side here is equivalent to σ, as we wished to show. Q.E.D.

Definition 2.10 (see [Springer]). Suppose G is a complex connected reductive algebraic group, B ⊂ G is a Borel subgroup, and T ⊂ G is a maximal torus. The based root datum for G defined by B and T is the quadruple

Ψ_{0}(G, B, T) = (X^{∗}(T), Δ(B, T), X_{∗}(T), Δ^{∨}(B, T)).

Here X^{∗}(T) is the lattice of rational characters of T, Δ(B, T) ⊂ X^{∗}(T) is the set of simple roots of T in 𝔟, X_{∗}(T) is the lattice of rational one-parameter subgroups of T, and Δ^{∨}(B, T) ⊂ 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 based root datum for G, Ψ_{0}(G), as the projective limit over all such T ⊂ B:

Ψ_{0}(G) = (X^{∗}, Δ, X_{∗}, Δ^{∨}) = lim_{B,T} Ψ_{0}(G, B, T).

The structure of the based root datum consists of the lattice structures on X^{∗} and X_{∗}, the containments Δ ⊂ X^{∗} and Δ^{∨} ⊂ X_{∗}, and the perfect pairing

⟨, ⟩ : X^{∗} × X_{∗} → ℤ.

By an isomorphism or automorphism of based root data we will understand a map preserving these structures.

Proposition 2.11 (see [Springer], Corollary 2.14.) Suppose G is a complex connected reductive algebraic group. Write Aut(G) for the (complex) group of rational (equivalently, holomorphic) automorphisms of G, and Aut(Ψ_{0}(G)) for the (discrete) group of automorphisms of the based root datum of G. Then there is a natural short exact sequence

1 → Int(G) → Aut(G) →^{Ψ_{0}} Aut(Ψ_{0}(G)) → 1.

This sequence splits (but not canonically), as follows. Choose a Borel subgroup B of G, a maximal torus T ⊂ B, and a set of basis vectors {X_{α}} for the simple root spaces of T in the Lie algebra 𝔟; and define Aut(G, B, T, {X_{α}}) to be the set of holomorphic automorphisms of G preserving B, T, and {X_{α}} as sets. Then the restriction of Ψ_{0} to Aut(G, B, T, {X_{α}}) is an isomorphism.

For the reader’s convenience we recall how the map Ψ_{0} is defined. Fix τ ∈ Aut(G), and choose any pair T ⊂ B as in Definition 2.10. (We assume no relationship between τ and B or T.) If λ is any rational character of T, then λ ∘ τ^{−1} is a rational character of τ(T). This defines an isomorphism

X^{∗}(T) → X^{∗}(τ(T))

carrying Δ(B, T) to Δ(τ(B), τ(T)). Similarly, composition with τ carries oneparameter subgroups of T to one-parameter subgroups of τ(T). Assembling these maps, we get an isomorphism

Ψ_{0}(τ, B, T) : Ψ_{0}(G, B, T) → Ψ_{0}(G, τ(B), τ(T)).

Since both range and domain are canonically isomorphic to Ψ_{0}(G), this map provides the automorphism Ψ_{0}(τ) that we want.

An automorphism belonging to one of the sets Aut(G, B, T, {X_{α}}) is called 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.

Proposition 2.12. 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 Ψ_{0}(G).

Proof. Suppose σ is any antiholomorphic automorphism of G; we will show how to define an automorphism Ψ_{0}(σ) of Ψ_{0}(G). (This is not immediately handled by Proposition 2.11, since σ is not a holomorphic automorphism of G.) Fix a Borel subgroup B of G, and T ⊂ B a maximal torus. If λ is a holomorphic character of T, then λ ∘ σ^{−1} is an antiholomorphic character of σ(T); so \overline{λ ∘ σ^{−1}} is a holomorphic character of σ(T). The map sending λ to \overline{λ ∘ σ^{−1}} is an isomorphism

X^{∗}(T) → X^{∗}(σ(T)),

and it carries Δ(B, T) to Δ(σ(B), σ(T)). Continuing in this way, we find that σ induces an isomorphism

Ψ_{0}(σ, B, T) : Ψ_{0}(G, B, T) → Ψ_{0}(G, σ(B), σ(T)).

Since both of these objects are canonically isomorphic to Ψ_{0}(G), we have the automorphism we want. If σ′ is another antiholomorphic automorphism of G, then it is clear from the definitions that

Ψ_{0}(σ ∘ (σ′)^{−1}) = Ψ_{0}(σ) ∘ Ψ_{0}(σ′)^{−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 Ψ_{0}(σ) = Ψ_{0}(σ′) if and only if σ and σ′ 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 Ψ_{0}(G) of order 2, and data B, T, and {X_{α}} as in Proposition 2.8. We must find a real form σ of G with Ψ_{0}(σ) = a. The theory of Chevalley forms (forms of a reductive group defined and split over any field) shows that there is a real form σ_{s} of G such that

σ_{s}(B) = B,  σ_{s}(T) = T,  σ_{s}(X_{α}) = X_{α},

and σ_{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,  t_{a}(T) = T,  t_{a}(X_{α}) = X_{a(α)},

and t_{a} induces the automorphism a on X^{∗}(T) and X_{∗}(T). The composition σ = t_{a} ∘ σ_{s} is an antiholomorphic automorphism of G satisfying

σ(B) = B,  σ(T) = T,  σ(X_{α}) = X_{t(α)},

and σ induces the automorphism a on X^{∗}(T) and X_{∗}(T). This implies first of all that Ψ_{0}(σ) = a. Since a has order 2, it follows that σ^{2} is a holomorphic automorphism of G acting trivially on T and fixing the various X_{α}. Such an automorphism is trivial by Proposition 2.8; so σ^{2} = 1, and σ 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 Ψ_{0}(G). We need to consider at the same time representations of various real forms of G. It is natural therefore to consider pairs (π, σ), with σ a real form of G (say in the specified inner class) and π a representation of G(ℝ, σ). It is natural to define two such pairs (π, σ) and (π′, σ′) to be equivalent if there is an element g ∈ G such that σ′ = Ad(g) ∘ σ ∘ Ad(g^{−1}), and π ∘ Ad(g^{−1}) (which is a representation of G(ℝ, σ′) on the space of π) is equivalent to π′. The difficulty with this definition first appears in the example of SL(2, ℝ), discussed after (2.3). If we let π be a holomorphic discrete series representation of G(ℝ, σ) = SL(2, ℝ), and π′ the corresponding antiholomorphic discrete series representation, then this definition makes (π, σ) equivalent to (π′, σ). 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.

Definition 2.13. Suppose G is a connected reductive complex algebraic group. A (weak) extended group containing G is a real Lie group G^{Γ} subject to the following conditions.

(a1) G^{Γ} contains G as a subgroup of index two. (a2) Every element of G^{Γ} − G acts on G (by conjugation) as an antiholomorphic

automorphism.

Condition (a1) may be rephrased as follows. Write Γ = Gal(ℂ/ℝ) for the Galois group.

(a1′) There is a short exact sequence

1 → G → G^{Γ} → Γ → 1.

A strong real form of G^{Γ} is an element δ ∈ G^{Γ} − G such that δ^{2} ∈ Z(G) has finite order. The associated real form for δ is the (antiholomorphic involutive) automorphism σ(δ) of G defined by conjugation by δ:

σ(δ)(g) = δgδ^{−1}.

The group of real points of δ is defined to be the group of real points of σ(δ):

G(ℝ, δ) = { g ∈ G | δgδ^{−1} = g }.

Two strong real forms δ and δ′ of G^{Γ} are called equivalent if they are conjugate by G:

δ ∼ δ′ if and only if δ′ = gδg^{−1} for some g ∈ G.

Thus the set of strong real forms equivalent to a fixed strong real form δ is a homogeneous space G/H; the isotropy group is

H = { g ∈ G | gδg^{−1} = δ } = G(ℝ, δ)

(cf. (2.3)).

A representation of a strong real form of G^{Γ} is a pair (π, δ), subject to

(a) δ is a strong real form of G^{Γ} ; and

(b) π is an admissible representation of G(ℝ, δ). Two such representations (π, δ) and (π′, δ′) are said to be (infinitesimally) equivalent if there is an element g ∈ G such that gδg^{−1} = δ′, and π∘Ad(g^{−1}) is (infinitesimally) equivalent to π′. (In particular, this is possible only if the strong real forms are equivalent.) Finally, define

Π(G^{Γ}) = Π(G/ℝ)

to be the set of (infinitesimal) equivalence classes of irreducible representations of strong real forms of G^{Γ}. (Here when we use the notation on the right, we must have in mind a particular weak extended group G^{Γ}.)

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^{Γ} 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 W needed to complete the definition of an extended group plays no rôle in this lemma.) Surjectivity of the map is clear; what must be established is injectivity. So suppose that π and π′ are irreducible representations of G(ℝ, δ_{s}) and G(ℝ, δ_{s′}), respectively, and that the pairs (π, δ_{s}) and (π′, δ_{s′}) are equivalent. We must show that δ_{s} = δ_{s′}, and that π is equivalent to π′. The hypothesis means that there is an element g ∈ G such that

gδ_{s}g^{−1} = δ_{s′}, π ∘ Ad(g^{−1}) ∼ π′

The first condition says that the strong real forms δ_{s} and δ_{s′} are equivalent. Since they belong to a set of representatives for the equivalence classes, they must in fact coincide. Consequently g commutes with δ_{s}, and so belongs to G(ℝ, δ_{s}). But this means that π ∘ Ad(g^{−1}) is equivalent to π, the equivalence being implemented by the operator π(g) on the space of π. The second condition therefore implies that π′ is equivalent to π as representations of G(ℝ, δ_{s}), as we wished to show. Q.E.D.

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

Proposition 2.14. Suppose G^{Γ} is a weak extended group. Then the set of real forms of G associated to strong real forms of G^{Γ} constitutes exactly one inner class of real forms.

Proof. The conjugation action of any element δ of G^{Γ} − G defines an antiholomorphic automorphism σ(δ) of G. This automorphism preserves Z(G), and in fact its restriction σ_{Z} to Z(G) is independent of the choice of δ.

The proof of Proposition 2.12 attaches to each element δ of G^{Γ} − G an automorphism a = Ψ_{0}(δ) of the based root datum of G. The proof also shows that a is independent of the choice of δ, and that the various conjugation actions σ(δ) give all antiholomorphic automorphisms σ of G such that Ψ_{0}(σ) = a. In particular, Proposition 2.12 implies that for any real form σ in the inner class defined by a, there is a δ_{1} ∈ G^{Γ} − G with σ(δ_{1}) = σ. Since σ is an involution, this implies that δ_{1}^{2} = z_{1} ∈ Z(G). To complete the proof, we must show that our choice of δ_{1} can be modified to make z_{1} have finite order. We first compute

σ_{Z}(z_{1}) = δ_{1}zδ_{1}^{−1} = δ_{1}δ_{1}^{2}δ_{1}^{−1} = δ_{1}^{2} = z_{1}.

That is, z_{1} ∈ Z^{σ_{Z}}. We can now apply the following elementary lemma.

Lemma 2.15. Suppose Z is a (possibly disconnected) complex reductive abelian algebraic group, and σ is an antiholomorphic involutive automorphism of Z. Put

(1 + σ)Z = { zσ(z) | z ∈ Z }.

Then the quotient group Z^{σ}/(1 + σ)Z is finite, and each coset has a representative of finite order in Z.

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} ∈ Z(G) such that z = z_{1}z_{2}σ_{Z}(z_{2}) has finite order. Set

δ = z_{2}δ_{1}.

Then σ(δ) = σ, and δ^{2} = z has finite order; so δ is the required strong real form. Q.E.D.

Here is the classification of weak extended groups.

Corollary 2.16. Suppose G is a connected reductive complex algebraic group.

a) Fix a weak extended group G^{Γ} for G. Let σ_{Z} be the antiholomorphic involution of Z(G) defined by the conjugation action of any element of G^{Γ} − G. We can attach to G^{Γ} two invariants. The first of these is an involutive automorphism

a ∈ Aut(Ψ_{0}(G))

of the based root datum of G. The second is a class

\overline{z} ∈ Z(G)^{σ_{Z}} /(1 + σ_{Z})Z(G).

b) Suppose G^{Γ} and (G^{Γ})′ 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^{Γ} to (G^{Γ})′.

c) Suppose a ∈ Aut(Ψ_{0}(G)) is an involutive automorphism. Write σ_{Z} for the antiholomorphic involution of Z(G) defined by the action of any real form σ in the inner class corresponding to a (Proposition 2.12); and suppose

\overline{z} ∈ Z(G)^{σ_{Z}} /(1 + σ_{Z})Z(G).

Then there is a weak extended group G^{Γ} with invariants (a, \overline{z}).

Proof. For (a), any element δ of G^{Γ} − G defines by conjugation an antiholomorphic automorphism σ(δ) of G. By Proposition 2.12 and its proof, the corresponding automorphism a = Ψ_{0}(σ(δ)) is independent of the choice of δ. To define \overline{z}, fix a quasisplit real form σ_{q} in the inner class defined by a, and choose δ_{q} ∈ G^{Γ} − G so that

σ(δ_{q}) = σ_{q}

(as is possible by Proposition 2.14.) Since σ_{q}^{2} = 1,

δ_{q}^{2} = z ∈ Z(G);

and using δ_{q} to compute σ_{Z}, we see that

σ_{Z}(z) = z.

By Proposition 2.7, any other choice of σ′_{q} differs from σ_{q} by conjugating with an element Ad(g); so any other δ′_{q} is of the form

δ′_{q} = z_{1}gδ_{q}g^{−1}.

We compute immediately that

(δ′_{q})^{2} = z_{1}σ_{Z}(z_{1})δ_{q}^{2},

so the class \overline{z} is independent of all choices.

For (b), fix a quasisplit real form σ_{q} of G as in (a), and choose elements δ_{q} ∈ G^{Γ} − G, δ_{q}′′ ∈ (G^{Γ})′ so that

σ(δ_{q}) = σ_{q}, σ(δ_{q}′′) = σ_{q}.

Write z = δ_{q}^{2}, z′ = (δ_{q}′′)^{2}. By the hypothesis on the invariants of the two weak extended groups, there is an element z_{1} ∈ Z(G) so that z = z′ (z_{1}σ_{Z}(z_{1})). Set δ′_{q} = z_{1}δ_{q}′′; then

σ(δ′_{q}) = σ(δ_{q}) = σ_{q},  δ_{q}^{2} = (δ′_{q})^{2} = z.  (2.17)(a)

Now the group G^{Γ} is the disjoint union of G and the coset Gδ_{q}; these are multiplied according to the rules

(g_{1}δ_{q})(g_{2}δ_{q}) = g_{1}σ_{q}(g_{2})z,  (g_{1}δ_{q})(g_{2}) = g_{1}σ_{q}(g_{2})δ_{q}  (2.17)(b)

and the obvious rules for the other two kinds of product. We can define a bijection from G^{Γ} to (G^{Γ})′ by using the identity on G, and sending gδ_{q} to gδ′_{q}. By (2.17), this bijection is a group homomorphism, proving (b).

For (c), choose a quasisplit real form σ_{q} of G in the inner class corresponding to a (Proposition 2.12), and a representative z ∈ Z(G) of the class \overline{z}. Define G^{Γ} to consist of the union of G and the set of formal symbols gδ_{q} (topologized as the union of two copies of G). Introduce a multiplication on G^{Γ} by the rules in (2.17)(b); then it is a simple matter to check that G^{Γ} is a weak extended group with the desired invariants. Q.E.D.

3. Structure theory: extended groups and Whittaker models.

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, ℝ), 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) 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(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(ℝ), but not necessarily by an inner automorphism. It is essentially known from [Kostant] that each tempered Lpacket for G(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.

Definition 3.1. Suppose G(ℝ) is a quasisplit real form of a complex connected reductive algebraic group. Fix T ⊂ B Cartan and Borel subgroups of G defined over ℝ, and write A ⊂ T for the maximal split torus in T. (Thus the identity component of A(ℝ) is an “Iwasawa A” for G(ℝ).) Write N for the unipotent radical of B. List the simple (restricted) roots of A in N as {α_{1}, …, α_{l}}, and write 𝔤_{α_{j}} ⊂ 𝔫 for the corresponding restricted root spaces. Then

𝔫/[𝔫, 𝔫] ≃ ∑_{j=1}^{l} 𝔤_{α_{j}}.  (3.1)(a)

All the spaces here are defined over ℝ, so we get a corresponding decomposition of 𝔫(ℝ)/[𝔫(ℝ), 𝔫(ℝ)]. In particular, the space of (one-dimensional) unitary characters of N(ℝ) is isomorphic (by taking differentials) to

∑_{j=1}^{l} i𝔤_{α_{j}}(ℝ)^{∗},  (3.1)(b)

the space of imaginary-valued linear functionals on the simple restricted real root spaces. A unitary character χ of N(ℝ) is called non-degenerate if its restriction to each simple restricted root subgroup is non-trivial.

Suppose χ is a non-degenerate unitary character of N(ℝ). Write ℂ_{χ} for the one-dimensional space on which χ acts, and

L_{χ} = G(ℝ) ×_{N(ℝ)} ℂ_{χ}  (3.1)(c)

for the corresponding line bundle on G(ℝ)/N(ℝ). The Whittaker model W h(χ) for G(ℝ) defined by χ is the space of smooth sections of L_{χ}:

W h(χ) ≃ { f ∈ C^{∞}(G(ℝ)) | f(gn) = χ(n)^{−1}f(g) }  (3.1)(d).

We make W h(χ) into a smooth representation of G(ℝ) by left translation.

A Hilbert space representation (π, H_{π}) of G(ℝ) is said to admit a Whittaker model of type χ if there is a non-zero continuous map

H_{π}^{∞} → W h(χ)  (3.1)(e)

respecting the action of G(ℝ). (Equivalently, π should admit a non-zero distribution vector transforming according to the character χ under N(ℝ).)

We consider first the uniqueness of non-degenerate characters. The following lemma is well illustrated by the examples of GL(2, ℝ) (which has up to conjugacy only one kind of Whittaker model) and SL(2, ℝ) (which has two). The element t of (c) below can always be chosen in T (ℝ) in the first case, but not in the second.

Lemma 3.2([Kostant], Lemma 6.2.1). Suppose G(ℝ) is a quasisplit real form of a complex connected reductive algebraic group, and T ⊂ B are Cartan and Borel subgroups defined over ℝ.

a) The decomposition (3.1)(a) of 𝔫/[𝔫, 𝔫] is invariant under Ad(B). Consequently the adjoint action of B(ℝ) on characters of N(ℝ) preserves the set of nondegenerate unitary characters. In particular, the notion of non-degenerate unitary character of N(ℝ) is independent of the choice of T.

b) Suppose t ∈ T. Then the automorphism Ad(t) of G is defined over ℝ if and only if it preserves each of the simple real restricted root spaces 𝔤_{α_{j}}(ℝ).

c) Suppose χ and χ′ are non-degenerate unitary characters of N(ℝ). Then there is an element t ∈ T such that Ad(t) is defined over ℝ (so that Ad(t) defines an automorphism of N(ℝ)), and t · χ is equal to χ′ (the action on characters being defined by composition of the character with the inverse of the automorphism of N(ℝ)).

d) In (c), the coset tZ(G)is uniquely determined by χ and χ′. In particular, χ is conjugate to χ′ by G(ℝ) if and only if t ∈ T (ℝ)Z(G).

Proof. For (a), Ad(N) acts trivially on 𝔫/[𝔫, 𝔫]; so we need only consider 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 (ℝ) are conjugate by N(ℝ).

For (b), the condition is obviously necessary for Ad(t) to be defined over ℝ; so suppose that it holds. Write σ for the complex conjugation on G. Because σ preserves T, it must permute the root spaces in 𝔤; we write σ for the corresponding permutation of the roots. Then Ad(t) is defined over ℝ if and only if for every root β of T in G, we have

β(σt) = \overline{(σβ)(t)}.  (3.3)(a)

It suffices to verify this condition for simple roots β. Write α for the restriction of β to A; then the real restricted root space 𝔤_{α} consists of elements of the form

X + σX (X ∈ 𝔤_{β}).

Now

Ad(t)(X + σX) = β(t)X + (σβ(t))σX.  (3.3)(b)

It follows that Ad(t) preserves the real restricted root space if and only if

(σβ(t))σX = σ(β(t)X).

Since σ is conjugate-linear, this is equivalent to (3.3)(a).

For (c), we regard the complexified differential of χ as a character of the Lie algebra 𝔫. As such, it has a complex value dχ(X) ∈ ℂ on any X ∈ 𝔫. The condition that χ be unitary is equivalent to dχ taking purely imaginary values on each simple restricted real root space. By (3.3)(a), this is equivalent to

dχ(X) = −\overline{dχ(σX)}  (X ∈ 𝔤_{β})  (3.4)(a)

for every simple root β. The non-degeneracy condition is

dχ(X) ≠ 0  (X ∈ 𝔤_{β} − 0).  (3.4)(b)

Suppose now that χ and χ′ are as in (c) of the lemma. Since the simple roots are linearly independent, we can find t ∈ T with

β(t) = dχ(X)/dχ′(X)  (X ∈ 𝔤_{β} − 0).  (3.4)(c)

Combining (3.4)(c) with (3.4)(a) gives exactly the condition in (3.3)(a) for Ad(t) to be defined over ℝ. To compute the action of t on χ, it is enough to evaluate the differential at an element X ∈ 𝔤_{β}. This is

d(t · χ)(X) = dχ(Ad(t^{−1})X) = dχ(β(t^{−1})X) = β(t)^{−1}dχ(X) = dχ′(X).

It follows that t · χ = χ′, as we wished to show.

For (d), the proof of (c) shows that the choice (3.4)(c) of β(t) (for every simple β) is forced by the requirement that t·χ = χ′. This gives the first assertion. For the second, χ is conjugate to χ′ by G(ℝ) if and only if it is conjugate by the normalizer B(ℝ) = T (ℝ)N(ℝ) of N(ℝ). Since N(ℝ) fixes χ, (d) follows. Q.E.D.

Recall now from Definition 1.12 the notion of an extended group (G^{Γ}, W) for G. The invariants of the extended group are the automorphism a of the based root datum Ψ_{0}(G) attached to the underlying weak extended group G^{Γ} (Corollary 2.16), and the element

z = δ^{2} ∈ Z(G)^{σ_{Z}} .  (3.5)

Here δ is any element of a triple (δ, N, χ) ∈ W (Definition 1.12). The element z is independent of the choice of δ, since W is a single conjugacy class under G.

Here is the classsification of extended groups.

Proposition 3.6. Suppose G is a connected reductive complex algebraic group.

a) Suppose (G^{Γ}, W) and ((G^{Γ})′ , W′) 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^{Γ} to (G^{Γ})′ carrying W to W′. Any two such extensions differ by an inner automorphism of G^{Γ} from Z(G).

b) Fix a weak extended group G^{Γ} for G with invariants (t, \overline{z}). If (G^{Γ}, W) is an extended group, then its second invariant is a representative for the class of \overline{z}. Conversely, if z ∈ Z(G)^{σ_{Z}} is an element of finite order representing the class of \overline{z}, then there is an extended group structure on G^{Γ} with second invariant z.

Proof. For (a), fix (δ, N, χ) ∈ W. Since δ^{2} = z ∈ Z(G), conjugation by δ defines an antiholomorphic involutive automorphism σ = σ(δ) of G, and thus a real form G(ℝ). By condition (b)(2) in Definition 1.12, G(ℝ) is quasisplit. The real forms defined analogously using W′ are also quasisplit, and inner to G(ℝ). By Proposition 2.12, at least one of them must coincide with G(ℝ). That is, we can find (δ′, N′, χ′) ∈ W′ so that

σ(δ′) = σ.  (3.7)(a)

Now N and N′ are maximal unipotent subgroups of G defined over ℝ. They are therefore conjugate by an element g ∈ G(ℝ). After replacing (δ′, N′, χ′) by their conjugates by g (which does not change δ′, and therefore preserves (3.7)(a)) we may assume that

N′ = N.  (3.7)(b)

Now χ and χ′ are non-degenerate unitary characters of N(ℝ). By Lemma 3.2, there is an element t ∈ G normalizing N(ℝ), carrying χ′ to χ, and with Ad(t) defined over ℝ. This last condition means that

σ(t) = tw

for some w ∈ Z(G). We finally replace (δ′, N′, χ′) by their conjugates by t. This replaces δ′ by w^{−1}δ′, 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,

χ′ = χ.  (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 δ to δ′.

For the uniqueness, any other such isomorphism must send (δ, N, χ) to a Gconjugate g · (δ′, N, χ) of (δ′, N, χ) (since it carries W to W′). We wish to show that g ∈ G(ℝ)Z(G). Since the isomorphism is the identity on G, we deduce three facts: the conjugation actions of δ and g(δ′)g^{−1} must agree on G; gNg^{−1} = N; and g · χ = χ. The first of these facts means that Ad(g) is defined over ℝ. The second fact implies that g = tn ∈ B. Since Ad(g) is defined over ℝ, it follows that n ∈ N(ℝ) and that Ad(t) is defined over ℝ. Because of Lemma 3.2(d), the third fact guarantees that t ∈ 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 δ of G^{Γ} − G with σ(δ) a quasisplit real form of G, and δ^{2} = z. Let N be any maximal unipotent subgroup of G defined over ℝ, and let χ be any non-degenerate unitary character of N(ℝ). Then the G-conjugacy class W of the triple (δ, N, χ) is evidently an extended group structure with second invariant z. Q.E.D.

Corollary 3.8. 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)^{σ_{Z}} (cf. Corollary 2.16).

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 rôle, as we will see.

4. Structure theory: L-groups.

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 Ψ_{0} = (X^{∗}, Δ, X_{∗}, Δ^{∨}) is the based root datum of a reductive group if and only if ^{∨}Ψ_{0} = (X_{∗}, Δ^{∨}, X^{∗}, Δ) is as well. We call ^{∨}Ψ_{0} the dual based root datum to Ψ_{0}. Notice that the automorphism groups of dual based root data are canonically isomorphic:

Aut(Ψ_{0}) ≃ Aut(^{∨}Ψ_{0}).  (4.1)

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

∨Ψ_{0}(G) ≃ Ψ_{0}(^{∨}G).

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

Definition 4.3 (cf. [AV2], Definition 10.5). Suppose G is a complex connected reductive algebraic group. A weak E-group for G is an algebraic group ^{∨}G^{Γ} containing a dual group ^{∨}G for G as a subgroup of index 2. That is, there is a short exact sequence

1 → ^{∨}G → ^{∨}G^{Γ} → Γ → 1.

Here Γ = Gal(ℂ/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^{Γ} is a weak extended group for G (Definition 2.13). Recall that there is attached to G^{Γ} 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^{Γ}. A weak E-group for G^{Γ} is a weak E-group for G with second invariant a (Proposition 4.4(a)). We may also call this a weak E-group for G and the specified inner class of real forms, or even a weak E-group for G(ℝ) (with G(ℝ) a real form in the specified inner class).

Finally, we will occasionally need to use this terminology in a less specific way. We will say that a weak E-group is an algebraic group H^{Γ} containing a complex connected reductive algebraic subgroup H of index two. That is, there is a short exact sequence

1 → H → H^{Γ} → Γ → 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.

The notion of dual group is symmetric, in the sense that if ^{∨}G is a dual group for G then G is a dual group for ^{∨}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.

Proposition 4.4. Suppose G is a connected reductive complex algebraic group.

a) Fix a weak E-group ^{∨}G^{Γ} for G. Let θ_{Z} be the holomorphic involution of Z(^{∨}G) defined by the conjugation action of any element of ^{∨}G^{Γ} −^{∨}G. We can attach to ^{∨}G^{Γ} two invariants. The first of these is an involutive automorphism

a ∈ Aut(Ψ_{0}(G))

of the based root datum of G. The second is a class

\overline{z} ∈ Z(^{∨}G)^{θ_{Z}} /(1 + θ_{Z})Z(^{∨}G).

b) Suppose ^{∨}G^{Γ} and (^{∨}G^{Γ})′ are weak E-groups for G with the same invariants (a, \overline{z}). Then any one of the canonical isomorphisms from ^{∨}G to ^{∨}G′ described after Definition 4.2 extends to an isomorphism from ^{∨}G^{Γ} to (^{∨}G^{Γ})′.

c) Suppose a ∈ Aut(Ψ_{0}(G)) is an involutive automorphism, and ^{∨}G is a dual group of G. Write θ_{Z} for the involution of Z(^{∨}G) defined by the action of any automorphism σ corresponding to a (Proposition 2.11 and (4.1)); and suppose

\overline{z} ∈ Z(^{∨}G)^{θ_{Z}} /(1 + θ_{Z})Z(^{∨}G).

Then there is a weak E-group ^{∨}G^{Γ} with invariants (a, \overline{z}).

(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^{∨}δ be any element of^{∨}G^{Γ}−^{∨}G such that the conjugation action of ^{∨}δ on ^{∨}G is a distinguished automorphism (as defined after Proposition 2.11). (Such elements necessarily exist.) Then

∨δ^{2} = z ∈ Z(^{∨}G)  (4.5)

is a representative of \overline{z}.

Definition 4.6. Suppose G is a complex connected reductive algebraic group. An E-group for G is a pair (^{∨}G^{Γ}, D), subject to the following conditions.

(a) ^{∨}G^{Γ} is a weak E-group for G (Definition 4.3).

(b) D is a ^{∨}G-conjugacy class of elements of finite order in ^{∨}G^{Γ} −^{∨}G.

(c) Suppose ^{∨}δ ∈ D. Then conjugation by ^{∨}δ is a distinguished involutive automorphism of ^{∨}G (see the definition after Proposition 2.11). The invariants of the E-group are the automorphism a attached to ^{∨}G^{Γ} by

Proposition 4.4, and the element

z = ^{∨}δ^{2} ∈ Z(^{∨}G)^{θ_{Z}};

here ^{∨}δ is any element of D.

An L-group for G is an E-group whose second invariant is equal to 1. That is, we replace condition (b) above by

(b)′ D is a ^{∨}G-conjugacy class of elements of order two in ^{∨}G^{Γ} −^{∨}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.

Here is the classification of E-groups.

Proposition 4.7. Suppose G is a connected reductive complex algebraic group.

a) Suppose (^{∨}G^{Γ}, D) and ((^{∨}G^{Γ})′ , D′) are E-groups for G with the same invariants

(a, z). Then any one of the canonical isomorphisms from ^{∨}G to ^{∨}G′ described after Definition 4.2 extends to an isomorphism from ^{∨}G^{Γ} to (^{∨}G^{Γ})′ carrying D to D′.

b) In the setting of (a), suppose τ_{1} and τ_{2} are isomorphisms from ^{∨}G^{Γ} to (^{∨}G^{Γ})′ with the property that τ_{i} |_{^{∨}G} is one of the canonical isomorphisms described after Definition 4.2, and that τ_{i}(D) = D′. Then there is an element g ∈ ^{∨}G such that τ_{1} = τ_{2} ∘ Ad(g).

c) Fix a weak E-group ^{∨}G^{Γ} for G with invariants (a, \overline{z}). If (^{∨}G^{Γ}, D) is an E-group, then its second invariant (Definition 4.6) is a representative for the class of \overline{z}. Conversely, if z ∈ Z(^{∨}G)^{θ_{Z}} is an element of finite order representing the class of \overline{z}, then there is an E-group structure on ^{∨}G^{Γ} with second invariant z.

The argument follows that given for Proposition 3.6, and we leave it to the reader.

Corollary 4.8. 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 (^{∨}G^{Γ}, D) and ((^{∨}G^{Γ})′, D′) are isomorphic. We may choose this isomorphism τ so that τ(D) = D′, and the restriction of τ to ^{∨}G is one of the canonical isomorphisms described after Definition 4.2. This choice of τ is unique up to composition with an inner automorphism from ^{∨}G.

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.

Definition 4.9. Suppose G is a complex connected reductive algebraic group, B is a Borel subgroup, and T ⊂ B is a maximal torus. Write

χ(2^{∨}ρ) : ℂ^{×} → T  (4.9)(a)

for the sum of the positive coroots, an element of X_{∗}(T). Set

m(^{∨}ρ) = χ(2^{∨}ρ)(i) ∈ T, z(^{∨}ρ) = m(^{∨}ρ)^{2} = χ(2^{∨}ρ)(−1).  (4.9)(b)

The familiar fact that the sum of the positive coroots takes the value 2 on each simple root α means that

Ad(m(^{∨}ρ))(X_{α}) = i^{2}X_{α} = −X_{α}, Ad(z(^{∨}ρ))(X_{α}) = X_{α}  (4.9)(c)

for any element X_{α} of the α root space. In particular, z(^{∨}ρ) ∈ Z(G); this element is independent of the choice of B and T, so it is preserved by all automorphisms of G.

Suppose θ is an involutive (holomorphic) automorphism of G preserving a Borel subgroup B. Fix a θ-stable maximal torus T ⊂ B (as is possible). We say that the pair (θ, B) is large if θ acts by −1 on each θ-stable simple root space. We say that (θ, B) is distinguished if θ acts by +1 on each θ-stable simple root space. (Since T is unique up to conjugation by the θ-invariants in B, these definitions do not depend on the choice of T.)

Lemma 4.10. Suppose G is a complex connected reductive algebraic group, B ⊂ G is a Borel subgroup, and θ is an involutive holomorphic automorphism of G preserving B. Fix a θ-stable maximal torus T ⊂ B.

a) The automorphism θ fixes the map χ(2^{∨}ρ) of Definition 4.9. In particular, it fixes the element m(^{∨}ρ); so

θ′ = Ad(m(^{∨}ρ)) ∘ θ

is another involutive automorphism of G preserving B.

b) The pair (θ, B) is large (respectively distinguished) if and only if (θ′, B) is distinguished (respectively large).

c) The pair (θ, B) is distinguished if and only if θ is a distinguished automorphism (as defined after Proposition 2.11).

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 (θ, B) is distinguished in the sense of Definition 4.9, and T is a θ-stable maximal torus in B. To see that θ is distinguished in the sense of section 2, we must find a set of simple root vectors {X_{α}} permuted by θ. Begin with any set {X′_{α}} of simple root vectors. If α is fixed by θ, then the assumption that (θ, B) is distinguished means that θX′_{α} = X′_{α}. For such roots α we define X_{α} = X′_{α}. The remaining simple roots occur in pairs interchanged by θ; by choosing one representative of each pair, we can list them as {β_{1}, …, β_{r}, θβ_{1}, …, θβ_{r}}. We now define

X_{β_{i}} = X′_{β_{i}}, X_{θβ_{i}} = θX′_{β_{i}}.

Then θ ∈ Aut(G, B, T, {X_{α}}), as we wished to show. Q.E.D.

Lemma 4.11. Suppose G is a complex connected reductive algebraic group, and θ is a distinguished involutive holomorphic automorphism of G. Then any two θ-stable Borel subgroups of G are conjugate by the invariants of θ in G.

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 θ in G, and K_{0} for its identity component. Suppose B and B′ are θ-stable. Choose θ-stable tori T and T′ in B and B′; then T ∩ K_{0} and T′ ∩ K_{0} are maximal tori in K_{0}, contained in Borel subgroups B ∩ K_{0} and B′ ∩ 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 ∩ K_{0} = T′ ∩ K_{0}, B ∩ K_{0} = B′ ∩ K_{0}.

Now T ∩ K_{0} contains regular elements of G (for example in the image of the map χ(2^{∨}ρ)), 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 θ is an involutive automorphism. A root of T in G is B-simple if and only if its restriction to T ∩ K_{0} is simple for B ∩ K_{0}; this is a consequence of the fact that (θ, 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.

Definition 4.12. Suppose G is a complex connected reductive algebraic group. An E-group for G is a pair (^{∨}G^{Γ}, E), subject to the following conditions.

(a) ^{∨}G^{Γ} is a weak extended group for G (Definition 4.3).

(b) E is a ^{∨}G-conjugacy class of pairs (^{∨}δ,^{d}B), with ^{∨}δ an element of finite order in ∨G^{Γ} −^{∨}G, and ^{d}B a Borel subgroup of ^{∨}G.

(c) Suppose (^{∨}δ,^{d}B) ∈ E. Then conjugation by ^{∨}δ is a distinguished involutive automorphism of ^{∨}G (see the definition after Proposition 2.11) preserving ^{d}B.

To see that Definition 4.12 is equivalent to Definition 4.6, suppose first that (^{∨}G^{Γ}, E) is an E-group in the sense of Definition 4.12. Set

D = {^{∨}δ | (^{∨}δ,^{d}B) ∈ E }.  (4.13)(a)

Then (^{∨}G^{Γ}, D) is an E-group in the sense of Definition 4.6. Conversely, suppose (^{∨}G^{Γ}, D) is an E-group in the sense of Definition 4.6. Set

E = { (^{∨}δ,^{d}B) | ^{∨}δ ∈ D, and ^{d}B is a ^{∨}δ-invariant Borel subgroup of ^{∨}G }.
(4.13)(b)

Lemma 4.11 implies that E is a single^{∨}G-conjugacy class, so (^{∨}G^{Γ}, 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.

Definition 4.14 ([AV2], Definition 10.5). Suppose G is a complex connected reductive algebraic group. An E-group for G is a pair (^{∨}G^{Γ}, S), subject to the following conditions.

(a) ^{∨}G^{Γ} is a weak extended group for G (Definition 4.3).

(b) S is a ^{∨}G-conjugacy class of pairs (^{∨}δ,^{d}B), with ^{∨}δ an element of finite order in ∨G^{Γ} −^{∨}G, and ^{d}B a Borel subgroup of ^{∨}G.

(c) Suppose (^{∨}δ,^{d}B) ∈ S. Then conjugation by ^{∨}δ defines an involutive automorphism θ of ^{∨}G preserving ^{d}B. The pair (θ,^{d}B) is large (Definition 4.9). The invariants of the E-group are the automorphism a attached to ^{∨}G^{Γ} by

Proposition 4.5, and the element

z = z(ρ)^{∨}δ^{2} ∈ Z(^{∨}G)^{θ_{Z}};

here (^{∨}δ,^{d}B) is any element of S, and z(ρ) ∈ Z(^{∨}G) is the distinguished element described in Definition 4.9.

An L-group for G is an E-group whose second invariant is equal to 1. That is, the elements ^{∨}δ appearing in (b) are required to satisfy ^{∨}δ^{2} = z(ρ).

We recall from section 9 of [AV2] the proof that Definitions 4.12 and 4.14 are equivalent. Suppose that (^{∨}G^{Γ}, E) is an E-group in the sense of Definition 4.12. Fix (^{∨}δ,^{d}B) ∈ E, and write θ for the automorphism of ^{∨}G defined by conjugation by ^{∨}δ. Choose a θ-stable maximal torus ^{d}T ⊂ ^{d}B, and define m(ρ) ∈ ^{d}T as in Definition 4.9. Set

∨δ′ = m(ρ)^{−1}^{∨}δ;  (4.15)(a)

the corresponding automorphism of ^{∨}G is

θ′ = Ad(m(ρ))θ.  (4.15)(b)

Changing the choice of ^{d}T changes ^{∨}δ′ only by conjugation by ^{d}B; so the conjugacy class

S = Ad(^{∨}G)(^{∨}δ′,^{d}B)  (4.15)(c)

is well-defined. By hypothesis, the pair (θ,^{d}B) is distinguished, so by (4.15)(b) and Lemma 4.10 the pair (θ′,^{d}B) is large. Consequently (^{∨}G^{Γ}, 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.

5. Langlands parameters and L-homomorphisms.

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 rôle 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 less complicated than L-groups.) In any case we will need to have E-groups available when we discuss endoscopy.

Definition 5.1 ([Langlands], section 2). Suppose ^{∨}H^{Γ} and ^{∨}G^{Γ} are weak Egroups (Definition 4.3). An L-homomorphism from ^{∨}H^{Γ} to ^{∨}G^{Γ} is a morphism ϵ : ^{∨}H^{Γ} → ^{∨}G^{Γ} of algebraic groups, with the property that the diagram

∨H^{Γ} ⟶^{ϵ} ^{∨}G^{Γ}
↘  ↙

Γ

(cf. Definition 4.3) commutes. (Even when ^{∨}H^{Γ} and ^{∨}G^{Γ} are E-groups, we do not require that ϵ should respect the distinguished conjugacy classes D, E, or S of Definitions 4.6, 4.12, and 4.14.) Two such morphisms ϵ and ϵ′ are said to be equivalent if they are conjugate by the action of ^{∨}G; that is, if there is an element g ∈ ^{∨}G so that ϵ′ = Ad(g) ∘ ϵ.

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 ℝ.

Definition 5.2 (see [Langlands], [Borel], or [Tate]). The Weil group of ℝ is the real Lie group W_{ℝ} generated by ℂ^{×} and a distinguished element j, subject to the relations

j^{2} = −1 ∈ ℂ^{×}, jzj^{−1} = \overline{z}.  (5.2)(∗)

We define a homomorphism from W_{ℝ} to the Galois group Γ of ℂ/ℝ by sending ℂ^{×} to the identity and j to complex conjugation. (This gives the exact sequence

1 → ℂ^{×} → W_{ℝ} → Γ → 1

of (1.4)(b).) Again we emphasize that the Weil group is not a complex Lie group.

Suppose^{∨}G^{Γ} is a weak E-group. A quasiadmissible homomorphism (or Langlands parameter) ϕ from W_{ℝ} to ^{∨}G^{Γ} is a continuous group homomorphism satisfying

(a) the diagram

W_{ℝ} ⟶^{ϕ} ^{∨}G^{Γ}
↘  ↙

Γ

is commutative; and

(b) ϕ(ℂ^{×}) consists of semisimple elements of ^{∨}G.

Two such homomorphisms are said to be equivalent if they are conjugate by the action of ^{∨}G. The set of Langlands parameters for ^{∨}G^{Γ} is written

P (^{∨}G^{Γ}) = { ϕ : W_{ℝ} → ^{∨}G^{Γ} | ϕ is quasiadmissible }.

We make ^{∨}G act on P (^{∨}G^{Γ}) by conjugation on the range of a homomorphism. The set of equivalence classes of Langlands parameters — that is, the set of orbits of ∨G on P (^{∨}G^{Γ}) — is written

Φ(^{∨}G^{Γ}) = {^{∨}G orbits on P (^{∨}G^{Γ}) }.

Definition 5.3 (see [Langlands], [Borel]). Suppose G is a connected complex reductive algebraic group endowed with an inner class of real forms. If (^{∨}G^{Γ}, D) is any L-group for G (Definition 4.6), then we write

Φ(G/ℝ) = Φ(^{∨}G^{Γ}).

The omission of ^{∨}G^{Γ} from the notation is justified by Corollary 4.8: if ((^{∨}G^{Γ})′, D′) is any other L-group for the same inner class of real forms, then the corollary provides a canonical bijection from Φ(^{∨}G^{Γ}) to Φ((^{∨}G^{Γ})′). A little more generally, if (^{∨}G^{Γ}, D) is an E-group with second invariant z (Definition 4.6), then we write

Φ^{z}(G/ℝ) = Φ(^{∨}G^{Γ}).

The set of Langlands parameters behaves well under L-homomorphisms.

Proposition 5.4. Suppose ^{∨}H^{Γ} and ^{∨}G^{Γ} are weak E-groups, and ϵ : ^{∨}H^{Γ} → ∨G^{Γ} is an L-homomorphism. Then composition with ϵ defines a map

P (ϵ) : P (^{∨}H^{Γ}) → P (^{∨}G^{Γ})

on Langlands parameters, which descends to a map

Φ(ϵ) : Φ(^{∨}H^{Γ}) → Φ(^{∨}G^{Γ})

on equivalence classes.

This is obvious. We begin now the proof of Proposition 1.6.

Lemma 5.5 Suppose H is a complex Lie group, with Lie algebra 𝔥. Then the set of continuous homomorphisms ϕ from ℂ^{×} into H is in one-to-one correspondence with the set of pairs (λ, μ) ∈ 𝔥 × 𝔥, subject to the following conditions:

a) [λ, μ] = 0; and

b) exp(2πi(λ − μ)) = 1.

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 ϕ, however; it is

ϕ(e^{t}) = exp(tλ + \overline{t}μ)  (t ∈ ℂ).  (5.5)(a)

This is usually written more succinctly (if perhaps a little less clearly) as

ϕ(z) = z^{λ}\overline{z}^{μ}  (z ∈ ℂ^{×}).  (5.5)(b)

We restate Proposition 1.6 in our present slightly more general setting.

Proposition 5.6. Suppose ^{∨}G^{Γ} is a weak E-group (Definition 4.3). The set P (^{∨}G^{Γ}) of quasiadmissible homomorphisms from W_{ℝ} into ^{∨}G^{Γ} (Definition 5.2) may be identified with the set of pairs (y, λ) satisfying the following conditions:

a) y ∈ ^{∨}G^{Γ} −^{∨}G, and λ ∈ ^{∨}𝔤 is a semisimple element;

b) y^{2} = exp(2πiλ); and

c) [λ, Ad(y)λ] = 0.

Proof. Suppose ϕ is a quasiadmissible homomorphism. By Lemma 5.4, the restriction of ϕ to ℂ^{×} determines two commuting elements λ and μ of ^{∨}𝔤 such that

ϕ(z) = z^{λ}\overline{z}^{μ}  (z ∈ ℂ^{×}).  (5.7)(a)

By Definition 5.2(b), the elements λ and μ are semisimple. Since j acts on ℂ^{×} by complex conjugation, we have

Ad(ϕ(j))(λ) = μ.  (5.7)(b)

Define

y = exp(πiλ)ϕ(j) ∈ ^{∨}G^{Γ} −^{∨}G;  (5.7)(c)

the last inclusion is a consequence of Definition 5.2(a). This gives (a) of the Proposition. Since Ad(exp(πiλ)) fixes λ, we have

Ad(y)(λ) = μ.  (5.7)(d)

Since λ and μ commute, this gives (c) of the proposition. Now

y^{2} = exp(πiλ)ϕ(j)exp(πiλ)ϕ(j)
= exp(πiλ)ϕ(j)exp(πiλ)ϕ(j)^{−1}ϕ(j^{2})
= exp(πiλ)exp(πiμ)ϕ(−1) (by (5.7)(b) and (5.2)(∗))
= exp(πi(λ + μ))exp(πi(λ − μ)) (by (5.5)(a))
= exp(2πiλ),

as required by (b) of the proposition.

Conversely, suppose we are given y and λ satisfying (a)–(c) of the proposition. Define μ by (5.7)(d); then λ and μ are commuting semisimple elements of ^{∨}𝔤. By (b) of the proposition,

exp(2πiλ) = exp(2πiAd(y)(λ)).

So λ and μ satisfy the conditions of Lemma 5.5, and we can define a homomorphism ϕ from ℂ^{×} into ^{∨}G by (5.7)(a). We extend this to W_{ℝ} by setting

ϕ(j) = exp(−πiλ)y.  (5.7)(e)

That the relations (5.2)(∗) are preserved, and that ϕ 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, λ, and ϕ are related as in Proposition 5.6 — that is, when (5.7)(a)–(e) hold — we write

ϕ = ϕ(y, λ), y = y(ϕ), λ = λ(ϕ)  (5.8)(a)

It will also be convenient to write (following (1.7)(c))

e(λ) = exp(2πiλ) ∈ H  (5.8)(b)

whenever H is a complex Lie group and λ ∈ 𝔥.

Corollary 5.9. Suppose ^{∨}G^{Γ} is a weak E-group, ϕ ∈ P (^{∨}G^{Γ}) is a Langlands parameter, and y, λ are as in Proposition 5.6. Define

∨G(λ) = centralizer in ^{∨}G of e(λ)
L(λ) = { g ∈ ^{∨}G | Ad(g)λ = λ }
L(λ, y · λ) = { g ∈ ^{∨}G | Ad(g)λ = λ, Ad(g)(y · λ) = y · λ }
∨G_{ϕ} = centralizer in ^{∨}G of ϕ(W_{ℝ})
K(y) = centralizer in ^{∨}G of y.

a) We have

∨G_{ϕ} ⊂ L(λ, y · λ) ⊂ L(λ) ⊂ ^{∨}G(λ).

All four groups are complex reductive algebraic subgroups of ^{∨}G; the middle two are Levi subgroups of parabolics (and therefore connected).

b) Conjugation by y defines an involutive automorphism θ_{y} of ^{∨}G(λ) preserving L(λ, y · λ).

c) The group of fixed points of θ_{y} on ^{∨}G(λ) is K(y). We have

∨G_{ϕ} = K(y) ∩ L(λ, y · λ) = K(y) ∩ L(λ).

In particular,

d) ^{∨}G_{ϕ} may be described as the set of fixed points of an involutive automorphism of the connected reductive (Levi) subgroup L(λ, y · λ) of ^{∨}G.

This is immediate from Proposition 5.6.

Recall now from Definition 1.16 the algebraic universal cover ^{∨}G^{alg} of ^{∨}G. We write

1 → π_{1}(^{∨}G)^{alg} → ^{∨}G^{alg} → ^{∨}G → 1.  (5.10)(a)

(Recall that π_{1}(^{∨}G)^{alg} is the projective limit of the finite quotient groups of π_{1}(^{∨}G).) If H is any subgroup of ^{∨}G, then we write H^{alg} for its inverse image in ^{∨}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,^{∨}G}.) This gives an exact sequence

1 → π_{1}(^{∨}G)^{alg} → H^{alg} → H → 1.  (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 δ^{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 δ^{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 ^{∨}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).

Definition 5.11. Suppose ^{∨}G^{Γ} is a weak E-group, and ϕ ∈ P (^{∨}G^{Γ}). Use the notation of Corollary 5.9. Define

A_{ϕ}^{loc} = ^{∨}G_{ϕ}/ (^{∨}G_{ϕ})_{0} ,

the Langlands component group for ϕ. Similarly, set

A_{ϕ}^{loc,alg} = ^{∨}G_{ϕ}^{alg}/ (^{∨}G_{ϕ}^{alg})_{0} ,

the universal component group for ϕ. (We will occasionally need to write A_{ϕ,^{∨}G}^{loc,alg} to avoid ambiguity. When ^{∨}G^{Γ} is an E-group for G, we simplify this to A_{ϕ,G}^{loc,alg}.) It is a consequence of Corollary 5.9(d) that A_{ϕ}^{loc} is a (finite) product of copies of ℤ/2ℤ, and that A_{ϕ}^{loc,alg} 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

π_{1}(^{∨}G)^{alg} → A_{ϕ}^{loc,alg} → A_{ϕ}^{loc} → 1.

(The kernel of the first map is the intersection of π_{1}(^{∨}G)^{alg} with the identity component of ^{∨}G_{ϕ}^{alg}.) A complete Langlands parameter for ^{∨}G^{Γ} is a pair (ϕ, τ), with τ an irreducible representation of A_{ϕ}^{loc,alg}. Two such parameters are called equivalent if they are conjugate under the obvious action of ^{∨}G^{alg}. We write Ξ(^{∨}G^{Γ}) 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

Ξ(G/ℝ) = { 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 (^{∨}G^{Γ}, D) is an E-group for G^{Γ} with second invariant z (Definition 4.6) then we write Ξ^{z}(G/ℝ) for Ξ(^{∨}G^{Γ}). If ((^{∨}G^{Γ})′, D′) is another E-group with second invariant z, then Proposition 4.7(b) provides a canonical bijection from Ξ(^{∨}G^{Γ}) onto Ξ((^{∨}G^{Γ})′). This justifies the omission of the E-group from the notation.

Although we will make no use of it, the following proposition may provide a little motivation for this definition.

Proposition 5.12. Suppose ^{∨}G is a weak E-group. Then there is a natural one-to-one correspondence between equivalence classes of complete Langlands parameters for ^{∨}G^{Γ} (Definition 5.11), and irreducible ^{∨}G^{alg}-equivariant local systems on ^{∨}G-orbits on P (^{∨}G^{Γ}) (Definition 5.2).

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 → Q → ^{∨}G^{Q} → ^{∨}G → 1.  (5.13)(a)

If ϵ : H → ^{∨}G is any morphism of algebraic groups, we can pull back the extension (5.13)(a) to

\begin{aligned}1 & → & Q & → & H^{Q} & → & H & → & 1 \\ & & ↓ & & ↓ ϵ^{Q} & & ↓ ϵ \\ 1 & → & Q & → & ^{∨}G^{Q} & → & ^{∨}G & → & 1 \end{aligned}  (5.13)(b)

(The case of (5.10) has Q = π_{1}(^{∨}G)^{alg}, and ϵ the inclusion of a subgroup. In almost all the examples we consider, Q will be a quotient of π_{1}(^{∨}G)^{alg}.) If ϕ ∈ P (^{∨}G^{Γ}), we define the Q-component group for ϕ by

A_{ϕ}^{loc,Q} = ^{∨}G_{ϕ}^{Q}/(^{∨}G_{ϕ}^{Q})_{0}.  (5.13)(c)

If we need to emphasize the group, we write A_{ϕ,^{∨}G}^{loc,Q}. There is a right exact sequence

Q → A_{ϕ}^{loc,Q} → A_{ϕ}^{loc} → 1.  (5.13)(d)

A complete Langlands parameter for ^{∨}G^{Γ} of type Q is a pair (ϕ, τ), with ϕ a Langlands parameter and τ an irreducible representation of A_{ϕ}^{loc,Q}. Two such parameters are called equivalent if they are conjugate by the action of ^{∨}G^{Q}. We write Ξ(^{∨}G^{Γ})^{Q} for the set of equivalence classes of complete Langlands parameters of type Q.

Suppose now that we have an L-homomorphism

ϵ : ^{∨}H^{Γ} → ^{∨}G^{Γ},  (5.14)(a)

(Definition 5.1), and that we are given compatible pro-finite extensions

\begin{aligned}1 & → & Q_{H} & → & ^{∨}H^{Q_{H}} & → & ^{∨}H & → & 1 \\ & & ↓ & & ↓ ϵ^{•} & & ↓ ϵ \\ 1 & → & Q & → & ^{∨}G^{Q} & → & ^{∨}G & → & 1 \end{aligned}  (5.14)(b)

(In our examples, ^{∨}G^{Q} will usually be the algebraic universal covering of ^{∨}G, and ∨H^{Γ} will be (roughly) the centralizer in ^{∨}G^{Γ} of a semsimple element. We will take Q_{H} = Q, and ^{∨}H^{Q} the induced extension (cf. (5.13)(b)). In this case ^{∨}H^{Q} will be a quotient of the algebraic universal cover of ^{∨}H.) Suppose now that ϕ ∈ P (^{∨}H^{Γ}), so that ϵ ∘ ϕ ∈ P (^{∨}G^{Γ}) (Proposition 5.4). Then the map

ϵ^{•} : ^{∨}H^{Q_{H}} → ^{∨}G^{Q}  (5.14)(c)

carries the centralizer of the image of ϕ to the centralizer of the image of ϵ ∘ ϕ; so we get an induced homomorphism of component groups

A^{loc}(ϵ) : A_{ϕ,^{∨}H}^{loc,Q_{H}} → A_{ϵ∘ϕ,^{∨}G}^{loc,Q}.  (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 ^{∨}H^{Γ} induces a Langlands parameter for ^{∨}G^{Γ}, 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.

6. Geometric parameters.

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 𝔥, and λ ∈ 𝔥 is a semisimple element. Set

𝔥(λ)_{n} = { μ ∈ 𝔥 | [λ, μ] = nμ } (n ∈ ℤ)  (6.1)(a)
𝔥(λ) = ∑_{n∈ℤ} 𝔥(λ)_{n}  (6.1)(b)

𝔩(λ) = 𝔥(λ)_{0} = centralizer of λ in 𝔥  (6.1)(c)
𝔫(λ) = ∑_{n=1,2,...} 𝔥(λ)_{n}  (6.1)(d)

𝔭(λ) = 𝔩(λ) + 𝔫(λ)  (6.1)(e)

Next, define

e(λ) = exp(2πiλ) ∈ H.  (6.2)(a)
H(λ) = centralizer in H of e(λ)  (6.2)(b)
L(λ) = centralizer in H of λ  (6.2)(c)
N(λ) = connected unipotent subgroup with Lie algebra 𝔫(λ)  (6.2)(d)
P (λ) = L(λ)N(λ)  (6.2)(e)

Lemma 6.3. Suppose H is a complex reductive algebraic group, and λ ∈ 𝔥 is a semisimple element. Use the notation of (6.1) and (6.2).

a) H(λ) is a reductive subgroup of H with Lie algebra 𝔥(λ).

b) L(λ) is a reductive subgroup of H(λ) with Lie algebra 𝔩(λ). It is connected if H is.

c) N(λ) is a connected unipotent subgroup of H(λ) with Lie algebra 𝔫(λ).

d) P (λ) is a parabolic subgroup of H(λ) with Lie algebra 𝔭(λ), and Levi decomposition P (λ) = L(λ)N(λ). It is connected if H is. e) Ad(P (λ)) · λ = Ad(N(λ)) · λ = λ + 𝔫(λ). More precisely, the adjoint action of

N(λ) defines an isomorphism of varieties

N(λ) ≃ λ + 𝔫(λ), n ↦ Ad(n)λ.

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:

F(λ) = Ad(P (λ)) · λ = Ad(N(λ)) · λ = λ + 𝔫(λ).  (6.4)

Proposition 6.5. Suppose H is a complex reductive algebraic group, and λ ∈ 𝔥 is a semisimple element. Use the notation of (6.1)–(6.4), and suppose λ′ ∈ F(λ). Then

a) λ′ is a semisimple element of 𝔥, conjugate to λ by Ad(N(λ));

b) e(λ) = e(λ′), H(λ) = H(λ′), N(λ) = N(λ′), P (λ) = P (λ′), F(λ) = F(λ′); and c) the stabilizer in H of the set F(λ) is P (λ). More precisely, if Ad(h) · F(λ) has

a non-trivial intersection with F(λ), then h ∈ P (λ).

Proof. Part (a) is immediate from (6.4). Write λ′ = Ad(n) · λ, for some n ∈ N(λ) ⊂ H(λ). Obviously Ad(n) carries objects defined in terms of λ to those defined in terms of λ′. Since e(λ) is central in H(λ), the first claim of (b) follows. For the next three, use the fact that N(λ) normalizes H(λ), N(λ), and P (λ). For the last claim of (b), use the fact that F(λ) is already a homogeneous space for N(λ). For (c), suppose h ∈ H carries λ to another element λ′ of F(λ). By (a), there is an element n ∈ N(λ) with λ′ = Ad(n) · λ. Since n and h carry λ to the same element, the element l = n^{−1}h belongs to the stabilizer L(λ) of λ. Hence h = nl belongs to P (λ), as we wished to show. Q.E.D.

Suppose now that Λ is a canonical flat in 𝔥. We define

e(Λ) = e(λ), H(Λ) = H(λ), N(Λ) = N(λ), P (Λ) = P (λ)  (6.6)

for any λ ∈ Λ; this is well-defined by Proposition 6.5(b). We write 𝔥(Λ), etc., for the Lie algebras.

Proposition 6.7. Suppose H is a complex reductive algebraic group, and Λ ⊂ 𝔥 is a canonical flat. Use the notation of (6.6).

a) Λ is an affine space for the vector space 𝔫(Λ), and a homogeneous space for P (Λ). These structures are compatible (via the adjoint action of P (Λ) on 𝔫(Λ)). We can therefore form the induced bundle

H ×_{P (Λ)} Λ → H/P (Λ),

which is an affine bundle for the vector bundle

H ×_{P (Λ)} 𝔫(Λ) → H/P (Λ).

b) The inclusion of Λ in 𝔥 induces a map

H ×_{P (Λ)} Λ → 𝔥, (h, λ) ↦ Ad(h) · λ.

This map is an isomorphism onto the conjugacy class containing Λ. The fibers of the bundle structure map to the canonical flats in the conjugacy class.

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, λ), with h ∈ H and λ ∈ Λ; the equivalence relation is (hp, λ) ∼ (h, p · λ) for p ∈ P (Λ).) 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 λ consists of the equivalence classes of pairs (h, λ′) with Ad(h) · λ′ = λ. By (c) of Proposition 6.5, it follows that h ∈ P (Λ), and thus that (h, λ′) is equivalent to (e, λ). 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 (Λ) maps to the canonical flat Λ. The general statement then follows from the H-equivariance of the map. Q.E.D.

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 λ, we have P (λ) = L(λ) = T, a Cartan subgroup of H; F(λ) = {λ}; the unipotent group N(λ) is trivial; the bundles of Proposition 6.7 are trivial on H/T; and Proposition 6.7(b) simply identifies the conjugacy class of λ with H/T.

At the other extreme, suppose λ has all of its eigenvalues integral. Then H(λ) contains the identity component of H, and P (λ) is a parabolic subgroup of H. The partition of the Ad(H) orbit of λ 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, ℂ). Write {h, e, f} for the standard basis of the Lie algebra, and choose

λ = 1/2h = ( \begin{matrix}1/2 & 0 \\ 0 & −1/2 \end{matrix} )  (6.8)(a)

Then

𝔥(λ)_{−1} = ℂf, 𝔥(λ)_{0} = ℂh, 𝔥(λ)_{1} = ℂe  (6.8)(b)
𝔥(λ) = 𝔥, 𝔩(λ) = ℂh, 𝔫(λ) = ℂe, 𝔭(λ) = ℂh + ℂe  (6.8)(c)
e(λ) = −I ∈ SL(2, ℂ), H(λ) = SL(2, ℂ)  (6.8)(d)
F(λ) = \{ ( \begin{matrix}1/2 & t \\ 0 & −1/2 \end{matrix} ) | t ∈ ℂ \}  (6.8)(e)

The orbit of λ is the quadric

Ad(H) · λ = \{ ( \begin{matrix}a & b \\ c & −a \end{matrix} ) | a^{2} + bc = 1/4 \};  (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).

Definition 6.9. Suppose ^{∨}G^{Γ} is an E-group. A geometric parameter for ^{∨}G^{Γ} is a pair (y, Λ) satisfying

(a) y ∈ ^{∨}G^{Γ} −^{∨}G;

(b) Λ ⊂ ^{∨}𝔤 is a canonical flat (Definition 1.7); and

(c) y^{2} = e(Λ) (notation (6.6)). The set of geometric parameters for ^{∨}G^{Γ} is written X(^{∨}G^{Γ}). We make ^{∨}G act

on X(^{∨}G^{Γ}) by conjugation. Two geometric parameters are called 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 Φ(^{∨}G^{Γ}).)

If O is a ^{∨}G orbit of semisimple elements in ^{∨}𝔤, then as in Lemma 1.9 we define

X(O,^{∨}G^{Γ}) = { (y, Λ) ∈ X(^{∨}G^{Γ}) | Λ ⊂ O }.

We begin now our analysis of the spaces X(O,^{∨}G^{Γ}). Fix a^{∨}G orbit of semisimple elements

O ⊂ ^{∨}𝔤,  (6.10)(a)

and write F(O) for the set of canonical flats in O. If we fix such a flat Λ, then Proposition 6.5(c) provides an isomorphism

F(O) ≃ ^{∨}G/P (Λ).  (6.10)(b)

We also need to consider the conjugacy class of e(Λ) in ^{∨}G. This is

C(O) = { ge(Λ)g^{−1} | g ∈ ^{∨}G }.  (6.10)(c)

By (6.2)(b), there is an isomorphism

C(O) ≃ ^{∨}G/^{∨}G(Λ).  (6.10)(d)

Because e(Λ) is a semisimple element, the conjugacy class C(O) is closed in ^{∨}G. From the form of Definition 1.8, it is clear that we also need to consider the subvariety

I(O) = { y ∈ ^{∨}G^{Γ} −^{∨}G | y^{2} ∈ C(O) }  (6.10)(e)

It will be convenient also to consider

I(Λ) = { y ∈ ^{∨}G^{Γ} −^{∨}G | y^{2} = e(Λ) }.  (6.10)(f)

Finally, for y ∈ I(O), we write

K(y) = centralizer of y in ^{∨}G.  (6.10)(g)

Now the definition of the map e (cf. (6.2)(a) and (6.6)) on F(O) involves the exponential map, and so is not algebraic in nature. Nevertheless, we have

Lemma 6.11. In the setting of (6.10), the map

e : F(O) → C(O), Λ ↦ e(Λ)

is a smooth projective algebraic morphism.

Proof. By (6.10)(b) and (d), the map is just the natural quotient map of algebraic homogeneous spaces

∨G/P (Λ) → ^{∨}G/^{∨}G(Λ).

Since P (Λ) is parabolic in ^{∨}G(Λ), the map is projective. Q.E.D.

The next lemma will be the key to the finiteness claim of Lemma 1.9.

Lemma 6.12. Suppose H is a complex reductive algebraic group (possibly disconnected), and z ∈ H centralizes the identity component H_{0}. Consider the set

I = { y ∈ H | y^{2} = z }.

Then I is the union of finitely many orbits under the conjugation action of H_{0}.

Proof. The conjugation action of y ∈ I on H_{0} defines an involutive automorphism θ_{y}. Fix a connected component C of H, and consider the subset I ∩ C of I. Since H has finitely many components, it suffices to show that I ∩ C has finitely many H_{0} orbits. The involutions in the set

Θ(C) = {θ_{y} | y ∈ I ∩ C}

all differ by inner automorphisms (since C is a coset of H_{0}). In particular, they have a common restriction θ_{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 Θ(C) is a finite union of H_{0} orbits. To complete the proof, we must study the fibers of the map

I ∩ C → Θ(C), y ↦ θ_{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 θ_{y} is clearly

{ yw | w ∈ Z(H_{0}), wθ_{Z}(w) = e } = yZ_{1}.

On the other hand, the Z(H_{0}) conjugacy class of y is

{ yw | w = vθ_{Z}(v)^{−1}, some v ∈ Z(H_{0}) } = yZ_{2}.

The two subgroups of Z(H_{0}) appearing here are algebraic, and Z_{2} ⊂ Z_{1}. Since they have the same Lie algebra (namely the −1 eigenspace of the differential of θ_{Z}), it follows that Z_{2} has finite index in Z_{1}, as we wished to show. Q.E.D.

Proposition 6.13. In the setting (6.10), the orbits of the conjugation action of ^{∨}G on I(O) are in one-to-one correspondence (by intersection with I(Λ)) with ∨G(Λ) orbits on I(Λ). All of these orbits are closed, and there are finitely many of them.

Fix a single orbit I_{1}(O) of ^{∨}G on I(O). Then the natural map

I_{1}(O) → C(O), y ↦ y^{2}

is a smooth algebraic morphism.

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 ^{∨}G^{Γ}. For the finiteness, we consider the reductive group

H = centralizer of e(Λ) in ^{∨}G^{Γ}.

Obviously any element y of I(Λ) belongs to H (since y always commutes with y^{2}). Furthermore z = e(Λ) is central in H. It follows that the set I of Lemma 6.12 contains I(Λ). Since ^{∨}G(Λ) contains the identity component of H, the finiteness we need follows from Lemma 6.12.

For the last claim, fix an element y_{1} ∈ I_{1}(O) with y_{1}^{2} = e(Λ). Then

I_{1}(O) ≃ ^{∨}G/K(y_{1}),  (6.14)(a)

and the morphism in question is just the quotient map

∨G/K(y_{1}) → ^{∨}G/^{∨}G(Λ).  (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.

Lemma 6.15. 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 ×_{G/H} G/B → G/H.

On the other hand, H acts on H/A and on H/B, so it acts on the product H/A × H/B. We can therefore form the induced bundle

Y = G ×_{H} (H/A × H/B) → G/H.

Finally, A acts on the homogeneous space H/B, so we can form the induced bundle

Z = G ×_{A} H/B → G/A → G/H

(and similarly with A and B exchanged).

a) X, Y, and Z are smooth bundles over G/H.

b) Y is isomorphic to X by the map sending the equivalence class of (g, (h_{1}A, h_{2}B)) ∈ Y to (gh_{1}A, gh_{2}B) ∈ X. This map is a G-equivariant isomorphism of fiber bundles from Y to X.

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.

d) The isomorphism in (b) induces a natural bijection from the orbits of H on (H/A × H/B) onto the orbits of G on X.

e) The isomorphism in (c) induces a natural bijection from the orbits of A on H/B onto the orbits of G on X. f) The following four sets are in natural one-to-one correspondence: the orbits of

H on (H/A × H/B); the (A, B) double cosets in H; the orbits of A on H/B; and the orbits of B on H/A.

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 × 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 × 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.

Proposition 6.16. Suppose ^{∨}G^{Γ} is a weak E-group. Fix an orbit O of ^{∨}G on the semisimple elements of ^{∨}𝔤, and use the notation of (6.6) and (6.10). In particular, we have smooth algebraic maps

F(O) → C(O), I(O) → C(O).

The variety X(O,^{∨}G^{Γ}) of Definition 6.9 is the fiber product

X(O,^{∨}G^{Γ}) = F(O) ×_{C(O)} I(O).

A little more explicitly, it has the following structure. List the orbits of ^{∨}G on I(O) as I_{1}(O), …, I_{r}(O) (Proposition 6.13). Fix a canonical flat Λ ∈ F(O). This gives a reductive group ^{∨}G(Λ) and a parabolic subgroup P (Λ) (cf. (6.6) and Lemma 6.3). For each i, choose a point

y_{i} ∈ I_{i}(O), y_{i}^{2} = e(Λ).

Then conjugation by y_{i} defines an involutive automorphism θ_{i} of ^{∨}G(Λ) with fixed point set K_{i} = K(y_{i}) (cf. (6.10)(g)). The variety X(O,^{∨}G^{Γ}) is the disjoint union of r closed subvarieties

X_{i}(O,^{∨}G^{Γ}) = F(O) ×_{C(O)} I_{i}(O).

The ith subvariety looks like

\begin{aligned}X_{i}(O,^{∨}G^{Γ}) & ≃ ^{∨}G ×_{^{∨}G(Λ)} (^{∨}G(Λ)/K_{i} ×^{∨}G(Λ)/P (Λ)) \\ & ≃ ^{∨}G ×_{K_{i}} (^{∨}G(Λ)/P (Λ)). \end{aligned}

In particular, the orbits of ^{∨}G on X_{i}(O,^{∨}G^{Γ}) are in one-to-one correspondence with the orbits of K_{i} on the partial flag variety ^{∨}G(Λ)/P (Λ). These orbits are finite in number. The isotropy group of the action of ^{∨}G at the point x = (y, Λ) is

∨G_{x} = K(y) ∩ P (Λ).

We will see later (Proposition 7.14) that the correspondence between K_{i} orbits on ^{∨}G(Λ)/P (Λ) and ^{∨}G orbits on X_{i}(O) not only is a bijection of sets, but also preserves the closure relations and the nature of the singularities of closures.

Proof. Everything but the last finiteness assertion is immediate from Lemma 6.15 and Proposition 6.13. (The description of X(O,^{∨}G^{Γ}) 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 ⊂ 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.

Proposition 6.17. Suppose ^{∨}G^{Γ} is a weak E-group. Then there is a natural ∨G-equivariant map p from the set of Langlands parameters for ^{∨}G^{Γ} (Definition 5.2) onto the set of geometric parameters for ^{∨}G^{Γ} (Definition 6.9). In terms of the description of Langlands parameters in Proposition 5.6, this map is

p : P (^{∨}G^{Γ}) → X(^{∨}G^{Γ}), p(ϕ(y, λ)) = (y, F(λ))

(notation (5.8), (6.4)). The map p induces a bijection from equivalence classes of Langlands parameters (that is, ^{∨}G orbits on P (^{∨}G^{Γ})) onto equivalence classes of geometric parameters (that is, ^{∨}G orbits on X(^{∨}G^{Γ})). If x = p(ϕ), then the isotropy group ^{∨}G_{ϕ} (Corollary 5.9) is a Levi subgroup of the isotropy group ^{∨}G_{x} (Proposition 6.16). In particular, the fiber p^{−1}(x) is a principal homogeneous space for the unipotent radical of ^{∨}G_{x}.

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

Lemma 6.18. Suppose G is a connected complex reductive algebraic group, P ⊂ G is a parabolic subgroup, and θ is an involutive automorphism of G. (We do not assume that θ preserves P.) Write K ⊂ G for the group of fixed points of θ.

a) There is a maximal torus T ⊂ P such that θT = T.

¿From now on we fix such a torus. There is a unique Levi decomposition P = LN with T ⊂ L.

b) The group L_{θ} = L∩θL is a connected θ-stable reductive subgroup of G containing T.

c) The group N_{θ} generated by N ∩ θP and P ∩ θN is a connected θ-stable unipotent subgroup of G, normalized by L_{θ}.

d) The group P_{θ} = P ∩ θP is a connected algebraic subgroup of G, with θ-stable Levi decomposition P_{θ} = L_{θ}N_{θ}.

e) We have

L ∩ K = L_{θ} ∩ K, P ∩ K = P_{θ} ∩ K = (L_{θ} ∩ K)(N_{θ} ∩ K).

The last formula is a Levi decomposition of the algebraic group P ∩ K.

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 θ preserves T, θ acts on R as an automorphism of order 2. Write

R(P, T) = R(L, T) ∪ R(N, T)

for the corresponding sets of roots. Fix an element λ ∈ 𝔱 with the property that

α(λ) = 0 (α ∈ R(L, T)), α(λ) > 0 (α ∈ R(N, T)).

Then θλ also belongs to 𝔱, and has the corresponding properties for θP. For (b), L is the centralizer in G of λ, so

L ∩ θL = { g ∈ G | Ad(g)λ = λ, Ad(g)(θλ) = θλ }.

Since λ and θλ are commuting semisimple elements of 𝔤, it follows that L ∩ θL is connected and reductive. The rest of (b) is clear.

For (c) and (d), one computes easily that the roots of T in L_{θ} and P_{θ} are

R(L_{θ}, T) = { α ∈ R | α(λ) = 0 and α(θλ) = 0 }
R(P_{θ}, T) = { α ∈ R | α(λ) ≥ 0 and α(θλ) ≥ 0 }.

The groups N ∩ θP and P ∩ θN are obviously unipotent algebraic, and therefore connected. The corresponding roots are

R(N ∩ θP, T) = { α ∈ R | α(λ) > 0 and α(θλ) ≥ 0 }
R(P ∩ θN, T) = { α ∈ R | α(λ) ≥ 0 and α(θλ) > 0 }.

Now set

R_{θ}^{+} = R(N ∩ θP, T) ∪ R(P ∩ θN, T).

Because this set is easily seen to be closed under addition, it is the set of roots of T in the group N_{θ}. To prove (c), we must show that N_{θ} is unipotent. This is a consequence of the fact that all the roots in R_{θ}^{+} lie on one side of the hyperplane defined by λ + θλ.

For (d), observe first that

R(P_{θ}, T) = R(L_{θ}, T) ∪ R(N_{θ}, T),

a disjoint union. It follows from (b) and (c) that L_{θ}N_{θ} is a Levi decomposition of the identity component of P_{θ}; it remains to show that P_{θ} is connected. So suppose p ∈ P_{θ}; we want to show that p is in the identity component. Now pT p^{−1} is a maximal torus in P_{θ}; so by the conjugacy of maximal tori in connected algebraic groups, we can find an element p_{0} in the identity component of P_{θ} conjugating pT p^{−1} to T. After replacing p by p_{0}p, we may therefore assume that p normalizes T. Now the normalizer of T in P is contained in L; so p ∈ L. Similarly, the normalizer of T in θP is contained in θL, so p ∈ θL. Therefore p ∈ L_{θ}, which we already know is a connected subgroup of P_{θ}.

Part (e) is clear: since θ respects the Levi decomposition in P_{θ}, an element of P_{θ} ∩ K must have both its Levi components in K. Q.E.D.

Proof of Proposition 6.17. That p is a well-defined ^{∨}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, Λ) ∈ X(^{∨}G^{Γ}). Obviously

p^{−1}(x) = { ϕ(y, λ) | λ ∈ Λ, [λ, y · λ] = 0 }.  (6.19)(a)

Use the notation of (6.6); then P (Λ) is a parabolic subgroup of^{∨}G(Λ) (Lemma 6.3), and the conjugation action of y defines an involutive automorphism θ of ^{∨}G(Λ), with fixed points K(y). Apply Lemma 6.18 to this situation. We conclude first of all that P (Λ) contains a θ-stable maximal torus T. The semisimple P (Λ)-conjugacy class Λ ⊂ 𝔭(Λ) must meet the Lie algebra 𝔱 in some element λ. Then λ and y · λ both belong to 𝔱, so they commute; so ϕ(y, λ) ∈ 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 (Λ) ∩ K(y)  (6.19)(b)

(cf. Proposition 6.16) acts transitively on the fiber p^{−1}(x). Now any element of G_{x} is fixed by θ, so it must also belong to θP (Λ). Consequently

G_{x} = P_{θ}(Λ) ∩ K(y) = (L_{θ}(λ) ∩ K(y))(N_{θ}(Λ) ∩ K(y)).  (6.19)(c)

So suppose ϕ(y, λ′) is another point in the fiber. By Lemma 6.3(e),

λ′ = Ad(n)λ  (6.20)(a)

for a unique n ∈ N(Λ). By (6.19)(a), we have

[n · λ, θ(n · λ)] = 0.  (6.20)(b)

Now the centralizer of λ in ^{∨}𝔤 is 𝔩(λ) ⊂ 𝔭(Λ). It follows that the centralizer of θ(n · λ) is

θ(n · 𝔩(λ)) ⊂ θ(n · 𝔭(Λ)) = θ𝔭(Λ).

¿From this and (6.20)(b), we deduce that

n · λ ∈ 𝔭(Λ) ∩ θ𝔭(Λ).

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 · λ ∈ (λ + 𝔫(Λ)) ∩ (𝔩_{θ}(λ) + 𝔫_{θ}(Λ)) = λ + (𝔫(Λ) ∩ 𝔫_{θ}(Λ)).

Now the isomorphism in Lemma 6.3(e) respects T -stable subgroups of N(Λ); so the conclusion we draw (using the uniqueness in Lemma 6.3(e)) is

n ∈ N(Λ) ∩ N_{θ}(Λ).

We now rewrite (6.20)(b) by applying θ(n^{−1}) to each term in the bracket. The conclusion is

[(θ(n^{−1})n) · λ, θλ] = 0.  (6.20)(c)

Since N_{θ}(Λ) is preserved by θ, the element m = θ(n^{−1})n belongs to N_{θ}(Λ). The discussion of roots in Lemma 6.18 leads to a factorization

N_{θ}(Λ) = (N(Λ) ∩ θL(λ))(N(Λ) ∩ θN(Λ))(L(λ) ∩ θN(Λ));

write m = m_{1}m_{2}m_{3} accordingly. Then m_{3} fixes λ, and m_{1} fixes θλ, so (6.20)(c) leads to

[m_{2} · λ, θλ] = 0.

But Lemma 6.3(e) tells us that the first term here is of the form λ + X_{2}, with X_{2} ∈ 𝔫(Λ)∩θ𝔫(Λ). Such an element commutes with θλ only if X_{2} = 0, which implies that m_{2} = e. On the other hand, the definition of m shows that θm = m^{−1}, from which we conclude that m_{1} = θ(m_{3}^{−1}). Define n′ = nm_{3}^{−1}; then θ((n′)^{−1})n′ = e. Assembling all of this, we find

n′ ∈ N_{θ}(Λ) ∩ K(y),  n′ · λ = λ′.  (6.20)(d)

Clearly n′ · ϕ(y, λ) = ϕ(y, λ′). 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.

Corollary 6.21. Suppose ^{∨}H^{Γ} and ^{∨}G^{Γ} are weak E-groups, and ϵ : ^{∨}H^{Γ} → ^{∨}G^{Γ} is an L-homomorphism (Definition 5.1). Then there is a natural map

X(ϵ) : X(^{∨}H^{Γ}) → X(^{∨}G^{Γ})

on geometric parameters, compatible with the maps P (ϵ), Φ(ϵ) of Proposition 5.4 via the maps p of Proposition 6.17. Explicitly,

X(ϵ)(y, Λ) = (ϵ(y), ϵ(Λ)).

(Here ϵ(Λ) denotes the unique canonical flat containing dϵ(Λ).)

Fix a ^{∨}H orbit O ⊂ ^{∨}𝔥 of semisimple elements, and define ϵ(O) to be the unique ∨G orbit containing dϵ(O). Then X(ϵ) restricts to a morphism of algebraic varieties

X(O, ϵ) : X(O,^{∨}H^{Γ}) → X(ϵ(O),^{∨}G^{Γ}).

If ϵ is injective, then X(O, ϵ) is a closed immersion.

Proof. The first point that is not quite obvious is that the image under dϵ of a canonical flat Λ is contained in a single canonical flat for ^{∨}𝔤. To see this, fix λ ∈ Λ, and use the notation of (6.1)–(6.4). Write λ′ = dϵ(λ), Λ′ = F(λ′). Clearly

∨𝔥(λ)_{n} = (dϵ)^{−1}^{∨}𝔤(λ′)_{n}

for every n. From this it follows that

Λ = λ + ∑_{n>0}^{∨}𝔥(λ)_{n} ⊂ (dϵ)^{−1}(λ′) + ∑_{n>0}(dϵ)^{−1}^{∨}𝔤(λ′)_{n} = (dϵ)^{−1}(Λ′).

It follows that dϵ(Λ) ⊂ Λ′, as we wished to show. Notice also that if dϵ is injective, this containment becomes an equality:

Λ = (dϵ)^{−1}(Λ′).

For the last assertion, we first verify that X(ϵ) is injective if ϵ is. So suppose

X(ϵ)(y_{1}, Λ_{1}) = X(ϵ)(y_{2}, Λ_{2}) = (y′, Λ′).

This means first of all that ϵ(y_{i}) = y, and therefore that y_{1} = y_{2}. Next, the last observation in the preceding paragraph shows that Λ_{i} = (dϵ)^{−1}(Λ′), and therefore that Λ_{1} = Λ_{2}. This proves that X(ϵ) is injective. Essentially the same argument proves that X(O, ϵ) is a closed immersion; we leave the remaining details to the reader. Q.E.D.

Example 6.22. Suppose G = GL(3, 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

(ℤ^{3}, {(1, −1, 0), (0, 1, −1)}, ℤ^{3}, {(1, −1, 0), (0, 1, −1)}),

with the standard pairing between the two copies of ℤ^{3}. The automorphism t of Proposition 2.12 associated to the inner class of real forms acts on ℤ^{3} by t(a, b, c) = (−c, −b, −a). As a dual group for G we can choose ^{∨}G = GL(3, ℂ). Define an automorphism θ of ^{∨}G by

θg = Ad ( \begin{matrix}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix} ) (^{t}g^{−1}).

Then θ is an involutive automorphism preserving the standard Cartan and Borel subgroups ^{d}T ⊂ ^{d}B; the induced automorphism of the based root datum is t. We can therefore construct a weak E-group ^{∨}G^{Γ} generated by ^{∨}G and an element ^{∨}δ of order two, subject to the relations

(^{∨}δ)g(^{∨}δ^{−1}) = θg (g ∈ ^{∨}G).

(If we take as the distinguished set S of Definition 4.14 the conjugacy class of (^{∨}δ,^{d}B), then ^{∨}G^{Γ} is an L-group for G.)

Consider the elements

λ = ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix} ) ,  y = ^{∨}δ,  y · λ = ( \begin{matrix}0 & 0 & 0 \\ 0 & −1 & 0 \\ 0 & 0 & −1 \end{matrix} ) .

Notice that e(λ) = e = y^{2}, and λ commutes with y · λ; so there is a Langlands parameter ϕ = ϕ(y, λ). Since e(λ) = e, ^{∨}G(λ) = ^{∨}G. The corresponding parabolic P (λ) is the standard parabolic in GL(3) with Levi subgroup GL(2) × GL(1), and the canonical flat is

Λ = \{( \begin{matrix}1 & 0 & a \\ 0 & 1 & b \\ 0 & 0 & 0 \end{matrix} ) | a, b ∈ ℂ\} .

Of course x = (y, Λ) is a geometric parameter. The Levi subgroup L(λ, y · λ) is the diagonal subgroup; its intersection with K(y) (which is a form of O(3)) is

∨G_{ϕ} = \{( \begin{matrix}z & 0 & 0 \\ 0 & ±1 & 0 \\ 0 & 0 & z^{−1} \end{matrix} ) | z ∈ ℂ^{×}\} .

The Langlands component group A_{ϕ} (Definition 5.11) therefore has order 2. The group P_{θ}(Λ) of Lemma 6.18 is just the standard Borel subgroup. The isotropy group of the geometric parameter x has unipotent radical equal to the θ-invariant upper triangular unipotent matrices; this group is

K(y) ∩ N_{θ}(Λ) = \{( \begin{matrix}1 & a & −a^{2}/2 \\ 0 & 1 & −a \\ 0 & 0 & 1 \end{matrix} ) | a ∈ ℂ\} .

The full stabilizer of x in ^{∨}G is the semidirect product of ^{∨}G_{ϕ} with this unipotent group, namely

∨G_{x} = \{( \begin{matrix}z & ±a & −z^{−1}a^{2}/2 \\ 0 & ±1 & −z^{−1}a \\ 0 & 0 & z^{−1} \end{matrix} ) | a ∈ ℂ, z ∈ ℂ^{×}\} .

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 ^{∨}G(Λ) need not be connected. The only subtlety in the variant is the definition

K(y)^{0} = K(y) ∩^{∨}G(Λ)_{0}  (6.23)(a)

for y ∈ I(Λ) (cf. (6.10)). In the notation of Lemma 6.12, this is the group of fixed points of the involution θ_{y} on the connected group ^{∨}G(Λ)_{0}. We have

K(y)_{0} ⊂ K(y)^{0} ⊂ K(y),  (6.23)(b)

and both of these inclusions may be proper.

Proposition 6.24. Suppose ^{∨}G^{Γ} is a weak E-group. Fix an orbit O of ^{∨}G on the semisimple elements of ^{∨}𝔤, and use the notation of (6.6), (6.10), and (6.23). Fix a canonical flat Λ ⊂ O, and write P^{0}(Λ) for the ^{∨}G(Λ)_{0} orbit of Λ in the set of canonical flats. List the orbits of ^{∨}G(Λ)_{0} on I(Λ) as I_{1}^{0}(Λ), …, I_{s}^{0}(Λ) (notation (6.10)(f) and Lemma 6.12). For each j, choose a point y_{j} ∈ I_{j}^{0}(Λ). Then conjugation by y_{j} defines an involutive automorphism θ_{j} of ^{∨}G(Λ)_{0} with fixed point set K_{j}^{0} = K(y_{j})^{0} (cf. (6.23)). ¿From the chosen base points Λ and y_{j} we get isomorphisms

P^{0}(Λ) ≃ ^{∨}G(Λ)_{0}/P (Λ), I_{j}^{0}(Λ) ≃ ^{∨}G(Λ)_{0}/K_{j}^{0}.

The variety X(O,^{∨}G^{Γ}) is the disjoint union of s closed connected smooth subvarieties X_{j}(O,^{∨}G^{Γ}). The jth subvariety looks like

X_{j}(O,^{∨}G^{Γ}) ≃ ^{∨}G ×_{^{∨}G(Λ)_{0}} (I_{j}^{0}(Λ) × P^{0}(Λ))
≃ ^{∨}G ×_{^{∨}G(Λ)_{0}} (^{∨}G(Λ)_{0}/K_{j}^{0} ×^{∨}G(Λ)_{0}/P (Λ))
≃ ^{∨}G ×_{K_{j}^{0}} (^{∨}G(Λ)_{0}/P (Λ)).

In particular, the orbits of ^{∨}G on X_{j}(O,^{∨}G^{Γ}) are in one-to-one correspondence with the orbits of K_{j}^{0} on the partial flag variety ^{∨}G(Λ)_{0}/P (Λ). These orbits are finite in number.

The proof is identical to that given for Proposition 6.16. (It is helpful to keep in mind that P (Λ) is connected (Lemma 6.3(d)) and therefore contained in ^{∨}G(Λ)_{0}.) As with Proposition 6.16, we will eventually want the stronger information about the orbit correspondence contained in Proposition 7.14(c).

7. Complete geometric parameters and perverse sheaves.

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 ^{∨}G^{alg} on X(O,^{∨}G^{Γ}).

Definition 7.1. Suppose Y is a complex algebraic variety on which the proalgebraic group H acts with finitely many orbits. A geometric parameter for H acting on Y is a (closed) point of Y. Two such parameters are called 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 Φ(Y, H).

Suppose y ∈ Y; write H_{y} for the isotropy group of the action at y. The (local) equivariant fundamental group at y is

A_{y}^{loc} = H_{y}/(H_{y})_{0},  (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 ∈ H with h · y = y′. It follows that conjugation by h carries H_{y} isomorphically onto H_{y′}, so we get an isomorphism

A_{y}^{loc} ≃ A_{y′}^{loc}.  (7.1)(b)

The coset hH_{y} is uniquely determined, so this isomorphism is unique up to inner automorphisms.

Suppose S ∈ Φ(Y, H) is an orbit of H on Y. The equivariant fundamental group of S is the pro-finite group

A_{S}^{loc} = A_{y}^{loc}  (y ∈ S).  (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 (local) complete geometric parameter for H acting on Y is a pair (y, τ) with y ∈ Y and τ ∈ Â_{y}^{loc} an irreducible representation. Two such parameters (y, τ) and (y′, τ′) are called equivalent if there is an element of H carrying y to y′ so that the induced isomorphism (7.1)(b) carries τ to τ′. The set of equivalence classes of (local) complete geometric parameters is written Ξ(Y, H).

A complete geometric parameter for H acting on Y is a pair (S, V), with S ∈ Φ(Y, H) an orbit of H on Y and 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).

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} | i ∈ 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} → H_{j}  (i ≥ j)  (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 → H_{j}  (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} ⊂ I, the various {H_{i} | i ∈ 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} ≃ lim_{⟵} H_{i,y}/(H_{i,y})_{0}.  (7.2)(c)

Again we have surjective maps

a_{i} : A_{y}^{loc} → H_{i,y}/(H_{i,y})_{0}.  (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 ≥ 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.

Lemma 7.3. In the setting of Definition 7.1, suppose y ∈ Y. Write S = H · y for the orbit of y.

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 π_{1}(S; y) of homotopy classes of loops in S based at y.

b) More generally, there is always a natural map

π_{1}(S; y) → 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 π_{1}(S; y). c) Suppose (τ, V) is a representation of A_{y}^{loc}. Regard τ as a representation of H_{y}

trivial on the identity component. Then the induced bundle

V = H ×_{H_{y}} V → S

carries an H-invariant flat connection.

d) Suppose V is an H-equivariant vector bundle on S carrying an H-invariant flat connection. Write V for the fiber of V at y, and τ for the isotropy action of H_{y} on V. Then τ is trivial on (H_{y})_{0}, and so factors to a representation of A_{y}^{loc}.

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.

We will use Lemma 7.3(e) to identify the set Ξ(Y, H) of equivalence classes of local complete geometric parameters. Thus if ξ ∈ Ξ(Y, H), we will write

ξ = (S, V) = (S_{ξ}, V_{ξ}),  (7.4)

and say that a pair (y, τ) in the equivalence class “represents ξ.” The assertions about fundamental groups are included to explain how H-equivariant local systems are related to arbitrary local systems.

Proof. We begin by defining the map in (b). Suppose

γ : [0, 1] → S, γ(0) = γ(1) = y

is a loop. Identify S with the quotient H/H_{y}; then γ can be lifted to a continuous map

γ̃ : [0, 1] → H, γ̃(0) = e.

The element h = γ̃(1) must map to y, so h ∈ H_{y}. Any two liftings γ̃ 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} ∈ A_{y}^{loc} is independent of the lifting. Similarly, a homotopy of the loop γ 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 ∈ 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 γ is a loop at y mapping to the class of e. This means that the lifting γ̃ satisfies γ̃(1) = h ∈ (H_{y})_{0}. Choose a path connecting h to the identity in (H_{y})_{0}; modifying γ̃ by this path gives a new lifting γ̃′ satisfying γ̃′(1) = e. This new lifting is therefore a loop. Since H is simply connected, γ̃′ is homotopic to a trivial map. This homotopy descends to a homotopy from γ 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 τ is really a representation of some component group H_{i,y}/(H_{i,y})_{0} (cf. (7.2)). Then the algebraic vector bundle

V = H_{i} ×_{H_{i,y}} V → S

carries an action of H_{i}, and thus compatible actions of all H_{j} with j ≥ i. Therefore it is an “H-equivariant vector bundle.” Conversely, to say that 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.

Lemma 7.5. Suppose ^{∨}G^{Γ} is a weak E-group, ϕ ∈ P (^{∨}G^{Γ}) is a Langlands parameter, and x = p(ϕ) ∈ X(^{∨}G^{Γ}) is a geometric parameter (cf. Proposition 6.17). Then the natural inclusion of isotropy groups

∨G_{ϕ} ↪ ^{∨}G_{x}

induces an isomorphism of component groups

∨G_{ϕ}/(^{∨}G_{ϕ})_{0} ≃ ^{∨}G_{x}/(^{∨}G_{x})_{0}.

If we consider isotropy groups in the algebraic universal cover ^{∨}G^{alg} (Definition 1.16 and (5.10)), we get

∨G_{ϕ}^{alg}/(^{∨}G_{ϕ}^{alg})_{0} ≃ ^{∨}G_{x}^{alg}/(^{∨}G_{x}^{alg})_{0}.

Similarly, in the setting (5.13),

∨G_{ϕ}^{Q}/(^{∨}G_{ϕ}^{Q})_{0} ≃ ^{∨}G_{x}^{Q}/(^{∨}G_{x}^{Q})_{0}.

Proof. The inclusion of isotropy groups comes from the equivariance of the map p. By Proposition 6.17, ^{∨}G_{x} is the semidirect product of ^{∨}G_{ϕ} and a connected unipotent group U_{x}. It follows that (^{∨}G_{x})_{0} must be the semidirect product of (^{∨}G_{ϕ})_{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 ∨G_{x}^{Q} must be the semidirect product of ^{∨}G_{ϕ}^{Q} and U_{x}. Now we can argue as for the first claim to get the isomorphism of component groups. Q.E.D.

Definition 7.6. Suppose ^{∨}G^{Γ} is a weak E-group, and X(^{∨}G^{Γ}) 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(O,^{∨}G^{Γ}) (see Proposition 6.16). For x ∈ X(^{∨}G^{Γ}), we consider the (local) equivariant fundamental groups for the actions of ^{∨}G and ^{∨}G^{alg},

A_{x}^{loc} = ^{∨}G_{x}/(^{∨}G_{x})_{0}, A_{x}^{loc,alg} = ^{∨}G_{x}^{alg}/(^{∨}G_{x}^{alg})_{0}.

By Lemma 7.5, there are natural isomorphisms

A_{x}^{loc} ≃ A_{ϕ}^{loc}, A_{x}^{loc,alg} ≃ A_{ϕ}^{loc,alg}

for any Langlands parameter ϕ ∈ p^{−1}(x) (cf. Definition 5.11). We therefore refer to these groups as the Langlands component group for x and the universal component group for x. If S is an orbit on X(^{∨}G^{Γ}), we write following Definition 7.1

A_{S}^{loc} = A_{x}^{loc},  A_{S}^{loc,alg} = A_{x}^{loc,alg}  (x ∈ S);

again these groups are defined up to inner automorphism.

A (local) complete geometric parameter for ^{∨}G^{Γ} is a pair (x, τ) with x ∈ X(^{∨}G^{Γ})

and τ ∈ { \atop A_{x}^{loc,alg}} an irreducible representation. Two such parameters are called equivalent if they are conjugate by ^{∨}G^{alg}. Lemma 7.5 provides a natural map from complete Langlands parameters for ^{∨}G^{Γ} onto (local) complete geometric parameters, inducing a bijection on equivalence classes. We may therefore write Ξ(^{∨}G^{Γ}) for the set of equivalence classes (cf. Definition 5.11).

As in (7.4), we identify Ξ(^{∨}G^{Γ}) with the set of pairs (S, V), with S an orbit on X(^{∨}G^{Γ}) and V an irreducible ^{∨}G^{alg}-equivariant local system on S; we call such a pair a complete geometric parameter for X(^{∨}G^{Γ}). If ξ ∈ Ξ(^{∨}G^{Γ}), we write

ξ = (S_{ξ}, V_{ξ})

as in (7.4). A (local) complete geometric parameter (x, τ) in the corresponding equivalence class will sometimes be called a representative of ξ.

All of this discussion can be extended to the setting (5.13). We write

A_{x}^{loc,Q} = ^{∨}G_{x}^{Q}/(^{∨}G_{x}^{Q})_{0},

the Q-component group for x. The set of complete geometric parameters for ^{∨}G^{Q} acting on X(^{∨}G^{Γ}) is written Ξ(^{∨}G^{Γ})^{Q}; we call these complete geometric parameters of type Q.

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

Definition 7.7. Suppose Y is a smooth complex algebraic variety on which the pro-algebraic group H acts with finitely many orbits. Write D_{Y} for the sheaf of algebraic differential operators on Y ([BorelD], VI.1). Define

C(Y, H) = 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

P(Y, H) = 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 π_{1} factoring through finite quotients.) This will be crucial for our eventual application of the results of [LV].

Finally, we will consider

D(Y, H) = category of H-equivariant regular holonomic sheaves of 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 D_{Y} -modules on Y is automatically regular holonomic.)

Each of these three categories is abelian, and every object has finite length. The corresponding Grothendieck groups are written

KC(Y, H),  KP(Y, H),  KD(Y, H).

Each is a free abelian group with a basis corresponding to the (finite) set of irreducible objects in the corresponding category.

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.)

Lemma 7.8 ([BBD].) In the setting of Definition 7.7, suppose P is an Hequivariant perverse sheaf on Y. Then the cohomology sheaves H^{i}P 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

χ : Ob P(Y, H) → KC(Y, H), χ(P) = ∑(−1)^{i}H^{i}P.

The map χ induces an isomorphism

χ : KP(Y, H) → KC(Y, H)

of Grothendieck groups.

Theorem 7.9 (the Riemann-Hilbert correspondence — see [BorelD], Theorem VIII.14.4). In the setting of Definition 7.7, there is an equivalence of categories (the “de Rham functor”)

DR : D(Y, H) → P(Y, H).

This induces an isomorphism

DR : KD(Y, H) → KP(Y, H)

of Grothendieck groups.

We will use the isomorphisms of Lemma 7.8 and Theorem 7.9 to identify the three Grothendieck groups, writing simply

K(Y, H) ≃ KC(Y, H) ≃ KP(Y, H) ≃ KD(Y, H)  (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 Ξ(Y, H) of complete geometric parameters (Definition 7.1). Fix ξ = (S, V) ∈ Ξ. We need a little notation from [BBD], section 1.4. Write

j = j_{S} : S → \overline{S},  i = i_{S} : \overline{S} → Y  (7.10)(b)

for the inclusion of S in its closure, and the inclusion of the closure in Y. Write d = d(S) = d(ξ) for the dimension of S. Regard the local system 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

μ(ξ) = i_{∗}j_{!}V ∈ Ob C(Y, H),  (7.10)(c)

the extension of ξ 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 V[−d] consisting of the single sheaf 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 (ξ) = i_{∗}j_{!∗}V[−d] ∈ Ob P(Y, H),  (7.10)(d)

the perverse extension of ξ. Its irreducibility, and the fact that every irreducible has this form, are consequences of Théorème 4.3.1 of [BBD]. Finally, we define

D(ξ) = DR^{−1}(P (ξ)) ∈ Ob D(Y, H).  (7.10)(e)

It is a consequence of Theorem 7.9 that every irreducible H-equivariant D_{Y} -module on Y is of this form. (Of course one can define D(ξ) somewhat more directly as a D_{Y} -module. The starting point is the sheaf M of germs of sections of V on S. Since V has an H-equivariant flat connection, M has the structure of a an H-equivariant D_{S}-module ([BorelD], section VI.1.6). One now applies the (subtle) notion of direct image for D-modules to get a 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 {μ(ξ) | ξ ∈ Ξ}, {P (ξ) | ξ ∈ Ξ}, and {D(ξ) | ξ ∈ Ξ} 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 not identified by the isomorphism of Lemma 7.8. As in Definition 1.22, we can write in K(Y, H)

μ(ξ) = (−1)^{d(ξ)} ∑_{γ∈Ξ(Y,H)} m_{g}(γ, ξ)P (γ)  (7.11)(a)

with m_{g}(γ, ξ) an integer. The matrix m_{g} is called the geometric multiplicity matrix. It follows easily from the definitions that

m_{g}(ξ, ξ) = 1,  m_{g}(γ, ξ) ≠ 0 only if S_{γ} ⊂ (\overline{S_{ξ}} − S_{ξ}) (γ ≠ ξ).  (7.11)(b)

It is also useful to consider the inverse matrix c_{g} = m_{g}^{−1}, the geometric character matrix. By its definition, c_{g} satisfies

P (γ) = ∑_{ξ∈Ξ(Y,H)}(−1)^{d(ξ)}c_{g}(ξ, γ)μ(ξ).  (7.11)(c)

Because of the form of the isomorphism in Lemma 7.8, we have

c_{g}(ξ, γ) = (−1)^{d(ξ)} ∑_{i}(−1)^{i}(multiplicity of V_{ξ} in H^{i}P (γ)|_{S_{ξ}}).  (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}(γ, γ) = 1,  c_{g}(ξ, γ) ≠ 0 only if S_{ξ} ⊂ (\overline{S_{γ}} − S_{γ}) (ξ ≠ γ).  (7.11)(e)

Because of (7.11)(b), we can define the Bruhat order on Ξ(Y, H) as the smallest partial order with the property that

m_{g}(γ, ξ) ≠ 0 only if γ ≤ ξ.  (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_{ξ} is affinely embedded in its closure. (For Y = X(O,^{∨}G^{Γ}) as in Definition 6.9, this turns out to be automatic if the orbit O consists of regular elements.) Regarded as an element of the derived category concentrated in degree −d, the constructible sheaf μ(ξ)[−d] is then perverse ([BBD], Corollary 4.1.3), and the map of Lemma 7.8 is

χ(μ(ξ)[−d]) = (−1)^{d}μ(ξ).

On the other hand, as a perverse sheaf μ(ξ)[−d] must have a finite composition series. The integer m_{g}(γ, ξ) is just the multiplicity of the irreducible perverse sheaf P (γ) in this composition series. In particular, it is non-negative. One can give a parallel discussion in terms of 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 S^{•} of sheaves on Y; we will write H^{i}(S^{•}) for the ith 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}(C(Y))  (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_{rh}^{b}(D(Y))  (7.12)(b)

for the bounded derived category of sheaves of quasicoherent D_{Y} -modules having regular holonomic cohomology sheaves (cf. [BorelD], Corollary VII.12.8).

Definition 7.13. Suppose ^{∨}G^{Γ} is a weak E-group, and X(^{∨}G^{Γ}) is the corresponding set of geometric parameters (Definition 6.9). Define

C(X(^{∨}G^{Γ}),^{∨}G^{alg})

to be the direct sum over semisimple orbits O ⊂ ^{∨}𝔤 of the categories C(X(O,^{∨}G^{Γ}),^{∨}G^{alg}) of Definition 7.7. We call the objects of this category^{∨}G^{alg}-equivariant constructible sheaves on X(^{∨}G^{Γ}). Similarly we define

P(X(^{∨}G^{Γ}),^{∨}G^{alg}), D(X(^{∨}G^{Γ}),^{∨}G^{alg}),

the^{∨}G^{alg}-equivariant perverse sheaves and the^{∨}G^{alg}-equivariant coherent D-modules on X(^{∨}G^{Γ}). 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(^{∨}G^{Γ}) for the common Grothendieck group of the three categories. The irreducible objects in each category are parametrized by the set Ξ(^{∨}G^{Γ}) of complete geometric parameters (Definition 7.6); we write μ(ξ), P (ξ), and D(ξ) as in (7.10). We get matrices m_{g} and c_{g} satisfying (7.11), the geometric multiplicity matrix and the geometric character matrix for ^{∨}G^{Γ}.

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

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.

Proposition 7.14. Suppose Y is a smooth complex algebraic variety on which the pro-algebraic group H acts with finitely many orbits. Suppose G is a proalgebraic group containing H. Consider the induced bundle

X = G ×_{H} Y.

a) The inclusion

i : Y → X, i(y) = equivalence class of (e, y)

induces a bijection

Φ(Y, H) → Φ(X, G), S ↦ G · i(S)

from H-orbits on Y to G-orbits on X.

b) If y ∈ 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_{y}^{loc} = A_{i(y)}^{loc}

(notation as in Definition 7.1). Consequently there is an isomorphism

A_{S}^{loc} ≃ A_{Φ(i)(S)}^{loc},

canonically defined up to inner automorphisms.

c) The maps of (a) and (b) combine to give a natural bijection

i : Ξ(Y, H) → Ξ(X, G).

In the language of (7.4), the H-equivariant local system V on S corresponds to

i(V) = G ×_{H} V

on G · i(S).

d) There are natural equivalences of categories

C(Y, H) ≃ C(X, G),  P(Y, H) ≃ P(X, G),  D(Y, H) ≃ D(X, G).

These equivalences are compatible with the parametrizations of irreducibles by Ξ(Y, H) ≃ Ξ(X, H) ((7.10) and (c) above) and with the isomorphisms of Grothendieck groups coming from Lemma 7.8 and Theorem 7.9.

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).

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 Dmodule case is handled by Theorem 7.9. It is helpful to introduce the intermediate space

Z = G × Y,

on which the group G × H acts by

(g, h) · (g′, y) = (gg′h^{−1}, h · y).

It is easy to see that Ξ(Z, G×H) is in one-to-one correspondence with Ξ(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 → 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 → 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 P(Y, H) (respectively P(X, G)) into P(Z, G × 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.

Proposition 7.15. Suppose X and Y are smooth algebraic varieties on which the pro-algebraic group H acts with finitely many orbits, and

f : X → Y

is a surjective smooth equivariant morphism having connected geometric fibers of dimension d.

a) There is a natural inclusion

f^{∗} : Ξ(Y, H) ↪ Ξ(X, H)

from complete geometric parameters for H acting on Y to complete geometric parameters for H acting on X.

b) The functor f^{∗}[d] is a fully faithful exact functor from P(Y, H) into P(X, H), carrying the irreducible P (ξ) to P (f^{∗}ξ) (notation (7.10)(d)).

c) The inclusion of (a) respects the geometric character matrix of (7.11):

c_{g}(f^{∗}ξ, f^{∗}γ) = c_{g}(ξ, γ) (ξ, γ ∈ Ξ(Y, H).

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^{∗} : Φ(Y, H) → Φ(X, H).  (7.16)(a)

The image of f^{∗} consists of orbits S′ ⊂ X satisfying any of the three equivalent properties

S′ is open in f^{−1}(f(S′)),  dim f(S′) = dim S′−d,  dim H_{f(x)} = dim H_{x}+d (x ∈ S′).
(7.16)(b)

Suppose S′ satisfies these conditions. A typical fiber of the map S′ → f(S′) is dense in the corresponding fiber of f, and is therefore connected. Consequently

H_{f(x)}/H_{x} is connected  (x ∈ S′ ∈ f^{∗}(Φ(Y, H)).  (7.16)(c)

For the natural map on component groups, it follows that

A_{x}^{loc} → A_{f(x)}^{loc} is surjective  (x ∈ S′ ∈ f^{∗}(Φ(Y, H)).  (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, τ) for which x and its orbit S′ satisfy (7.16)(b), and the representation τ is trivial on the kernel of the map (7.16)(d). On the level of local systems, the map f^{∗} sends (S, V) to (S′, V′), with

S′ open in f^{−1}S, V′ = (f^{∗}V) |_{S′} .  (7.16)(e)

For the rest of the argument, we refer to [BBD], Proposition 4.2.5. Q.E.D.

It is 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

ϵ : (Y, H) → (X, G).  (7.17)(a)

This means that ϵ : Y → X is a morphism of varieties, ϵ : H → G is a morphism of pro-algebraic groups, and

ϵ(h) · ϵ(y) = ϵ(h · y)  (h ∈ H, y ∈ Y).  (7.17)(b)

It follows immediately that ϵ induces a map from the set of orbits of H on Y to the set of orbits of G on X:

Φ(ϵ) : Φ(Y, H) → Φ(X, G),  Φ(ϵ)(H · y) = G · ϵ(y)  (7.17)(c)

(cf. Definition 7.1). Evidently ϵ carries the isotropy group H_{y} into G_{ϵ(y)}, so we get a homomorphism on the level of equivariant fundamental groups

A^{loc}(ϵ) : A_{y}^{loc} → A_{ϵ(y)}^{loc}  (7.17)(d)

(Definition 7.1).

Proposition 7.18. In the setting of (7.17), there is an exact functor

ϵ^{∗} : C(X, G) → C(Y, H).

The stalk (ϵ^{∗}C)_{y} at a point y ∈ Y is naturally identified with C_{ϵ(y)}. In particular, the representation of A_{y}^{loc} on this stalk is obtained from the representation of A_{ϵ(y)}^{loc} on C_{ϵ(y)} by composition with the homomorphism A^{loc}(ϵ) of (7.17)(d).

The functor ϵ^{∗} induces a homomorphism of Grothendieck groups

ϵ^{∗} : K(X, G) → K(Y, H)

(notation (7.10)(a)).

This is obvious: the functor ϵ^{∗} is just the usual inverse image for constructible sheaves. This functor behaves very well on the derived category D^{b}(C(X)), but not on the abelian subcategory of perverse sheaves. If P is a G-equivariant perverse sheaf on X, then ϵ^{∗}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 ϵ^{∗} 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 ϵ : ^{∨}H^{Γ} → ^{∨}G^{Γ}, and compatible pro-finite extensions ^{∨}H^{Q_{H}} and ^{∨}G^{Q}. The maps X(ϵ) of Corollary 6.21 and ϵ^{•} of (5.14)(c) give

ϵ = (X(ϵ), ϵ^{•}) : (X(^{∨}H^{Γ}),^{∨}H^{Q_{H}})) → (X(^{∨}G^{Γ}),^{∨}G^{Q})  (7.19)(a)

as in (7.17). In particular, we have the correspondence of orbits

Φ(ϵ) : Φ(^{∨}H^{Γ}) → Φ(^{∨}G^{Γ})  (7.19)(b)

and the homomorphisms of equivariant fundamental groups

A^{loc}(ϵ) : A_{y}^{loc,Q_{H}} → A_{ϵ(y)}^{loc,Q}  (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

ϵ^{∗} : KX(^{∨}G^{Γ})^{Q} → KX(^{∨}H^{Γ})^{Q_{H}} .  (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 ϵ.

8. Perverse sheaves on the geometric parameter space.

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 D_{Y} of algebraic differential operators on Y. We write

D_{Y} = ΓD_{Y}  (8.1)(a)

for the algebra of global sections of D_{Y}, the 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

ψ_{Y} : U(𝔤) → D_{Y}  (8.1)(b)

called the operator representation of U(𝔤) (see for example [BorBI], section 3.1). The kernel of ψ_{Y} is a two-sided ideal

I_{Y} = ker ψ_{Y} ⊂ U(𝔤).  (8.1)(c)

If M is any sheaf of D_{Y} -modules, then the vector space M = ΓM is in a natural way a D_{Y} -module, and therefore (via ψ_{Y}) a module for U(𝔤)/I_{Y}. The functor sending M to the U(𝔤)/I_{Y} -module M is called the global sections functor. In the other direction, if M is any module for U(𝔤)/I_{Y}, then we may form the tensor product

M = D_{Y} ⊗_{ψ_{Y} (U(𝔤)/I_{Y} )} M.  (8.1)(d)

This is a sheaf of D_{Y} -modules on Y. The functor sending M to M is called localization. For formal reasons, there is an adjoint relationship

Hom_{D_{Y}} (D_{Y} ⊗ M, N) ≃ Hom_{U(𝔤)/I_{Y}} (M, ΓN).  (8.1)(e)

We need to recall a little algebraic representation theory at this point.

Definition 8.2. A compatible pair (𝔤, H) consists of a Lie algebra 𝔤 and a pro-algebraic group H, endowed with an algebraic action

Ad : H → Aut(𝔤),

and an injective Lie algebra homomorphism

i : 𝔥 → 𝔤

compatible with the differential of Ad. A compatible (𝔤, H)-module is a 𝔤-module M endowed with an algebraic representation π of H, satisfying the compatibility conditions

π(h)(X · m) = (Ad(h)(X)) · π(h) · m

and

dπ(Z) · m = i(Z) · m

for h ∈ H, X ∈ 𝔤, m ∈ M, and Z ∈ 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 ⊂ U(𝔤). The category of finite length compatible (𝔤, H)-modules annihilated by I is denoted

F(𝔤, H, I).

This idea is essentially due to Harish-Chandra; the definition was formalized by Lepowsky.

Theorem 8.3 (Beilinson-Bernstein localization theorem — see [BB], [BorBI], Theorem 3.8, and [BorBIII], Theorem 1.9). Suppose G is a complex connected algebraic group, and Y is a complete homogeneous space for G.

a) The operator representation ψ_{Y} : U(𝔤) → D_{Y} of (8.1)(b) is surjective.

b) The global section and localization functors provide an equivalence of categories between quasicoherent sheaves of D_{Y} -modules on Y and modules for U(𝔤)/I_{Y}.

c) Suppose (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 D(Y, H) (Definition 7.7) and F(𝔤, H, I_{Y}) (Definition 8.2).

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 D_{Y} -module is automatically regular holonomic ([BorelD], Theorem VII.12.11).

For the rest of this section, we will fix

∨G^{Γ},  O ⊂ ^{∨}𝔤  (8.4)(a)

a weak E-group and a semisimple orbit in its Lie algebra. Fix a canonical flat Λ ⊂ O, and define

e(Λ) ∈ ^{∨}G,  P (Λ) ⊂ ^{∨}G(Λ)  (8.4)(b)

as in (6.6). Recall (Lemma 6.3) that P (Λ) is a connected parabolic subgroup of the reductive group ^{∨}G(Λ). As in Proposition 6.24, write

P^{0}(Λ) = ^{∨}G(Λ)_{0} · Λ ⊂ O  (8.4)(c)

for the indicated orbit of Λ in the variety of all canonical flats in O. Proposition 6.5 identifies P^{0}(Λ) naturally with the variety of conjugates of P (Λ) in ^{∨}G(Λ)_{0} (by sending each flat in P^{0}(Λ) to its stabilizer). In particular,

P^{0}(Λ) ≃ ^{∨}G(Λ)_{0}/P (Λ).  (8.4)(d)

This space is a complete homogeneous space for^{∨}G(Λ)_{0}, and we will apply Theorem 8.3 to it. Recall the orbit decomposition

I(Λ) = I_{1}^{0}(Λ) ∪ … ∪ I_{s}^{0}(Λ)  (8.4)(e)

of Proposition 6.24, and choose a representative y_{j} ∈ I_{j}^{0}(Λ) for each orbit. Define K_{j}^{0} as in Proposition 6.24, so that

I_{j}^{0}(Λ) ≃ ^{∨}G(Λ)_{0}/K_{j}^{0}.  (8.4)(f)

Recall from (5.10) that we write K_{j}^{0,alg} for the preimage of K_{j}^{0} in ^{∨}G^{alg}

Theorem 8.5. Suppose ^{∨}G^{Γ} is a weak E-group and O ⊂ ^{∨}𝔤 a semisimple orbit in its Lie algebra. Use the notation of (8.4). Then the category of ^{∨}G^{alg}- equivariant perverse sheaves on X(O,^{∨}G^{alg}) (Definitions 6.9, 7.7, and 7.13) is equivalent to the direct sum of the categories F(^{∨}𝔤(Λ), K_{j}^{0,alg}, I_{P^{0}(Λ)}) (Definition 8.2) of compatible modules of finite length annihilated by the kernel I_{P^{0}(Λ)} of the operator representation on P^{0}(Λ) (see (8.1)).

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 Λ. For that we need a kind of “translation functor” for these categories.

Definition 8.6. In the setting of (8.5), suppose O′ is another orbit of semisimple elements. A translation datum from O to O′ is a ^{∨}G-conjugacy class T of pairs (Λ_{1}, Λ′_{1}), subject to the following conditions.

(a) Λ_{1} ⊂ O and Λ′_{1} ⊂ O′ are canonical flats.

(b) P (Λ′_{1}) ⊂ P (Λ_{1}), and e(Λ_{1}) = e(Λ′_{1}).

There are several other ways of specifying the same data. If we fix a canonical flat Λ ⊂ O, then it is equivalent to specify a single canonical flat Λ′ ⊂ O′, subject to the conditions

(c) P (Λ′) ⊂ P (Λ), and e(Λ) = e(Λ′). (Necessarily Λ′ will be contained in 𝔭(Λ).) In this case two canonical flats define the same translation datum if and only if they are conjugate by P (Λ).

If we fix a point λ ∈ O, then it is equivalent to specify another point λ′ ∈ O′, subject to

(d) P (λ′) ⊂ P (λ), L(λ′) ⊂ L(λ), and e(λ) = e(λ′). (Necessarily λ′ will belong to 𝔩(λ).) In this case two points λ′_{1}, λ′_{2} define the same translation family if and only if they are conjugate by L(λ).

Finally, we could fix a pair (λ, T), with T a maximal torus and λ ∈ O ∩ 𝔱. Then the translation datum is determined by the choice of another point λ′ ∈ O′ ∩ 𝔱, subject to

(e) For every root α of T in ^{∨}G,

α(λ) ∈ ℕ − {0} ⇒ α(λ′) ∈ ℕ − {0},

and

λ − λ′ ∈ X_{∗}(T).

(Of course the lattice on the right here is the kernel of the normalized exponential mapping e on 𝔱.) Two such points λ′_{1}, λ′_{2} determine the same translation datum if and only if they are conjugate by the Weyl group of T in L(λ).)

Lemma 8.7. Suppose ^{∨}G^{Γ} is a weak E-group and O ⊂ ^{∨}𝔤 a semisimple orbit in its Lie algebra. Then there is a regular semisimple orbit O′ ⊂ ^{∨}𝔤 and a translation datum T from O to O′ (Definition 8.6)

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 ^{∨}G containing the roots of T in N(λ) (notation (6.1) and (6.2)). Let μ be any element of X_{∗}(T) on which the positive roots take strictly positive values; for example, the sum of the positive coroots. Then λ′ = λ + μ is a regular semisimple element satisfying the conditions of Definition 8.6(e), so the corresponding conjugacy class

T = ^{∨}G · (F(λ), F(λ′))

is the translation datum we need. Q.E.D.

Proposition 8.8. Suppose ^{∨}G^{Γ} is a weak E-group, O, O′ are semisimple orbits in its Lie algebra, and T is a translation datum from O to O′.

a) The set T is the graph of a ^{∨}G-equivariant smooth proper morphism

f_{T} : F(O′) → F(O)

(notation (6.10)).

b) The morphism in (a) induces an equivariant smooth proper morphism

f_{T} : X(O′,^{∨}G^{Γ}) → X(O,^{∨}G^{Γ}).

Consequently (Proposition 7.15) the category of^{∨}G^{alg}-equivariant perverse sheaves on X(O,^{∨}G^{Γ}) is equivalent to a full subcategory of ^{∨}G^{alg}-equivariant perverse sheaves on X(O′,^{∨}G^{Γ}). This inclusion respects the geometric character matrix.

Fix a pair (Λ, Λ′) ∈ T .

c) The objects of (8.4) for Λ and Λ′ are related by

e(Λ) = e(Λ′), ^{∨}G(Λ)_{0} = ^{∨}G(Λ′)_{0}, I(Λ) = I(Λ′).

The morphism of (a) restricts to a smooth ^{∨}G(Λ)_{0}-equivariant proper morphism

f_{T} : P^{0}(Λ′) → P^{0}(Λ).

d) The kernel I_{P^{0}(Λ)} of the operator representation of U(^{∨}𝔤(Λ)) contains the kernel I_{P^{0}(Λ′)}. Consequently (for any y ∈ I(Λ)) the category F(^{∨}𝔤(Λ), K^{0,alg}(y), I_{P^{0}(Λ)}) (Definition 8.2) is a full subcategory of F(^{∨}𝔤(Λ′), K^{0,alg}(y), I_{P^{0}(Λ′)}).

e) The inclusions of categories in (b) and (d) correspond under the equivalences of Theorem 8.5.

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 (Λ′) is parabolic in ^{∨}G(Λ) (Lemma 6.3), and therefore also in P (Λ) ⊂ ^{∨}G(Λ).) The second condition in Definition 8.6(b) implies that C(O) = C(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 ∨G(Λ)_{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 Ξ(O,^{∨}G^{Γ}) 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

Ξ(O,^{∨}G^{Γ}) = ∐_{j} Ξ(P^{0}(Λ), K_{j}^{0,alg}).  (8.9)(a)

This decomposition respects the geometric multiplicity and character matrices. (In particular, if ξ and ξ′ correspond to different values of j on the right, then m_{g}(ξ, ξ′) = c_{g}(ξ, ξ′) = 0.)

Suppose now that O is regular. Then the groups L(λ) of Lemma 6.3 are all maximal tori, so P (Λ) is a Borel subgroup of ^{∨}G(Λ)_{0}, and the variety P^{0}(Λ) may be identified with the variety B of Borel subgroups of ^{∨}G(Λ)_{0}. The parameter set corresponding to j on the right in (8.9)(a) is now

D_{j} = Ξ(B, K_{j}^{0,alg}).  (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 (ξ, ξ′) of elements of D_{j}, Theorem 1.11 of [LV] provides a polynomial

P_{g}(ξ, ξ′) ∈ ℤ[u],  (8.9)(c)

the Kazhdan-Lusztig polynomial. The algorithm of [LV] for computing this polynomial involves an action of the Hecke algebra associated to the Weyl group of ∨G(Λ)_{0} on the free ℤ[u, u^{−1}]-module with basis D_{j}. Theorem 1.12 of [LV] asserts that these polynomials compute the geometric character matrix:

c_{g}(ξ, ξ′) = (−1)^{d(ξ)−d(ξ′)}P_{g}(ξ, ξ′)(1).  (8.9)(d)

To be a little more precise: (7.11)(d) says that c_{g}(ξ, ξ′) 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}(ξ, ξ′). We will recall some additional details in section 16, in connection with the proof of Theorem 1.24.

9. The Langlands classification for tori.

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) = Hom_{alg}(ℂ^{×}, T),  (9.1)(a)

the lattice of (rational) one-parameter subgroups of T. We can use the exponential map to identify the Lie algebra of ℂ^{×} with ℂ. A one-parameter subgroup is determined by its differential, which is a complex-linear map from ℂ to 𝔱. Such a map is in turn determined by its value at 1 ∈ ℂ; so X_{∗}(T) is identified with a lattice in 𝔱. As is well-known ([AV2], Proposition 2.2)

X_{∗}(T) ≃ (1/2πi) ker(exp) = ker(e) ⊂ 𝔱.  (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) ⊗_{ℤ} ℂ^{×} ≃ T, ϕ ⊗ z ↦ ϕ(z).  (9.1)(c)

On the level of Lie algebras, this gives

X_{∗}(T) ⊗_{ℤ} ℂ ≃ 𝔱, ϕ ⊗ z ↦ dϕ(z)  (9.1)(d)
𝔱^{∗} ≃ Hom_{ℤ}(X_{∗}(T), ℂ), λ ↦ (ϕ ↦ λ(dϕ(1))).  (9.1)(e)

Dually, write

X^{∗}(T) = Hom_{alg}(T, ℂ^{×}),  (9.2)(a)

the lattice of (rational) characters of T. A character is determined by its differential, which is a linear functional on 𝔱^{∗}; the resulting lattice may be identified as

X^{∗}(T) = { λ ∈ 𝔱^{∗} | λ(X_{∗}(T)) ⊂ ℤ }.  (9.2)(b)

This exhibits the natural pairing between X_{∗}(T) and X^{∗}(T) as the restriction of the pairing between 𝔱 and 𝔱^{∗}. The analogue of (9.1)(c) is

T ≃ Hom_{ℤ}(X^{∗}(T), ℂ^{×}), t ↦ (τ ↦ τ(t)).  (9.2)(c)

Similarly,

𝔱 ≃ Hom_{ℤ}(X^{∗}(T), ℂ), X ↦ (τ ↦ dτ(X))  (9.2)(d)
X^{∗}(T) ⊗_{ℤ} ℂ ≃ 𝔱^{∗}, τ ⊗ z ↦ (X ↦ zdτ(X)).  (9.2)(e)

By the remark after Definition 4.2, a dual group ^{∨}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

∨T = X^{∗}(T) ⊗_{ℤ} ℂ^{×} or ^{∨}T = Hom_{ℤ}(X_{∗}(T), ℂ^{×}).  (9.3)(a)

In any case Definition 4.2, (9.1)(d)–(e), and (9.2)(d)–(e) provide natural isomorphisms

∨𝔱 ≃ 𝔱^{∗},  ^{∨}𝔱^{∗} ≃ 𝔱.  (9.3)(b)

Proposition 9.4 (Langlands — see [Langlands], Lemma 2.8, or [AV2], Theorem 4.4.) Suppose T is an algebraic torus defined over ℝ. Fix an L-group (^{∨}T^{Γ}, D) for T (Definition 4.6). Then there is a natural bijection

ϕ → π(ϕ)

from the set Φ(T/ℝ) = Φ(^{∨}T^{Γ}) of equivalence classes of Langlands parameters (Definitions 5.2 and 5.3) onto the set Π(T (ℝ)) of equivalence classes of irreducible representations (that is, continuous homomorphisms into ℂ^{×}) of T (ℝ). In this bijection, the differential dπ(ϕ) (an element of 𝔱^{∗}) is identified by (9.3)(b) with the parameter λ(ϕ) ∈ ^{∨}𝔱 (cf. (5.8)(a)). The trivial character of T (ℝ) is attached to the parameters ϕ(^{∨}δ, 0) with ^{∨}δ ∈ D (notation (5.8)(a)); these constitute a single equivalence class.

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 not as it stands a special case of Theorem 1.18, for it mentions neither strong real forms of T (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.

Lemma 9.5. 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 ⊂ T corresponds the sublattice

L = { τ ∈ X^{∗}(T) | τ(s) = 1, all s ∈ S }.  (a)

Dually, to a sublattice L corresponds the algebraic subgroup

S = { s ∈ T | τ(s) = 1, all τ ∈ L }.  (b)

Suppose the subgroups S_{1} ⊂ S_{2} correspond to sublattices L_{1} ⊃ L_{2}. Then restriction of characters from T to S_{1} defines a natural isomorphism

Hom_{alg}(S_{1}/S_{2}, ℂ^{×}) ≃ L_{2}/L_{1}.  (c)

Suppose the subgroup S corresponds to the sublattice L. Then the identity component S_{0} ⊂ S corresponds to the lattice

L^{0} = { τ ∈ X^{∗}(T) | nτ ∈ L for some positive integer n } ⊃ L.  (d)

Suppose θ is an algebraic automorphism of T, and a is the transpose automorphism of X^{∗}(T) (so that τ(θt) = (aτ)(t) for all τ ∈ X^{∗}(T) and t ∈ T). Then the algebraic subgroup T^{θ} of fixed points of θ corresponds to the sublattice

(1 − a)X^{∗}(T) = { τ − aτ | τ ∈ X^{∗}(T) }.  (e)

Dually, the sublattice X^{∗}(T)^{a} of fixed points of a corresponds to the (connected) subgroup

(1 − θ)T = { t(θt)^{−1} | t ∈ T }.  (f)

This is well-known and elementary; we omit the proof.

We turn now to the description of complete geometric parameters for tori.

Proposition 9.6. Suppose ^{∨}T^{Γ} 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 θ for the automorphism of ^{∨}T defined by any element of ^{∨}T^{Γ}−^{∨}T. Suppose x ∈ X(^{∨}T^{Γ}) is a geometric parameter (Definition 6.9).

a) The stabilizer ^{∨}T_{x} of x for the action of ^{∨}T on geometric parameters is the group ∨T^{θ} of fixed points of θ.

b) The Langlands component group A_{x}^{loc} is

∨T^{θ}/^{∨}T _{0}^{θ} = ^{∨}T^{θ}/(1 + θ)^{∨}T

.

c) The group of characters of the Langlands component group is given by a natural

isomorphism

Â_{x}^{loc} ≃ X_{∗}(T)^{−a}/(1 − a)X_{∗}(T).

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.

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 ^{ ˜∨}T is a finite covering of an algebraic torus ^{∨}T. The lattice of rational characters of ^{ ˜∨}T must contain X^{∗}(^{∨}T) as a sublattice of finite index:

X^{∗}(^{∨}T) ⊂ X^{∗}(^{ ˜∨}T).  (9.7)(a)

We may therefore regard X^{∗}(^{ ˜∨}T) as a lattice in the rational vector space generated by X^{∗}(^{∨}T):

X^{∗}(^{ ˜∨}T) ⊂ X^{∗}(^{∨}T) ⊗_{ℤ} ℚ.  (9.7)(b)

Conversely, any lattice in this vector space containing X^{∗}(^{∨}T) as a subgroup of finite index is the character lattice of a finite covering of ^{∨}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 ^{∨}T^{alg}. Extending the notation of (9.2), we write

X^{∗}(^{∨}T^{alg}) = X^{∗}(^{∨}T) ⊗_{ℤ} ℚ.  (9.7)(c)

Equivalently,

X^{∗}(^{∨}T^{alg}) = X_{∗}(T) ⊗_{ℤ} ℚ.  (9.7)(d)

Proposition 9.8. Suppose we are in the setting of Proposition 9.6; use the notation there and in (9.7). Write as usual ^{∨}T _{x}^{alg} for the preimage of ^{∨}T_{x} in ∨T^{alg}, and (^{∨}T _{x}^{alg})_{0} for its identity component.

a) The group of rational characters of^{∨}T^{alg} trivial on^{∨}T _{x}^{alg} is identified via (9.7)(d) with (1 − a)X_{∗}(T).

b) The group of rational characters of ^{∨}T^{alg} trivial on (^{∨}T _{x}^{alg})_{0} is identified via (9.7)(d) with (X_{∗}(T) ⊗_{ℤ} ℚ)^{−a}. c) The group of characters of the universal component group (Definition 7.6) is

given by a natural isomorphism

Â_{x}^{loc,alg} ≃ (X_{∗}(T) ⊗_{ℤ} ℚ)^{−a}/(1 − a)X_{∗}(T).

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.

Lemma 9.9. 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

𝔱_{ℚ} = X_{∗}(T) ⊗_{ℤ} ℚ ⊂ 𝔱

(cf. (9.1)(d)).

a) The normalized exponential mapping e, given by e(τ) = exp(2πiτ) (cf. (6.2)(a))

defines an isomorphism

𝔱/X_{∗}(T) ≃ T.

b) The preimage of T^{fin} under the isomorphism in (a) is precisely 𝔱_{ℚ}, so we have

a natural isomorphism

𝔱_{ℚ}/X_{∗}(T) ≃ T^{fin}.

c) Suppose σ is an antiholomorphic automorphism of T. Then the isomorphism of

(a) carries −dσ to σ:

e(−dσ(τ)) = σ(e(τ)).

d) In the setting of (c), write a for the automorphism of X_{∗}(T) induced by σ (Proposition 2.12). Write a also for the corresponding (ℚ-linear) automorphism of 𝔱_{ℚ}. Then the isomorphism of (b) carries −a to σ:

e(−a(τ)) = σ(e(τ)).

In particular, a = dσ on 𝔱_{ℚ}.

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 ∈ ℂ^{×} has finite order, then \overline{z} = z^{−1}. Q.E.D.

Proposition 9.10. Suppose T^{Γ} is a weak extended group for T (Definition 2.13). Write σ for the antiholomorphic involution on T defined by conjugation by any element of T^{Γ} − T. Consider the three subgroups

T^{−σ,fin} = { t ∈ T | tσ(t) has finite order }
T^{−σ} = { t ∈ T | tσ(t) = 1 }
T_{0}^{−σ} = identity component of T^{−σ} = { sσ(s^{−1}) | s ∈ T }.

a) Left multiplication defines a natural simply transitive action of T^{−σ,fin} on the set of strong real forms of T^{Γ} (Definition 2.13). b) Two strong real forms are equivalent precisely when they lie in the same coset

of T_{0}^{−σ}. Consequently there is a natural simply transitive action of the quotient T^{−σ,fin}/T_{0}^{−σ} on the set of equivalence classes of strong real forms.

c) The mapping τ ↦ e(τ/2) (cf. Lemma 9.9) maps the −1 eigenspace 𝔱_{ℚ}^{−a} into T^{−σ,fin}. The preimage of the subgroup T^{−σ} is the lattice X_{∗}(T)^{−a}. The preimage of the subgroup T_{0}^{−σ} is the lattice (1 − a)X_{∗}(T).

d) After composition with the quotient, the map of (c) becomes a surjection onto T^{−σ,fin}/T_{0}^{−σ}.

e) There are natural isomorphisms

𝔱_{ℚ}^{−a}/(1 − a)X_{∗}(T) ≃ T^{−σ,fin}/T_{0}^{−σ}
X_{∗}(T)^{−a}/(1 − a)X_{∗}(T) ≃ T^{−σ}/T_{0}^{−σ}.

Proof. The first thing to prove is the equivalence of the two definitions of T_{0}^{−σ}. The Lie algebra of {sσ(s^{−1})} is {τ − dσ(τ)}. Since σ has order two, this is equal to the −1-eigenspace of dσ, which in turn is the Lie algebra of T^{−σ}. Since {sσ(s^{−1})} is clearly connected, the claim follows.

Now consider (a). Suppose t ∈ T, and δ is a strong real form. Then

(tδ)^{2} = t(δtδ^{−1})δ^{2} = tσ(t)δ^{2}.

Since δ^{2} is assumed to have finite order, it follows that tδ is a strong real form if and only if t ∈ T^{−σ,fin}. A similar calculation shows that

s(tδ)s^{−1} = t(sσ(s^{−1}))δ,

from which (b) follows.

For (c), suppose that τ ∈ 𝔱_{ℚ}^{−a}. Then

e(τ/2)σ(e(τ/2)) = e(τ/2)e(−aτ/2) = e(τ/2)e(τ/2) = e(τ).

Here we use successively Lemma 9.9(d) and the assumption on τ. 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_{0}^{−σ}T_{0}^{σ}.  (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 σ has order 2.

For the third part of (c), suppose that τ ∈ 𝔱_{ℚ}^{−a} and that e(τ/2) = sσ(s^{−1}). Because of (9.11), we may as well assume that s ∈ T_{0}^{−σ}. By Lemma 9.9(c), s = e(β), with β in the +1-eigenspace of dσ. The assumption on s says (in light of Lemma 9.9(a) and (c)) that for some γ ∈ X_{∗}(T),

τ/2 = β/2 + dσ(β/2) + γ = β + γ.

The first conclusion is that β ∈ 𝔱_{ℚ}. By Lemma 9.9(c), β is in the +1-eigenspace of a. Consequently τ/2 is the projection of γ on the −1-eigenspace of a. That is,

τ/2 = (γ − aγ)/2,

which gives the third part of (c).

For (d), (9.11) and Lemma 9.9(c) imply that every coset in T^{−σ,fin}/T_{0}^{−σ} has a representative t = e(τ/2) with τ in the −1-eigenspace of dσ. As in (a) we compute tσ(t) = e(τ); so τ must actually belong to the −1-eigenspace of dσ on 𝔱_{ℚ}. By Lemma 9.9(d), τ belongs to 𝔱_{ℚ}^{−a}, as we wished to show.

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

Corollary 9.12. Suppose ^{∨}T^{Γ} is a weak E-group for the algebraic torus T (Definition 4.3), and T^{Γ} is a weak extended group for the corresponding real form (Definition 2.13). Suppose x ∈ X(^{∨}T^{Γ}) is a geometric parameter (Definition 6.9). Then there is a natural simply transitive action of the group Â^{loc,alg} (of characters of the universal component group) on the set of equivalence classes of strong real forms of T^{Γ}.

Proof. Combine Proposition 9.8(c) with Proposition 9.10(d). Q.E.D.

Proof of Theorem 1.18 for tori. The extra datum 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.

10. Covering groups and projective representations.

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 “ρ-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.

Definition 10.1. Suppose G^{Γ} is a weak extended group for G (Definition 2.13). A connected finite covering group

1 → F → G̃ → G → 1

is said to be distinguished if the following two conditions are satisfied.

(a) For every x ∈ G^{Γ} − G, the conjugation action σ_{x} of x on G lifts to an automorphism σ̃_{x} of G̃.

It is equivalent to require this condition for a single x. As x varies, σ̃_{x} changes by inner automorphisms; so its restriction σ̃_{Z} to Z(G̃) is independent of the choice of x. Necessarily σ̃_{Z} preserves the subgroup F of Z(G̃). We can now formulate the second condition on G̃.

(b) The automorphism σ̃_{Z} sends every element of F to its inverse.

Write G^{sc} for the simply-connected covering group of G, and π_{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 σ_{Z}^{sc} of Z(G^{sc}), which preserves the subgroup π_{1}(G). We may regard F as a quotient of π_{1}(G):

1 → K → π_{1}(G) → F → 1.

Then condition (a) amounts to the requirement that K should be preserved by σ_{Z}^{sc}, and condition (b) to

K ⊃ { y · σ_{Z}^{sc}(y) | y ∈ π_{1}(G) }.

The 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 → π_{1}(G)^{can} → G^{can} → G → 1.

The group π_{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 π_{1}(G).

Lemma 10.2. Suppose G is a connected complex reductive algebraic group, and ∨G is a dual group (Definition 4.2).

a) The group of complex characters of π_{1}(G) is naturally isomorphic to the center

of ^{∨}G:

Z(^{∨}G) ≃ Hom(π_{1}(G), ℂ^{×}).

Write this isomorphism as z ↦ χ_{z}.

Suppose G is endowed with an inner class of real forms. Write σ_{Z}^{sc} for the induced action of any of these real forms on π_{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 θ_{Z} for the automorphism of Z(^{∨}G) induced by any automorphism of ^{∨}G corresponding to a (cf. Proposition 4.4).

b) The automorphisms σ_{Z}^{sc} and θ_{Z} composed with inversion are transposes of each other with respect to the isomorphism in (a). That is, if z ∈ Z(^{∨}G) and p ∈ π_{1}(G), then

χ_{z}(p) = χ_{θ_{Z}(z)^{−1}}(σ_{Z}^{sc}(p)).

c) The character χ_{z} is trivial on the subgroup {pσ_{Z}^{sc}(p)} of π_{1}(G) if and only if θ_{Z}(z) = z.

d) The group of continuous characters of π_{1}(G)^{can} (Definition 10.1) is naturally isomorphic to the group of elements of finite order in Z(^{∨}G)^{θ_{Z}}.

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 π_{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.

Definition 10.3 (cf. Definition 2.13). Suppose G^{Γ} is a weak extended group for G; use the notation of Definition 10.1 and Proposition 10.2. If δ is a strong real form of G^{Γ} (Definition 2.13), write G(ℝ, δ)^{can} for the preimage of G(ℝ, δ) in G^{can}. There is a short exact sequence

1 → π_{1}(G)^{can} → G(ℝ, δ)^{can} → G(ℝ, δ) → 1.

A canonical projective representation of a strong real form of G^{Γ} is a pair (π, δ), subject to

(a) δ is a strong real form of G^{Γ}; and

(b) π is an admissible representation of G(ℝ, δ)^{can}. Equivalence is defined as in Definition 2.13. Suppose z ∈ Z(^{∨}G)^{θ_{Z}}. We say that (π, δ) is of type z if the restriction of π to π_{1}(G)^{can} is a multiple of χ_{z} (Proposition 10.2(a)). (Notice that every irreducible canonical projective representation has a unique type.) Finally, define

Π^{z}(G^{Γ}) = Π^{z}(G/ℝ)

to be the set of infinitesimal equivalence classes of irreducible canonical projective representations of type z.

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.)

Theorem 10.4. Suppose (G^{Γ}, W) is an extended group for G (Definition 1.12), and (^{∨}G^{Γ}, 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 Π^{z}(G/ℝ) of equivalence classes of irreducible canonical projective representations of strong real forms of G of type z (Definition 10.3), and the set Ξ^{z}(G/ℝ) of complete geometric parameters for ^{∨}G^{Γ} (Definition 7.6 and Definition 5.11).

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 ℝ. A calculation analogous to (9.7) gives

X^{∗}(T^{can}) = { λ ∈ X_{∗}(^{∨}T) ⊗_{ℤ} ℚ | λ − θλ ∈ X_{∗}(^{∨}T)}.  (10.5)(a)

Following Lemma 9.9, we write

∨𝔱_{ℚ} = X_{∗}(^{∨}T) ⊗_{ℤ} ℚ.  (10.5)(b)

Now Lemma 9.9(b) says that the normalized exponential mapping e provides an isomorphism

∨𝔱_{ℚ}/X_{∗}(^{∨}T) ≃ ^{∨}T^{fin}.  (10.5)(c)

Clearly X^{∗}(T^{can}) is precisely the inverse image of the θ-fixed elements ^{∨}T^{fin} under this map; so

X^{∗}(T^{can})/X^{∗}(T) ≃ (^{∨}T^{fin})^{θ}.  (10.5)(d)

Of course the group on the left may be identified with the characters of π_{1}(T)^{can}, and we recover the isomorphism in Lemma 10.2(d).

Proposition 10.6 ([AV2], Theorem 5.11.) Suppose T is an algebraic torus defined over ℝ. Fix an E-group (^{∨}T^{Γ}, W) for T with second invariant z (Definition 4.6). Then there is a natural bijection

ϕ → π(ϕ)

from the set Φ(^{∨}T^{Γ}) of equivalence classes of Langlands parameters (Definition 5.2) onto the set Π^{z}(T (ℝ)) of equivalence classes of irreducible canonical projective representations of T (ℝ) of type z (Definition 10.3). In this bijection, the differential dπ(ϕ) (an element of 𝔱^{∗}) is identified by (9.3)(b) with the parameter λ(ϕ) ∈ ^{∨}𝔱 (cf. (5.8)(a)).

Suppose λ ∈ ^{∨}𝔱_{ℚ} satisfies e(λ) = z. By (10.5), λ may be identified with a rational character of T^{can} of type z. By restriction to T (ℝ)^{can}, λ defines an irreducible canonical projective representation of T (ℝ) of type z. Its Langlands parameters are the various ϕ(^{∨}δ, λ) (notation (5.8)(a)) with ^{∨}δ ∈ D; these constitute a single equivalence class.

As in Proposition 9.4, the last assertion is included primarily to show how the classification theorem forces the choice of the class D in addition to the weak Egroup structure.

Corollary 10.7. Theorem 10.4 is true if G is an algebraic torus.

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.

Definition 10.8. In the setting of Theorem 10.4, suppose ξ ∈ Ξ^{z}(G/ℝ) is a complete geometric parameter. A representative for the corresponding equivalence class of representation and strong real form will be written as

(π(ξ), δ(ξ)) ∈ Π^{z}(G/ℝ)

(Definition 10.3). If we regard ξ as represented by a complete Langlands parameter (ϕ, τ) (Definition 5.11), we may write instead (π(ϕ, τ), δ(ϕ, τ)).

Conversely, suppose (π, δ) ∈ Π^{z}(G/ℝ). The corresponding complete geometric parameter ξ will be written as

ξ(π, δ) = (S(π, δ), V(π, δ)).

Here S is an orbit of ^{∨}G on the geometric parameter space, and V is a ^{∨}G^{alg}- equivariant local system on S. A representative for the corresponding equivalence class of complete Langlands parameters is (ϕ(π, δ), τ(π, δ)) (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(π) or d(π, δ) for the dimension of the orbit S(π, δ) (cf. (7.10)), and e(π) or e(π, δ) for the Kottwitz sign attached to ξ(π, δ) (Definition 15.8 below).

Suppose now that ϕ ∈ Φ^{z}(G/ℝ) (Definition 5.3), and let S_{ϕ} ⊂ X(^{∨}G^{Γ}) be the corresponding orbit (Proposition 6.17). The L-packet attached to ϕ is the set of irreducible (canonical projective) representations of strong real forms parametrized by complete geometric parameters supported on S:

Π^{z}(G/ℝ)_{ϕ} = { (π(ξ), δ(ξ)) | ξ = (S_{ϕ}, V) }
= { (π, δ) | ϕ(π, δ) is equivalent to ϕ }.

Perhaps the least satisfactory aspect of Theorem 10.4 is the appearance of the algebraic universal covering^{∨}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.

Lemma 10.9. Suppose we are in the setting of Lemma 10.2.

a) The group of continuous characters of π_{1}(^{∨}G)^{alg} is naturally isomorphic to the group Z(G)^{fin} of elements of finite order in Z(G). Write χ_{z} for the character corresponding to z.

b) In the setting of Theorem 10.4, write z_{0} ∈ Z(G)^{fin} for the second invariant of the extended group (G^{Γ}, W) (cf. (3.5)). Suppose that the complete geometric parameter ξ corresponds to the irreducible canonical projective representation (π(ξ), δ(ξ)). Set z_{1} = δ^{2} ∈ Z(G)^{fin}. Then π_{1}(^{∨}G)^{alg} acts on ξ by the character χ_{z_{0}z_{1}}.

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).

Definition 10.10. Suppose ^{∨}G^{Γ} is a weak E-group, and Q is any quotient of π_{1}(^{∨}G)^{alg}:

1 → K_{Q} → π_{1}(^{∨}G)^{alg} → Q → 1  (10.10)(a)

(We assume that K_{Q} is closed; equivalently, that Q is the inverse limit of its finite quotients.) Define

∨G^{Q} = ^{∨}G^{alg}/K_{Q},  (10.10)(b)

a pro-finite covering of ^{∨}G:

1 → Q → ^{∨}G^{Q} → ^{∨}G → 1.  (10.10)(c)

(Notice in particular that ^{∨}G^{π_{1}(^{∨}G)^{alg}} is just ^{∨}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, V) with the property that K_{Q} acts trivially on V, so

Ξ(^{∨}G^{Γ})^{Q} ⊂ Ξ(^{∨}G^{Γ}).  (10.10)(c)

In the setting of Theorem 10.4, we may write Ξ^{z}(G/ℝ)^{Q} for Ξ(^{∨}G^{Γ})^{Q}.

Suppose (G^{Γ}, W) is an extended group with second invariant z_{0} (cf. (3.5)), and J ⊂ Z(G)^{fin} is any subgroup. A strong real form of G of type J is an element δ ∈ G^{Γ} − G such that δ^{2} ∈ z_{0}J. (The shift by z_{0} guarantees that the real forms in W are of type J for every J.) We write

Π(G/ℝ)_{J} = { (π, δ) ∈ Π(G/ℝ) | δ^{2} ∈ z_{0}J }

for the set of equivalence classes of irreducible representations of strong real forms of G of type J, and extend other notation analogously.

Theorem 10.11. In the setting of Theorem 10.4, suppose Q is the quotient of π_{1}(^{∨}G)^{alg} by a closed subgroup K_{Q} ⊂ π_{1}(^{∨}G)^{alg}. Define J ⊂ 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 ∈ Z(G)^{fin} | χ_{z}(k) = 1, all k ∈ K_{Q} }.

Then the bijection of Theorem 10.4 restricts to a bijection

Ξ^{z}(G/ℝ)^{Q} ↔ Π^{z}(G/ℝ)_{J}.

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 Q̂ of characters of Q, and conversely. The conclusion of Theorem 10.11 may therefore be written as

Ξ^{z}(G/ℝ)^{Q} ↔ Π^{z}(G/ℝ)_{Q},  (10.12)(a)

or as

Π^{z}(G/ℝ)_{J} ↔ Ξ^{z}(G/ℝ)_{J}.  (10.12)(b)

11. The Langlands classification without L-groups.

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(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_{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 are linear.

So suppose G is a connected reductive complex algebraic group defined over ℝ; and suppose G_{ℝ} is a real Lie group endowed with a homomorphism

G_{ℝ} → G(ℝ)  (11.1)(a)

having finite kernel and cokernel. We use the homomorphism to identify the Lie algebra 𝔤_{ℝ} with 𝔤(ℝ), and its complexification with 𝔤. A Cartan subgroup of G_{ℝ} is by definition the centralizer in G_{ℝ} of a Cartan subalgebra of 𝔤_{ℝ}. (Such a subgroup T_{ℝ} is the preimage of the real points T (ℝ) of a unique maximal torus defined over ℝ:

T_{ℝ} → T (ℝ).  (11.1)(b)

(We should remark that T_{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(ℝ) and its subgroups back to G_{ℝ}, getting for example a central extension

1 → π_{1}(G)^{can} → G_{ℝ}^{can} → G_{ℝ} → 1.  (11.1)(c)

We may therefore speak of a “canonical projective representation of G_{ℝ} of type z” as in Definition 10.3, and we write

Π^{z}(G_{ℝ}) = { infinitesimal equivalence classes of such irreducible representations }.

(11.1)(d)

(Here z is an element of finite order in Z(^{∨}G)^{θ_{Z}} if we have a dual group^{∨}G available; otherwise we can just think of z directly as a character of π_{1}(G)^{can} as in Lemma 10.2(d).) If H_{ℝ} is a subgroup of G_{ℝ}, we write H_{ℝ}^{can} or H_{ℝ}^{can,G} for its preimage in G_{ℝ}^{can}. This is a central extension

1 → π_{1}(G)^{can} → H_{ℝ}^{can,G} → H_{ℝ} → 1.  (11.1)(e)

Definition 11.2 (cf. [Under], Definition 2.6, and [AV2], Definition 8.18). In the setting of (11.1), suppose T_{ℝ} is a Cartan subgroup of G_{ℝ}. Fix an element z ∈ Z(^{∨}G)^{θ_{Z}} of finite order. We may choose as a dual group for T a maximal torus ∨T ⊂ ^{∨}G; then

Z(^{∨}G)^{θ_{Z}} ⊂ ^{∨}T^{θ_{Z}} ,

and this embedding is independent of all choices. It therefore makes sense to regard z as an element of finite order in ^{∨}T^{θ_{Z}}. Recall the element z(ρ) ∈ ^{∨}T^{θ_{Z}} constructed in Definition 4.9. (Actually z(ρ) even belongs to Z(^{∨}G)^{θ_{Z}}, but this fact is a bit of an accident in the present context, and is better ignored.) In any case zz(ρ) ∈ ^{∨}T^{θ_{Z}} is an element of finite order, so it makes sense to speak of canonical projective representations of T_{ℝ} of type zz(ρ).

A G_{ℝ}-limit character of T_{ℝ} of type z is a triple Λ = (Λ^{can}, R_{iℝ}^{+}, R_{ℝ}^{+}) subject to the following conditions. The first term is an irreducible canonical projective representation

Λ^{can} ∈ Π^{zz(ρ)}(T_{ℝ}).  (11.2)(a)

This means that Λ^{can} is an irreducible representation of T_{ℝ}^{can,T} and that the restriction of Λ^{can} to π_{1}(T)^{can} is a multiple of χ_{zz(ρ)}^{T}. Write

λ = dΛ^{can} ∈ 𝔱^{∗}.  (11.2)(b)

Next,

R_{ℝ}^{+},  R_{iℝ}^{+}  (11.2)(c)

are positive root systems for the real and imaginary roots of T in G. Finally, we assume that

⟨α, λ⟩ ≥ 0,  (α ∈ R_{iℝ}^{+}).  (11.2)(d)

If these inequalities are strict (for example if λ is regular), the choice of R_{iℝ}^{+} is forced by (11.2)(d), and we may sometimes omit it in writing the limit character.

The limit character is called G-regular if ⟨α, λ⟩ ≠ 0 for any root α of T in G. Attached to each limit character Λ there is a standard limit representation M(Λ),

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 quotients. Thus we define

π(Λ) = largest completely reducible quotient of M(Λ),  (11.2)(e)

the Langlands quotient of M(Λ). Occasionally we will need the standard representation having a Langlands subrepresentation; this is written

M̃(Λ) ⊃ π(Λ).  (11.2)(f)

This representation has exactly the same composition factors and multiplicities as M(Λ).

The infinitesimal character of M(Λ) or π(Λ) is given in the Harish-Chandra parametrization (see for example [Green], Corollary 0.2.10, or [Hump], p. 130) by the weight λ = dΛ^{can} ∈ 𝔱^{∗}. In particular, the infinitesimal character is regular if and only if the limit character is G-regular.

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_{ℝ} of G_{ℝ} so that the corresponding Cartan involution preserves T_{ℝ}. This gives rise to a direct product decomposition

T_{ℝ} = (T_{ℝ} ∩ K_{ℝ})(A_{ℝ})  (11.3)(a)

with the second factor a vector group. This decomposition in turn leads to

T_{ℝ}^{can} = (T_{ℝ} ∩ K_{ℝ})^{can}(A_{ℝ}).  (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

ρ(R^{+}) = ρ(R_{iℝ}^{+}) on the Lie algebra of T_{ℝ} ∩ K_{ℝ}.  (11.3)(c)

Now it follows from (10.5) that ρ(R^{+}) may be regarded as a rational character of T^{can,T} of type z(ρ). Consequently

Λ^{can} ⊗ ρ(R^{+}) ∈ Π^{z}(T (ℝ))  (11.3)(d)

may be regarded as a character of T_{ℝ}^{can,T} of type z. Since the type is in Z(^{∨}G)^{θ_{Z}}, we can now replace T_{ℝ}^{can,T} by T_{ℝ}^{can,G}. After tensoring with the sum of the negative compact imaginary roots and an appropriate character of the vector group A_{ℝ}, we therefore get a character Λ̃ ∈ Π^{z}(T (ℝ)) satisfying

dΛ̃ = λ + ρ(R_{iℝ}^{+}) − 2ρ(R_{iℝ,compact}^{+})  (11.3)(e)

It is not difficult to show that Λ̃ is independent of the choice of R^{+}; the argument is contained in Lemma 11.5 below. The pair (Λ̃, λ) is what is called (Γ, \overline{γ}) in Definition 2.4 of [Under]. Write L_{ℝ} for the centralizer of A_{ℝ} in G_{ℝ}. The standard representation M(Λ) may be constructed by parabolic induction from a limit of discrete series representation on L_{ℝ}^{can} having lowest L_{ℝ} ∩ K_{ℝ}-type of highest weight Λ̃.

As is explained in [AV2], the dependence of M(Λ) on R_{ℝ}^{+} is very mild. We recall the result. Suppose (R_{ℝ}^{+})′ is another set of positive real roots for T in G. Define

𝔫(R_{ℝ}^{+}, (R_{ℝ}^{+})′) = span(X_{α} | α ∈ R_{ℝ}^{+} − (R_{ℝ}^{+})′).  (11.4)(a)

The Cartan subgroup T_{ℝ} acts on 𝔫(R_{ℝ}^{+}, (R_{ℝ}^{+})′), and the determinant of this action is a real-valued character of T_{ℝ}. We may therefore define a character τ(R_{ℝ}^{+}, (R_{ℝ}^{+})′) taking values in {±1} by

τ(R_{ℝ}^{+}, (R_{ℝ}^{+})′)(t) = sgn (det(Ad(t) on 𝔫(R_{ℝ}^{+}, (R_{ℝ}^{+})′)) .  (11.4)(b)

Lemma 11.5 ([AV2], Lemma 8.24). Suppose Λ = (Λ^{can}, R_{iℝ}^{+}, R_{ℝ}^{+}) is a limit character of T_{ℝ} (Definition 11.2) and (R_{ℝ}^{+})′ is another set of positive real roots of T in G. Choose sets of positive roots R^{+} ⊃ R_{iℝ}^{+}, R_{ℝ}^{+} and (R^{+})′ ⊃ R_{iℝ}^{+}, (R_{ℝ}^{+})′ for T in G as in (11.3), and use them to construct characters Λ̃ and Λ̃′.

a) ρ(R^{+}) = ρ((R^{+})′) ⊗ τ(R_{ℝ}^{+}, (R_{ℝ}^{+})′) on (T_{ℝ} ∩ K_{ℝ})^{can,T}.

b) Λ̃ = Λ̃′ ⊗ τ(R_{ℝ}^{+}, (R_{ℝ}^{+})′).

c) Write

Λ′ = (Λ^{can} ⊗ τ(R_{ℝ}^{+}, (R_{ℝ}^{+})′), R_{iℝ}^{+}, (R_{ℝ}^{+})′).

Then there is an isomorphism of standard limit representations M(Λ) ≃ M(Λ′).

Proof. With notation analogous to (11.4), it is easy to check that (as rational characters of T^{can,T})

ρ(R^{+}) = ρ((R^{+})′) ⊗ (det(Ad on 𝔫(R^{+}, (R^{+})′))).

That is,

ρ(R^{+})(t) = ρ((R^{+})′)(t) ∏_{{α∈R^{+} \atop α∉(R^{+})′}} α(t).

Assume now that t ∈ (T_{ℝ} ∩ K_{ℝ})^{can,T}. The roots (like all characters of T_{ℝ}) have absolute value 1 on the compact group T_{ℝ} ∩K_{ℝ}; so the real roots contribute exactly τ(R_{ℝ}^{+}, (R_{ℝ}^{+})′)(t) to the product. Therefore

ρ(R^{+})(t) = ρ((R^{+})′)(t)τ(R_{ℝ}^{+}, (R_{ℝ}^{+})′)(t) ∏_{{α∈R^{+} complex \atop α∉(R^{+})′}} α(t).

The roots in this last product occur in complex conjugate pairs, since the nonimaginary roots in R^{+} and (R^{+})′ were assumed to be stable under complex conjugation. Such a pair contributes |α(t)|^{2} = 1 to the product. This proves (a). Part (b) is immediate from (a) and the definition of Λ̃. 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.

Definition 11.6. In the setting of (11.1) and Definition 11.2, suppose (Λ^{can}, R_{iℝ}^{+}, R_{ℝ}^{+}) is a limit character of T_{ℝ}, and ((Λ^{can})′, (R_{iℝ}^{+})′, (R_{ℝ}^{+})′) is a limit character of T′_{ℝ}. We say that these limit characters are equivalent if there is a g ∈ G_{ℝ} that conjugates T′_{ℝ} to T_{ℝ} and (R_{iℝ}^{+})′ to R_{iℝ}^{+}, and has the following additional property. Write (R_{ℝ}^{+})′′ = Ad(g)(R_{ℝ}^{+})′ (a set of positive real roots for T in G), and

(Λ^{can})′′ = Ad(g)(Λ^{can})′,

a canonical projective representation of T_{ℝ}. Then our final requirement is

(Λ^{can})′′ = Λ^{can} ⊗ τ(R_{ℝ}^{+}, (R_{ℝ}^{+})′′).

Because of Lemma 11.5(c), equivalent limit characters define equivalent standard limit representations.

With this (rather subtle) notion of equivalence in hand, we can formulate an important special case of the Langlands classification.

Theorem 11.7 ([Langlands]) Suppose G_{ℝ} is a real reductive group as in (11.1). Then the infinitesimal equivalence classes of irreducible admissible representations of G_{ℝ} with regular infinitesimal character are in one-to-one correspondence with the equivalence classes of G-regular limit characters, as follows.

a) Suppose Λ is a G-regular limit character of a Cartan subgroup T_{ℝ} (Definition 11.2). Then the corresponding standard representation M(Λ) is non-zero, and has a unique irreducible quotient. In particular, the Langlands quotient representation π(Λ) is irreducible.

b) Suppose π is an irreducible canonical projective representation of G_{ℝ} of type z (Definition 11.2), having regular infinitesimal character. Then there is a Gregular limit character Λ of type z with the property that π is infinitesimally equivalent to π(Λ).

c) Suppose Λ and Λ′ are G-regular limit characters, and that π(Λ) is infinitesimally equivalent to π(Λ′). Then Λ is equivalent to Λ′ (Definition 11.6).

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 Miličić’ 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(Λ) may be zero. The simplest example occurs with G_{ℝ} = SU(2), and T_{ℝ} a compact torus. In that case z(ρ) = 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 Λ^{can} to be the trivial character of T_{ℝ}, and R_{iℝ}^{+} 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 ρ 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_{ℝ} = SL(2, ℝ), and T_{ℝ} ≃ ℝ^{×} the diagonal subgroup. Again all the coverings are trivial. We take Λ^{can} to be the trivial character of T_{ℝ}, and R_{ℝ}^{+}. Because the restriction to T_{ℝ}∩K_{ℝ} = {±1} of the rational character ρ is non-trivial, we find that the character Λ̃ of (11.3) is the signum character of ℝ^{×}. The standard representation is induced from Λ̃, and is therefore equal to the reducible unitary principal series representation of SL(2, 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_{ℝ} = GL(2, ℝ), T_{ℝ} = split Cartan subgroup, and (T_{ℝ})′ = fundamental Cartan subgroup. The standard representation we want is an irreducible unitary principal series whose restriction to SL(2, 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, ℂ), and the element z(ρ) is −I. The complex torus T consists of diagonal matrices in GL(2, ℂ), so

T ≃ { (z_{1}, z_{2}) | z_{i} ∈ ℂ^{×} }  (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(ρ) factors through the double cover

T̃ = { (z_{1}, z_{2}, w) | z_{i}, w ∈ ℂ^{×}, w^{2} = z_{1}z_{2}^{−1} }.  (11.8)(b)

(This is the “square root of ρ double cover” considered in Definition 1.33 of [Orange], or Definition 8.3.11 of [AV2].) Its pullback to T_{ℝ} is

T̃_{ℝ} = { (x_{1}, x_{2}, w) | x_{i} ∈ ℝ^{×}, w ∈ ℂ^{×}, w^{2} = x_{1}x_{2}^{−1} }.  (11.8)(c)

We take Λ^{can} to be the character

Λ^{can}(x_{1}, x_{2}, w) = |w|sgn(x_{1})/w,  (11.8)(d)

which takes values in {±1, ±i}. Let R_{ℝ}^{+} be the positive root corresponding to the upper triangular matrices. Then the rational character ρ on T̃ is ρ(z_{1}, z_{2}, w) = w. The character Λ̃ of (11.3) from which we induce is now easily computed to be

Λ̃(x_{1}, x_{2}) = sgn(x_{1}).  (11.8)(e)

Now it is clear that M(Λ) is an irreducible unitary principal series whose restriction to SL(2, ℝ) is the reducible principal series of the preceding example. We leave to the reader the construction of the corresponding limit character on (T_{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 ℂ/ℝ) that nothing essentially more complicated than these examples can happen. The problem of vanishing is particularly simple.

Proposition 11.9. In the setting of Definition 11.2, suppose Λ = (Λ^{can}, R_{iℝ}^{+}, R_{ℝ}^{+}) is a limit character. Then M(Λ) = 0 if and only if there is a simple root α ∈ R_{iℝ}^{+} such that α is compact, and ⟨α, λ⟩ = 0.

(Notice that the first condition provides a subgroup of G_{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.)

Proof. The standard limit representation may be constructed by induction from a parabolic subgroup P_{ℝ} = L_{ℝ}N_{ℝ}, so it will be zero if and only if the inducing representation is zero. This provides a reduction to the case L_{ℝ} = G_{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_{ℝ} = G_{ℝ}; that is, that R_{iℝ}^{+} is a full set of positive roots of T in G. Fix a dominant regular rational character μ of T, and consider the limit character

(Λ^{can} ⊗ μ, R_{iℝ}^{+}, R_{ℝ}^{+}).

We denote this by the symbol Λ + μ. The corresponding weight is λ + dμ, which is dominant regular by (11.2)(d) and the choice of μ. Then M(Λ + μ) is a (non-zero irreducible) discrete series representation. The standard limit representation M(Λ) is obtained from M(Λ + μ) by applying a Jantzen-Zuckerman translation functor (namely the one denoted ψ_{λ+μ}^{λ} in [Green]). By [Green], Corollary 7.3.23, it follows that M(Λ) = 0 if and only if there is a root α in τ(M(Λ + μ)) (the Borho-Jantzen-Duflo τ-invariant) satisfying ⟨α, λ⟩ = 0. Now the τ-invariant is a subset of the set of simple roots for the system of λ-integral roots. By (11.2)(d), a simple λ-integral singular root α must be simple in all of R^{+}. By hypothesis, α must be noncompact. By [Green], Corollary 8.4.7, α does not lie in the τ-invariant; so M(Λ) ≠ 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 α orthogonal to λ. 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 π′ of SL(2, ℝ) of infinitesimal character zero. In the notation of the second example after Theorem 11.7, π′ is the unique Langlands quotient of — in fact is equal to — the standard limit representation M(Λ′), with (Λ′)^{can} the signum character of T_{ℝ}. Furthermore π′ occurs in no other standard limit representation. To get a good classification theorem, we need to keep the limit character Λ′ but discard Λ. What distinguishes these two limit characters is their behavior on the non-identity component of T_{ℝ}. The appropriate generalization of this is provided by the “parity condition.”

Definition 11.10 ([IC1], Proposition 4.5, or [Green], Definition 8.3.11). Suppose G_{ℝ} is as in (11.1), T_{ℝ} is a Cartan subgroup, and α is a real root of T in G. Choose a root subgroup homomorphism

ϕ_{α} : SL(2) → G  (11.10)(a)

defined over ℝ, so that ϕ_{α}(diagonal matrices) ⊂ T. Since SL(2) is simply connected, this lifts to

ϕ_{α}^{can} : SL(2) → G^{can}.  (11.10)(b)

Since SL(2, ℝ) is connected, this map restricts to

ϕ_{α}^{can} : SL(2, ℝ) → image of G_{ℝ}^{can} ⊂ G(ℝ)^{can}.  (11.10)(c)

The mapping (11.1)(a) therefore induces an extension of SL(2, ℝ). Passing to its identity component, we get

ϕ̃_{α}^{can} : S̃L(2, ℝ) → G_{ℝ}^{can},  (11.10)(d)

with

S̃L(2, ℝ) → SL(2, ℝ)  (11.10)(e)

a connected cover. Define

m̃ = exp ( \begin{matrix}0 & π \\ −π & 0 \end{matrix} ) ∈ S̃L(2, ℝ),  m̃_{α} = ϕ̃_{α}^{can}(m̃) ∈ G_{ℝ}^{can}.  (11.10)(f)

(If G_{ℝ} is linear, then S̃L(2, ℝ) = SL(2, ℝ), and m̃ = −I. In general m̃ is a generator for Z(S̃L(2, ℝ)), and it may have any (finite) order.) The homomorphism ϕ_{α} is not quite uniquely determined by the conditions we have imposed on it. It is easy to check that changing the choice replaces m̃_{α} by m̃_{α}^{±1}.

Suppose Λ = (Λ^{can}, R_{iℝ}^{+}, R_{ℝ}^{+}) is a limit character of T_{ℝ}. Choose any set R^{+} of positive roots for T in G. As in (11.3), regard Λ^{can} ⊗ ρ as a representation of T_{ℝ}^{can,G}. It may then be applied to the element m̃_{α} defined above. We say that α satisfies the parity condition for Λ if the eigenvalues of the operator (Λ^{can} ⊗ρ)(m̃_{α}) are contained in the set

− exp (±iπ⟨α^{∨}, λ + ρ − ρ(R_{ℝ}^{+})⟩) .  (11.10)(g)

(Because of the ± in the exponent, this condition is unchanged if we replace m̃_{α} by m̃_{α}^{−1}.) Here it is important to understand that ρ(R_{ℝ}^{+}) refers to the set of positive real roots fixed in Λ, but that ρ refers to the arbitrary set R^{+} of positive roots. We do not assume that R_{ℝ}^{+} is contained in R^{+}. If G_{ℝ} is linear (so that m̃_{α} is a central element of order 2 in T_{ℝ}^{can,G}) this condition amounts to

⟨α^{∨}, λ + ρ − ρ(R_{ℝ}^{+})⟩ = n ∈ ℤ, and (Λ^{can} ⊗ ρ)(m̃_{α}) = (−1)^{n+1}.  (11.10)(h)

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 α^{∨} ∈ X_{∗}(T) for the coroot corresponding to α. By a calculation in SL(2), the image of m̃_{α} in G is

m_{α} = α^{∨}(−1) ∈ T.  (11.11)(a)

It follows that if γ is any rational character of T, then

γ(m_{α}) = (−1)^{⟨α^{∨},γ⟩}.  (11.11)(b)

In particular, if R^{+} and (R^{+})′ are two systems of positive roots,

(ρ(R^{+}) ⊗ ρ((R^{+})′)^{−1}) (m_{α}) = (−1)^{⟨α^{∨},ρ(R^{+})−ρ((R^{+})′)⟩}.  (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.

Proposition 11.12 (The Hecht-Schmid character identity — see [Schmid] and [IC1], Proposition 4.5.) In the setting of Definition 11.2, suppose there is a real root α such that α satisfies the parity condition for Λ (Definition 11.10), and ⟨α, λ⟩ = 0. Then the standard limit representation M(Λ) is isomorphic to a direct sum of one or two standard limit representations attached to limit characters on a Cartan subgroup (T_{ℝ})′ having a strictly smaller split part.

Definition 11.13 ([Under], Definition 2.6.) In the setting of Definition 11.2, the limit character Λ = (Λ^{can}, R_{iℝ}^{+}, R_{ℝ}^{+}) is called final if it satisfies the following two conditions.

(a) Suppose α is a simple root in R_{iℝ}^{+}, and ⟨α, λ⟩ = 0. Then α is noncompact.

(b) Suppose α is a real root, and ⟨α, λ⟩ = 0. Then α does not satisfy the parity condition for Λ (Definition 11.10).

We write

L^{z}(G_{ℝ}) = { equivalence classes of final limit characters of type z }.

The standard limit representations attached to final limit characters are called final standard limit representations.

Theorem 11.14 ([KZ]; see [Under], Theorem 2.6). Suppose G_{ℝ} is a real reductive group as in (11.1). Then the set Π^{z}(G_{ℝ}) of infinitesimal equivalence classes of irreducible admissible representations of G_{ℝ} is in one-to-one correspondence with the set L^{z}(G_{ℝ}) of equivalence classes of final limit characters, as follows.

a) Suppose Λ is a final limit character of a Cartan subgroup T_{ℝ} (Definition 11.13). Then the corresponding standard representation M(Λ) is non-zero, and has a unique irreducible quotient. In particular, the Langlands quotient representation π(Λ) is irreducible.

b) Suppose π is an irreducible canonical projective representation of G_{ℝ} of type z (Definition 11.2). Then there is a final limit character Λ of type z with the property that π is infinitesimally equivalent to π(Λ).

c) Suppose Λ and Λ′ are final limit characters, and that π(Λ) is infinitesimally equivalent to π(Λ′). Then the limit characters are equivalent (Definition 11.6). Any standard limit representation of G_{ℝ} (Definition 11.2) is isomorphic to a

direct sum of final standard limit representations.

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

λ_{a} ∈ 𝔱_{a}^{∗}.  (11.15)(a)

We study representations of infinitesimal character λ_{a}. Fix also a system of positive roots R_{a}^{+} for T_{a} in G, chosen so that

⟨α, λ_{a}⟩ > 0 ⇒ α ∈ R_{a}^{+}.  (11.15)(b)

Fix a regular dominant rational character μ_{a} ∈ X^{∗}(T_{a}). By (11.15)(b),

λ′_{a} = λ_{a} + μ_{a}  (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

ψ = ψ_{λ_{a}+μ_{a}}^{λ_{a}}  (11.15)(d)

from modules with the (regular) infinitesimal character λ_{a} + μ_{a} to modules with infinitesimal character λ_{a}. (The functor is defined by tensor product with the finite-dimensional representation of lowest weight −μ_{a}, followed by projection on the infinitesimal character λ_{a}.) Finally, define

Δ_{a}^{0} = simple roots for R_{a}^{+} orthogonal to λ_{a}.  (11.15)(e)

Any irreducible representation π′ of infinitesimal character λ′_{a} has a Borho-Jantzen-Duflo τ-invariant, τ(π′) which is a subset of the simple roots for the positive λ′_{a}- integral roots (cf. [Green], Definition 7.3.8.) This set of simple roots includes Δ_{a}^{0}, so we can define

τ^{0}(π′) = τ(π′) ∩ Δ_{a}^{0}.  (11.15)(f)

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

Proposition 11.16 ([Green], Corollary 7.3.23; cf. also [Dix], Proposition 7.7). Suppose G_{ℝ} is a real reductive group as in (11.1), and suppose we are in the setting (11.15). Then the translation functor ψ of (11.15)(d) is an exact functor from canonical projective representations of G_{ℝ} of type z and infinitesimal character λ′_{a} to canonical projective representations of type z and infinitesimal character λ_{a}. It has the following additional properties.

a) Suppose π′ is an irreducible representation of infinitesimal character λ′_{a}. Then ψ(π′) is irreducible or zero. The first possibility occurs if and only if τ^{0}(π′) (cf. (11.15)(f)) is empty.

b) Suppose π is an irreducible representation of infinitesimal character λ_{a}. Then there is a unique irreducible representation π′ with ψ(π′) ≃ π.

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 τ^{0}(π′) in terms of the classification of Theorem 11.7. Suppose then that Λ′ = ((Λ′)^{can}, R_{iℝ}^{+}, R_{ℝ}^{+}) is a G_{ℝ}- limit character of T_{ℝ} of infinitesimal character λ′_{a}. Write λ′ for the differential of Λ′. Since λ′ and λ′_{a} are regular and define the same infinitesimal character, there is a unique isomorphism

i(λ′_{a}, λ′) : T_{a} → T  (11.17)(a)

induced by an element of G and carrying λ′_{a} to λ′. We also write i(λ′_{a}, λ′) for the induced isomorphism from the root system of T_{a} in G to that of T, and so on. Put

R^{+} = i(λ′_{a}, λ′)(R_{a}^{+}),  (11.17)(b)

a system of positive roots for T in G. Because of (11.15)(b) and (11.2)(d), we have

R^{+} ⊃ R_{iℝ}^{+}.  (11.17)(c)

Similarly, define

μ = i(λ′_{a}, λ′)(μ_{a}) ∈ X^{∗}(T).  (11.17)(d)

Then we can define

Λ′ − μ = ((Λ′)^{can} ⊗ μ^{−1}, R_{iℝ}^{+}, R_{ℝ}^{+}).  (11.17)(e)

Because of (11.15)(b) and (11.17)(c), this is a G_{ℝ}-limit character of T_{ℝ} of infinitesimal character λ_{a}.

Proposition 11.18 Suppose we are in the setting of (11.15), and Λ′ is a G_{ℝ}- limit character of infinitesimal character λ′_{a}. Use the notation of (11.17).

a) The translation functor ψ satisfies

ψ(M(Λ′)) ≃ M(Λ′ − μ).

b) Write Δ^{0} for the image of Δ_{a}^{0} under the bijection i(λ′_{a}, λ′). Then the τ invariant τ^{0}(π(Λ′)) corresponds to the subset of roots in Δ^{0} satisfying one of the following three conditions:

i) α is compact imaginary; ii) α is complex, and its complex conjugate \overline{α} is positive; or iii) α is real and satisfies the parity condition (Definition 11.10).

c) Suppose τ^{0}(π(Λ′)) is empty. Then ψ(π(Λ′)) is the unique irreducible quotient of

M(Λ′ − μ). That is,

ψ(π(Λ′)) = π(Λ′ − μ).

Proof. Part (a) is proved for linear groups in [Green], Proposition 7.4.1; the argument is unchanged in general. For (b), the τ-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_{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 π_{1} is an irreducible quotient of M(Λ). By Proposition 11.16, there is an irreducible representation π′_{1} of infinitesimal character λ′_{a} with ψ(π′_{1}) ≃ π_{1}. The translation functor ψ has a two-sided adjoint

ϕ = ψ_{λ_{a}}^{λ_{a}+μ_{a}}  (11.19)(a)

([Green], Proposition 4.5.8). Consequently

Hom(M(Λ′ − μ), π_{1}) ≃ Hom(ψ(M(Λ′)), ψ(π′_{1})) ≃ Hom(M(Λ′), ϕψ(π′_{1})).
(11.19)(b)

(The Hom spaces are infinitesimal homomorphisms; that is, (𝔤, K_{ℝ})-module maps for an appropriate maximal compact K_{ℝ}.) Now it is a formal consequence of the adjointness of ϕ and ψ that any irreducible subrepresentation π′_{2} of ϕψ(π′_{1}) must be a preimage of π_{1} under ψ. By Proposition 11.16(b),

Hom(π′_{2}, ϕψ(π′_{1})) = \{ \begin{matrix}ℂ, & ifπ′_{2} ≃ π′_{1} \\ 0 & otherwise. \end{matrix}  (11.19)(c)

On the other hand, (11.19)(b) implies that the unique irreducible quotient π′ of M(Λ′) must be a composition factor of ϕψ(π′_{1}). But the only composition factor of ϕψ(π′_{1}) having τ^{0} empty is π′_{1} itself ([Green], Proposition 7.3.2(b) and Corollary 7.3.21). Consequently π′_{1} ≃ π′, and π_{1} ≃ ψ(π′). The last group in (11.19)(b) is therefore

Hom(M(Λ′), ϕψ(π′)).  (11.19)(d)

The image P′ of a non-zero map in this space must have π′ as its unique irreducible quotient (since P′ is a quotient of M(Λ′)) and as its unique irreducible subrepresentation (by (11.19)(c)). But π′ occurs only once as a composition factor of M(Λ′), so it follows that P′ = π′. Consequently

Hom(M(Λ′), ϕψ(π′)) ≃ Hom(π′, ϕψ(π′)) ≃ ℂ.  (11.19)(e)

In conjunction with (11.19)(b), this shows that ψ(π′) occurs exactly once as a quotient of M(Λ), as we wished to show. Q.E.D.

Propositions 11.16 and 11.18 explain rather explicitly how to describe representations of infinitesimal character λ_{a} in terms of representations of (regular) infinitesimal character λ′_{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.

Proposition 11.20. In the setting (11.15), there is a bijection from the set

{equivalence classes of limit characters Λ′ of infinitesimal character λ′_{a} with τ^{0}(π(Λ′)) empty }

onto the set

{equivalence classes of final limit characters Λ of infinitesimal character λ_{a}}.

In the notation of (11.17), this bijection sends the class of Λ′ to the class of Λ′ − μ.

Proof. Suppose first that τ^{0}(π(Λ′)) is empty; we want to show that Λ = Λ′ − μ is final (Definition 11.13). First, Proposition 11.18 guarantees that M(Λ) ≠ 0. By Proposition 11.9, it follows that Λ satisfies condition (11.13)(a). To verify condition (11.13)(b), we need to understand a little about the root system

R^{0} = { α ∈ R(G, T) | ⟨α, λ⟩ = 0 }.  (11.21)(a)

First, it has (R^{0})^{+} = R^{+} ∩ R^{0} (notation (11.17)(c)) as a set of positive roots, with Δ^{0} the corresponding simple roots. Define

Δ_{ℝ}^{0} = Δ^{0} ∩ R_{ℝ},  R_{ℝ}^{0} = R^{0} ∩ R_{ℝ}.  (11.21)(b)

What we need to show is that

Δ_{ℝ}^{0} is a set of simple roots for R_{ℝ}^{0}.  (11.21)(c)

Assume for a moment that we have established this. From Definition 11.10 one can check that

if α and β in R_{ℝ}^{0} do not satisfy the parity condition, then neither does s_{α}(β).

(11.21)(d)

(For linear groups one can say even more: because of (11.11)(a), the set of coroots for roots in R_{ℝ}^{0} 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 Λ′, the roots in Δ_{ℝ}^{0} do not satisfy the parity condition. By (11.21)(c) and (d), it follows that no root in R_{ℝ}^{0} satisfies the parity condition. This is (11.13)(b).

To prove (11.21)(c), observe first that

if α ∈ (R^{0})^{+}, then ⟨−\overline{α}, λ⟩ ≥ 0.  (11.22)(a)

For it is enough to prove this for α ∈ Δ^{0}. If α is real or imaginary, the inner product is zero. If α is complex, then the hypothesis on Λ′ and Proposition 11.18(b)(ii) guarantee that −\overline{α} ∈ R^{+}; so the inequality follows from (11.15)(b) (and (11.17)). Now define

R^{00} = \{ α ∈ R^{0} | −\overline{α} ∈ R^{0} \}
= \{ α ∈ R | ⟨α, λ⟩ = ⟨−\overline{α}, λ⟩ = 0 \}.  (11.22)(b)

This is a root system; we define

(R^{00})^{+} = R^{00} ∩ R^{+},  Δ^{00} = Δ^{0} ∩ R^{00}.  (11.22)(c)

Because of (11.22)(a),

Δ^{00} is a set of simple roots for (R^{00})^{+}.  (11.22)(d)

Now α ↦ −\overline{α} is an involutive automorphism of R^{00}, so we can apply to it Lemma 8.6.1 of [Green]. Suppose α is a complex simple root for (R^{00})^{+}. By (11.22)(d), α ∈ Δ^{0}; so by Proposition 11.18(b)(ii) and the hypothesis on Λ′, −\overline{α} ∈ (R^{00})^{+}. Lemma 8.6.1 of [Green] therefore implies that

the set of non-real roots in (R^{00})^{+} is stable under α ↦ −\overline{α}.  (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

Λ = (Λ^{can}, R_{iℝ}^{+}, R_{ℝ}^{+}) is a final limit character of T_{ℝ} of infinitesimal character λ_{a}.

(11.23)(a)

Write R for the set of roots of T in G. Define

R^{0} = { α ∈ R | ⟨α, λ⟩ = 0 },  (11.23)(b)
R^{00} = \{ α ∈ R^{0} | −\overline{α} ∈ R^{0} \}.  (11.23)(c)

Choose a set of positive roots (R^{00})^{+} for R^{00} with the property that

the set of non-real roots in (R^{00})^{+} is stable under α ↦ −\overline{α}.  (11.23)(d)

(This is certainly possible.) Possibly after modifying this system by reflections in imaginary roots, we may assume that

R_{iℝ}^{+} ∩ R^{00} ⊂ (R^{00})^{+}.  (11.23)(e)

Now define a positive root system for R^{0} by

(R^{0})^{+} = (R^{00})^{+} ∪ \{ α ∈ R^{0} | ⟨−\overline{α}, λ⟩ > 0 \}.  (11.23)(f)

Write

Δ^{0} = simple roots for (R^{0})^{+}.  (11.23)(g)

Since λ and λ_{a} define the same infinitesimal character, there is a unique isomorphism

i(λ_{a}, λ; Δ_{a}^{0}, Δ^{0}) : T_{a} → T  (11.24)(a)

induced by an element of G, carrying λ_{a} to λ and Δ_{a}^{0} to Δ^{0}. Just as in (11.17), we now define

R^{+} = i(λ_{a}, λ; Δ_{a}^{0}, Δ^{0})(R_{a}^{+}),  (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^{+} ⊃ R_{iℝ}^{+}.  (11.24)(c)

Put

μ = i(λ_{a}, λ; Δ_{a}^{0}, Δ^{0})(μ_{a}) ∈ X^{∗}(T).  (11.24)(d)

Then we can define

Λ′ = (Λ^{can} ⊗ μ, R_{iℝ}^{+}, R_{ℝ}^{+}).  (11.24)(e)

Because of (11.23)(f) and (11.24)(c), this is a G_{ℝ}-limit character of T_{ℝ} of infinitesimal character λ′_{a}. We are again in the situation (11.17), and Λ = Λ′ − μ. We need to check that τ^{0}(π(Λ′)) is empty. So fix a root α ∈ Δ^{0}. If α is imaginary, then (11.24)(c) implies that it is simple in R_{iℝ}^{+}; so since Λ is final, α is noncompact. If α is complex, then (11.23)(d) and (f) imply that −\overline{α} is positive. If α is real, then since Λ is final, α does not satisfy the parity condition. By Proposition 11.18(b), α is not in the τ-invariant. This proves the surjectivity of the map in Proposition 11.20.

For the injectivity, suppose Λ′_{1} and Λ′_{2} both map to the equivalence class of Λ. By Proposition 11.18(c),

ψ(π(Λ′_{1})) ≃ π(Λ) ≃ ψ(π(Λ′_{2})).

By Proposition 11.16(b), it follows that π(Λ′_{1}) ≃ π(Λ′_{1}). By Theorem 11.7(c), Λ′_{1} is equivalent to Λ′_{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 λ′_{a} in the setting (11.15)). Q.E.D.

12. Langlands parameters and Cartan subgroups.

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.

Definition 12.1. Suppose G^{Γ} is a weak extended group for G, ^{∨}G is a dual group for G, and z ∈ Z(^{∨}G)^{θ_{Z}} (cf. Definition 10.3). A Cartan subgroup of G^{Γ} is a weak extended group T^{Γ} ⊂ G^{Γ} with identity component a maximal torus T ⊂ G. The conjugation action of any element δ ∈ T^{Γ} − T on T is a real form of T, independent of the choice of δ; we write T (ℝ) for the group of real points.

A G-limit character of T^{Γ} of type z is a pair (δ, Λ) subject to the following conditions. First,

δ ∈ T^{Γ} − T is a strong real form of G  (12.1)(a)

(Definition 2.13). Thus T (ℝ) is a Cartan subgroup of G(ℝ, δ). We can therefore impose the second requirement

Λ is a G(ℝ, δ)-limit character of T (ℝ) of type z  (12.1)(b)

(Definition 11.2). We say that (δ, Λ) is final if Λ is (Definition 11.13). We say that (δ, Λ) is equivalent to the G-limit character (δ′, Λ′) of (T′)^{Γ} if there is an element g ∈ G that conjugates δ′ to δ, so that g · Λ′ and Λ are equivalent as limit characters of G(ℝ, δ) (Definition 11.6). Write

L^{z}(G/ℝ) = { equivalence classes of final limit characters (δ, Λ) of type z }. (12.1)(c)

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^{Γ} of type z is a quadruple

(δ, Λ^{can}, R_{iℝ}^{+}, R_{ℝ}^{+}),  (12.2)(a)

subject to the conditions below. First,

δ ∈ T^{Γ} − T, and δ^{2} ∈ Z(G).  (12.2)(b)

Write T^{can,T} for the corresponding canonical cover (Definition 10.1), and T^{can,T} (ℝ) for the preimage of T (ℝ) in this cover (Definition 10.3):

1 → π_{1}(T)^{can} → T (ℝ)^{can,T} → T (ℝ) → 1.  (12.2)(c)

Then zz(ρ) ∈ ^{∨}T^{θ_{Z}} (Definition 11.2) defines a character χ_{zz(ρ)}^{T} of π_{1}(T)^{can}. The second condition is

Λ^{can} is a character of T (ℝ)^{can,T}, and Λ^{can}|π_{1}(T)^{can} = χ_{zz(ρ)}^{T}.  (12.2)(d)

(Notice that this condition does not involve δ.) Write R_{iℝ} for the set of roots assuming imaginary values on 𝔱(ℝ), and R_{ℝ} for those assuming real values. Then

R_{iℝ}^{+} is a set of positive roots for R_{iℝ}, and R_{ℝ}^{+} for R_{ℝ}.  (12.2)(e)

Finally, write λ = dΛ^{can} ∈ 𝔱^{∗}. Then we require

⟨α, λ⟩ ≥ 0,  (α ∈ R_{iℝ}^{+}).  (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.

Theorem 12.3. Suppose G^{Γ} is a weak extended group for G (Definition 2.13). Then there is a natural bijection between the set Π^{z}(G/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/ℝ) of equivalence classes of final limit characters of type z of Cartan subgroups in G^{Γ} (Definition 12.1).

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.

Definition 12.4. Suppose ^{∨}G^{Γ} is a weak E-group (Definition 4.3). A Cartan subgroup of ^{∨}G^{Γ} is a weak E-group^{d}T^{Γ} ⊂ ^{∨}G^{Γ} such that the identity component^{d}T is a maximal torus in ^{∨}G, and the inclusion of ^{d}T^{Γ} in ^{∨}G^{Γ} is an L-homomorphism (Definition 5.1). Conjugation by any element y of ^{d}T^{Γ} −^{d}T defines an involutive automorphism θ of ^{d}T, which is independent of y. Write

A(^{d}T^{Γ}) = ^{d}T^{θ}/(^{d}T^{θ})_{0}  (12.4)(a)

for the group of connected components of the fixed points of θ. We call this group the Langlands component group for ^{d}T^{Γ}. As in (5.10) we can form such groups as

dT^{alg,^{∨}G} (which is a quotient of the algebraic universal covering ^{d}T^{alg,^{d}T}). Define

A(^{d}T^{Γ})^{alg,^{∨}G} = (^{d}T^{θ})^{alg,^{∨}G}/((^{d}T^{θ})^{alg,^{∨}G})_{0},  (12.4)(b)

the universal component group for ^{d}T^{Γ} with respect to ^{∨}G^{Γ}.

Since ^{d}T^{Γ} is a weak E-group in its own right, we can speak of Langlands parameters for ^{d}T^{Γ} (Definition 5.2). A Langlands parameter ϕ is said to be ^{∨}G-regular if for every root α of ^{d}T in ^{∨}G, we have

α(λ(ϕ)) ≠ 0  (12.4)(c)

(notation (5.8)). In general the roots that fail to satisfy (12.4)(c) are called λ(ϕ)- singular or ϕ-singular.

A complete Langlands parameter for ^{d}T^{Γ} with respect to ^{∨}G^{Γ} is a pair (ϕ, τ_{1}) with ϕ a Langlands parameter for ^{d}T^{Γ}, and τ_{1} an irreducible representation of A(^{d}T^{Γ})^{alg,^{∨}G} (cf. Definition 5.12). Such a parameter is said to be equivalent to a complete parameter (ϕ′, τ′_{1}) for (^{d}T^{Γ})′ if the triple (^{d}T^{Γ}, ϕ, τ_{1}) is conjugate by ^{∨}G to ((^{d}T^{Γ})′, ϕ′, τ′_{1}).

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 θ extends (although not uniquely) to an automorphism of ^{∨}G, it must permute the roots and coroots of dT in ^{∨}G. We write

dR = R(^{∨}G,^{d}T)  (12.5)(a)
dR_{ℝ} = { α ∈ ^{d}R | θα = −α }  (12.5)(b)
dR_{iℝ} = { α ∈ ^{d}R | θα = α },  (12.5)(c)

the real and imaginary roots of ^{d}T^{Γ}. Suppose now that α is a real root. The corresponding coroot α^{∨} is a homomorphism from ℂ^{×} to^{d}T (or even into^{d}T^{alg,^{∨}G}). Since α is real, it satisfies

θ(α^{∨}(z)) = α^{∨}(z^{−1}).  (12.6)(a)

In particular, we get a distinguished element of order two

m_{α} = α^{∨}(−1) ∈ ^{d}T^{θ}.  (12.6)(b)

Similarly, we write

m_{α}^{alg} = α^{∨}(−1) ∈ (^{d}T^{θ})^{alg,^{∨}G}.  (12.6)(c)

These elements define elements of order two in the component groups of Definition 12.4, which we write as

\overline{m_{α}^{alg}} ∈ A(^{d}T^{Γ})^{alg,^{∨}G}  (α ∈ ^{d}R_{ℝ}).  (12.6)(d)

Next, suppose α is an imaginary root, and y ∈ ^{d}T^{Γ} −^{d}T. Choose root vectors X_{±α} ∈ ^{∨}𝔤. Since θ fixes α, Ad(y) must send each root vector to a multiple of itself:

Ad(y)X_{±α} = z_{±α}X_{α}.  (12.7)(a)

The bracket of these two root vectors is a multiple of the derivative of the coroot α^{∨}, and is therefore fixed by θ = Ad(y). It follows that

z_{α} = (z_{−α})^{−1}.  (12.7)(b)

Similarly, if α, β and α + β are all imaginary roots,

z_{α+β} = z_{α}z_{β}.  (12.7)(c)

Now y^{2} belongs to ^{d}T, so α(y^{2}) is defined. Clearly

(z_{α})^{2} = α(y^{2}).  (12.7)(d)

We say that α is y-compact if z_{α} = 1, and y-noncompact if z_{α} = −1. As a consequence of (12.7)(d), every imaginary root in the centralizer of y^{2} is either compact or noncompact.

Definition 12.8. Suppose ϕ is a Langlands parameter for ^{d}T^{Γ}. As in (5.8), we associate to ϕ elements y = y(ϕ) ∈ ^{d}T^{Γ} −^{d}T and λ = λ(ϕ) ∈ ^{d}𝔱. An imaginary root α is called ϕ-compact (respectively ϕ-noncompact) if it is y(ϕ)-compact (respectively y(ϕ)-noncompact) in the sense of (12.7). By Proposition 5.6(b) and (12.7)(d), every ϕ-singular imaginary root is either ϕ-compact or ϕ-noncompact.

Suppose next that (ϕ, τ_{1}) is a complete Langlands parameter for^{d}T^{Γ} with respect to ^{∨}G^{Γ} (Definition 12.4). We say that the pair (ϕ, τ_{1}) is final if it satisfies the following two conditions.

(a) Every ϕ-singular imaginary root of ^{d}T in ^{∨}G is ϕ-compact.

(b) If α is a ϕ-singular real root of ^{d}T^{Γ} in ^{∨}G^{Γ}, then

τ_{1}(\overline{m_{α}^{alg}}) = 1

(notation 12.6)(c).)

Theorem 12.9. Suppose ^{∨}G^{Γ} is a weak E-group (Definition 4.3). Then there is a one-to-one correspondence between the set Ξ(^{∨}G^{Γ}) of equivalence classes of complete Langlands parameters for ^{∨}G^{Γ} (Definition 5.11) and the set of equivalence classes of final complete Langlands parameters for Cartan subgroups of ^{∨}G^{Γ} (Definition 12.8).

Proof. Suppose (ϕ, τ) is a complete Langlands parameter for ^{∨}G^{Γ}. We want to construct a Cartan subgroup ^{d}T^{Γ} containing the image of ϕ. To do this, use the notation of Corollary 5.9. There we defined a Levi subgroup L(λ, y · λ) of ^{∨}G, on which conjugation by y acts as an involutive automorphism θ_{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.)

Lemma 12.10 Suppose G is a connected complex reductive algebraic group, and θ 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 θ, with the property that the automorphisms of G in Ad(K) all commute with θ. Then there is a θ-stable maximal torus T ⊂ G, determined up to conjugation by K_{0} by either of the following properties. Write 𝔞 for the −1 eigenspace of θ on 𝔱, and M for the centralizer of 𝔞 in G.

a) The Lie algebra 𝔞 is a maximal semisimple abelian subalgebra in the −1 eigenspace of θ on 𝔤.

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). In addition, T has the following properties.

c) T ∩ K meets every connected component of K. That is, the natural map (T ∩ K)/(T ∩ K)_{0} → K/K_{0} is surjective.

d) The kernel T ∩ K_{0} of the map in (c) is generated by (T ∩ K)_{0} and the elements m_{α} ∈ T attached to real roots α (cf. (12.6)). In particular, the characters of the component group K/K_{0} are in one-to-one

correspondence with the characters of (T ∩ K)/(T ∩ K)_{0} trivial on all the elements m_{α} (for α real).

A torus with the properties in the lemma will be called a maximally θ-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 θ_{y}-split torus

dT ⊂ L(λ, y · λ).  (12.11)(a)

Since L(λ, y · λ) is a Levi subgroup of ^{∨}G, it follows that ^{d}T is a maximal torus in ∨G. Furthermore the Lie algebra ^{d}𝔱 must contain the central elements λ and y · λ of 𝔩(λ, y · λ), so

ϕ|_{ℂ^{×}} : ℂ^{×} → ^{d}T .

Define

dT^{Γ} = group generated by y and ^{d}T,  (12.11)(b)

so that

ϕ : W_{ℝ} → ^{d}T^{Γ}.  (12.11)(c)

Since θ_{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^{Γ}. It follows easily that dT is a weak E-group, and so a Cartan subgroup of ^{∨}G^{Γ} (Definition 12.4).

A real or imaginary root is λ-singular if and only if it is y · λ-singular. It follows that the real or imaginary ϕ-singular roots are exactly the real or imaginary roots of ^{d}T in L(λ, y · λ). By Lemma 12.10(b), this means that

every ϕ-singular imaginary root of ^{d}T in ^{∨}G is ϕ-compact.  (12.11)(d)

As in Corollary 5.9, write K(y) for the centralizer of y in ^{∨}G. The component group of which the datum τ is a character is

A_{ϕ}^{loc,alg} = [K(y) ∩ L(λ, y · λ)]^{alg}/ ([K(y) ∩ L(λ, y · λ)]^{alg})_{0}

(Corollary 5.9(c)). Lemma 12.10(c) (applied to the group L(λ, y ·λ)^{alg,^{∨}G}) provides a surjective map

A(^{d}T^{Γ})^{alg,^{∨}G} → A_{ϕ}^{loc,alg}.  (12.11)(e)

We use this map to pull τ back to a character τ_{1} of A(^{d}T^{Γ})^{alg,^{∨}G}. By Lemma 12.10(d),

τ_{1}(m_{α}^{alg}) = 1  (α real and ϕ-singular).  (12.11)(f)

By (12.11)(d) and (12.11)(f), the pair (ϕ, τ_{1}) is a final complete Langlands parameter for ^{d}T^{Γ} with respect to ^{∨}G^{Γ}. 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 ϕ in ^{∨}G. It follows easily that the map from parameters for ∨G^{Γ} 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 ^{∨}G^{Γ} from a complete final parameter for a Cartan subgroup. It follows that the correspondence is bijective. Q.E.D.

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 = centralizer of (T ∩ 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 ∩ K)_{0} is also a torus in K_{0}, it follows that L ∩ K_{0} is also connected. It follows that

T ∩ K_{0} = T ∩ (L ∩ K_{0}) = T ∩ (L ∩ 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 θ-stable direct product decomposition. After such reductions, we may assume that G is simple and simply connected, and that θ acts by −1 on 𝔱. Then T^{θ} 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_{α} generate the full group of elements of order 2 in T. Consequently

T ∩ K_{0} ⊂ T^{θ} = group generated by the m_{α}.

The other containment (that is, that m_{α} ∈ K_{0}) follows from a standard calculation in SL(2). Q.E.D.

13. Pairings between Cartan subgroups and the proof of Theorem 10.4.

Theorem 12.3 describes representations in terms of Cartan subgroups of G^{Γ}, and Theorem 12.9 describes Langlands parameters in terms of Cartan subgroups of ∨G^{Γ}. To complete the proof of Theorem 10.4, we need to relate Cartan subgroups of G^{Γ} and ^{∨}G^{Γ}. 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.

Lemma 13.1. Suppose G is a complex connected reductive algebraic group, and ∨G is a dual group for G (Definition 4.2). Fix maximal tori T ⊂ G and ^{d}T ⊂ ^{∨}G. Write ^{∨}T for a dual torus to T (cf. (9.3)). Suppose Δ ⊂ X^{∗}(T) is a set of simple roots for T in G, and ^{d}Δ ⊂ X^{∗}(^{d}T) a set of simple roots for ^{d}T in ^{∨}G. Write Δ^{∨} ⊂ X_{∗}(T) and ^{d}Δ^{∨} ⊂ X_{∗}(^{d}T) for the corresponding sets of simple coroots.

a) There is a natural isomorphism

ζ(Δ,^{d}Δ) : ^{∨}T → ^{d}T .

b) The induced map on one-parameter subgroup lattices

ζ_{∗}(Δ,^{d}Δ) : (X_{∗}(^{∨}T) = X^{∗}(T)) → X_{∗}(^{d}T)

carries Δ onto ^{d}Δ^{∨}.

c) The induced map (in the other direction) on character lattices

ζ^{∗}(Δ,^{d}Δ) : X^{∗}(^{d}T) → (X^{∗}(^{∨}T) = X_{∗}(T))

carries ^{d}Δ onto Δ^{∨}.

d) If sets of simple roots are not fixed, we obtain a finite family of natural isomor-

phisms

ζ : ^{∨}T → ^{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 ^{∨}G. The characteristic property of the isomorphism ζ(Δ,^{d}Δ) is that the induced maps

described in (b) and (c) implement the natural isomorphism from the dual

∨Ψ_{0}(G) = (X_{∗}(T), Δ^{∨}, X^{∗}(T), Δ)

of the based root datum for G, onto the based root datum

Ψ_{0}(^{∨}G) = (X^{∗}(^{d}T),^{d}Δ, X_{∗}(^{d}T),^{d}Δ^{∨}).

With the rôles of G and ^{∨}G reversed, (a) provides also an isomorphism

ζ(^{d}Δ, Δ) : ^{∨}^{d}T → T.

It coincides with the one obtained from ζ(Δ,^{d}Δ) 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.

Definition 13.2 (cf. [AV2], Definition 9.11). Suppose G^{Γ} is a weak extended group, and ^{∨}G^{Γ} is a weak E-group for G^{Γ} (Definition 4.3). Fix Cartan subgroups T^{Γ} ⊂ G^{Γ} and ^{d}T^{Γ} ⊂ ^{∨}G^{Γ} (Definitions 12.1 and 12.4). Write σ for the real form of T defined by any element of T^{Γ} − T, and θ for the involutive automorphism of ^{d}T defined by any element of ^{d}T^{Γ} −^{d}T. By Proposition 2.12, σ defines an involutive automorphism

a_{T} ∈ Aut(Ψ_{0}(T)) = 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 ℤ. Similarly (Proposition 2.11 or Proposition 4.4) the involution θ defines an involutive automorphism

a_{^{d}T} ∈ Aut(^{∨}Ψ_{0}(^{d}T)) = Aut(X_{∗}(^{d}T), X^{∗}(^{d}T)).

A weak pairing between T^{Γ} and ^{d}T^{Γ} is an identification of ^{d}T^{Γ} with a weak Egroup for T^{Γ}, subject to one additional condition that we now describe. According to Definition 4.3, the identification in question amounts to an isomorphism

ζ : ^{∨}T → ^{d}T  (13.2)(a)

of the dual torus ^{∨}T for T (cf. (9.3)) with ^{d}T. Such an identification is just an isomorphism

ζ : (X^{∗}(T), X_{∗}(T)) → (X_{∗}(^{d}T), X^{∗}(^{d}T))  (13.2)(b)

In order to make ^{d}T^{Γ} an E-group for T^{Γ}, this isomorphism must satisfy

ζ carries a_{T} to a_{^{d}T}.  (13.2)(c)

The additional requirement that we impose for a weak pairing is

ζ is one of the natural isomorphisms of Lemma 13.1(d).  (13.2)(d)

This condition brings the groups G^{Γ} and ^{∨}G^{Γ} into the definition.

Lemma 13.3. Suppose we are in the setting of Definition 13.2; use the notation there.

a) The isomorphism ζ induces an identification of Weyl groups

ζ : W (G, T) → W (^{∨}G,^{d}T).

b) The isomorphism of (a) carries the action of the involution σ on W (G, T) to the action of θ on W (^{∨}G,^{d}T). In particular, we have an isomorphism

ζ : W (G, T)^{σ} → W (^{∨}G,^{d}T)^{θ}.

c) Suppose ζ′ is another isomorphism from ^{∨}T to ^{d}T (Lemma 13.1(d)). Then ζ′ is a weak pairing between the Cartan subgroups if and only if ζ′ = w ∘ ζ for some w ∈ W (^{∨}G,^{d}T)^{θ}.

This is obvious.

Proposition 13.4 ([AV2], Lemma 9.16). Suppose G^{Γ} is a weak extended group, and ^{∨}G^{Γ} is a weak E-group for G^{Γ} (Definition 4.3).

a) Suppose T^{Γ} is a Cartan subgroup of G^{Γ} (Definition 12.1). Then there is a Cartan subgroup ^{d}T^{Γ} of ^{∨}G^{Γ} and a weak pairing between T^{Γ} and ^{d}T^{Γ} (Definition 13.2). The Cartan subgroup ^{d}T^{Γ} is unique up to conjugation by ^{∨}G.

b) Suppose ^{d}T^{Γ} is a Cartan subgroup of ^{∨}G^{Γ} (Definition 12.4). Then there is a Cartan subgroup T^{Γ} of G^{Γ} and a weak pairing between T^{Γ} and ^{d}T^{Γ} (Definition 13.2). The Cartan subgroup T^{Γ} is unique up to conjugation by G.

Proof. For (a), let^{d}T be any maximal torus in^{∨}G, and y an element of^{∨}G^{Γ}−^{∨}G normalizing ^{d}T. (To find such an element y, start with any y_{0} in ^{∨}G^{Γ} − ^{∨}G, and modify it by an element of ^{∨}G conjugating Ad(y_{0})(^{d}T) back to ^{d}T.) Then conjugation by y defines an automorphism θ_{y} of ^{d}T, and so an automorphism a_{y} of ^{∨}Ψ_{0}(^{d}T).

Fix one of the natural isomorphisms ζ from ^{∨}T to ^{d}T (Lemma 13.1). This isomorphism carries the automorphism a_{T} to an involutive automorphism a_{^{d}T} of ∨Ψ_{0}(^{d}T). Since ^{∨}G^{Γ} is an E-group for G^{Γ}, it follows from Definition 4.3 that a_{y} and a_{^{d}T} must differ by the action of an element w ∈ W (^{∨}G,^{d}T):

a_{^{d}T} = w ∘ a_{y}.

Now let n ∈ ^{∨}G be any representative of w, and set y′ = ny. Then y′ ∈ ^{∨}G^{Γ} − ∨G still normalizes ^{d}T, and the conjugation action of y′ defines the involutive automorphism θ of ^{d}T corresponding to a_{^{d}T}. The group ^{d}T^{Γ} generated by y′ and dT is therefore a Cartan subgroup of ^{∨}G^{Γ} paired with T^{Γ}.

For the uniqueness, suppose (^{d}T _{1}^{Γ})′ is another Cartan subgroup paired by ζ_{1} with T^{Γ}. After conjugating by ^{∨}G, we may assume that ^{d}T_{1} = ^{d}T and that ζ_{1} = ζ. Then the automorphism of ^{d}T defined by any element y of ^{d}T^{Γ} − ^{d}T coincides with the automorphism defined by any element y′ of (^{d}T^{Γ})′ −^{d}T. It follows that y^{d}T = y′^{d}T, so that ^{d}T^{Γ} = (^{d}T^{Γ})′ 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).

Definition 13.5. Suppose (G^{Γ}, W) is an extended group for G (Definition 1.12). A based Cartan subgroup of (G^{Γ}, W) is a quadruple T^{Γ} = (T^{Γ}, W(T^{Γ}), R_{iℝ}^{+}, R_{ℝ}^{+}), subject to the following conditions.

(a) T^{Γ} is a Cartan subgroup of G^{Γ} (Definition 12.1).

(b) W(T^{Γ}) is an extended group structure on T^{Γ} (Definition 1.12). That is, W(T^{Γ}) is a T -conjugacy class of elements δ ∈ T^{Γ} − T, with the property that δ^{2} has finite order.

(c) W(T^{Γ}) is a subset of W, in the sense that for every δ ∈ W(T^{Γ}) there are N and χ so that (δ, N, χ) ∈ W.

(d) R_{iℝ}^{+} is a set of positive imaginary roots of T in G, and R_{ℝ}^{+} is a set of positive real roots.

(e) The extended group structure and the positive imaginary roots are compatible in the following sense. Fix δ ∈ W(T^{Γ}), and N and χ so that (δ, N, χ) ∈ W. Write G(ℝ) = G(ℝ, δ) for the corresponding (quasisplit) real form of G. Suppose Λ^{can} is an irreducible canonical projective representation of T (ℝ)^{can,T} of type zz(ρ), and that

Λ = (Λ^{can}, R_{iℝ}^{+}, R_{ℝ}^{+}))

is a G(ℝ)-limit character of type z (Definition 11.2). Then what we require is that the corresponding standard limit representation M(Λ) admit a Whittaker model of type χ (Definition 3.1). Two based Cartan subgroups are called equivalent if they are conjugate by G.

Proposition 13.6. Suppose T^{Γ} is a Cartan subgroup of the extended group (G^{Γ}, W), and (R_{iℝ}^{+}, R_{ℝ}^{+}) are systems of imaginary and real positive roots. Then there is a unique extended group structure W(T^{Γ}) on T^{Γ} making the quadruple (T^{Γ}, W(T^{Γ}), R_{iℝ}^{+}, R_{ℝ}^{+}) a based Cartan subgroup.

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].

Definition 13.7. Suppose (^{∨}G^{Γ}, S) is an E-group for G (Definition 4.14). A based Cartan subgroup of (^{∨}G^{Γ}, S) is a quadruple ^{d}T^{Γ} = (^{d}T^{Γ}, S(^{d}T^{Γ}),^{d}R_{iℝ}^{+},^{d}R_{ℝ}^{+}), subject to the following conditions.

(a) ^{d}T^{Γ} is a Cartan subgroup of ^{∨}G^{Γ} (Definition 12.4).

(b) S(^{d}T^{Γ}) is an E-group structure on ^{d}T^{Γ} (Definition 4.14). That is, S(^{d}T^{Γ}) is a dT -conjugacy class of elements ^{∨}δ ∈ T^{Γ} − T, with the property that ^{∨}δ^{2} has finite order.

(c) S(^{d}T^{Γ}) is a subset of S, in the sense that for every ^{∨}δ ∈ S(^{d}T^{Γ}) there is a Borel subgroup ^{d}B so that (^{∨}δ,^{d}B) ∈ S.

(d) ^{d}R_{iℝ}^{+} is a set of positive imaginary roots of ^{d}T in ^{∨}G, and ^{d}R_{ℝ}^{+} is a set of positive real roots (cf. (12.5)).

(e) The E-group structure and the positive imaginary roots are compatible in the following sense. Fix ^{∨}δ ∈ S(^{d}T^{Γ}), and define ^{d}B to be the set of Borel subgroups dB so that (^{∨}δ,^{d}B) ∈ S). (The set ^{d}B is a single orbit under the action of the centralizer K of ^{∨}δ in ^{∨}G.) Then what we require is that ^{d}R_{iℝ}^{+} be special with respect to ^{d}B ([AV2], Definition 6.29). Two based Cartan subgroups are called equivalent if they are conjugate by ^{∨}G.

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.

Proposition 13.8. Suppose ^{d}T^{Γ} is a Cartan subgroup of the E-group (^{∨}G^{Γ}, S), and (^{d}R_{iℝ}^{+},^{d}R_{ℝ}^{+}) are systems of imaginary and real positive roots. Then there is a unique E-group structure S(^{d}T^{Γ}) on^{d}T^{Γ} making the quadruple (^{d}T^{Γ}, W(^{d}T^{Γ}), R_{iℝ}^{+}, R_{ℝ}^{+}) a based Cartan subgroup.

Again we postpone the proof until the next section.

Definition 13.9. Suppose (G^{Γ}, W) is an extended group for G (Definition 1.12), and (^{∨}G^{Γ}, S) is a corresponding E-group (Definition 4.14). Fix based Cartan subgroups T^{Γ} for G^{Γ} (Definition 13.5) and^{d}T^{Γ} for^{∨}G^{Γ} (Definition 13.7). A pairing between T^{Γ} and ^{d}T^{Γ} is a weak pairing

ζ : ^{∨}T → ^{d}T  (13.9)(a)

(Definition 13.2), subject to the additional conditions

ζ carries the positive imaginary roots R_{iℝ}^{+} onto the positive real coroots ^{d}R_{ℝ}^{∨,+}

(13.9)(b)

(cf. Lemma 13.1(b)), and

ζ carries the positive real roots R_{ℝ}^{+} onto the positive imaginary coroots ^{d}R_{iℝ}^{∨,+}.

(13.9)(c)

In particular, this identifies (^{d}T^{Γ}, S(^{d}T^{Γ})) as an E-group of (T^{Γ}, W(T^{Γ})).

Proposition 13.10. Suppose (G^{Γ}, W) is an extended group for G (Definition 1.12), and (^{∨}G^{Γ}, S) is a corresponding E-group (Definition 4.14).

a) Suppose T^{Γ} is a based Cartan subgroup for G^{Γ} (Definition 13.5). Then there is a based Cartan subgroup ^{d}T^{Γ} for ^{∨}G^{Γ} and a pairing between them (Definition 13.9). ^{d}T^{Γ} is uniquely determined up to conjugation by ^{∨}G^{Γ}.

b) Suppose ^{d}T^{Γ} is a based Cartan subgroup of ^{∨}G^{Γ} (Definition 13.7). Then there is a based Cartan subgroup T^{Γ} for G^{Γ} and a pairing between them. T^{Γ} is unique up to conjugation by G.

c) Suppose ζ is a pairing between based Cartan subgroups as above. If ζ′ is another isomorphism from ^{∨}T to ^{d}T, then ζ′ is a pairing if and only if ζ′ = w ∘ ζ for some w ∈ W (^{∨}G,^{d}T)^{θ} preserving ^{d}R_{iℝ}^{+} and ^{d}R_{ℝ}^{+}.

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)^{σ}. If σ′ is any real form of G extending σ on T, then w has a representative in G(ℝ, σ′).

Proof. For (a), use Proposition 13.4(a) to find a Cartan subgroup^{d}T^{Γ} of^{∨}G^{Γ} and a weak pairing ζ between it and T^{Γ}. By (13.2)(c), ζ carries the positive imaginary roots R_{iℝ}^{+} onto some set ^{d}R_{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}T^{Γ}. 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.

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

Suppose ^{d}T^{Γ} is a based Cartan subgroup of the E-group (^{∨}G^{Γ}, S) (Definition 13.7). A complete Langlands parameter for ^{d}T^{Γ} (Definition 12.4) is said to be compatible with ^{d}T^{Γ} if

⟨α, λ⟩ ≥ 0  (α ∈ ^{d}R_{ℝ}^{+}).  (13.11)

Here λ is constructed from the Langlands parameter as in (5.8). We do not assume any relationship between λ and the positive imaginary roots.

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

A little more precisely, each Langlands parameter ϕ for ^{d}T^{Γ} gives rise to a canonical projective character Λ^{can}(ϕ) of T (ℝ) of type zz(ρ) (Proposition 10.6). Each irreducible representation τ_{1} of A(^{d}T^{Γ})^{alg} gives rise to a T -conjugacy class in T^{Γ}−T, say with a representative δ(τ_{1}) (Corollary 9.12). We have

a) The element δ(τ_{1}) defines a strong real form of G — that is, δ(τ_{1})^{2} belongs to Z(G) — if and only if τ_{1} factors through the quotient group A(^{d}T^{Γ})^{alg,^{∨}G}.

b) The positivity requirement (12.2)(f) on Λ^{can}(ϕ) is equivalent to the compatibility requirement (13.11).

c) A simple imaginary root α for T is noncompact with respect to δ(τ_{1}) if and only the corresponding simple real root ^{d}α for ^{d}T satisfies

τ_{1}(\overline{m_{^{d}α}^{alg}}) = 1.

d) A real root β for T satisfies the parity condition for Λ^{can}(ϕ) if and only if the corresponding imaginary root ^{d}β for ^{d}T is ϕ-noncompact (Definition 12.8).

Proof. Because of the definitions, it suffices to prove the assertions (a)–(d). For (a), fix an element δ ∈ W(T^{Γ}). The element δ(τ_{1}) is obtained by multiplying δ by an appropriate element t = e(τ/2), with τ ∈ 𝔱_{ℚ}^{−a} (Proposition 9.10); here τ represents τ_{1} in the isomorphism of Proposition 9.8(c). One can check easily that τ_{1} factors through A(^{d}T^{Γ})^{alg,^{∨}G} if and only if the roots of T in G take integer values on the element τ. (To see what the root lattice has to do with the problem, recall that π_{1}(^{d}T) may be identified with X_{∗}(^{d}T) ≃ X^{∗}(T). The inclusion of ^{d}T in ^{∨}G induces a surjection π_{1}(^{d}T) → π_{1}(^{∨}G), and the kernel of this map is the lattice of roots of T in G.) On the other hand, we compute

δ(τ_{1})^{2} = tσ(t)δ^{2} = e(τ)δ^{2} = e(τ)z(ρ)

(cf. proof of Proposition 9.10). So δ(τ_{1}) is a strong real form if and only if e(τ) ∈ Z(G). Of course this condition is also equivalent to the roots of T in G taking integral values on τ, 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

τ_{1}(\overline{m_{^{d}α}^{alg}}) = exp(2πiα(τ)) = (−1)^{α(τ/2)}.

So the condition on τ_{1} is equivalent to

α(τ/2) is an even integer.

On the other hand, every simple imaginary root α of T in G is noncompact with respect to δ (see the proof of Proposition 13.6 in section 14). Consequently α is nocompact with respect to δ(τ_{1}) = tδ if and only if α(t) = 1. Since t = e(τ/2), this is equivalent to α(τ/2) being an even integer.

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

Proposition 13.13. Suppose (G^{Γ}, W) is an extended group for G, and (^{∨}G^{Γ}, 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^{Γ} (Definition 12.1) and equivalence classes of final complete limit parameters of Cartan subgroups of ^{∨}G^{Γ} (Definition 12.8).

Proof. Suppose (ϕ, τ_{1}) is a complete limit parameter for ^{d}T^{Γ}. To construct the corresponding limit character, we need to make ^{d}T^{Γ} a based Cartan subgroup; so we must choose certain sets of positive roots. Let ^{d}R_{ℝ}^{+} be a set of positive real roots for ^{d}T making λ(ϕ) dominant (cf. (12.5) and Definition 13.11); and let ^{d}R_{iℝ}^{+} be an arbitrary set of positive imaginary roots. Let ^{d}T^{Γ} be the corresponding based Cartan subgroup (Proposition 13.8). Fix a based Cartan subgroup T^{Γ} for G^{Γ} and a pairing

ζ : ^{∨}T → ^{d}T  (13.14)

with ^{d}T^{Γ}. Proposition 13.12 gives a final limit character (δ(τ_{1}), Λ(ϕ)) of T^{Γ}. 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_{ℝ}^{+})′ of positive real roots making λ(ϕ) dominant. These two positive systems differ by a Weyl group element w that is a product of simple reflections in real roots vanishing on λ. Let n be a representative of w. Changing the sytem of positive real roots changes the based Cartan subgroup ^{d}T^{Γ} by conjugation by n, and replaces ζ by w ∘ ζ. (Evidently this implies that the final limit character (δ(τ_{1}), Λ(ϕ)) is unchanged.) We claim that w has a representative n fixing (ϕ, τ_{1}). It is enough to check this for a simple reflection s_{α}, with α a real root vanishing on λ. Choose a root subgroup ϕ_{α} : SL(2) → ^{∨}G as usual (cf. Definition 11.10)) with the additional property that ϕ_{α} carries the inverse transpose involution on SL(2) to the action of θ_{y}. We may take

n_{α} = ϕ_{α} ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )

as a representative of s_{α}. Evidently n_{α} belongs to K(y) ∩ L(λ, y · λ) (cf. Corollary 5.9); that is, to the centralizer of ϕ. To compute its effect on τ_{1}, recall that τ_{1} is a character of (a certain quotient of) (^{d}T^{θ})^{alg,^{∨}G}. If t is in this group, then α(t) = ±1 since α is real. Consequently

s_{α}(t) = tα^{∨}(α(t^{−1})) = tα^{∨}(±1) = t or tm_{α}^{alg}.

Since (ϕ, τ_{1}) is final, τ_{1}(m_{α}^{alg}) = 1; so we find that s_{α}(τ_{1}) = τ_{1}, as we wished to show.

Next, consider a second choice (^{d}R_{iℝ}^{+})′ 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}T^{Γ})′, we must also conjugate S(^{d}T^{Γ}) by a representative n of w (because of Definition 13.7(e)). This conjugation has the effect of multiplying S(^{d}T^{Γ}) by an element t = nθ(n^{−1}); this element satisfies θ(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_{ℝ}^{+})′ corresponding to (^{d}R_{iℝ}^{+})′. The pairing ζ is then unchanged. The canonical character (Λ^{can})′ attached to these new choices differs from Λ^{can} only because of the change in the E-group structure on ^{d}T^{Γ}. The proof of Lemma 9.28 of [AV2] shows that the effect of this change is to twist Λ^{can} by the character τ(R_{ℝ}^{+}, (R_{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 (λ, τ_{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.

14. Proof of Propositions 13.6 and 13.8.

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.

Lemma 14.1. Suppose T is a maximal torus in the connected reductive algebraic group G, and σ is a real form of T permuting the roots of T in G. Suppose that σ_{1} and σ_{2} are two real forms of G extending σ, and that they determine the same sets of compact (and noncompact) imaginary roots. Then σ_{1} and σ_{2} are conjugate by an element of T.

Proof. Choose a set R^{+} of positive roots of T in G having the property that σ 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 σ on the rational span of the coroots.) In the argument below, we will usually write β for an imaginary root, γ for a non-imaginary root, and α for a general root. For any root in R^{+}, we have either

β is imaginary, and σβ = −β  (14.2)(a)

or

γ is not imaginary, and σγ is positive.  (14.2)(b)

For simple roots in R^{+}, one can say even more.

Lemma 14.3. In the setting of Lemma 14.1, suppose R^{+} is a set of positive roots for T in G such that σ preserves the non-imaginary positive roots. Then every simple root of R^{+} falls into exactly one of the following three categories.

a) β is imaginary, and σβ = −β.

b) γ is not imaginary, and

σγ = γ + ∑_{β simple imaginary} n_{β}(γ)β

with n_{β}(γ) a non-negative integer.

c) γ is not imaginary, and there is a distinct non-imaginary root γ′ so that

σγ = γ′ + ∑_{β simple imaginary} n_{β}(γ)β

with n_{β}(γ) a non-negative integer. In this case

σγ′ = γ + ∑_{β simple imaginary} n_{β}(γ)β.

Proof. Suppose γ is a non-imaginary simple root. Then σγ 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 γ′ = γ′(γ), for otherwise σγ (and therefore γ) would be imaginary. Thus

σγ = γ′ + (other non-imaginary simple roots) + ∑ n_{β}β.

Now apply σ 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 γ, since σ is an involution. It follows that the second term is zero, and that σγ′ involves γ. 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_{α}} for all the roots. As usual we may choose these so that

[X_{α}, X_{−α}] = H_{α}  (14.4)(a)

(the usual coroot), and

[X_{α}, X_{α′}] = q(α, α′)X_{α+α′}  (14.4)(b)

(with q(α, α′) rational) whenever α + α′ is a root. Define complex constants c_{i}(α) by

σ_{i}(X_{α}) = c_{i}(α)X_{σα}.  (14.4)(c)

Since σ_{1} and σ_{2} agree on T, they differ by the adjoint action of an element s of T:

c_{2}(α)/c_{1}(α) = α(s).  (14.4)(d)

In particular,

c_{2}/c_{1} extends to a multiplicative character of the root lattice.  (14.4)(e)

Replacing σ_{1} by tσ_{1}t^{−1} replaces c_{1}(α) by

[(σα)(t)/\overline{α(t)}]c_{1}(α).  (14.5)(a)

It follows that t conjugates σ_{1} to σ_{2} if and only if

(σα)(t)/\overline{α(t)} = c_{2}(α)/c_{1}(α)  (14.5)(b)

for every root α. Because of (14.4)(e), it suffices to verify (14.5)(b) for any set of roots α generating the root lattice. What we must show is that our hypotheses on σ_{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

S_{1} = simple imaginary roots;
S_{2} = simple roots as in Lemma 14.3(b);
S_{3} = one simple root γ from each pair (γ, γ′) as in Lemma 14.3(c); and
S_{4} = the roots σγ, with γ ∈ S_{3}.

(14.6)

Lemma 14.3 shows that S is actually a basis of the root lattice. We may therefore choose t so that α(t) has any value we specify for α ∈ S; we must show that this can be done so that (14.5)(b) holds for α ∈ S.

Applying the antiholomorphic involution σ_{i} to (14.4)(c), we find that

\overline{c_{i}(α)}c_{i}(σα) = 1.  (14.7)(a)

Applying σ_{i} to (14.4)(b) gives

c_{i}(α)c_{i}(α′) = q(α, α′)c_{i}(α + α′)  (14.7)(b)

whenever α + α′ is a root.

Suppose now that β is an imaginary root. Then σH_{β} = −H_{β}. Applying σ_{i} to (14.4)(a) gives

c_{i}(β)c_{i}(−β) = 1.  (14.8)(a)

On the other hand, (14.7)(a) gives

\overline{c_{i}(β)}c_{i}(−β) = 1.  (14.8)(b)

Consequently

c_{i}(β) is real, and c_{i}(−β) = c_{i}(β)^{−1}.  (14.8)(c)

Calculation in SL(2) shows that

c_{i}(β) is positive if and only if β is noncompact for σ_{i}.  (14.8)(d)

The hypothesis of Lemma 14.1 therefore guarantees that c_{1}(β)/c_{2}(β) is positive for every imaginary root β. We therefore require of our element t that

β(t) is a square root of c_{1}(β)/c_{2}(β)  (14.8)(e)

for every β in S_{1}. Condition (14.5)(b) follows for such β, and then for all imaginary roots by (14.4)(e).

Suppose next that γ belongs to S_{2}. The requirement (14.5)(b) on t may be written as

(c_{2}/c_{1})(γ) = (γ(t)/\overline{γ(t)})(σγ − γ)(t).  (14.9)(a)

This is the same as

γ(t)/\overline{γ(t)} = (c_{2}/c_{1})(γ)(γ − σγ)(t).  (14.9)(b)

We can choose γ(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 σ_{2} by the same equation for σ_{1}, to obtain

\begin{aligned}1 & = \overline{(c_{2}/c_{1})(γ)}(c_{2}/c_{1})(σγ) \\ & = \overline{(c_{2}/c_{1})(γ)}(c_{2}/c_{1})(γ)(c_{1}/c_{2})(γ − σγ) \\ & = |(c_{2}/c_{1})(γ)|^{2}(c_{1}/c_{2})(γ − σγ). \end{aligned}  (14.9)(c)

Now Lemma 14.3 shows that γ − σγ 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 (γ − σγ)(t)^{2}. Consequently

1 = |(c_{2}/c_{1})(γ)|^{2}(γ − σγ)(t)^{2}.  (14.9)(d)

This guarantees the existence of a solution γ(t) to (14.9)(b).

Finally, suppose γ ∈ S_{3}. We require of t that

γ(t) = 1,  σγ(t) = (c_{2}/c_{1})(γ).  (14.10)

Then (14.5)(b) is automatically satisfied for α = γ. To check it for α = σγ, 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.

Proof of Proposition 13.6. The real form σ_{T} of T (defined by conjugation by any element y of T^{Γ} − T) is an antiholomorphic involution preserving the roots. It may therefore be extended to a quasisplit real form σ_{G} of G, with the property that every simple root of T in R_{iℝ}^{+} is noncompact. (A proof of this fact may be found in [IC4], Lemma 10.9.) Clearly σ_{G} belongs to the inner class defined by G^{Γ}, so by Proposition 2.12 there is a triple (δ_{1}, N_{1}, χ_{1}) ∈ W with Ad(δ_{1}) = σ_{G}. In particular, δ_{1} and y both act by σ_{T} on T, so δ_{1} ∈ yT = T^{Γ} − T. If we write W_{1}(T^{Γ}) for the T -conjugacy class of δ_{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 (δ_{1}, N, χ_{1}) as above, and write G(ℝ, δ_{1}) for the corresponding real form. Define A_{ℝ} ⊂ T (ℝ) as in (11.3) (the identity component of the split part of T (ℝ)), and put

M = centralizer of A_{ℝ} in G.

Fix a σ_{G}-stable parabolic subgroup P = MU of G with Levi subgroup M. After replacing (N, χ_{1}) by a conjugate under G(ℝ, δ_{1}), we may assume that

P N is open in G.

It follows that N_{M} = N ∩ M is a maximal unipotent subgroup of M normalized by δ_{1}, and that χ_{M,1} = χ_{1}|_{N_{M} (ℝ)} is a non-degenerate unitary character (Definition 1.12). We apply to this situation the following lemma.

Lemma 14.11. Suppose σ is a quasisplit real form of a complex connected reductive algebraic group G, and P = MU is a σ-stable Levi decomposition of a parabolic subgroup of G. Suppose (N, χ) is the data for a Whittaker model for G (Definition 3.1). Suppose also that P N is open in G, so that if we write N_{M} = N ∩ M, χ_{M} = χ|_{N_{M} (ℝ)}, then (N_{M}, χ_{M}) is the data for a Whittaker model for M.

Suppose π_{M} is a finite length admissible Hilbert space representation of M(ℝ), and π = Ind_{P (ℝ)}^{G(ℝ)}π_{M}. Then π admits a Whittaker model of type χ if and only if π_{M} admits a Whittaker model of type χ_{M}.

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 χ_{1} if and only if these limits of discrete series admit Whittaker models of type χ_{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(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 π_{M} of M(ℝ). By the construction of σ_{G}, every simple imaginary root of T in M is noncompact. It follows that the annihilator of π_{M} in the enveloping algebra of 𝔪 is a minimal primitive ideal ([GKDim], Theorem 6.2). By Theorem 6.7.2 of [Kostant], there is a non-degenerate unitary character χ_{M} of N_{M}(ℝ) so that π_{M} admits a Whittaker model of type χ_{M}. It is clear from Definition 3.1 that there is a non-degenerate unitary character χ of N(ℝ) restricting to χ_{M} on M. By Lemma 14.11, our standard limit representations all admit Whittaker models of type χ. By Lemma 3.2, there is a t ∈ G normalizing N, so that Ad(t) is defined over ℝ, and t · χ_{1} = χ. Consider now the triple

(tδ_{1}t^{−1}, N, t · χ_{1}) = (δ, N, χ).

Since W is a G-orbit, this triple belongs to W. Since Ad(t) is defined over ℝ, conjugation by δ defines the same real form σ_{G} as δ_{1}. Define

W(T^{Γ}) = T -conjugacy class of δ.

This satisfies the requirements of Definition 13.5.

For the uniqueness, suppose W′(T^{Γ}) is another extended group structure on T^{Γ} satisfying the requirements of Definition 13.5. Fix a triple (δ′, N′, χ′) ∈ W so that δ′ belongs to W′(T^{Γ}). 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_{iℝ}^{+} must be noncompact for the real form σ′_{G} defined by conjugation by δ′. By the construction of σ_{G}, it follows that σ_{G} and σ′_{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 W′(T^{Γ}) is a T -conjugacy class, we may therefore assume that δ′ is chosen so that

the conjugation action σ′_{G} of δ′ is equal to σ_{G}.  (14.12)(a)

This means that the real form G(ℝ) defined by δ′ coincides with the one defined by δ. The groups N and N′ are therefore conjugate by G(ℝ); after applying such a conjugation to N′ (which does not change δ′) we may assume that N = N′. Fix a maximal torus T_{s} ⊂ M normalizing N and defined over ℝ; we write

B_{s} = T_{s}N,  B_{s}(ℝ) = T_{s}(ℝ)N_{s}(ℝ).  (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 ℝ may be decomposed as

(B_{s}/Z(G))(ℝ) = (T_{s}/Z(G))(ℝ)N_{s}(ℝ).  (14.12)(c)

Recall now that (δ, N, χ) and (δ′, N, χ′) both belong to 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))(ℝ)). In particular,

t · χ = χ′,  tδt^{−1} = δ′.  (14.13)

On the other hand, Definition 13.5(e) and Lemma 14.11 provide a single limit of discrete series representation for M(ℝ) admitting Whittaker models of types χ_{M} and χ′_{M}. To this situation we can apply

Lemma 14.14. Suppose G(ℝ) is a quasisplit real form of a complex connected reductive algebraic group, and π is an irreducible representation in the limits of (relative) discrete series. If π admits Whittaker models of types χ and χ′, then χ and χ′ must be conjugate under G(ℝ).

Proof. Fix a Borel subgroup B_{s} = T_{s}N defined over ℝ, and a maximally compact maximal torus T (ℝ). We may as well assume that χ and χ′ are non-degenerate unitary characters of N(ℝ). Consider the group (Ad G)(ℝ) of inner automorphisms of G defined over ℝ. It contains the image (Ad(G(ℝ)) of G(ℝ) as a subgroup of finite index. Define

Q(G(ℝ)) = (Ad G)(ℝ)/(Ad(G(ℝ)).  (14.15)(a)

This finite group acts on the set of equivalence classes of representations of G(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(ℝ)) meets T_{s}. It therefore follows from Lemma 3.2 that

Q(G(ℝ)) acts simply transitively on conjugacy classes of Whittaker models for G(ℝ).

(14.15)(b)

At the same time, the existence of representatives in T_{s} shows that

Q(G(ℝ)) acts trivially on the set of principal series representations for G(ℝ).

(14.15)(c)

On the other hand, the essential uniqueness of the maximally compact torus T shows that every coset in Q(G(ℝ)) meets the normalizer of T in G. Define

W_{2}(G, T) = Weyl group of T in (Ad G)(ℝ).

Since T (ℝ) is connected, it follows that

Q(G(ℝ)) ≃ W_{2}(G, T)/W (G(ℝ), T (ℝ)).

¿From this it follows that

Q(G(ℝ)) acts without fixed points on limits of discrete series representations of G(ℝ).

(14.15)(d)

To prove the lemma, realize π as a quotient of a principal series representation ρ. By (14.15)(b), there is an element q ∈ Q(G(ℝ)) with q·χ is conjugate by G(ℝ) to χ′. It follows formally that π′ = q · π has a Whittaker model of type χ′. By (14.15)(c), q · ρ is equivalent to ρ, so π′ is also a quotient of ρ. The quotient representations π and π′ of ρ now both have Whittaker models of type χ′. Because ρ has a unique Whittaker model ([Kostant], Theorem 6.6.2) it follows that π must be equivalent to π′. By (14.15)(d), q = 1, so χ is conjugate to χ′. Q.E.D.

Returning to the proof of Proposition 13.6, we deduce that χ_{M} is conjugate to χ′_{M} by an element of M(ℝ); by Lemma 3.2, we may as well choose this element t_{M} in T_{s}(ℝ). We can replace χ′ by t_{M} ·χ′, and t by tt_{M}; then (14.13) remains true, and we have in addition that χ_{M} = χ′_{M}. From this we deduce that Ad(t) acts trivially on each of the simple restricted root spaces in 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 δ and δ′ 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.

Lemma 14.16. Suppose T is a maximal torus in the connected reductive algebraic group G, and θ is a (holomorphic) involutive automorphism of T permuting the roots of T in G. Suppose θ_{1} and θ_{2} are two involutive automorphisms of G extending θ, and that they determine the same sets of compact (and noncompact) imaginary roots. Then θ_{1} and θ_{2} are conjugate by an element of T.

(Recall from (12.5) – (12.7) that a root is called imaginary if it is fixed by θ, 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.

Proof of Proposition 13.8. By Lemma 9.17 of [AV2], there is an element ^{∨}δ_{1} ∈ S ∩^{d}T^{Γ}. Write K_{1} for the centralizer of ^{∨}δ_{1} in ^{∨}G, and

dB_{1} = {^{d}B|(^{∨}δ_{1},^{d}B) ∈ S }

This is evidently a single K_{1}-orbit of Borel subgroups. By Proposition 6.30 of [AV2], there is a set ^{d}R_{iℝ,1}^{+} of positive imaginary roots special with respect to ^{d}B_{1}. Now there is a unique element w of the Weyl group of imaginary roots carrying ^{d}R_{iℝ,1}^{+} to ^{d}R_{iℝ}^{+}. The action of w on ^{d}T (as a product of reflections in imaginary roots) commutes with the action of ^{∨}δ_{1} on ^{d}T. Choose an element n of the normalizer of ^{d}T in ^{∨}G representing w, and define ^{∨}δ = n(^{∨}δ_{1})n^{−1}. Clearly ^{∨}δ belongs to S. The action of ^{∨}δ on ^{d}T agrees with that of ^{∨}δ_{1} (since the latter commutes with w); so ^{∨}δ = ^{∨}δ_{1}t for some t ∈ ^{d}T; so ^{∨}δ belongs to S ∩^{d}T^{Γ}. The corresponding set ^{d}B of Borel subgroups is obtained from ^{d}B_{1} by conjugating by n; so it follows immediately from the definition that ^{d}R_{iℝ}^{+} = w(^{d}R_{iℝ,1}^{+}) is special with respect to

dB. The ^{d}T -conjugacy class S(^{d}T^{Γ}) of ^{∨}δ therefore satisfies the requirements of the proposition.

For the uniqueness, suppose S′(^{d}T^{Γ}) is another E-group structure satisfying the requirements of the proposition. Fix ^{∨}δ′ ∈ S′(^{d}T^{Γ}); what we are trying to show is that ^{∨}δ′ is conjugate to ^{∨}δ by ^{d}T. Write θ and θ′ for the involutive automorphisms of ^{∨}G defined by conjugation by ^{∨}δ and ^{∨}δ′. By Proposition 6.30(a) of [AV2], every simple root in ^{d}R_{iℝ}^{+} is noncompact with respect both to θ and to θ′. By Lemma 14.16, it follows that θ and θ′ are conjugate by ^{d}T. After replacing ^{∨}δ′ by a conjugate, we may therefore assume that θ = θ′.

Now ^{∨}δ and ^{∨}δ′ belong to S, which is a single orbit of ^{∨}G. It follows that there is an element x of ^{∨}G conjugating ^{∨}δ to ^{∨}δ′. 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 Ad(x) commutes with θ, so the coset xZ(^{∨}G) belongs to the group \overline{K} of fixed points of θ in Ad(^{∨}G). Now Ad(x) carries ^{d}T to another θ-stable maximal torus ^{d}T′, so these two tori are conjugate by \overline{K}. We apply to Ad(^{∨}G) the following easy lemma.

Lemma 14.17. Suppose G is a connected reductive algebraic group, and θ is an involutive automorphism of G. Write K for the group of fixed points of θ. Suppose T and T′ are θ-stable maximal tori in G, conjugate by K. Then they are conjugate by the identity component K_{0}.

We omit the elementary proof.

In our situation, we conclude that there is an element x_{0} ∈ 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 ∈ W (^{∨}G,^{d}T) commuting with θ. Proposition 3.12 of [IC4] describes the Weyl group elements commuting with θ. 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 ^{∨}δ to ^{∨}δ′, it follows that w(^{d}R_{iℝ}^{+}) must (like ^{d}R_{iℝ}^{+}) be special with respect to ^{d}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 ∈ ^{d}T, as we wished to show. Q.E.D.

15. Multiplicity formulas for representations.

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_{ℝ}) of equivalence classes of final limit characters of type z. For Λ ∈ L^{z}(G_{ℝ}), write

M(Λ) = standard limit representation attached to Λ  (15.1)(a)

(cf. (11.2)), and

π(Λ) = Langlands quotient of M(Λ).  (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, π(Λ) is irreducible, and this correspondence establishes a bijection

L^{z}(G_{ℝ}) ↔ Π^{z}(G_{ℝ}),  Λ ↔ π(Λ).  (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_{ℝ} of G_{ℝ}. (This is unique up to conjugation by G_{ℝ}.) One can then consider

M^{z}(𝔤, K_{ℝ}),  (15.2)(a)

the category of finite-length canonical projective (𝔤, K_{ℝ})-modules of type z, or Harish-Chandra modules. (Such a module is a representation simultaneously of 𝔤 and of K_{ℝ}^{can,G} (cf. (11.1)(e)), with π_{1}(G)^{can} acting by the character parametrized by z.) Sometimes it is convenient to obscure the choice of K_{ℝ}, writing instead

M_{HC}^{z}(G_{ℝ})  (15.2)(b)

for this category. Alternatively, one can consider the category

M_{∞}^{z}(G_{ℝ})  (15.2)(c)

for which a typical object is the space of smooth vectors in a finite-length canonical projective representation of G_{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_{ℝ}-finite vectors) to M^{z}(𝔤, K_{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 −∞, ω, or −ω). 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_{ℝ}) are significant. Recall also that the critical Definition 13.5(e) really makes sense only on M_{∞}^{z} (cf. Definition 3.1).

In any case, the categories are all canonically equivalent, so we can safely define

KΠ^{z}(G_{ℝ}) = Grothendieck group of M_{HC}^{z}(G_{ℝ}).  (15.2)(d)

Suppose Θ and Λ belong to L^{z}(G_{ℝ}). As in (1.21), define

m_{r}(Θ, Λ) = multiplicity of π(Θ) in M(Λ).  (15.3)(a)

We call m_{r} the representation-theoretic multiplicity matrix. In the Grothendieck group this definition amounts to

M(Λ) = ∑_{Θ∈L^{z}(G_{ℝ})} m_{r}(Θ, Λ)π(Θ).  (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 representation-theoretic character matrix. Explicitly,

π(Θ) = ∑_{Λ∈L^{z}(G_{ℝ})} c_{r}(Λ, Θ)M(Λ)  (15.3)(c)

in KΠ^{z}(G_{ℝ}). In particular, the final standard limit representations M(Θ) 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 π(Θ); hence the term “character matrix.” In the case of regular infinitesimal character, the standard representation M(Λ) is characterized by the appearance of one particular term in the formula for its distribution character (on the Cartan subgroup T_{ℝ} corresponding to Λ). The entries c_{r}(Λ, Θ) may therefore be interpreted as certain coefficients in the character formula for π(Θ).) In analogy with (7.11)(f), we define the Bruhat order on L^{z}(G_{ℝ}) to be the smallest partial order with the property that

m_{r}(Θ, Λ) ≠ 0 only if Θ ≤ Λ.  (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.

Lemma 15.4. Suppose G is a complex connected reductive algebraic group, and ∨G is a dual group for G. Write 𝔤 for the Lie algebra of G, and

ℨ(𝔤) = center of U(𝔤).

Then there is a natural one-to-one correspondence between the set of algebra homomorphisms

χ : ℨ(𝔤) → ℂ

and the set of semisimple orbits O of ^{∨}G on ^{∨}𝔤.

Proof. Fix maximal tori T ⊂ G and ^{d}T ⊂ ^{∨}G. Harish-Chandra’s theorem (see [Hump]) parametrizes the characters of ℨ(𝔤) by orbits of the Weyl group on the dual 𝔱^{∗} of the Lie algebra of T. By (9.3)(b), 𝔱^{∗} is naturally isomorphic to the Lie algebra ^{∨}𝔱 of the dual torus ^{∨}T. By Lemma 13.1, there is an isomorphism of ^{∨}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 ^{∨}G on ^{∨}𝔤 meets ^{d}𝔱 in a unique Weyl group orbit. Q.E.D.

We sometimes write

χ_{O} : ℨ(𝔤) → ℂ  (15.5)(a)

for the character corresponding to the semisimple orbit O in the bijection of Lemma 15.4. If T is a maximal torus in G and λ ∈ 𝔱^{∗}, then we may also write

χ_{λ} : ℨ(𝔤) → ℂ  (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}(O, G_{ℝ}) = { Λ ∈ L^{z}(G_{ℝ}) | π(Λ) has infinitesimal character χ_{O} } (15.5)(c)

M_{HC}^{z}(O, G_{ℝ}) = Harish-Chandra modules of generalized infinitesimal character χ_{O}

(15.5)(d)

KΠ^{z}(O, G_{ℝ}) = Grothendieck group of M_{HC}^{z}(O, G_{ℝ}).  (15.5)(e)

The set L^{z}(O, G_{ℝ}) is finite, so KΠ^{z}(O, G_{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_{ℝ}. So suppose for a moment that (G^{Γ}, W) is an extended group for G. As in Lemma 1.15, choose a set { δ_{s} | s ∈ Σ } of representatives for the equivalence classes of strong real forms of G. Recall from Definition 12.1 that L^{z}(G/ℝ) is the set of G-conjugacy classes of pairs (Λ, δ), with δ a strong real form of G (Definition 1.13) and Λ ∈ L^{z}(G(ℝ, δ)) a final limit character. As in Lemma 1.15, there is a natural identification

L^{z}(G/ℝ) ≃ ∐_{s∈Σ} L^{z}(G(ℝ, δ_{s})).  (15.6)(a)

Theorem 12.3 provides a bijection between these parameters and the set Π^{z}(G/ℝ) of Definition 10.3; and the proof of Lemma 1.15 shows that

Π^{z}(G/ℝ) ≃ ∐_{s∈Σ} Π^{z}(G(ℝ, δ_{s})).  (15.6)(b)

Just as in (15.5), we can restrict attention to a single infinitesimal character corresponding to a ^{∨}G-orbit O, writing for example L^{z}(O, G/ℝ). Finally, let (^{∨}G^{Γ}, S) be an E-group for G^{Γ} with second invariant z. Theorem 10.4 identifies Π^{z}(G/ℝ) with

Ξ^{z}(G/ℝ) = Ξ(^{∨}G^{Γ}).  (15.6)(c)

This set also decomposes according to the semisimple orbits of ^{∨}G on ^{∨}𝔤, and we have

Ξ^{z}(O, G/ℝ) = { complete geometric parameters for ^{∨}G^{alg} acting on X(O,^{∨}G^{Γ}) }

(15.6)(d)

It is clear from the definitions that Theorem 10.4 provides an identification

Π^{z}(O, G/ℝ) ≃ Ξ^{z}(O, G/ℝ).  (15.6)(e)

In analogy with Definition 7.13, we can form the direct sum of abelian categories

M_{HC}^{z}(G/ℝ) = ⨁_{s∈Σ} M_{HC}^{z}(G(ℝ, δ_{s})).  (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/ℝ) or Ξ^{z}(G/ℝ). For ξ ∈ Ξ^{z}(O, G/ℝ), we may write as in (15.3)

M(ξ) = ∑_{γ∈Ξ^{z}(O,G/ℝ)} m_{r}(γ, ξ)π(γ)  (15.7)(b)

and so forth; this identity is in the Grothendieck group

KΠ^{z}(O, G/ℝ) = ⨁_{s∈Σ} KΠ^{z}(O, G(ℝ, δ_{s})).  (15.7)(c)

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

Definition 15.8. Suppose ^{∨}G^{Γ} is a weak E-group. Recall from Definition 4.9 the element z(ρ) ∈ Z(^{∨}G^{alg}); it has order 2. Because it is canonically defined, it is fixed by the conjugation action of any element of ^{∨}G^{Γ}. If ξ = (ϕ, τ) ∈ Ξ(^{∨}G^{Γ}), it follows that z(ρ) belongs to ^{∨}G_{ϕ}^{alg}. Write \overline{z(ρ)} for its image in the universal component group A_{ϕ}^{loc,alg}; this is a central element of order 2. It therefore acts by ±1 in the irreducible representation τ, and we define

e(ξ) = τ(\overline{z(ρ)}) = ±1.

We also recall from (1.22) the notation

d(ξ) = dim S_{ξ},

with S_{ξ} the ^{∨}G-orbit on X(^{∨}G^{Γ}) corresponding to ξ (Definition 7.6).

Lemma 15.9. In the setting of Theorem 10.4, suppose ξ ∈ Ξ^{z}(G/ℝ) corresponds to an irreducible representation π of a real form G(ℝ, δ). Then the sign e(ξ) of Definition 15.8 is equal to the sign e(G(ℝ, δ)) defined in [Kott1]. In particular, it is equal to 1 if G(ℝ, δ) is quasisplit.

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

Definition 15.11. Suppose we are in the setting (15.6). The Grothendieck groups KΠ^{z}(O, G/ℝ) (cf. (15.7)(c)) and KX(O,^{∨}G^{Γ}) (Definition 7.13) are both free ℤ-modules on bases parametrized by Ξ^{z}(G/ℝ). It therefore makes sense to define the canonical perfect pairing

⟨, ⟩ : KΠ^{z}(O, G/ℝ) × KX(O,^{∨}G^{Γ}) → ℤ

by the requirement

⟨M(ξ), μ(γ)⟩ = e(ξ)δ_{ξ,γ}.

The last term on the right is a Kronecker delta.

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

Theorem 15.12. Suppose (G^{Γ}, W) is an extended group for G (Definition 1.12), and (^{∨}G^{Γ}, S) is an E-group for the corresponding inner class of real forms, having second invariant z (Definition 4.14). Fix a semisimple orbit O of ^{∨}G on ^{∨}𝔤. Write KΠ^{z}(O, G/ℝ) for the Grothendieck group of canonical projective representations of type z and infinitesimal character χ_{O} of strong real forms of G (cf. (15.7)(c)), and KX(O,^{∨}G^{Γ}) for the Grothendieck group of ^{∨}G^{alg}-equivariant constructible sheaves on the geometric parameter space X(O,^{∨}G^{Γ}) (Definitions 6.9 and 7.13). Fix complete geometric parameters ξ, γ ∈ Ξ^{z}(O, G/ℝ) (cf. (15.6)(d)), and write π(ξ), P (γ) for the corresponding irreducible representation and perverse sheaf (Theorem 10.4 and Definition 7.13). Then the canonical pairing of Definition 15.11 satisfies

⟨π(ξ), P (γ)⟩ = e(ξ)(−1)^{d(ξ)}δ_{ξ,γ}.

Here e(ξ) is defined in (15.8).

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.

Corollary 15.13. In the setting of Theorem 15.12, fix η and γ in Ξ^{z}(G/ℝ). Then the geometric and representation-theoretic multiplicity and character matrices (cf. (7.11) and (15.3)) satisfy

a) c_{g}(η, γ)(−1)^{d(η)−d(γ)} = m_{r}(γ, η).

b) c_{r}(η, γ)(−1)^{d(η)−d(γ)} = m_{g}(γ, η).

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

\begin{aligned}⟨π(ξ), P (γ)⟩ & = ⟨∑_{ξ′} c_{r}(ξ′, ξ)M(ξ′), ∑_{γ′} c_{g}(γ′, γ)μ(γ′)(−1)^{d(γ′)}⟩ \\ & = ∑_{ξ′,γ′} c_{r}(ξ′, ξ)c_{g}(γ′, γ)(−1)^{d(γ′)}⟨M(ξ′), μ(γ′)⟩. \end{aligned}

By Definition 15.11, only the terms with γ′ = ξ′ contribute; we get

∑_{η} c_{r}(η, ξ)c_{g}(η, γ)(−1)^{d(η)}e(η).

Now the central element z(ρ) clearly acts by e(γ) on all the stalks of the perverse sheaf P (γ). So whenever the second factor is non-zero, we must have e(η) = e(γ). This leads to

e(γ)(−1)^{d(γ)} ∑_{η} c_{r}(η, ξ)c_{g}(η, γ)(−1)^{d(η)−d(γ)}.

Comparing with the formula in Theorem 15.12, we find

∑_{η} c_{r}(η, ξ)c_{g}(η, γ)(−1)^{d(η)−d(γ)} = δ_{ξ,γ}.

Since the inverse of the matrix c_{r}(γ, ξ) is by definition m_{r}(γ, ξ), it follows that

m_{r}(γ, η) = c_{g}(η, γ)(−1)^{d(η)−d(γ)},

which is (a).

For (b), write

c_{g}(η, γ) = m_{r}(γ, η)(−1)^{d(η)−d(γ)}.

Multiply by m_{g}(ξ, η) and sum over η. Since c_{g} is by definition the inverse of m_{g}, we get δ_{ξ,γ} on the left. The right side can be written as

(−1)^{d(γ)−d(ξ)} ∑_{η} m_{r}(γ, η)m_{g}(ξ, η)(−1)^{d(ξ)−d(η)}.

Since the left side is zero unless γ = ξ, 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}(η, ξ) = m_{g}(ξ, η)(−1)^{d(ξ)−d(η)}.

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

Theorem 15.12 is equivalent to Corollary 15.13(a).  (15.14)

16. The translation principle, the Kazhdan-Lusztig algorithm, and Theorem 1.24.

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.

Definition 16.1. In the setting of Lemma 15.4, suppose O and O′ are semisimple orbits of ^{∨}G on ^{∨}𝔤, and T is a translation datum from O to O′ (Definition 8.6). Fix maximal tori T ⊂ G and ^{d}T ⊂ ^{∨}G, and construct μ = λ′ − λ ∈ X_{∗}(^{d}T) as in Definition 8.6(e); this element is defined up to the Weyl group. By Lemma 13.1(b), μ corresponds to a weight (also called μ) 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_{T} of G having extremal weight −μ. Write P_{O} for the functor of projection on the generalized infinitesimal character χ_{O}; this is defined on all representations of 𝔤 on which ℨ(𝔤) acts in a locally finite way.

The Jantzen-Zuckerman translation functor

ψ_{T} : M^{z}(O′, G_{ℝ}) → M^{z}(O, G_{ℝ})

is defined by

ψ_{T} (M) = P_{O}(M ⊗ F_{T}).

(Of course it is also defined on M^{z}(O′, G/ℝ).) In the notation of (11.15), this is ψ_{λ_{a}+μ_{a}}^{λ_{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”)

ϕ_{T} : M^{z}(O, G_{ℝ}) → M^{z}(O′, G_{ℝ})

defined by

ϕ_{T} (M) = P_{O′}(M ⊗ F_{T}^{∗}).

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.

Lemma 16.2. Suppose G is a complex connected reductive algebraic group, ^{∨}G is a dual group, and O and O′ are semisimple orbits of ^{∨}G on ^{∨}𝔤. Then a translation datum T from O to O′ (Definition 8.6) may be identified with a G-conjugacy class of triples (T, λ, λ′) subject to the conditions below.

i) T is a maximal torus in G, and λ and λ′ belong to 𝔱^{∗}. ii) The infinitesimal characters χ_{λ} and χ′_{λ} are equal to χ_{O} and χ_{O′} respectively

(notation (15.5)). iii) Suppose α is a root of T in 𝔤, and ⟨λ, α^{∨}⟩ is a positive integer. Then ⟨λ′, α^{∨}⟩ is

a positive integer. iv) The weight μ = λ′ − λ belongs to X^{∗}(T).

Any triple (T, λ, λ′) satisfying (i)–(iv) determines a unique translation datum.

This is clear from Definition 8.6 and Lemma 13.1. In the language of Definition 4.5.7 of [Green], our (𝔱, λ, λ′) corresponds to (𝔥^{s}, ξ + \overline{λ_{2}}, ξ + \overline{λ_{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 ψ_{T} is called ψ_{ξ+λ_{2}}^{ξ+λ_{1}} in [Green], and our ϕ_{T} is ψ_{ξ+λ_{1}}^{ξ+λ_{2}}.

In the setting of Definition 16.1, suppose Λ′ = ((Λ^{can})′, R_{iℝ}^{+}, R_{ℝ}^{+}) is a limit character of infinitesimal character O′ (cf. (11.2)). When G is linear we can define a new limit character

Λ = ψ_{T} (Λ′)  (16.3)(a)

as follows (cf. (11.17)). First, the Cartan subgroup T_{ℝ} and the positive root systems are unchanged. Identify the translation datum T as in Lemma 16.2, and choose a triple (T, λ, λ′) with T the complexification of T_{ℝ} and λ′ the differential of (Λ′)^{can}; this is possible by our hypothesis on the infinitesimal character of Λ′. By the definition of a translation datum (see Lemma 16.2(iii)), the stabilizer of λ′ in the Weyl group W (G, T) also stabilizes λ. It follows that λ is uniquely determined by T and λ′; so the weight μ = λ′ − λ ∈ X^{∗}(T) is well-defined. (This step is not symmetric in Λ and Λ′; we cannot reverse it to define “ϕ_{T} (Λ).”) Via the map (11.1)(b), we can regard μ as a character of T_{ℝ}; so we can define

Λ^{can} = ((Λ)^{can})′ ⊗ μ^{−1}.  (16.3)(b)

The positivity assumption in Lemma 16.2(iii) guarantees that Λ inherits from Λ′ 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 μ dominant.)

Theorem 16.4. Suppose G is a complex connected reductive algebraic group, and G_{ℝ} is a linear real Lie group as in (11.1). Fix infinitesimal characters O and O′ (Lemma 15.4) and a translation datum T from O to O′.

a) Suppose Λ′ is a limit character with infinitesimal character O′ write Λ = ψ_{T} (Λ′)

(cf. (16.3)). Then

ψ_{T} (M(Λ′)) = M(Λ)

(notation (11.2) and (16.1).)

b) Suppose π′ ∈ Π^{z}(G_{ℝ}) is an irreducible canonical projective representation of infinitesimal character O′. Then ψ_{T} (π′) is irreducible or zero.

c) In the setting of (a), suppose in addition that Λ′ is final and that ψ_{T} (π(Λ′)) ≠ 0.

Then Λ is also final, and

ψ_{T} (π(Λ′)) = π(Λ).

In this way ψ_{T} defines a bijection from a subset of Π^{z}(O′, G_{ℝ}) onto Π^{z}(O, G_{ℝ}).

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

ψ_{T}^{−1} : Π^{z}(O, G_{ℝ}) → Π^{z}(O′, G_{ℝ})  (16.5)(a)

for the injective map inverting the correspondence of (c). This is not so easy to describe explicitly; one characterization is

ψ_{T}^{−1}(π) = unique irreducible subrepresentation of ϕ_{T} (π).  (16.5)(b)

Because of Theorem 11.14, there is a corresponding injective map on equivalence classes of final limit characters

ϕ_{T} : L^{z}(O, G_{ℝ}) → L^{z}(O′, G_{ℝ}).  (16.5)(c)

As a consequence of Theorem 16.4, this map respects the representation-theoretic multiplicity matrix:

m_{r}(ϕ_{T} (Θ), ϕ_{T} (Λ)) = m_{r}(Θ, Λ)  (Θ, Λ ∈ L^{z}(O, G_{ℝ})).  (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.

Proposition 16.6. In the setting of (15.6), suppose T is a translation datum from O to O′. Then the diagram

L^{z}(O, G/ℝ) ⟶^{ϕ_{T}} L^{z}(O′, G/ℝ)
↓  ↓
Ξ^{z}(O,^{∨}G^{Γ}) ⟶^{f_{T}^{∗}} Ξ^{z}(O′,^{∨}G^{Γ})

commutes. Here the vertical arrows are the bijections of Theorem 10.4.

Proof. The map in the top row has been described rather explicitly in terms of Cartan subgroups of G^{Γ} 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^{∨}G^{Γ} (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

O ⊂ ^{∨}𝔤 is a regular semisimple orbit of ^{∨}G.  (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}, λ_{1}) and (^{d}T_{2}, λ_{2}) are pairs with ^{d}T_{i} a maximal torus in ∨G and λ_{i} ∈ ^{d}T_{i} ∩ O. Just as in (11.17), there is a unique isomorphism

j(λ_{1}, λ_{2}) : ^{d}T_{1} → ^{d}T_{2}  (16.7)(b)

induced by an element of ^{∨}G and carrying λ_{1} to λ_{2}. (It is the assumption that O is regular that makes the isomorphism unique.) Define

(^{∨}T_{O}, λ_{O}) = lim (^{d}T , λ),  (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_{O}, λ_{O}), say by

j(λ_{O}, λ) : ^{∨}T_{O} → ^{d}T  (16.7)(d)

Write

T_{O} = dual torus to ^{∨}T_{O}.  (16.8)(a)

The dual of the Lie algebra of T_{O} may be identified with ^{∨}𝔱_{O}, so we may write

λ_{O} ∈ 𝔱_{O}^{∗}.  (16.8)(b)

By inspection of the proof of Lemma 15.4, it is clear that we may identify

(T_{O}, λ_{O}) = lim (T, λ).  (16.8)(c)

The projective limit is taken over pairs (T, λ) with T a maximal torus in G, and λ ∈ 𝔱^{∗} a weight defining the infinitesimal character χ_{O}, using isomorphisms analogous to those of (16.7)(b). In particular, we get isomorphisms

i(λ_{O}, λ) : T_{O} → T  (16.8)(d)

as in (16.7)(d).

The torus ^{∨}T_{O} inherits a root system R(^{∨}G,^{∨}T_{O}) ⊂ X^{∗}(^{∨}T_{O}), a Weyl group W (^{∨}G,^{∨}T_{O}), and so on. Similarly (using (16.8)(c)) we can define R(G, T_{O}). The identification (16.8)(a) makes R(G, T_{O}) and R(^{∨}G,^{∨}T_{O}) into dual root systems, so that the roots for one may be regarded as coroots for the other. We define the set of O-integral coroots

R^{∨}(O) = { α^{∨} ∈ R(^{∨}G,^{∨}T_{O})|α^{∨}(λ_{O}) ∈ ℤ }.  (16.9)(a)

Equivalently,

R^{∨}(O) = { α^{∨} ∈ R^{∨}(G, T_{O})|λ_{O}(α^{∨}) ∈ ℤ }.  (16.9)(b)

Clearly an isomorphism j(λ_{O}, λ) as in (16.7)(d) identifies R^{∨}(O) with the set of roots of ^{d}T in ^{∨}G(λ) (cf. (6.2)(b)). In G, i(λ_{O}, λ) identifies R(O) with the set of λ-integral roots of T in G. There is a natural positive root system

R^{+}(O) = { α ∈ R(O)|α^{∨}(λ_{O}) > 0 };  (16.9)(c)

j(λ_{O}, λ) identifies the positive coroots with the roots of ^{d}T in the Borel subgroup P (λ) of ^{∨}G(λ) (cf. (6.2)(e)). We write

R(λ) = i(λ_{O}, λ)(R(O)) = { α ∈ R(G, T)|α^{∨}(λ) ∈ ℤ }, R^{+}(λ) = i(λ_{O}, λ)(R^{+}(O)),

(16.9)(d)

and similarly for R^{∨}. Define

Δ(O) = set of simple roots of R^{+}(O), W (O) = Weyl group of R(O).

(16.9)(e)

We can regard W (O) as a group of automorphisms of T_{O} or of ^{∨}T_{O}. The corresponding set of simple reflections is written

S(O) = { s_{α}|α ∈ Δ(O) }.  (16.9)(f)

Of course (W (O), S(O)) is a Coxeter group. We may therefore attach to it a Hecke algebra H(O) (cf. [KL0]). The Hecke algebra is a free ℤ[u^{1/2}, u^{−1/2}]-algebra with basis

{ T_{w}|w ∈ W (O) }.  (16.10)(a)

It is characterized by the relations

T_{x}T_{y} = T_{xy}  (x, y ∈ W (O), l(x) + l(y) = l(xy))  (16.10)(b)

and

(T_{s} + 1)(T_{s} − u) = 0  (s ∈ S(O)).  (16.10)(c)

(Here l(x) is the length function on the Coxeter group (W (O), S(O)); it need not be the restriction of any length function on the larger Weyl group W (G, T_{O}).) It follows that the operators T_{s} generate the Hecke algebra. The specialization to u^{1/2} = 1 of H(O) — that is, its quotient by the ideal generated by u^{1/2} − 1 — is naturally isomorphic to the group algebra of W (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 (O). Recall the map e from ^{∨}𝔤 to ^{∨}G (cf. (6.2)(a)). We use the notation of (6.10). Thus F(O) is the set of canonical flats in O, C(O) is the corresponding conjugacy class in ^{∨}G, and e is a smooth projective algebraic morphism from F(O) to C(O). Define

Z(O,^{∨}G) = F(O) ×_{C(O)} F(O)  (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

{ ^{∨}G-orbits on Z(O) } ↔ W (O).  (16.11)(b)

Here the diagonal orbit (which is just F(O)) corresponds to the identity element of W (O). The length function on W (O) corresponds to the dimension of orbits (shifted by the dimension of F(O)). Now one can realize the group algebra of W (O) (and even the Hecke algebra H(O)) as an algebra of correspondences, endowed with a natural geometric action on KX(O,^{∨}G^{Γ}). 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 ^{∨}G-orbit on F(O) ×_{C(O)} F(O′).)

We return now to our description of the Kazhdan-Lusztig algorithms. Suppose G_{ℝ} is a linear Lie group as in (15.1)–(15.2). Define

KΠ^{z}(O, G_{ℝ}) = KΠ^{z}(O, G_{ℝ}) ⊗_{ℤ} ℤ[u^{1/2}, u^{−1/2}],  (16.12)(a)

(cf. (15.7)), the free ℤ[u^{1/2}, u^{−1/2}]-module with basis the irreducible (or standard) representations of G_{ℝ}. We may refer to this as the 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

KΠ^{z}(O, G/ℝ) = KΠ^{z}(O, G/ℝ) ⊗_{ℤ} ℤ[u^{1/2}, u^{−1/2}],  (16.12)(b)

Similarly, we define

KX(O,^{∨}G^{Γ}) = KX(O,^{∨}G^{Γ}) ⊗_{ℤ} ℤ[u^{1/2}, u^{−1/2}],  (16.12)(c)

the 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{K}Π^{z}(O, G/ℝ) = ℤ[u^{1/2}, u^{−1/2}]-linear combinations of elements of Π^{z}(O, G/ℝ); (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{K} the formal mixed Grothendieck group of representations.

Proposition 16.13. In the setting (16.7) – (16.12), there are natural actions of the Hecke algebra H(O) on the mixed Grothendieck groups KΠ^{z}(O, G_{ℝ}) (or KΠ^{z}(O, G/ℝ), or \overline{K}Π^{z}(O, G/ℝ)) and KX(O,^{∨}G^{Γ}).

Proof. By (8.9), the ℤ[u^{1/2}, u^{−1/2}]-module KX(O,^{∨}G^{Γ}) 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 KΠ^{z}(O, G_{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 KΠ^{z}(O, G_{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.

Definition 16.14 (see [KL0]). Verdier duality is the unique algebra automorphism

D : H(O) → H(O)

satisfying

Du^{1/2} = u^{−1/2}, DT_{s} = u^{−1}(T_{s} + (1 − u)) (s ∈ S(O)).

The terminology arises from the geometric interpretation of H(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} (w ∈ W (O)).

Another useful characterization is

Du^{1/2} = u^{−1/2}, D(T_{s} + 1) = u^{−1}(T_{s} + 1) (s ∈ S(O)).

It follows from either characterization that D^{2} = 1. If M is a module for H(O), then a Verdier duality for M is a ℤ-linear involution

D_{M} : M → M

satisfying

D_{M}(a · m) = (Da) · (D_{M}m) (a ∈ H(O), m ∈ M).

Again several equivalent characterizations are possible, notably

D_{M}(u^{1/2}·m) = u^{−1/2}(D_{M}m), D_{M}((T_{s}+1)·m) = u^{−1}(T_{s}+1)·(D_{M}m) (m ∈ M, s ∈ S(O).

Proposition 16.15 ([LV], Theorem 1.10, and [IC3], Lemma 6.8). In the setting of Proposition 16.13, there is a natural Verdier duality D_{X} on the Hecke algebra module KX(O,^{∨}G^{Γ}). It is characterized uniquely by the requirement that the matrix of D_{X} with respect to the basis {μ(ξ)} of (7.10)(c) be upper triangular in the Bruhat order (cf. (7.11)(f)) with u^{−d(ξ)} on the diagonal.

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_{ℝ}) analogous to the function d(ξ) (dimension of the orbit) on Ξ(^{∨}G^{Γ}). In the cases of interest Theorem 10.4 provides a bijection between these two sets, so we could simply use d(ξ); 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.

Definition 16.16. Let G_{ℝ} be a linear Lie group as in (11.1). Fix a Langlands decomposition M_{ℝ}A_{ℝ}N_{ℝ} of a minimal parabolic subgroup of G_{ℝ}, and let B_{M} be a Borel subgroup of M. Define

c_{0}(G_{ℝ}) = 1/2(dim B_{M}).  (16.16)(a)

Suppose that Λ is a G_{ℝ}-limit character of a Cartan subgroup T_{ℝ}, with differential λ ∈ 𝔱_{ℝ} (Definition 11.2). Assume that Λ has infinitesimal character corresponding to O as in (16.7), so that we have a well-defined set R^{+}(λ) of positive integral roots (cf. (16.9)(d)). Choose a Cartan involution θ of G_{ℝ} preserving T_{ℝ}. Then θ acts on the roots of T in G, and this action preserves the integral roots. (For this we need the linearity of G_{ℝ}.) We can therefore define the integral length of Λ by

l^{I}(Λ) = −1/2 (|{ α ∈ R^{+}(λ)|θα ∈ R^{+}(λ) }| + dim(T_{ℝ}^{θ})) + c_{0}(G_{ℝ}). (16.16)(b)

Clearly this differs by a constant (depending on O and G_{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}(ξ) = l^{I}(Λ)  (16.16)(c)

whenever ξ ∈ Ξ^{z}(G/ℝ) corresponds to Λ ∈ L^{z}(G(ℝ, δ_{s})).

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

The construction of D_{Π} 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_{Π} is computable.

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

Proposition 16.18 ([IC3], Theorem 7.1). In the setting (16.7)–(16.12), there is for every geometric parameter γ ∈ Ξ(O,^{∨}G^{Γ}) an element

C_{g}(γ) = ∑_{ξ} P_{g}(ξ, γ)(u)μ(ξ) (P_{g}(ξ, γ) ∈ ℤ[u^{1/2}, u^{−1/2}])

of KX(O,^{∨}G^{Γ}) characterized by the following properties.

i) D_{X}C_{g}(γ) = u^{−d(γ)}C_{g}(γ) (cf. Proposition 16.15). ii) P_{g}(γ, γ) = 1. iii) P_{g}(ξ, γ) ≠ 0 only if ξ ≤ γ in the Bruhat order (cf. (7.11)(f)). iv) If ξ ≠ γ, then P_{g}(ξ, γ) is a polynomial in u of degree at most 1/2(d(γ)−d(ξ)−1).

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

Theorem 16.19 ([LV], Theorem 1.12). In the setting of Proposition 16.18, fix γ, ξ ∈ Ξ(O,^{∨}G^{Γ}). Recall from (7.10)–(7.11) the perverse sheaf P (γ) and the local system V_{ξ} on S_{ξ}.

a) H^{−d(γ)+i}P (γ) = 0 if i is odd.

b) The multiplicity of V_{ξ} in H^{−d(γ)+i}P (γ)|_{S_{ξ}} is the coefficient of u^{i/2} in P_{g}(ξ, γ).

c) c_{g}(ξ, γ) = (−1)^{d(ξ)−d(γ)}P_{g}(ξ, γ)(1).

d) The specialization C_{g}(γ)(1) ∈ KX(O,^{∨}G^{Γ}) is equal to (−1)^{d(γ)}P (γ).

Here the first two assertions are in [LV] (although our present definition of P (γ), 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.

Proposition 16.20 ([IC4], Lemma 12.15). Suppose we are in the setting (16.7)– (16.12) (so that in particular G_{ℝ} is a linear group). Then there is for every element Θ ∈ L^{z}(O, G_{ℝ}) an element

C_{r}(Θ) = ∑_{Λ}(−1)^{l^{I}(Θ)−l^{I}(Λ)}P_{r}(Λ, Θ)(u)M(Λ) (P_{r}(Λ, Θ) ∈ ℤ[u^{1/2}, u^{−1/2}])

of KΠ^{z}(O, G_{ℝ}) characterized by the following properties.

i) D_{Π}C_{r}(Θ) = u^{−l^{I}(Θ)}C_{r}(Θ) (cf. Proposition 16.17). ii) P_{r}(Θ, Θ) = 1. iii) P_{r}(Λ, Θ) ≠ 0 only if Λ ≤ Θ in the Bruhat order (cf. (15.3)(d)). iv) If Λ ≠ Θ, then P_{r}(Λ, Θ) is a polynomial in u of degree at most 1/2(l^{I}(Θ) −

l^{I}(Λ) − 1).

Again there is a better uniqueness theorem and an algorithm for computing the polynomials. We call the P_{r} 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.

Proposition 16.21. Suppose Λ = (Λ^{can}, R_{iℝ}^{+}, R_{ℝ}^{+}) is a G_{ℝ}-limit character of T_{ℝ} of infinitesimal character O (cf. Definition 11.2 and (16.7)). Choose a Cartan involution θ for G_{ℝ} preserving T_{ℝ}. Write λ for the differential of Λ^{can}, so that R^{+}(Λ) is the corresponding set of positive integral roots (cf. (16.9)(d)). After replacing Λ by an equivalent limit character, assume that R_{ℝ}^{+}(λ) ⊂ R_{ℝ}^{+} (Definition 11.6). Fix a system of positive roots

R^{+} ⊃ R^{+}(λ) ∪ R_{ℝ}^{+}

for T in G. Define

b = 1/2|{ α ∈ R^{+} − R^{+}(λ) | θα ∈ R^{+} }|,

a non-negative integer. Let 𝔫 be the nilpotent subalgebra spanned by the negative root vectors for R^{+}, so that Λ^{can} ⊗ ρ(𝔫) is a character of T_{ℝ}^{can,G} of type z. Recall from Definition 11.2 the dual standard representation M̃(Λ).

a) The weight Λ^{can} ⊗ ρ(𝔫) occurs in H_{i}(𝔫, M̃(Λ)) exactly once, in degree b − l^{I}(Λ).

b) If Θ is another standard limit character not equivalent to Λ, then Λ^{can} ⊗ ρ(𝔫) does not occur in H_{i}(𝔫, M̃(Θ)).

c) For any standard limit character Θ, the representation-theoretic character matrix is given by

c_{r}(Λ, Θ) = (−1)^{l^{I}(Λ)+b} ∑_{i}(−1)^{i}(multiplicity of Λ^{can} ⊗ ρ(𝔫) in H_{i}(𝔫, π(Θ))).

We use here the Harish-Chandra module version of the standard representations (cf. (15.2), taking for K_{ℝ} the fixed points of the Cartan involution θ chosen at the beginning of the proposition. The Lie algebra 𝔱 and the group T_{ℝ} ∩ K_{ℝ} both act on the Lie algebra homology groups, and the multiplicities are to be interpreted in the category of (𝔱, T_{ℝ} ∩ K_{ℝ})-modules. It is not difficult to reformulate the result without a choice of θ, using instead the complex conjugation on T coming from T_{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.

Proof. Since G_{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 α, θα 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 𝔫 homology is a well-defined map from the Grothendieck group KΠ^{z}(G_{ℝ}) to KΠ^{z}(T_{ℝ}).

Theorem 16.22. In the setting of Proposition 16.20, fix Λ and Θ in L^{z}(O, G_{ℝ}), and choose 𝔫 as in Proposition 16.21. a) Λ^{can} ⊗ ρ(𝔫) does not occur in H_{b−l^{I}(Θ)+i}(𝔫, π(Θ)) if i is odd.

b) The multiplicity of Λ^{can} ⊗ ρ(𝔫) in H_{b−l^{I}(Θ)+i}(𝔫, π(Θ)) is the coefficient of u^{i/2} in P_{r}(Λ, Θ).

c) c_{r}(Λ, Θ) = (−1)^{l^{I}(Λ)−l^{I}(Θ)}P_{r}(Λ, Θ)(1).

d) The specialization C_{r}(Θ)(1) ∈ KΠ^{z}(O, G_{ℝ}) is equal to π(Θ).

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

Definition 16.23. Suppose we are in the setting (15.6). The mixed Grothendieck groups KΠ^{z}(O, G/ℝ) and KX(O,^{∨}G^{Γ}) (cf. (16.12)) are both free ℤ[u^{1/2}, u^{−1/2}]- modules on bases parametrized by Ξ^{z}(O, G/ℝ). It therefore makes sense to define the canonical perfect pairing

⟨, ⟩ : KΠ^{z}(O, G/ℝ) × KX(O,^{∨}G^{Γ}) → ℤ[u^{1/2}, u^{−1/2}]

to be the ℤ[u^{1/2}, u^{−1/2}]-linear map satisfying

⟨M(ξ), μ(γ)⟩ = e(ξ)δ_{ξ,γ}u^{1/2(d(γ)+l^{I}(ξ))}

(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.

Theorem 16.24. Suppose we are in the setting (15.6). With respect to the pairing of Definition 16.23, the elements C_{r}(ξ) and C_{g}(γ) of Propositions 16.18 and 16.20 satisfy

⟨C_{r}(ξ), C_{g}(γ)⟩ = e(ξ)δ_{ξ,γ}u^{1/2(d(γ)+l^{I}(ξ))}.

The geometric and representation-theoretic Kazhdan-Lusztig matrices are essentially inverse transposes of each other:

∑_{η}(−1)^{l^{I}(ξ)−l^{I}(η)}P_{r}(η, ξ)P_{g}(η, γ) = δ_{ξ,γ}.

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.

17. Proof of Theorems 16.22 and 16.24.

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 Δ(O) of (16.9) is contained in a set of simple roots for R(G, T_{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(^{∨}G)^{θ_{Z}} to be trivial; this changes nothing.

To begin, we must choose a system of positive roots

R^{+}(G, T_{O}) ⊃ R^{+}(O)  (17.1)(a)

for the root system R(G, T_{O}). There is no distinguished choice for this positive system, and in fact the argument will use several different ones. We write

ρ_{O} = 1/2 ∑_{α∈R^{+}(G,T_{O})} α.  (17.1)(b)

Now classical intertwining operator methods (as for example in [SV]) show that c_{r}(Λ, Θ) is unchanged by a small modification of Θ that does not affect the integral roots. (Implicit here is the assertion that it is possible to make a corresponding modification of Λ.) After making such a modification, we may assume that the infinitesimal character is rational; that is, (in the notation of (16.8)) that there is a positive integer n with

n(λ_{O} − ρ_{O}) = γ_{O} ∈ X^{∗}(T_{O}).  (17.1)(c)

Define

B = variety of Borel subalgebras of 𝔤.  (17.2)(a)

If 𝔟 ∈ B is any Borel subalgebra with Cartan subalgebra 𝔱, then there is a natural isomorphism j : 𝔱_{O} → 𝔱 carrying R^{+}(G, T_{O}) to the roots of 𝔱 in 𝔤/𝔟. Using these isomorphisms, and the character γ_{O} of T_{O}, we can define an algebraic line bundle

L → B.  (17.2)(b)

(If the infinitesimal character is integral, then L has non-trivial sections.) The complexification K_{ℂ} of a maximal compact subgroup K_{ℝ} of G_{ℝ} acts algebraically on B and L, with a finite number of orbits on B. We make the multiplicative group ℂ^{×} act on L one fiber at a time, by

z · ξ = z^{n}ξ  (z ∈ ℂ^{×}, ξ ∈ L).  (17.2)(c)

Then the product group H = K_{ℂ} × ℂ^{×} acts on

L^{×} = L − zero section ,  (17.2)(d)

with finitely many orbits corresponding precisely to the orbits of K_{ℂ} on B.

It is convenient to develop this situation a little more generally at first. So suppose Y is a smooth complex algebraic variety, and L is an algebraic line bundle. Write L^{×} for L with the zero section removed:

π : L^{×} → Y  (17.3)(a)

a principal ℂ^{×} bundle. We make ℂ^{×} act on L^{×} as in (17.2)(c):

z · ξ = z^{n}ξ  (z ∈ ℂ^{×}, ξ ∈ L^{×}).  (17.3)(b)

We may therefore speak of ℂ^{×}-equivariant D_{L^{×}}-modules on L^{×}. If M is such a module, then its direct image on Y is graded by the action of ℂ^{×}:

π_{∗}(M) = ∑_{k∈ℤ} π_{∗}(M)(k)  (17.3)(c)

We say that M is genuine if the group of nth roots of unity in ℂ^{×} acts by the inverse of the tautological character:

ω · m = ω^{−1}m  (ω^{n} = 1, m ∈ M)  (17.3)(d)

Evidently this is equivalent to

π_{∗}(M) = ∑_{k≡−1 (mod n)} π_{∗}(M)(k)  (17.3)(e)

One can define a sheaf of algebras D_{Y} (L^{1/n}) on Y, as follows. Informally, D_{Y} (L^{1/n}) is the algebra of differential operators on sections of the (1/n)th power of L. The difficulty is that there is usually no such line bundle. Now if k is an integer, the space of sections of L^{⊗k} may be identified with the space of functions on L^{×} homogeneous of degree −k in the fiber variable; that is, with functions satisfying

f(z · ξ) = z^{−nk}f(ξ).  (17.4)(a)

Write E for the (Euler) vector field on L^{×}, induced by the vector field z \frac{d}{dz} on ℂ^{×} and the action (17.2)(c). This is a globally defined vector field on L^{×}. Clearly

{ functions homogeneous of degree −k } = { functions killed by E + nk }.

(17.4)(b)

It follows easily that the differential operators on sections of L^{⊗k} may be identified with the differential operators on L commuting with E, modulo the ideal generated by E + nk. This suggests defining (over every open set U ⊂ Y)

D_{Y} (L^{1/n})(U) = (D_{L^{×}}(π^{−1}U))^{E}/(E + 1)  (17.4)(c)

Here on the right we are dividing by the ideal generated by E + 1.

Proposition 17.5. In the setting (17.3)-(17.4), there is a natural equivalence of categories

( genuine ℂ^{×}-equivariant D_{L^{×}}-modules on L^{×} ) ↔ ( D_{Y} (L^{1/n})-modules on Y ).

In the notation of (17.3)(e), the equivalence is

M ↔ π_{∗}(M)(−1).

This is easy general nonsense.

We now return to the setting of (17.1) and (17.2). The action of G on B lifts to L^{×}, and so defines an operator representation

ψ_{L^{×}} : U(𝔤) → D_{L^{×}}.  (17.6)(a)

The action of G commutes with that of ℂ^{×}, so the image of U(𝔤) commutes with E. By (17.4)(c), we get

ψ_{B}(L^{1/n}) : U(𝔤) → D_{B}(L^{1/n}).  (17.6)(b)

We write

I_{B}(L^{1/n}) = ker ψ_{B}(L^{1/n}) ⊂ U(𝔤).  (17.6)(c)

Theorem 17.7 (Beilinson-Bernstein localization theorem — see [BB]). Suppose we are in the setting (17.1)–(17.2) (so that in particular the weight λ_{O} is regular). Use the notation of (17.3)–(17.4) and (17.6).

a) The operator representation (17.6)(b) is surjective, with kernel equal to the ideal generated by the kernel of the infinitesimal character χ_{O}.

b) The global sections and localization functors provide an equivalence of categories between quasicoherent sheaves of D_{B}(L^{1/n})-modules on B and 𝔤-modules of infinitesimal character χ_{O}.

Corollary 17.8. In the setting of Theorem 17.7, there is a (contravariant) equivalence of categories between finite-length (𝔤, K_{ℂ})-modules of infinitesimal character χ_{O}, and genuine H-equivariant perverse sheaves on L^{×}.

Here H = K_{ℂ} × ℂ^{×} acts on L^{×} 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 Hequivariant coherent D_{L^{×}}-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 L^{×} are parametrized by genuine H-equivariant local systems on orbits; so we must study such local systems.

Lemma 17.9. In the setting of Theorem 17.7, there is a natural bijection between genuine H-equivariant local systems on H-orbits on L^{×}, and equivalence classes of limit characters in L(O, G_{ℝ}) (cf. (15.5)). Suppose the limit character Λ 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}(Λ).

Proof. Fix a K_{ℂ}-orbit S_{0} on B, and write S for its preimage in L^{×} (an orbit of H). We can find a Borel subalgebra

𝔟 = 𝔱 + 𝔫 ∈ S_{0}  (17.10)(a)

with the property that 𝔱 is both θ-stable and defined over ℝ. The stabilizer B_{K} of 𝔟 in K has a Levi decomposition

B_{K} = T_{K}N_{K};  (17.10)(b)

the first factor is reductive, with compact real form

T_{K,ℝ} = T_{K} ∩ K_{ℝ} = T_{ℝ} ∩ K_{ℝ}.  (17.10)(c)

There is a natural map from B_{K} into the Borel subgroup B corresponding to 𝔟 (induced by restricting (11.1)(a) to K_{ℝ} and complexifying). On the other hand, the line bundle L gives a character γ of B, and so of B_{K}. The stabilizer in H of any point ξ of L^{×} over 𝔟 is then

B̃_{K} = { (b, z) | b ∈ B_{K}, z ∈ ℂ^{×}, γ(b) = z^{−n} }.  (17.10)(d)

By projection on the first factor, we see that B̃_{K} is a central extension of B_{K} by the nth roots of one. Projection on the second factor defines a genuine character of B̃_{K} of differential equal to ρ − λ. Tensoring with this character therefore defines a bijection

( characters of B_{K} with differential λ − ρ ) ↔ ( genuine characters of B̃_{K}/(B̃_{K})_{0} ).

(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_{ℝ} with differential λ − ρ. (The point is that T_{ℝ} is a direct product of T_{K,ℝ} with a vector group, so that characters with specified differential are determined by their restrictions to T_{K,ℝ}.) Such characters in turn give rise (by twisting by ρ as in (11.3)) to limit characters of differential λ, and so to classes in L(O, G_{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

Proposition 17.11. In the setting of Theorem 17.7, suppose V is a finitelength Harish-Chandra module for G_{ℝ} of infinitesimal character χ_{O}. Let P be the corresponding perverse sheaf on L^{×} (Theorem 17.7). Fix an orbit S and a genuine H-equivariant irreducible local system V on S. Choose corresponding T_{ℝ}, 𝔫, and limit character Λ as in the proof of Lemma 17.9. Then the multiplicity of V in H^{i}P |_{S} is equal to the multiplicity of the character Λ^{can} ⊗ ρ(𝔫) in H_{i+dim L^{×}}(𝔫, V).

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

Theorem 17.12. In the setting of Theorem 17.7, suppose P (Θ) is the irreducible perverse sheaf on L^{×} corresponding to the limit character Θ ∈ L(O, G_{ℝ}). Write d for the dimension of the underlying orbit. Suppose Λ is another limit character, corresponding to the local system V on the orbit S. Then the multiplicity of V in H^{−d+i}P (Θ)|_{S} is the coefficient of u^{i/2} in P_{r}(Λ, Θ).

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_{α}(X) are completely reducible (cf. [IC3], section 7). Here X is an irreducible representation of infinitesimal character χ_{O}, and α ∈ Δ(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_{α}(P) for the perverse sheaves corresponding to X and U_{α}(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_{α}(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_{O}) ⊃ R^{+}(O)  (17.13)(a)

with the additional property that

α is a simple root in (R^{+})′(G, T_{O}).  (17.13)(b)

Such a choice certainly exists; the difficulty is that we cannot make one choice for all α. This change replaces all the nilpotent algebras 𝔫 whose homology we considered by slightly different algebras 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 λ_{O} by some

λ_{O′} = λ_{O} + μ,  (17.13)(c)

with μ ∈ X^{∗}(T_{O}). What we require of the new infinitesimal character is

λ_{O′} is regular and dominant for R^{+}(O), and α^{∨}(λ_{O′}) = 1.  (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 O′.

After making all of these adjustments, we are reduced to the case when our simple integral root α is simple in R^{+}(G, T), and the coroot α^{∨} is one on λ_{O}. Because of (17.1)(c), we have

α^{∨}(γ_{O}) = 0  (17.14)(a)

Write

P_{α} = variety of parabolic subalgebras of type α  (17.14)(b)

(as for example in [LV], (3.2)). The condition (17.14)(a) is what is needed for the weight γ_{O} to define an equivariant algebraic line bundle

L_{α} → P_{α}.  (17.14)(c)

The natural projection π_{α} from B to P_{α} pulls L_{α} back to L; so it defines a (proper equivariant) morphism

π_{α} : L^{×} → L_{α}^{×}.  (17.14)(d)

The fibers of π_{α} are one-dimensional projective lines. The geometric U_{α} functor on the derived category of H-equivariant constructible sheaves on L^{×} is given by

U_{α,g}(C) = π_{α}^{∗}(π_{α})_{∗}(C)  (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 L^{×} as above. By the (very deep!) decomposition theorem of Beilinson, Bernstein, Deligne, and Gabber ([BBD], Theorem 6.2.5), the direct image (π_{α})_{∗}P (in an appropriate derived category) is a direct sum of irreducible (genuine H-equivariant) perverse sheaves on L_{α}^{×}, with various degree shifts. By Proposition 7.15, π_{α}^{∗}[1] carries irreducible perverse sheaves on L_{α}^{×} to irreducible perverse sheaves on L^{×}. It follows that U_{α,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_{α,g}(P) completely. The result agrees with the calculation of the Lie algebra homology of U_{α}(X) made in [Green] (compare the argument at (7.9) – (7.12) in [IC3]). It follows first of all that U_{α}(P) and U_{α,g}(P) have the same image in the Grothendieck group. Now an equivariant perverse sheaf in our setting that is 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_{α}(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 KX(O,^{∨}G^{Γ}), then to identify this dual with KΠ^{z}(O, G/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}(γ)}; an element of this basis must for formal reasons satisfy the requirements of Definition 16.20 characterizing the C_{r}(ξ). The first point is therefore to define a dual Hecke module.

Definition 17.15. In the setting of (16.10), suppose M is a module for H(O). Set

M^{∗} = Hom_{ℤ[u^{1/2},u^{−1/2}]}(M, ℤ[u^{1/2}, u^{−1/2}]).  (17.15)(a)

Any ℤ[u^{1/2}, u^{−1/2}]-linear map A on M defines a ℤ[u^{1/2}, u^{−1/2}]-linear map A^{t} on M^{∗}, by the requirement

(A^{t}μ)(m) = μ(Am)  (m ∈ M, μ ∈ M^{∗}).  (17.15)(b)

As usual, we have (AB)^{t} = B^{t}A^{t}. Since the Hecke algebra is non-commutative, we cannot make M^{∗} into a module for it just by transposing the action on M; we must twist by an anti-automorphism of H(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)).  (17.15)(c)

The invertibility of the other T_{w} then follows from (16.10)(b).) We can therefore make M^{∗} a module for the Hecke algebra by defining

T_{w} · μ = (−u)^{l(w)}(T_{w}^{−1})^{t} · μ.  (17.15)(d)

Using (17.15)(b)–(d), we can get three equivalent formulations of this definition:

\begin{aligned}(T_{s} · μ)(m) & = μ((−T_{s} − 1 + u) · m), or \\ ((T_{s} + 1) · μ)(m) & = −μ((T_{s} − u) · m), or \\ ((T_{s} − u) · μ)(m) & = −μ((T_{s} + 1) · m). \end{aligned}  (17.15)(e)

Suppose that M admits a Verdier duality D_{M}. We can define a ℤ-linear involution D_{M^{∗}} on M^{∗} by

(D_{M^{∗}}μ)(m) = \overline{μ(D_{M}m)}  (17.15)(f)

Here bar denotes the automorphism of ℤ[u^{1/2}, u^{−1/2}] defined by \overline{u^{1/2}} = u^{−1/2}. It is straightforward to check that D_{M^{∗}} is a Verdier duality for M^{∗} (Definition 16.14).

Proposition 17.16. Suppose we are in the setting of (15.6); use the notation of (16.12). Then the pairing of Definition 16.23 extends to

⟨, ⟩ : \overline{K}Π^{z}(O, G/ℝ) × KX(O,^{∨}G^{Γ}) → ℤ[u^{1/2}, u^{−1/2}]

On this level it provides an identification

\overline{K}Π^{z}(O, G/ℝ) ≃ Hom_{ℤ[u^{1/2},u^{−1/2}]}(KX(O,^{∨}G^{Γ}), ℤ[u^{1/2}, u^{−1/2}]) = KX(O,^{∨}G^{Γ})^{∗}.

The Hecke algebra actions provided by Proposition 16.13 and Definition 17.15 are identified by this isomorphism.

Proof. Because the representation-theoretic character matrix is a direct sum of invertible finite blocks, the formal mixed Grothendieck group \overline{K}Π^{z}(O, G/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))

⟨(T_{s} − u)M(ξ_{1}), μ(ξ_{2})⟩ = −⟨M(ξ_{1}), (T_{s} + 1)μ(ξ_{2})⟩  (17.17)

for every ξ_{1}, ξ_{2} ∈ Ξ^{z}(O, G/ℝ), and every simple reflection s = s_{α} ∈ S(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 ξ_{1} corresponds to a complete Langlands parameter (ϕ_{1}, τ_{1}) for a Cartan subgroup^{d}T_{1}^{Γ} of^{∨}G^{Γ} (Definition 12.4 and Theorem 12.9). Choose a compatible based Cartan subgroup structure

dT_{1}^{Γ} = (^{d}T _{1}^{Γ}, S(^{d}T _{1}^{Γ}),^{d}R_{iℝ,1}^{+},^{d}R_{ℝ,1}^{+})  (17.18)(a)

(Definition 13.11). Choose a based Cartan subgroup

T_{1}^{Γ} = (T_{1}^{Γ}, W(T_{1}^{Γ}), R_{iℝ,1}^{+}, R_{ℝ,1}^{+})  (17.18)(b)

for G^{Γ} and a pairing

ζ_{1} : ^{∨}T_{1} → ^{d}T_{1}  (17.18)(c)

between them (Definition 13.9 and Proposition 13.10(b)). Construct a limit character (δ_{1}, Λ_{1}) for T_{1}^{Γ} (Definition 12.1) as in Proposition 13.12. By the definition in Proposition 13.13, the standard representation M(ξ_{1}) is attached to the real form δ_{1} of G and the limit character Λ_{1}. The parameter λ_{1} ∈ ^{d}T_{1} attached to ϕ_{1} (Proposition 5.6) belongs to O, so (16.7)(d) provides natural isomorphisms

j(λ_{O}, λ_{1}) : ^{∨}T_{O} → ^{d}T_{1},  i(λ_{O}, λ_{1}) : T_{O} → T_{1}.  (17.18)(d)

Under these maps, the simple root α ∈ Δ(O) of (17.17) corresponds to

α_{1} ∈ Δ(λ_{1})  (17.18)(e)

(a root of T_{1} in G), and α^{∨} to a root

dα_{1} ∈ ^{d}Δ(λ_{1})  (17.18)(f)

of ^{d}T_{1} in ^{∨}G.

For our easy case, assume that α_{1} is a compact imaginary root of T_{1} in G(ℝ, δ_{1}), and that ξ_{1} = ξ_{2}. By [IC4], Definition 12.3,

T_{s}M(ξ_{1}) = uM(ξ_{1}).  (17.19)(a)

By Proposition 13.12(c), ^{d}α_{1} is a real root failing to satisfy the parity condition. By [LV], Lemma 3.5(e),

T_{s}μ(ξ_{2}) = −μ(ξ_{2}).  (17.19)(b)

It follows from (17.19) that both sides of (17.17) are zero.

For a hard case, assume that α_{1} is noncompact imaginary in G(ℝ, δ_{1}), and that the reflection s_{α_{1}} does not belong to the Weyl group of T_{1}(ℝ) in G(ℝ, δ_{1}). (In the terminology of [Green], α_{1} is “type I.” This case arises in SL(2, ℝ) but not in P GL(2, ℝ). Our assumption on ξ_{2} will be formulated after (17.20).) Define G(α_{1}) to be the subgroup generated by T_{1} and the image of the root subgroup ϕ_{α_{1}} (see (11.10)). The assumption that α_{1} is imaginary implies that G(α_{1}) is normalized by T_{1}^{Γ}, and in fact

G(α_{1})^{Γ} = G(α_{1})T_{1}^{Γ}  (17.20)(a)

is a weak extended group. By the proof of Proposition 13.6 (after (14.10)), α_{1} is also noncompact imaginary with respect to the real forms in W(T_{1}^{Γ}). We can therefore find an element

δ_{1}^{0} ∈ W(T_{1}^{Γ})  (17.20)(b)

that acts as δ_{1} does on G(α_{1}). Up to center, the common real form G(α_{1}, δ_{1}) = G(α_{1}, δ_{1}^{0}) is locally isomorphic to SL(2, ℝ), and T_{1}(ℝ) corresponds to a compact Cartan subgroup. Now let T_{2}(ℝ) be another Cartan subgroup of G(α_{1}, δ_{1}^{0}), corresponding to a split Cartan subgroup of SL(2, ℝ). Define

T_{2}^{Γ} = group generated by T_{2} and δ_{1}^{0},  W(T_{2}^{Γ}) = Ad(T_{2}) · δ_{1}^{0}.  (17.20)(c)

This is a Cartan subgroup of G^{Γ}, the Cayley transform of T_{1}^{Γ} through α_{1}. (Compare [Green], Definition 8.3.4.) We want to define on T_{2}^{Γ} a structure of based Cartan subgroup. To do that, we first fix an isomorphism

c : T_{1} → T_{2}  (17.20)(d)

given by conjugation by an element of G(α_{1}). This choice carries α_{1} to a real root α_{2} of T_{2} in G. It identifies the imaginary roots in T_{2} with the imaginary roots in T_{1} orthogonal to α_{1}, so we can define

R_{iℝ,2}^{+} = c(R_{iℝ,1}^{+}) ∩ R_{iℝ,2}.  (17.20)(e)

On the other hand, c^{−1} carries the real roots of T_{2} orthogonal to α_{2} onto the real roots of T_{1}. We choose a set of positive real roots subject to the two conditions

R_{ℝ,2}^{+} ⊃ c(R_{ℝ,1}^{+}),  α_{2} is simple in R_{ℝ,2}^{+}.  (17.20)(f)

This is certainly possible. We claim that

T_{2}^{Γ} = (T_{2}^{Γ}, W(T_{2}^{Γ}), R_{iℝ,2}^{+}, R_{ℝ,2}^{+})  (17.20)(g)

is a based Cartan subgroup of G^{Γ}. The only difficult condition is (e) of Definition 13.5. To check it, suppose Λ_{1}^{0} is a limit character for G(ℝ, δ_{1}^{0}) as in Definition 13.5(e). Form the Cayley transform of this limit character through the noncompact root α_{1} ([Green], Definition 8.3.6). This is a limit character Λ_{2}^{0} for the Cartan subgroup T_{2}(ℝ) of G(ℝ, δ_{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(Λ_{2}^{0}) contains M(Λ_{1}^{0}) as a subrepresentation. Since the latter is assumed to admit a Whittaker model, the former must as well; so T_{2}^{Γ} is a based Cartan subgroup. We define

Λ_{2} = (Λ_{2}^{can}, R_{iℝ,2}^{+}, R_{ℝ,2}^{+})  (17.20)(h)

to be the Cayley transform of Λ_{1} through the root α_{1} ([Green], Definition 8.3.6).

We assume (back in (17.17)) that ξ_{2} is the geometric parameter corresponding to (Λ_{2}, δ_{1}). Definition 12.3(c) of [IC4] expresses (T_{s} −u)M(ξ_{1}) in terms of standard representations. The coefficient of M(ξ_{2}) is 1. By Definition 16.23, the left side of (17.17) in this case is therefore e(ξ_{2})u^{1/2(d(ξ_{2})+l^{I}(ξ_{2}))}.

To continue with our calculation, we need to describe a based Cartan subgroup for ^{∨}G^{Γ} paired with T_{2}^{Γ}. Since α_{1} was assumed to be real, the corresponding root dα_{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_{2}^{Γ} of ^{d}T_{1}^{Γ} through ^{d}α_{1}. Specifically , the map ϕ_{1} of (17.18) provides an element

y ∈ ^{d}T_{1}^{Γ} −^{d}T_{1},  (17.21)(a)

and we can choose

∨δ_{1} ∈ S(^{d}T_{1}^{Γ})  (17.21)(b)

acting on ^{∨}G(^{d}α_{1}) as y does. We define

S(^{d}T_{2}^{Γ}) = Ad(^{d}T_{2}) ·^{∨}δ_{1},  (17.21)(c)

and fix an isomorphism ^{d}c between the maximal tori inner for ^{∨}G(^{d}α_{1}). As in (17.20)(e), this choice provides a noncompact imaginary root ^{d}α_{2}, and a natural choice ^{d}R_{ℝ,2}^{+} of positive real roots. It also carries λ_{1} ∈ ^{d}𝔱_{1} to λ_{2} ∈ ^{d}𝔱_{2}, and the pair (y, λ_{2}) defines a Langlands parameter

ϕ_{2} : W_{ℝ} → ^{d}T_{2}^{Γ}.  (17.21)(d)

We want to define a pairing between T_{2} and ^{d}T_{2}. We have a pairing ζ_{1} from (17.18)(c), and the map c of (17.20)(d) defines an isomorphism ^{∨}c from ^{∨}T_{2} to ^{∨}T_{1}. Put

ζ_{2} = ^{d}c ∘ ζ_{1} ∘^{∨}c : ^{∨}T_{2} → ^{d}T_{2},  (17.21)(e)

Finally, let ^{d}R_{iℝ,2}^{+} be the set of roots corresponding to R_{ℝ,2}^{+} under this pairing. By construction and (17.20)(f),

R_{iℝ,2}^{+} ⊃ ^{d}c(R_{iℝ,1}^{+}),  ^{d}α_{2} is simple in R_{iℝ,2}^{+}.  (17.21)(f)

We claim that R_{iℝ,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

dT_{2}^{Γ} = (^{d}T _{2}^{Γ}, S(^{d}T _{2}^{Γ}),^{d}R_{iℝ,2}^{+},^{d}R_{ℝ,2}^{+})  (17.21)(g)

is a based Cartan subgroup (Definition 13.7) paired with T_{2}^{Γ} by ζ_{2}. The strong real form δ_{1} of T_{2} (cf. (17.20)) defines a character τ_{2} of the canonical component group for ^{d}T_{2}^{Γ} with respect to ^{∨}G^{Γ} (Definition 12.4 and Proposition 13.12(a)). By Proposition 13.12, the pair (ϕ_{2}, τ_{2}) corresponds to the complete geometric parameter ξ_{2}.

By inspection of the definitions (particularly the proofs in section 9) one can find the following more direct description of τ_{2}. (Additional details may be found in [Green], section 8.3). Write

A_{i} = (^{d}T_{i}^{y})^{alg,^{∨}G}/(^{d}T_{i}^{y})_{0}^{alg,^{∨}G}  (17.22)(a)

By calculation in SL(2), one sees that

(^{d}T_{1}^{y})_{0}^{alg,^{∨}G} ⊂ (^{d}T_{2}^{y})_{0}^{alg,^{∨}G}.  (17.22)(b)

(That is, the compact part of the more compact Cartan subalgebra is larger.) Set

A_{12} = (^{d}T_{1}^{y} ∩^{d}T_{2}^{y})^{alg,^{∨}G}/(^{d}T_{1}^{y})_{0}^{alg,^{∨}G}.  (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}α_{1} (which takes values in {±1}). Similarly, the natural map from A_{12} to A_{2} is surjective; the kernel is generated by the element \overline{m_{^{d}α_{1}}^{alg}}. Now τ_{1} is a representation of A_{1}. We may therefore restrict it to a representation τ_{12} of A_{12}. Because of Proposition 13.12(c), τ_{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)μ(ξ_{2}) in terms of other extensions by zero of sheaves on orbits. The coefficient of μ(ξ_{1}) is −1. By Definition 16.23, the right side of (17.17) is therefore e(ξ_{1})u^{1/2(d(ξ_{1})+l^{I}(ξ_{1}))}. To compare this with the formula for the left side obtained after (17.20), notice first that τ_{1} and τ_{2} agree on the central element z(ρ) by (17.22). By Definition 15.8, e(ξ_{1}) = e(ξ_{2}). Next, the geometric description of orbits in [LV] shows that d(ξ_{1}) = d(ξ_{2})+1. Finally, it is elementary to show from Definition 16.16 and (17.20) that l^{I}(ξ_{1}) = l^{I}(ξ_{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.

Corollary 17.23. The identification

\overline{K}Π^{z}(O, G/ℝ) ≃ KX(O,^{∨}G^{Γ})^{∗}

of Proposition 17.16 identifies the Verdier duality D_{Π} of Proposition 16.17 with the “conjugate transpose” of D_{X} (Definition 17.15). Explicitly,

⟨D_{Π}M, μ⟩ = \overline{⟨M, D_{X}μ⟩}

for M ∈ \overline{K}Π^{z}(O, G/ℝ) and μ ∈ KX(O,^{∨}G^{Γ}).

Proof. Proposition 17.16 guarantees that the conjugate transpose of D_{X} is a Verdier duality. Proposition 16.17 provides a characterization of D_{Π} in terms of its action on the basis {M(ξ)}. 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}(ξ) form a basis of KX(O,^{∨}G^{Γ}). By Proposition 17.16, we may therefore define elements C′_{r}(ξ) of KΠ^{z}(O, G/ℝ) by the requirement

⟨C′_{r}(ξ), C_{g}(γ)⟩ = e(ξ)δ_{ξ,γ}u^{1/2(d(γ)+l^{I}(ξ))}.

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}(ξ) = C_{r}(ξ), 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 ξ has regular infinitesimal character. We may therefore place ourselves in the setting of (12.4)– (12.6), with ξ corresponding to some complete Langlands parameter (ϕ, τ_{1}) for^{d}T^{Γ}. We can choose a set ^{d}R^{+} of positive roots for ^{d}T in ^{∨}G with the property that

dR^{+} −^{d}R_{ℝ}^{+} is preserved by θ.  (17.24)(a)

For every root α, we can define an element

m_{α} = α^{∨}(−1)  (17.24)(b)

as in (12.6). For any positive root system, we have by (4.9)(b)

z(ρ) = ∏_{α∈^{d}R^{+}} m_{α}.  (17.24)(c)

By (17.24),

m_{α}m_{θα} = m_{α}(θm_{α}) ∈ ^{d}T_{0}^{θ}.  (17.24)(d)

Since the character τ_{1} is trivial on the identity component of ^{d}T^{θ}, it follows from (17.24)(c)–(d) that

τ_{1}(z(ρ)) = ∏_{α∈^{d}R_{ℝ}^{+}} τ_{1}(m_{α}).  (17.24)(e)

Proposition 13.12(c) allows us to translate this back to G(ℝ, δ) and a Cartan subgroup T (ℝ) dual to ^{d}T^{Γ}. Here is the result. Let χ be the character of the lattice of imaginary roots for T (ℝ) in G(ℝ, δ) which is 1 on noncompact simple roots and −1 on compact simple roots. Then

e(ξ) = ∏_{β∈R_{iℝ}^{+}} χ(β).  (17.24)(f)

At this point there are several ways to proceed. For one of them, let A(ℝ) be the maximal ℝ-split torus in T (ℝ), and M(ℝ) its centralizer in G(ℝ, δ). We have shown that e(ξ) may be calculated in M(ℝ); since the same is true of Kottwitz’s sign ([Kott1], Corollary (6) on page 295) we may assume M = G. Let χ_{0} be the character of the imaginary root lattice that is −1 on each simple root. Evidently χχ_{0} is −1 exactly on the noncompact roots, so

e(ξ) = (−1)^{(number of noncompact positive roots)} · e_{0},

where e_{0} does not depend on the real form. The exponent of −1 here is just q(G(ℝ, δ)), half the dimension of the symmetric space attached to G(ℝ); so

e(ξ) = (−1)^{q(G(ℝ,δ))} · e_{0}.

On the other hand, if G(ℝ, δ′) 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(ξ) = 1 in that case. Comparing with the last formula gives e_{0} = (−1)^{q(G(ℝ,δ′))}, so finally

e(ξ) = (−1)^{q(G(ℝ,δ))−q(G(ℝ,δ′))},  (17.24)(g)

which agrees with Kottwitz’s definition. Q.E.D.

18. Strongly stable characters and Theorem 1.29.

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 η is a finite-length canonical projective representation of type z of a strong real form δ of G^{Γ}. Thus η is in particular a representation of G(ℝ, δ)^{can}. The character of η is a generalized function Θ(η) on G(ℝ, δ)^{can}, defined as follows. Suppose f is a compactly supported smooth density on G(ℝ, δ)^{can}. Then η(f) is a well-defined operator on the space of η. (If we write f as a compactly supported smooth function times a Haar measure, f(g)dg, then η(f) is given by the familiar formula

η(f) = ∫_{G(ℝ,δ)^{can}} f(g)η(g)dg.)

The operator η(f) is trace class, and the value of the generalized function Θ(η) on the test density f is by definition the trace of this operator:

Θ(η)(f) = tr(η(f))  (18.1)(a)

Of course we can immediately define the trace of any virtual canonical projective representation (cf. (15.5)):

Θ : KΠ^{z}(G(ℝ, δ)) → ( generalized functions on G(ℝ, δ)^{can} ).  (18.1)(b)

This map is injective (since characters of inequivalent irreducible representations are linearly independent). Recall that an element g of G(ℝ, δ)^{can} is called 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(ℝ, δ)_{SR}^{can} ⊂ G(ℝ, δ)^{can}  (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

Θ_{SR} : KΠ^{z}(G(ℝ, δ)) ↪ C^{∞}(G(ℝ, δ)_{SR}^{can}).  (18.1)(d)

Of course the resulting functions are constant on conjugacy classes of G(ℝ, δ)^{can}.

Definition 18.2. ([Langlands2]). In the setting of (18.1), the virtual representation η is said to be stable if whenever g and g′ are strongly regular elements of G(ℝ, δ)^{can} conjugate under the complex group G^{can}, we have

Θ_{SR}(η)(g) = Θ_{SR}(η)(g′).

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(ℝ, ∗)^{can} = { (g, δ) | δ ∈ G^{Γ} − G a strong real form, g ∈ G(ℝ, δ)^{can} } (18.3)(a)

This set is analogous to P (^{∨}G^{Γ}) (Proposition 5.6), or to various fiber products appearing in section 6. The group G^{can} acts on it by conjugation. More generally, whenever D is a set of strong real forms of G, we can define

G(ℝ, D)^{can} = { (g, δ) | δ ∈ D, g ∈ G(ℝ, δ)^{can} }  (18.3)(b)

Suppose for example that D_{0} is the equivalence class of the strong real form δ_{0}. Then

G(ℝ, D_{0})^{can} ≃ G^{can} ×_{G(ℝ,δ_{0})^{can}} G(ℝ, δ_{0})^{can},  (18.3)(c)

with G(ℝ, δ_{0})^{can} acting by conjugation to define the induced bundle. (The base space of the bundle is G^{can}/G(ℝ, δ_{0})^{can}, which is just D_{0}.) The singular space G(ℝ, ∗)^{can} is undoubtedly an interesting place to do analysis, but we will confine our attention to the smooth open subset

G(ℝ, ∗)_{SR}^{can} = { (g, δ) ∈ G(ℝ, ∗)^{can}| g is strongly regular }.  (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).

Lemma 18.4. Suppose G^{Γ} is a weak extended group (Definition 2.13). In the setting (18.3), to specify a smooth G^{can}-invariant function on G(ℝ, ∗)_{SR}^{can} is equivalent to specifying (for one representative δ_{0} of each equivalence class of strong real forms of G) a smooth G(ℝ, δ_{0})^{can}-invariant function on G(ℝ, δ_{0})_{SR}^{can}.

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

Lemma 18.5. Suppose D_{0} is an equivalence class of strong real forms of the weak extended group G^{Γ}, δ_{0} ∈ D_{0}, and η is a virtual representation of G(ℝ, δ_{0})^{can}. Use the character of η to define a smooth G^{can}-invariant function Θ_{SR}(η, D_{0}) on G(ℝ, D_{0})_{SR}^{can} ((18.3)(b) and Lemma 18.4). Then η is stable (Definition 18.2) if and only if Θ_{SR}(η, D_{0}) is constant on the fibers of the first projection

p_{1} : G(ℝ, D_{0})_{SR}^{can} → G^{can}, p_{1}(g, δ) = g.

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(ℝ, ∗)_{SR}^{can}, by means of (18.1)(d) and Lemma 18.4. Here is the appropriate class of virtual representations.

Definition 18.6 (cf. Definition 1.27). Suppose G^{Γ} is a weak extended group (Definition 2.13). As in Corollary 1.26, write \overline{K}^{z}(G/ℝ) for the set of formal infinite integral combinations of irreducible canonical projective representations of strong real forms of G, the group of formal virtual representations. Such a formal virtual representation

η = ∑_{ξ∈Ξ^{z}(G/ℝ)} n(ξ)π(ξ)

is called locally finite if for each fixed strong real form δ there are only finitely many ξ with n(ξ) ≠ 0 and δ(ξ) (the strong real form parametrized by ξ) equivalent to δ. Write

\overline{K}_{f}Π^{z}(G/ℝ)  (18.6)(a)

for the set of locally finite formal virtual representations. Evidently the map of (18.1) extends to

Θ(·, δ) : \overline{K}_{f}Π^{z}(G/ℝ) → ( generalized functions on G(ℝ, δ)^{can} ).  (18.6)(b)

Finally, these maps can be assembled into a single map

Θ_{SR}(·, ∗) : \overline{K}_{f}Π^{z}(G/ℝ) → C^{∞}(G(ℝ, ∗)_{SR}^{can}))  (18.6)(c)

defined by

Θ_{SR}(η, ∗)(g, δ) = Θ(η, δ)(g).  (18.6)(d)

That is, given a strong real form δ and a (strongly regular) element g of it, we find the (finitely many) irreducible representations of G(ℝ, δ)^{can} appearing in η; add their distribution characters; and evaluate the sum at g.

The locally finite formal virtual representation η is said to be strongly stable if the function Θ_{SR}(η, ∗) is constant on the fibers of the first projection

p_{1} : G(ℝ, ∗)_{SR}^{can} → G^{can}, p_{1}(g, δ) = g.

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.

Theorem 18.7 ([Shelstad1]). Suppose G^{Γ} is a weak extended group for G (Definition 2.13), and δ_{0} is a strong real form of G. Suppose η_{0} is a stable virtual representation of G(ℝ, δ_{0})^{can} (Definition 18.2). Then there is a strongly stable locally finite formal virtual representation η for G (Definition 18.6) with the property that

Θ_{SR}(η, ∗)(g, δ_{0}) = Θ_{SR}(η_{0})(g)

for every strongly regular element g of G(ℝ, δ_{0})^{can} (cf. (18.6)(c) and (18.1)(d)). If δ_{0} is quasisplit, then η is unique.

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

(^{∨}G^{Γ}, D)  (18.8)

for G^{Γ} with second invariant z (Definition 4.6).

Definition 18.9 ([Shelstad1], section 3). In the setting of (18.1) and (18.8), suppose ϕ ∈ P (^{∨}G^{Γ}) is a Langlands parameter (Definition 5.2). The stable standard (virtual) representation attached to ϕ is by definition the sum of all the (inequivalent) standard final limit representations of G(ℝ, δ)^{can} parametrized by ϕ (Proposition 13.12); to simplify later definitions we insert the sign e(G(ℝ, δ)) attached to the real form by Kottwitz (Definition 15.8). It is written as η_{ϕ}^{loc}(δ). (Equivalently, η_{ϕ}^{loc}(δ) may be defined as the sum of the inequivalent standard representations induced from discrete series parametrized by ϕ.) In the notation of (1.19),

η_{ϕ}^{loc}(δ) = e(G(ℝ, δ)) ∑_{{ξ=(ϕ,τ)∈Ξ^{z}(G/ℝ) \atop δ(ξ)=δ}} M(ξ);

the sum is over complete Langlands parameters (Definition 5.11), with ϕ fixed.

The terminology of the definition is justified by

Lemma 18.10 ([Shelstad1], Lemma 5.2). In the setting of Definition 18.9, the virtual representation η_{ϕ}^{loc}(δ) is stable.

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

Lemma 18.11. In the setting of (18.1), the lattice of stable virtual representations has as a basis the set of non-zero η_{ϕ}^{loc}(δ) (as ϕ varies over Φ(^{∨}G^{Γ})).

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

Theorem 18.12 ([Shelstad1], Theorem 6.3). In the setting of Definition 18.9, suppose that δ and δ′ are strong real forms of G, and that

g ∈ G(ℝ, δ)^{can} ∩ G(ℝ, δ′)^{can}

is strongly regular. Then

Θ_{SR}(η_{ϕ}^{loc}(δ))(g) = Θ_{SR}(η_{ϕ}^{loc}(δ′))(g).

Here Θ_{SR} is as in (18.1).

Definition 18.13. In the setting of Definitions 18.6 and 18.9, suppose ϕ ∈ P (^{∨}G^{Γ}) is a Langlands parameter (Definition 5.2). The strongly stable standard formal virtual representation attached to ϕ is by definition the sum of all the (inequivalent) standard final limit representations parametrized by ϕ (Proposition 13.12), normalized by the signs e(ξ) (Definition 15.8). It is written as η_{ϕ}^{loc}. In the notation of (1.19),

η_{ϕ}^{loc} = ∑_{ξ=(ϕ,τ)∈Ξ^{z}(G/ℝ)} e(ξ)M(ξ);

the sum is over complete Langlands parameters (Definition 5.11). Equivalently,

η_{ϕ}^{loc} = ∑_{δ} η_{ϕ}^{loc}(δ);

the sum is over strong real forms. Finally, we can also write

η_{ϕ}^{loc} = ∑ ; τ∈(A_{ϕ}^{loc,alg})e(ϕ, τ)M(ϕ, τ),

a sum over the (necessarily one-dimensional) irreducible representations of the universal component group.

The terminology is justified by

Theorem 18.14. In the setting of Definition 18.12, the formal virtual representation η_{ϕ}^{loc} is strongly stable. The lattice of strongly stable formal virtual representations has as a basis the set of η_{ϕ}^{loc} (as ϕ varies over Φ(^{∨}G^{Γ})).

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 η_{ϕ}^{loc} in the isomorphism of Corollary 1.26; that is, as a ℤ-linear functional on the Grothendieck group KX(^{∨}G^{Γ}).

Lemma 18.15. In the setting of Definition 18.13, write S for the orbit of ^{∨}G on X(^{∨}G^{Γ}) corresponding to ϕ (Proposition 6.17). Then

η_{ϕ}^{loc} = ∑_{ξ=(S,V)∈Ξ^{z}(G/ℝ)} e(ξ)M(ξ).

Here we identify complete geometric parameters as in Definition 7.6; the sum is therefore over irreducible equivariant local systems V on S. In the isomorphism of Corollary 1.26, this formal virtual representation corresponds to the map

χ_{S}^{loc} : Ob C(X(^{∨}G^{Γ}),^{∨}G^{alg}) → ℕ

that assigns to a constructible equivariant sheaf the dimension of its fiber over S (Definition 1.28).

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 ^{∨}G^{alg} (Definition 12.4 and Theorem 12.9), proving the assertion.) We may therefore write

χ_{S}^{loc}(C) = ∑_{V}( multiplicity of V in C restricted to S ),  (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.

19. Characteristic cycles, micro-packets, and Corollary 1.32.

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 η_{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} : 𝔥 → T_{y}Y  (19.1)(a)

from the Lie algebra 𝔥 of H into the tangent space of Y at y. If y belongs to the H-orbit S, then

a_{y}(𝔥) = T_{y}S  (19.1)(b)

If we regard 𝔥 × Y as a trivial bundle over Y, we get a bundle map

a : 𝔥 × Y → T Y  (19.1)(c)

Of course the cotangent bundle of Y is dual to the tangent bundle, so we may define the conormal bundle to the H action as the annihilator of the image of a. Explicitly,

T_{H}^{∗}(Y) = { (λ, y) | λ ∈ T_{y}^{∗}(Y), λ(a_{y}(𝔥)) = 0 }  (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_{y}S in T_{y}^{∗}(Y)):

T_{H,y}^{∗}(Y) = T_{S,y}^{∗}(Y).  (19.1)(e)

Consequently

T_{H}^{∗}(Y) = ⋃ ; H-orbits S T_{S}^{∗}(Y).  (19.1)(f)

That is, the conormal bundle to the action is the union of the conormal bundles to the orbits.

Lemma 19.2. Suppose the proalgebraic group H acts on the smooth variety Y with finitely many orbits; use the notation of (19.1).

a) The conormal bundle T_{H}^{∗}(Y) is a closed Lagrangian cone in T^{∗}(Y).

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.

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 D_{Y} of algebraic differential operators has a natural filtration by order

O_{Y} = D_{Y} (0) ⊂ D_{Y} (1) ⊂ ⋯  (19.3)(a)

The terms here are coherent O_{Y} -modules, and the multiplication of differential operators satisfies

D_{Y} (p)D_{Y} (q) ⊂ D_{Y} (p + q),  (19.3)(b)

Write p : T^{∗}(Y) → Y for the natural bundle map. The symbol calculus for differential operators is an isomorphism of graded sheaves of algebras

gr D_{Y} ≃ p_{∗}(O_{T^{∗}(Y)}).  (19.3)(c)

Because p is an affine map, p_{∗} defines an equivalence of categories

( quasicoherent sheaves of O_{T^{∗}(Y)}-modules ) → ( quasicoherent sheaves of p_{∗}O_{T^{∗}(Y)}-modules ) (19.3)(d)

([Hartshorne], Exercise II.5.17(e)).

Suppose now that M is a quasicoherent D_{Y} -module. A filtration on M is a sequence

⋯ ⊂ M(p) ⊂ M(p + 1) ⊂ ⋯  (p ∈ ℤ)  (19.4)(a)

of quasicoherent O_{Y} -submodules, satisfying

D(p)M(q) ⊂ M(p + q), ⋃_{p} M(p) = M, ⋂_{p} M(p) = 0  (19.4)(b)

The first condition makes the associated graded sheaf gr M into a sheaf of modules for gr D_{Y}. By (19.3)(c) and (d), we may therefore regard gr M as a sheaf of modules on T^{∗}(Y). The filtration on M is called good if gr M is a coherent sheaf (of O_{T^{∗}(Y)}-modules). (This implies in particular that M(p) = 0 for p sufficiently negative.)

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

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 ^{∨}G^{alg}-equivariant D-modules on X(O,^{∨}G^{Γ}) admit global ^{∨}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.

Lemma 19.6 ([Ginsburg], Corollary 1.3; cf. [Assoc], Proposition 2.2). Suppose F and G are good filtrations of the coherent D_{Y} -module M. Then there are finite filtrations

gr(M, F)(0) ⊂ gr(M, F)(1) ⊂ ⋯ ⊂ gr(M, F)(N) = gr(M, F)
gr(M, G)(0) ⊂ gr(M, G)(1) ⊂ ⋯ ⊂ gr(M, G)(N) = gr(M, G)

by graded coherent sheaves of p_{∗}O_{T^{∗}(Y)}-modules ), with the property that the corresponding subquotients are isomorphic:

gr(M, F)(j)/gr(M, F)(j − 1) ≃ gr(M, G)(j)/gr(M, G)(j − 1) (0 ≤ j ≤ N).

(There is a constant N′, 0 ≤ N′ ≤ N, with the property that the jth isomorphism shifts the grading by j − N′.)

In particular, the class of gr(M) in the Grothendieck group of coherent sheaves of O_{T^{∗}(Y)}-modules is independent of the choice of good filtration.

Definition 19.7. Suppose V is a complex algebraic variety, and N is a coherent sheaf of O_{V} -modules. The associated variety (or support) of N is the set of (not necessarily closed) points v ∈ V at which the stalk of N is not zero:

V(N) = { v ∈ V | N_{v} ≠ 0 }.

It is a closed subvariety of V, defined by the sheaf of ideals Ann N ⊂ O_{V}. The set {W_{i}} of irreducible components of V(N) is the set of maximal points (that is, points not in the closure of other points) of V(N). (If V is affine, these correspond to the minimal prime ideals containing the annihilator of N.) To each such point we assign a positive integer

m_{W_{i}}(N),

the multiplicity of N along W_{i}. One way to define it is as the length of N_{W_{i}} as a module for the local ring O_{V,W_{i}}. The associated cycle of N is the formal sum

Ch(N) = ∑_{i} m_{W_{i}}(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 N is the union of the subvarieties appearing in the associated cycle.

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].

Lemma 19.8. Suppose N and N′ are coherent sheaves of modules on the complex algebraic variety V. Assume that N and N admit finite filtrations (as O_{V} - modules) with the property that gr N ≃ gr N′. Then Ch N = Ch N′.

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

Definition 19.9 Suppose M is a coherent D_{Y} -module on the smooth algebraic variety Y. Assume first that M admits a good filtration. The characteristic variety of M (or singular support) is the support of the O_{T^{∗}(Y)}-module gr M (cf. (19.4)):

SS(M) = V(gr M) ⊂ T^{∗}Y.

By Lemmas 19.6 and 19.8, this is independent of the choice of good filtration. Since gr M is graded, the characteristic variety is a closed cone in T^{∗}(Y). The characteristic cycle of M is the associated cycle of gr M:

Ch(M) = Ch(gr 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 M. Lemmas 19.6 and 19.8 guarantee that it is independent of the choice of good filtration.

In general (that is, even if 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).

Obviously the characteristic variety of M is contained in the inverse image under p (cf. (19.3)) of the support of M. In fact

SS(M) ∩ T_{y}^{∗}(Y) ≠ ∅ ⇔ y ∈ Supp(M).  (19.10)

In this sense the characteristic variety refines the notion of support for D-modules (or, via the Riemann-Hilbert correspondence of Theorem 7.9, for perverse sheaves).

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

Examples 19.11. If M is equal to D_{Y}, then the usual filtration by degree is good. The associated graded module is O_{T^{∗}(Y)}, so Ch(D_{Y}) = 1 · T^{∗}(Y).

Suppose M is equal to O_{Y}. Then we can define

M(p) = O_{Y} (p = 0, 1, 2, …).

(This module is regular holonomic.) The associated graded module is O_{Y}; that is, it is O_{T^{∗}(Y)} modulo the ideal defining the zero section of the cotangent bundle. Consequently Ch(O_{Y}) = 1 · Y

Suppose M is the D_{Y} module generated by a delta function supported at the point y; that is,

M = D_{Y} /D_{Y} I(y).

(Here we write I(y) for the ideal in O_{Y} defining the point y. Again the module is regular holonomic.) Then M inherits a good quotient filtration from D_{Y}. The associated graded module is the sheaf of functions on the cotangent space T_{y}^{∗}(Y), so Ch(M) = 1 · T_{y}^{∗}(Y).

We return now to the equivariant setting.

Proposition 19.12. Suppose we are in the setting of Lemma 19.2, and that M is an H-equivariant coherent D_{Y} -module.

a) For each X ∈ 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 M induced by the D_{Y} -module structure agrees with the action of X defined by differentiating the action of H on M.

b) The characteristic variety of M is contained in the conormal bundle to the H action (cf. (19.1)).

c) The H-components (Lemma 19.2) of SS(M) are closures of conormal bundles of H-orbits on Y. Consequently

Ch(M) = ∑ ; H-orbits S χ_{S}^{mic}(M)\overline{T_{S}^{∗}(Y)},

for some non-negative integers χ_{S}^{mic}(M).

d) The support of M is given by

Supp(M) = ⋃_{χ_{S}^{mic}(M)≠0} \overline{S}.

e) The functions χ_{S}^{mic} are additive for short exact sequences (of H-equivariant coherent D_{Y} -modules). Consequently they define ℤ-linear functionals

χ_{S}^{mic} : K(Y, H) → ℤ

(notation (7.10)).

Proof. Part (a) is a trivial exercise in understanding the definition of equivariant 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 ∈ 𝔥. We may regard the vector field a(X) as a first-order differential operator, and therefore

a(X) : M(p) → M(p + 1).

On the other hand, the action of H preserves degree in the filtration, so the differentiated action of 𝔥 does as well. By (a), it follows that

a(X) : M(p) → M(p).

It follows that the principal symbol of a(X) annihilates gr(M). This symbol is nothing but a(X), regarded as a function on T^{∗}(Y). It follows that

V(gr M) ⊂ { (λ, y) ∈ T^{∗}(Y) | λ(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.

Definition 19.13. In the setting of (19.1), fix S ∈ Φ(Y, H) (Definition 7.1); that is, an orbit of H on Y. To every complete geometric parameter ξ′ ∈ Ξ(Y, H) is associated an irreducible perverse sheaf P (ξ′) (cf. (7.10)(d)). The conormal bundle T_{S}^{∗}(Y) has a non-negative integral multiplicity χ_{S}^{mic}(ξ′) in the characteristic cycle Ch P (ξ′) (Proposition 19.12). We define the micro-packet of geometric parameters attached to S to be the set of ξ for which this multiplicity is non-zero:

Ξ(Y, H)_{S}^{mic} = { ξ | χ_{S}^{mic}(ξ′) ≠ 0 }.

Lemma 19.14. In the setting of Definition 19.13, suppose ξ′ = (S′, V′) is a geometric parameter (Definition 7.1).

a) If ξ′ ∈ Ξ(Y, H)_{S}^{mic}, then S ⊂ \overline{S′}.

b) if S′ = S, then ξ′ ∈ Ξ(Y, H)_{S}^{mic}. In this case χ_{S}^{mic}(ξ′) is equal to the rank of the local system V′; that is, to the dimension of the corresponding representation of A_{S}^{loc} (Definition 7.1).

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 D_{Y} -module corresponding to the perverse sheaf P (ξ′) ([BorelD]; some of the main points are contained in the proof of Kashiwara’s theorem VI.7.11 in [BorelD]). Q.E.D.

Definition 19.15. In the setting of Theorem 10.4, fix an equivalence class ϕ ∈ Φ^{z}(G/ℝ) of Langlands parameters (Definition 5.3), and write S = S_{ϕ} for the corresponding orbit of ^{∨}G on X(^{∨}G^{Γ}) (Definition 7.6). Then we can define the micro-packet of geometric parameters attached to ϕ as

Ξ^{z}(G/ℝ)_{ϕ}^{mic} = Ξ(X(^{∨}G^{Γ}),^{∨}G^{alg})_{S_{ϕ}}^{mic}

(notation as in Definition 19.13). The micro-packet of ϕ is the corresponding set of irreducible representations of strong real forms of G:

Π^{z}(G/ℝ)_{ϕ}^{mic} = { π(ξ′) | ξ′ ∈ Ξ^{z}(G/ℝ)_{ϕ}^{mic} }.

Corollary 19.16. In the setting of Theorem 10.4, fix an equivalence class ϕ ∈ Φ^{z}(G/ℝ) of Langlands parameters. Then the micro-packet Π^{z}(G/ℝ)_{ϕ}^{mic} contains the L-packet Π^{z}(G/ℝ)_{ϕ}. If π(ξ′) is any irreducible representation occurring in the micro-packet, then the corresponding orbit S′ on X(^{∨}G^{Γ}) contains S_{ϕ} in its closure. There is a strongly stable formal virtual character

η_{ϕ}^{mic} = ∑_{π′∈Π^{z}(G/ℝ)_{ϕ}^{mic}} e(π′)(−1)^{d(π′)−d(ϕ)}χ_{ϕ}^{mic}(π′)π′.

Here e(π′) = ±1 is Kottwitz’ sign attached to the real form of which π′ is a representation; d(π′) is the dimension of the orbit S_{π′} on X(^{∨}G^{Γ}); and the positive integer χ_{ϕ}^{mic}(π′) is the multiplicity of the conormal bundle to S_{ϕ} in the characteristic variety of the irreducible perverse sheaf corresponding to π′. This is therefore a sum over the full L-packet Π^{z}(G/ℝ)_{ϕ}^{mic}, together with “correction terms” taken from L-packets corresponding to larger orbits on X(^{∨}G^{Γ}).

The set { η_{ϕ}^{mic} } (as ϕ varies over Φ^{z}(G/ℝ)) is a basis of the lattice of strongly stable formal virtual representations.

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 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_{ϕ} in its support.

Example 19.17. Suppose that G is adjoint. In this case strong real forms are the same as real forms, and ^{∨}G is simply connected (so there are no coverings to consider). Take (^{∨}G^{Γ}, S) to be an L-group for G in the sense of Definition 4.14. Let O ⊂ ^{∨}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 C(O) of (6.10) consists of the single (central) element z(ρ). Since ρ is regular and integral, each canonical flat Λ ⊂ F(O) is preserved by a unique Borel subgroup ^{d}B(Λ) (notation (6.6)). We may therefore identify the geometric parameter space X(O,^{∨}G^{Γ}) with the set of pairs (y,^{d}B), with y ∈ ^{∨}G^{Γ} −^{∨}G, y^{2} = z(ρ), and ^{d}B a Borel subgroup of ^{∨}G (Definition 6.9).

Now by definition the L-group structure provides a distinguished orbit S_{0} = S of ∨G on X(O,^{∨}G^{Γ}). We write ϕ_{0} for the corresponding Langlands parameter. Fix a point (y,^{d}B) ∈ S_{0}. Since ^{∨}G is simply connected, the fixed point group K(y) of the involution θ_{y} is connected. Since θ_{y} preserves ^{d}B, the intersection ^{d}B ∩ K(y) is a Borel subgroup of K(y), so it is connected as well. Evidently ^{d}B ∩ K(y) is just the isotropy group of the action of^{∨}G; so it follows that the canonical component group A_{ϕ_{0}}^{loc,alg} is trivial. The L-packet Π_{ϕ_{0}} therefore consists of a single representation.

Inspecting the constructions in the proof of Theorem 10.4, we find that M(ϕ_{0}) is a spherical principal series representation of the quasisplit real form of G (in our fixed inner class), and that π(ϕ_{0}) is the trivial representation of this group. The strongly stable standard formal virtual representation η_{ϕ_{0}}^{loc} of Definition 18.13 is M(ϕ) alone. This representation has a great many irreducible composition factors, however, occurring with relatively high multiplicity; so as a “stabilization” of the trivial representation π(ϕ_{0}), M(ϕ_{0}) is not very useful. (Another way to say this is that the relationship between the characters of π(ϕ_{0}) and M(ϕ_{0}) is quite weak.)

We consider therefore the virtual representation η_{ϕ_{0}}^{mic}. To calculate it, we need to know the multiplicity of the conormal bundle T_{S_{0}}^{∗}(X(O,^{∨}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 ϕ_{0} consists of one trivial representation of each real form of G in our inner class. The multiplicities χ_{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

η_{ϕ_{0}}^{mic} = ∑_{real forms δ}( trivial representation of G(ℝ, δ) ).

This is a charming (if rather obvious) stabilization of the trivial representation. Notice that all real forms (in the inner class) appear, even though ϕ_{0} is relevant only for the quasisplit one.

20. Characteristic cycles and Harish-Chandra modules.

Recall that Theorem 8.5 related the category of equivariant perverse sheaves (or, equivalently, equivariant 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.

Proposition 20.1. Suppose X and Y are smooth algebraic varieties, and

f : X → Y

is a smooth surjective morphism having connected geometric fibers of dimension d. Suppose M is a D_{Y} -module.

a) There is a natural inclusion

f^{∗}(T^{∗}Y) ↪ T^{∗}X

identifying the pullback of the cotangent bundle of Y with a subbundle of T^{∗}X. This induces a bundle map

f̃: f^{∗}(T^{∗}Y) → T^{∗}Y

which is again a smooth surjective morphism having connected geometric fibers of dimension d.

b) The O_{X}-module f^{∗}M carries a natural D_{X}-module structure.

c) The functor f^{∗} is a fully faithful exact functor from coherent D_{Y} modules to coherent D_{X} modules.

d) Suppose M(p) is a good filtration of M. Then

(f^{∗}M)(p) = f^{∗}(M(p))

is a good filtration of f^{∗}M. The associated graded module gr(f^{∗}M) is defined on the subvariety f^{∗}(T^{∗}Y) of T^{∗}X, where it may be computed by

gr(f^{∗}M) ≃ f̃^{∗}gr(M).

The characteristic cycle of f^{∗}M is therefore

Ch(f^{∗}M) = f̃^{−1}(Ch(M)).

e) Suppose we are in the setting of Proposition 7.15 (that is, that a pro-algebraic group H acts on everything). Suppose M is an H-equivariant coherent 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

χ_{f^{∗}S}^{mic}(f^{∗}M) = χ_{S}^{mic}(M).

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 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 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×_{H} is an equivalence of categories from H-equivariant (quasi)-coherent O_{Y} -modules to G-equivariant (quasi)-coherent 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].)

Proposition 20.2. Suppose we are in the setting of Proposition 7.14. Regard 𝔤 × T Y as a vector bundle over Y, containing the bundle 𝔥 × Y of (19.1). We make H act on the bundle by the adjoint action on 𝔤. Write i : 𝔥 × Y → 𝔤 × Y for the inclusion, and consider the bundle map

i × a : 𝔥 × Y → 𝔤 × T Y.

Write Q for the quotient bundle: the fiber at y is

Q_{y} = (𝔤 × T_{y}Y)/{(X, a_{y}(X) | X ∈ 𝔥}.

a) The tangent bundle T X of X = G ×_{H} Y is naturally isomorphic to the bundle

on X induced by Q:

T X ≃ G ×_{H} Q.

b) The action mapping a_{x} : 𝔤 → T_{x}X (cf. (19.1)) may be computed as follows. Fix a representative (g, y) for the point x of G ×_{H} Y, and an element Z ∈ 𝔤. Then

a_{x}(Z) = class of (g, (Ad(g^{−1})Z, (y, 0))).

Here (y, 0) is the zero element of T_{y}Y, so the term paired with g on the right side represents a class in Q_{y}.

c) The conormal bundle to the G action on G ×_{H} Y is naturally induced by the conormal bundle to the H action on Y:

T_{G}^{∗}(G ×_{H} Y) ≃ G ×_{H} T_{H}^{∗}(Y).

Suppose M is an H-equivariant coherent D_{Y} -module admitting an H-invariant good filtration M(p). Let N be the corresponding G-equivariant D_{X}-module (Proposition 7.14(d)).

d) As a G-equivariant O_{X}-module, N is isomorphic to G ×_{H} M.

e) The filtration

N (p) = G ×_{H} M(p)

is a G-equivariant good filtration of N. In particular,

gr(N) ≃ G ×_{H} gr(M).

f) The characteristic cycle of N may be identified as a cycle in T_{G}^{∗}X (see (c) above)

as

Ch(N) = G ×_{H} Ch(M).

In particular, the multiplicities are given by

χ_{G×_{H}S}^{mic}(N) = χ_{S}^{mic}(M).

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 × Y → Y (projection on the second factor) and f_{H} : G × Y → G ×_{H} Y (dividing by the H action). Then f_{H}^{∗}M ≃ f_{G}^{∗}N; this is more or less the definition of 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 P of parabolic subgroups in G (or parabolic subalgebras in 𝔤). As in (19.1) or (8.1) there is a bundle map

a : 𝔤 × Y ↠ T Y.  (20.3)(a)

It is surjective because Y is a homogeneous space. The transpose of a is a bundle map

a^{∗} : T^{∗}Y ↪ 𝔤^{∗} × Y.  (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 moment map

μ : T^{∗}Y → 𝔤^{∗}  (20.3)(c)

([BorBI], 2.3). The first projection is a projective morphism because Y is a projective variety; so μ is also projective. Clearly

μ(T_{y}^{∗}(Y)) = (𝔤/𝔤_{y})^{∗},  (20.3)(d)

so that the image of μ consists of all linear functionals annihilating some parabolic subalgebra in the class P ([BorBI], Proposition 2.4). We write

N_{Y}^{∗} = μ(T^{∗}Y),  (20.3)(e)

an irreducible closed cone in 𝔤^{∗}.

Proposition 20.4 (Richardson — see [Rich] or [BorBI], 2.6). In the setting of (20.3), the cone N_{Y}^{∗} is contained in the cone of nilpotent elements of 𝔤^{∗}. It is the union of finitely many orbits under G. In particular there is a unique open orbit

Z_{Y} ⊂ N_{Y}^{∗} ,

the Richardson orbit attached to Y. The dimension of Z_{Y} is equal to the dimension of T^{∗}Y; so μ is generically finite. The preimage μ^{−1}(Z_{Y}) is an open orbit of G on T^{∗}Y.

The degree of μ may be computed as follows. Fix z ∈ Z_{Y}, and a point ζ ∈ T_{y}^{∗}(Y) with μ(ζ) = z. Write G_{z} for the isotropy group of z, G_{y} for the isotropy group of y (a parabolic subgroup in P), and G_{ζ} for the isotropy group of ζ. Then

G_{ζ} = G_{y} ∩ G_{z}, G_{z} ⊃ G_{ζ} ⊃ (G_{z})_{0}.

The degree of μ is equal to the index of G_{ζ} in G_{z}. Equivalently, it is the number of distinct parabolic subalgebras in P annihilated by the linear functional z ∈ 𝔤^{∗}.

We should say a word about the meaning of “nilpotent elements” of 𝔤^{∗}. The Lie algebra 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 𝔤^{∗}. A more intrinsic discussion may be found in section 5 of [Assoc].

Finally, suppose M is a finitely generated U(𝔤)-module. By an argument parallel to (but technically much simpler than) Definition 19.9, we can define the associated variety of M,

V(M) ⊂ 𝔤^{∗}  (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 associated cycle Ch(M) ([Assoc], Definition 2.4 or [BorBIII], 1.2).

Theorem 20.6 (Borho-Brylinski — see [BorBIII], Theorem 1.9). In the setting (20.3), suppose M is a finitely generated module for U(𝔤)/I_{Y} (cf. (8.1)(c)) and M is the corresponding D_{Y} -module (Theorem 8.3). Then

V(M) = μ(SS(M)).

Keeping track of multiplicities is a more delicate task, because the fibers of μ can be quite complicated.

Theorem 20.7 (Borho-Brylinski). In the setting of Theorem 20.6, fix good filtrations of M and M. Consider the higher direct images

R^{i}μ_{∗}(gr M) = N^{i},

which we regard as finitely generated graded S(𝔤)-modules. Then N^{i} is supported on

{ z ∈ N_{Y}^{∗} | dim(μ^{−1}(z) ∩ SS M) ≥ i } ⊂ V(M).

In the Grothendieck group of coherent modules supported on V(M), we have

gr M = ∑_{i}(−1)^{i}N^{i}.

Proof. That N^{i} is coherent follows from the fact that μ 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} ≃ H^{i}(T^{∗}Y, gr M)  (20.8)(a)

([Hartshorne, Proposition III.8.5). Here we are using (19.3) to regard gr M as a sheaf on T^{∗}Y. Elements of S(𝔤) define global functions on T^{∗}Y, and so act on this cohomology. Another is

N^{i} ≃ H^{i}(Y, gr M)  (20.8)(b)

([Hartshorne, Exercise III.8.2). Here we are regarding gr M as a quasicoherent sheaf on Y that happens to be also a module for p_{∗}O_{T^{∗}Y}. The extra structure is needed to make N^{i} an S(𝔤) module, but not to compute it as a vector space or as a module for some group.

Definition 20.9. In the setting of Proposition 20.4, we say that an element ζ ∈ T^{∗}Y is regular if it belongs to the inverse image of the G-orbit Z_{Y}. Similarly, an element z ∈ N_{Y}^{∗} is Y -regular (or simply regular) if it belongs to Z_{Y}. An irreducible subvariety W of T^{∗}Y or N_{Y}^{∗} is regular if it contains a regular element; equivalently, if its subset of regular elements is open and dense.

Corollary 20.10. In the setting of Proposition 20.4, suppose that W ⊂ T^{∗}Y is a regular irreducible subvariety (Definition 20.9). Then μ(W) is a Y -regular irreducible subvariety of N_{Y}^{∗}, and every Y -regular irreducible subvariety arises in this way. If in addition μ has degree 1, then this correspondence of regular irreducible subvarieties is bijective.

In the setting of Theorem 20.7, assume that μ has degree 1. Then μ defines a multiplicity-preserving bijection from the regular irreducible components of SS(M) onto the regular irreducible components of V(M).

Definition 20.11. A symmetric pair consists of a complex reductive Lie algebra 𝔤, an involutive automorphism θ of 𝔤, and a reductive pro-algebraic group K, with the following properties:

(a) (𝔤, K) is a compatible pair (Definition 8.2);

(b) 𝔨 = 𝔤^{θ}; and

(c) θ commutes with Ad(K). We will sometimes write 𝔰 for the −1-eigenspace of θ on 𝔤, so that

𝔤 = 𝔨 + 𝔰,

the Cartan decomposition.

In the setting of (20.3), we assume in addition now that

(𝔤, K) is a symmetric pair, and K acts compatibly on Y  (20.12)(a)

Since Y may be identified with a conjugacy class P of parabolic subalgebras, the action of K may be identified with

k · 𝔭 = Ad(k)(𝔭).  (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 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}^{∗} = (𝔤/(𝔤_{y} + 𝔨))^{∗}.  (20.12)(c)

Set

N_{Y,K}^{∗} = μ(T_{K}^{∗}(Y)) = ⋃_{𝔭∈P}(𝔤/(𝔭 + 𝔨))^{∗};  (20.12)(d)

the second description is immediate from (20.12)(c). This is a closed cone in N_{Y}^{∗}.

Lemma 20.13. In the setting (20.12), suppose M ∈ F(𝔤, K, I_{Y}) (Definition 8.2); that is, that M is a finite-length (𝔤, K)-module annihilated by the kernel of the operator representation on Y. Let M be the associated K-equivariant D_{Y} -module (Theorem 8.3)(c)), so that M may be identified with the space of global sections of M.

a) The associated variety V(M) is a closed K-invariant cone in N_{Y}^{∗}. b) The module M admits a K-invariant good filtration.

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 ≃ ΓM, and let M(0) be the (coherent) O_{Y} -submodule of M generated by M(0). Then M(0) generates M as a D_{Y} module (Theorem 8.3). It follows that

M(p) = D_{Y} (p)M(0)

is a K-invariant good filtration of M. Q.E.D.

Lemma 20.14 (Kostant-Rallis; see [KR] or [Assoc], Corollary 5.20). In the setting of (20.12), the group K has a finite number of orbits on N_{Y,K}^{∗}. If z ∈ N_{Y,K}^{∗}, then

dim K · z = 1/2 dim G · z.

In particular, z is regular (Definition 20.9) if and only if dim K · z = dim Y.

Theorem 20.15 (Borho-Brylinski) In the setting (20.12), suppose M ∈ F(𝔤, K, I_{Y}) (Definition 8.2); that is, that M is a finite-length (𝔤, K)-module annihilated by the kernel of the operator representation on Y. Let M be the associated K-equivariant D_{Y} -module (Theorem 8.3)(c)), so that M may be identified with the space of global sections of M. Then the following four conditions are equivalent:

1) the Gelfand-Kirillov dimension Dim M is equal to the dimension of Y;

2) the annihilator Ann M is equal to I_{Y};

3) the associated variety V(M) contains Y -regular elements (Definition 20.9); and 4) the singular support SS M contains regular elements (Definition 20.9).

Proof. We know that the Gelfand-Kirillov dimension of M is half that of U(𝔤)/Ann M ([GKDim], Theorem 1.1); so condition (1) is equivalent to

Dim U(𝔤)/Ann M = 2 dim Y.

It is well-known that the right side is equal to the Gelfand-Kirillov dimension of U(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 Ann M ⊃ I_{Y}. It follows that

Dim U(𝔤)/Ann M ≥ Dim U(𝔤)/I_{Y} ,

with equality only if 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 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.

Lemma 20.16. In the setting (20.12), suppose S is an orbit of K on Y. Then μ(\overline{T_{S}^{∗}Y}) is the closure of a single orbit Z_{S} of K on N_{Y,K}^{∗}. We have

dim Z_{S} = dim Y − fiber dimension of μ|_{T_{S}^{∗}Y} .

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 μ(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.

Definition 20.17. In the setting (20.12), an orbit S of K on Y is called regular if any of the following equivalent conditions is satisfied:

(a) T_{S}^{∗}Y is regular;

(b) μ(T_{S}^{∗}Y) is Y -regular;

(c) the restriction of μ to T_{S}^{∗}Y has finite degree; or

(d) dim μ(T_{S}^{∗}Y) = dim Y.

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

Theorem 20.18. In the setting (20.12), suppose that the moment map μ 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 Z_{S} of K on N_{Y,K}^{∗}. This bijection is multiplicity-preserving in the following sense. Suppose the K-equivariant D_{Y} -module M corresponds to the finite-length (𝔤, K)-module M (Theorem 8.3). Then for every regular K-orbit S on Y, the multiplicity χ_{S}^{mic}(M) (Proposition 19.12) is equal to the multiplicity of \overline{Z_{S}} in the associated cycle Ch(M).

This is a special case of Corollary 20.10. It is not very difficult to analyze the case when μ has degree greater than one. For example, if we fix a regular orbit Z_{0} of K on N_{Y,K}^{∗}, then

∑_{Z_{S}=Z_{0}} degree of μ on T_{S}^{∗}Y = degree of μ on T^{∗}Y .  (20.19)

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

Example 20.20. Suppose that Y is the variety of Borel subgroups of G. Then N_{Y}^{∗} is the nilpotent cone in 𝔤^{∗}, and Z_{Y} is the open orbit of principal nilpotent elements. Suppose S is a regular orbit. We claim that S consists of θ-stable Borel subalgebras having no compact simple roots. (These are the Borel subalgebras of large type for θ in the sense of Definition 4.9.) To see this, suppose z ∈ μ(T_{S,y}^{∗}Y) is a regular element. If y corresponds to the Borel subalgebra 𝔟, this means (by (20.12)(c)) that z ∈ (𝔤/(𝔟+𝔨))^{∗}; that is, that z annihilates 𝔟. Since θ𝔟 ⊂ 𝔟+𝔨, z also annihilates θ𝔟. The linear span of any two distinct Borel subalgebras must contain a larger parabolic subalgebra; so if 𝔟 ≠ θ𝔟, then there is a parabolic 𝔭 properly containing 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 𝔟 = θ𝔟. If some simple root α is compact, we find that z must also annihilate s_{α}𝔟, and reach a contradiction again. Conversely, if (θ, 𝔟) is large, then it is easy to find a principal nilpotent element z annihilating 𝔟 with θz = −z ([AV2], Proposition A.7). Then z annihilates 𝔨, and so is a regular element of μ(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.)

Example 20.21. Suppose G = Sp(4), and Y = ℙ^{3}, the variety of (isotropic) lines in ℂ^{4}; the Levi subgroup for the corresponding parabolic is GL(1) × Sp(2). It turns out that 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 ℂ^{4} = ℂ^{2} + ℂ^{2} by block diagonal matrices with blocks g and (g^{t})^{−1}. The two copies of ℂ^{2} are in natural duality by the symplectic form. We therefore think of ℂ^{4} as consisting of pairs (v, η), with v ∈ ℂ^{2} and η a linear functional on ℂ^{2}. Obviously there are four orbits of K on Y:

S_{1}^{+} = lines with v = 0,  S_{1}^{−} = lines with η = 0  (20.21)(a)
S_{2} = lines with η(v) = 0 not in S_{1}^{±},  S_{3} = lines with η(v) ≠ 0. (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 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.

Example 20.22. Take G = Sp(2n) and Y the variety of Lagrangian subspaces of ℂ^{2n}; the corresponding parabolic has Levi factor GL(n), and the moment map has degree 1. We take K = GL(n), acting on ℂ^{n} + (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} ⊂ V_{1} ⊂ 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 ∩ ℂ^{n},  V_{1} = { v ∈ ℂ^{n} | (v, λ) ∈ L for some λ ∈ (ℂ^{n})^{∗} }. (20.22)(a)

Suppose then that (v, λ) and (v′, λ′) belong to L. The quadratic form is defined by

Q(v, v′) = λ(v′) = λ′(v);  (20.22)(b)

these are equal because their difference is the symplectic pairing of (v, λ) and (v′, λ′). ¿From this description of Y, it is clear that the orbits of K are parametrized by integers p and q with p + q ≤ n: we can take p = dim V_{0}, q = dim(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 N_{Y,K}^{∗}.

21. The classification theorem and Harish-Chandra modules for the dual group.

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 ^{∨}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 ^{∨}G^{Γ}, and a semisimple orbit

O ⊂ ^{∨}𝔤  (21.1)(a)

We fix a canonical flat Λ ⊂ O, or sometimes just a point λ ∈ O. In either case we get as in (6.6) a well-defined semisimple element

e(Λ) = exp(2πiλ) ∈ ^{∨}G  (λ ∈ Λ)  (21.1)(b)

and subgroups

P (Λ) ⊂ ^{∨}G(Λ)_{0}  (21.1)(c)

The reductive group ^{∨}G(Λ)_{0} is the identity component of the centralizer of e(Λ). Its parabolic subgroup P (Λ) is the stabilizer of Λ in the action of ^{∨}G on F(O) (cf. (6.10)). Define P(Λ)^{0} as in Proposition 6.24 to be the ^{∨}G(Λ)_{0}-orbit of Λ. Then

P(Λ)^{0} ≃ ^{∨}G(Λ)_{0}/P (Λ),  (21.1)(d)

a flag manifold for ^{∨}G(Λ)_{0}. (This space will play the rôle of Y in section 20.) Recall from (8.1) the ideal

I_{P(Λ)^{0}} ⊂ U(^{∨}𝔤(Λ)),  (21.1)(e)

and from Proposition 20.4 the nilpotent orbit

Z_{P(Λ)^{0}} ⊂ ^{∨}𝔤(Λ)^{∗}.  (21.1)(f)

Recall from (6.10)(f) the set I(Λ), on which ^{∨}G(Λ)_{0} acts with finitely many orbits I_{1}^{0}(Λ), …, I_{s}^{0}(Λ); choose representatives

y_{j} ∈ I_{j}^{0}(Λ).  (21.1)(g)

Write θ_{j} for the involutive automorphism of ^{∨}G(Λ)_{0} defined by y_{j}, K_{j}^{0} for its group of fixed points, and

∨𝔤(Λ) = 𝔨_{j} + 𝔰_{j}  (21.1)(h)

for the corresponding Cartan decomposition. Finally, recall from Proposition 6.24 that the geometric parameter space X(O,^{∨}G^{Γ}) is the disjoint union of s smooth connected subvarieties X_{j}(O,^{∨}G^{Γ}), with

X_{j}(O,^{∨}G^{Γ}) ≃ ^{∨}G ×_{K_{j}^{0}} (^{∨}G(Λ)_{0}/P (Λ))  (21.1)(i)

In particular, the orbits of ^{∨}G on X_{j}(O,^{∨}G^{Γ}) are naturally in one-to-one correspondence with the orbits of K_{j}^{0} on ^{∨}G(Λ)_{0}/P (Λ).

We will sometimes abuse this notation in the following way. The element e(Λ), the group ^{∨}G(Λ)_{0}, and the sets I_{j}^{0}(Λ) depend only on the ^{∨}G(Λ)_{0} orbit P(Λ)^{0}, and not on Λ itself. We may therefore allow Λ to vary over P(Λ)^{0}, rather than keep it fixed.

Here is a reformulation of Theorem 8.5.

Theorem 21.2. Suppose we are in the setting (21.1). Then there are natural bijections among the following sets:

1) irreducible (^{∨}𝔤(Λ), K_{j}^{alg})-modules annihilated by I_{P(Λ)^{0}};

2) irreducible K_{j}^{alg}-equivariant D_{P(Λ)^{0}}-modules; and

3) irreducible ^{∨}G^{alg}-equivariant D-modules on X_{j}(O,^{∨}G^{Γ}).

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

4) equivalence classes of irreducible canonical projective representations of type z and infinitesimal character O of strong real forms of G.

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.

Proposition 21.3. Suppose we are in the setting (21.1). Regard x = (y_{j}, Λ) as a point of X(O,^{∨}G^{Γ}) (Definition 6.9), so that

S = ^{∨}G · x ⊂ X(O,^{∨}G^{Γ}), S_{K} = K_{j}^{0} · Λ ⊂ P(Λ)^{0}

are corresponding orbits.

a) The isotropy group of the action of ^{∨}G at x (or of the action of K_{j}^{0} at Λ) is P (Λ) ∩ K_{j}^{0}.

b) The conormal space to the orbit S at x (or to the orbit S_{K} at Λ) may be identified naturally (as a module for P (Λ) ∩ K_{j}^{0}) with

T_{S,x}^{∗}(X(O,^{∨}G^{Γ})) = T_{^{∨}G,x}^{∗}(X(O,^{∨}G^{Γ})) = (^{∨}𝔤(Λ)/(𝔭(Λ) + 𝔨_{j}))^{∗}.

c) Fix a non-degenerate ^{∨}G^{Γ}-invariant symmetric bilinear form on ^{∨}𝔤, and use it to identify ^{∨}𝔤(Λ)^{∗} with ^{∨}𝔤(Λ). Then the subspace appearing in (b) corresponds to

(^{∨}𝔤(Λ)/(𝔭(Λ) + 𝔨_{j}))^{∗} ≃ 𝔫(Λ) ∩ 𝔰_{j}.

Proof. Part (a) is obvious (and contained in Proposition 6.16). For (b), combine the last description of X(O,^{∨}G^{Γ}) in Proposition 6.24 with Proposition 20.2(a) and (20.12)(d). For (c), the left side is clearly identified with 𝔭(Λ)^{⊥} ∩ 𝔨_{j}^{⊥}; 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) Ad(y_{j}) respects the form. Q.E.D.

Proposition 21.4. Suppose we are in the setting (21.1).

a) Every ^{∨}G^{alg}-equivariant D-module on X(O,^{∨}G^{Γ}) admits a ^{∨}G^{alg}-invariant good filtration.

b) Suppose that the^{∨}G^{alg}-equivariant D-module N corresponds to the K_{j}^{0,alg}-equivariant D-module M in the equivalence of Theorem 21.2. Suppose S is any orbit of ^{∨}G on X(O,^{∨}G^{Γ}); write S_{K} for the corresponding orbit of K_{j}^{0,alg} on P^{0}(Λ). Then the multiplicities in the characteristic cycle are related by

χ_{S}^{mic}(N) = χ_{S_{K}}^{mic}(M).

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

μ : T^{∗}(P(Λ)^{0}) → N_{P(Λ)^{0}}^{∗} ⊂ ^{∨}𝔤(Λ)^{∗}  (21.5)(a)

and its restriction

μ_{K_{j}^{0}} : T_{K_{j}^{0}}^{∗}(P(Λ)^{0}) → N_{P(Λ)^{0},K_{j}^{0}}^{∗} ⊂ (^{∨}𝔤(Λ)/𝔨_{j})^{∗}.  (21.5)(b)

Write

Z_{P(Λ)^{0},K_{j}^{0}} = Z_{P(Λ)^{0}} ∩ (^{∨}𝔤(Λ)/𝔨_{j})^{∗},  (21.5)(c)

a finite union of orbits of K_{j}^{0} of dimension equal to dim P(Λ)^{0} (Lemma 20.14). Recall (Definition 20.17) that an orbit S_{K} of K_{j}^{0} on P(Λ)^{0} is called regular if and only if μ(T_{S_{K}}^{∗} (P(Λ)^{0})) meets Z_{P(Λ)^{0}}. We say that an orbit S of ^{∨}G on X_{j}(O,^{∨}G^{Γ}) is regular if the corresponding orbit of K_{j}^{0} is regular (cf. (21.1)(i)).

Theorem 21.6. Suppose we are in the setting (21.1); use the notation (21.5). Suppose that the irreducible (^{∨}𝔤(Λ), K_{j}^{alg})-module M, the irreducible K_{j}^{alg}-equivariant D_{P(Λ)^{0}}-module M, and the irreducible ^{∨}G^{alg}-equivariant D-module N correspond in the bijections of Theorem 21.2. Then the following conditions are equivalent.

a) The singular support SS(N) contains the conormal bundle of a regular orbit S (Definition 19.9 and (21.5)).

b) The singular support SS(M) contains the conormal bundle of a regular orbit S_{K}.

c) The associated variety V(M) meets Z_{P(Λ)^{0},K_{j}^{0}} ((20.5) and (21.5)).

d) The annihilator Ann M is equal to I_{P(Λ)^{0}} ((21.1)(e)).

e) The Gelfand-Kirillov dimension Dim M is equal to dim P(Λ)^{0}.

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.

Theorem 21.7. In the setting of (21.1), assume that the moment map μ of (21.5)(a) has degree one. Then there are natural bijections among the following three sets:

1) regular orbits S of ^{∨}G on X_{j}(O,^{∨}G^{Γ});

2) regular orbits S_{K} of K_{j}^{0} on P(Λ)^{0}; and

3) orbits Z_{S} of K_{j}^{0} on Z_{P(Λ)^{0},K_{j}^{0}} ((21.5)(c)).

This bijection is multiplicity-preserving in the following sense. Suppose S, S_{K}, and Z_{S} correspond as above, and N, M, and M correspond as in Theorem 21.2. Then χ_{S}^{mic}(N) = χ_{S_{K}}^{mic}(M), and these are equal to the multiplicity of \overline{Z_{S}} in the associated cycle Ch(M) (cf. (20.5)).

Proof. The first bijection is a definition, and the equality χ_{S}^{mic}(N) = χ_{S_{K}}^{mic}(M) is Proposition 21.4(b). The rest of the result is Theorem 20.18. Q.E.D.

Theorem 21.8 (cf. [BVunip], Proposition 3.24). Suppose we are in the setting (21.1). Then there is a natural order-reversing bijection

I → ^{∨}I

from the set of primitive ideals in U(𝔤) of infinitesimal character λ, onto the set of primitive ideals in U(^{∨}𝔤(Λ)) containing I_{P(Λ)^{0}}. In particular, the maximal primitive ideal J_{max}(λ) of infinitesimal character λ corresponds to I_{P(Λ)^{0}}:

∨J_{max}(λ) = I_{P(Λ)^{0}}.

In the setting of Theorem 21.2 and Theorem 10.4, suppose that the irreducible (^{∨}𝔤(Λ), K_{j}^{0})-module M corresponds to the irreducible canonical projective representation π of G(ℝ, δ)^{can}. Then the annihilators correspond:

∨(Ann π) = Ann M.

In particular, the annihilator of π is a maximal primitive ideal if and only if the annihilator of M is equal to I_{P(Λ)^{0}}.

Recall that Theorem 21.6 provides several characterizations of the modules M with annihilator equal to I_{P(Λ)^{0}}.

22. Arthur parameters.

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.

Definition 22.1. Suppose H is a complex algebraic group. An element h ∈ H is called elliptic if the closure in the analytic topology of the group ⟨h⟩ generated by h is compact. Similarly, an element X ∈ 𝔥 is called elliptic if the closure in the analytic topology of the one-parameter subgroup { exp(tX) | t ∈ ℝ } is compact.

Lemma 22.2. Suppose H is a complex algebraic group, h ∈ H, and X ∈ 𝔥.

a) If h is elliptic, then h is semisimple; and if X is elliptic, then X is semisimple.

b) The element h is elliptic if and only if h^{n} is elliptic for every non-zero integer n.

Suppose T ⊂ H is an algebraic torus. Define

𝔱_{ℝ} = X_{∗}(T) ⊗_{ℤ} ℝ ⊂ 𝔱

(cf. (9.1)(d) and Lemma 9.9), so that X_{∗}(T) is a lattice in the real vector space 𝔱_{ℝ}. Finally, define

T_{c} = e(𝔱_{ℝ}) ≃ 𝔱_{ℝ}/X_{∗}(T),

a compact analytic torus in T (cf. Lemma 9.9).

c) Suppose h ∈ T. Then h is elliptic if and only if h ∈ T_{c}.

d) Suppose X ∈ 𝔱. Then X is elliptic if and only if X ∈ i𝔱_{ℝ}.

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)).

Definition 22.3. Suppose ^{∨}G^{Γ} is a weak E-group. A Langlands parameter ϕ ∈ P (^{∨}G^{Γ}) (Definition 5.2) is said to be tempered if it satisfies any of the following equivalent conditions.

(a) The closure of ϕ(W_{ℝ}) in the analytic topology is compact.

(b) In the notation of Proposition 5.6, the Lie algebra element λ + Ad(y)λ is elliptic.

(c) Suppose ^{d}T^{Γ} is a Cartan subgroup of ^{∨}G^{Γ}, and that ϕ(W_{ℝ}) ⊂ ^{d}T^{Γ}. Fix an Egroup structure on ^{d}T^{Γ}, and let T be an algebraic torus defined over ℝ admitting dT^{Γ} as an E-group. Then the canonical projective character π(ϕ) of T (ℝ)^{can} associated to ϕ (Proposition 10.6) is unitary.

(d) Fix an E-group structure on ^{∨}G^{Γ}, and let (G^{Γ}, W) be an extended group admitting ^{∨}G^{Γ} as an E-group. Then at least one of the irreducible canonical projective representations of strong real forms of G parametrized by ϕ (Theorem 10.4) is tempered.

e) In the setting of (d), all the representations parametrized by ϕ are tempered. We write P_{temp}(^{∨}G^{Γ}) for the set of tempered Langlands parameters, and Φ_{temp}(^{∨}G^{Γ})

for the corresponding set of equivalence classes.

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 ℝ^{+} and a compact group; so (a) is equivalent to ellipticity of the generator of ϕ(ℝ^{+}). This generator is precisely λ+Ad(y)λ (cf. (5.5)(a)). Hence (a) and (b) are equivalent. In the setting of (c), T (ℝ) is the direct product of a compact group and a vector group ([AV2], Corollary 3.16), so π(ϕ) is unitary if and only if its restriction to the vector group is. Now λ+Ad(y)λ 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]).

Definition 22.4 (cf. [Arthur2], section 6). Suppose ^{∨}G^{Γ} is a weak E-group (Definition 4.3). An Arthur parameter is a homomorphism

ψ : W_{ℝ} × SL(2, ℂ) → ^{∨}G^{Γ}

(see Definition 5.2) satisfying

(a) the restriction of ψ to W_{ℝ} is a tempered Langlands parameter; and

(b) the restriction of ψ to SL(2, ℂ) is holomorphic. Two such parameters are called equivalent if they are conjugate by the action of ∨G. The set of Arthur parameters is written Q(^{∨}G^{Γ}); the set of equivalence classes is written Ψ(^{∨}G^{Γ}). In the setting of Definition 5.3, we may also write Ψ(G/ℝ).

Suppose ψ is an Arthur parameter. In analogy with Definition 5.11, we define

∨G_{ψ} = centralizer in ^{∨}G of ψ(W_{ℝ} × SL(2, ℂ)),

and

A_{ψ} = ^{∨}G_{ψ}/(^{∨}G_{ψ})_{0},

the Arthur component group for ψ. Similarly,

A_{ψ}^{alg} = ^{∨}G_{ψ}^{alg}/(^{∨}G_{ψ}^{alg})_{0},

the universal component group for ψ.

The associated Langlands parameter for ψ is the homomorphism

ϕ_{ψ} : W_{ℝ} → ^{∨}G, ϕ_{ψ}(w) = ψ(w, ( \begin{matrix}|w|^{1/2} & 0 \\ 0 & |w|^{−1/2} \end{matrix} )) (w ∈ W_{ℝ}).

Attached to ϕ_{ψ} is an orbit S_{ϕ_{ψ}} of ^{∨}G on X(^{∨}G^{Γ}) (Proposition 6.17); define

S_{ψ} = S_{ϕ_{ψ}}.

Write

P_{Arthur}(^{∨}G^{Γ}) = { ϕ_{ψ} | ψ ∈ Q(^{∨}G^{Γ}) } ⊂ P (^{∨}G^{Γ})

for the image of the map on parameters, and

Φ_{Arthur}(^{∨}G^{Γ}) ⊂ Φ(^{∨}G^{Γ})

for the corresponding set of equivalence classes.

Proposition 22.5 (Arthur [Arthur1], p.10). Suppose ^{∨}G^{Γ} is a weak E-group. The map ψ ↦ ϕ_{ψ} from Arthur parameters to Langlands parameters defines a bijection on the level of equivalence classes

Ψ(^{∨}G^{Γ}) ≃ Φ_{Arthur}(^{∨}G^{Γ}).

We have

Φ_{temp}(^{∨}G^{Γ}) ⊂ Φ_{Arthur}(^{∨}G^{Γ}) ⊂ Φ(^{∨}G^{Γ}).

If ψ is an Arthur parameter, there is an inclusion

∨G_{ψ}^{Γ} ⊂ ^{∨}G_{ϕ_{ψ}}^{Γ}.

The induced maps on component groups

A_{ψ} → A_{ϕ_{ψ}}^{loc}, A_{ψ}^{alg} → A_{ϕ_{ψ}}^{loc,alg}

are surjective.

Here is the definition of the “Arthur packets” discussed in the introduction.

Definition 22.6. In the setting of Theorem 10.4, suppose ψ ∈ Ψ(^{∨}G^{Γ}) is an Arthur parameter (Definition 22.4). The Arthur packet Π_{ψ}^{z} of ψ is by definition the micro-packet of the Langlands parameter ϕ_{ψ} attached to ψ (Definition 19.15, Definition 22.4):

Π^{z}(G/ℝ)_{ψ} = Π^{z}(G/ℝ)_{ϕ_{ψ}}^{mic}.

A little more explicitly , write S = S_{ψ} as in Definition 22.4. Then Π_{ψ}^{z} 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(^{∨}G^{Γ})) in its characteristic cycle (Proposition 19.12).

Theorem 22.7. Suppose (G^{Γ}, W) is an extended group for G (Definition 1.12), and (^{∨}G^{Γ}, D) is an E-group for the corresponding inner class of real forms, with second invariant z (Definition 4.6). Fix an Arthur parameter ψ ∈ Ψ^{z}(G/ℝ) (Definition 22.4). Define the Arthur packet Π_{ψ} as in Definition 22.6.

a) Π_{ψ} contains the L-packet Π_{ϕ_{ψ}}, and at most finitely many additional representations of each strong real form of G.

b) There is a strongly stable formal virtual character

η_{ψ} = ∑_{π′∈Π_{ψ}} e(π′)(−1)^{d(π′)−d(ϕ_{ψ})}χ_{ψ}(π′)π′.

Here the terms are as explained in Corollary 19.16. In particular, e(π′) = ±1 is Kottwitz’ sign attached to the real form of which π′ is a representation, and χ_{ψ}(π′) is a positive integer.

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 π′ ∈ Π_{ψ} a representation of A_{ψ}^{alg}, of dimension equal to χ_{ψ}(π′). This will turn out to be standard micro-local geometry. To see that, however, we need a micro-local interpretation of A_{ψ}^{alg}.

We turn therefore to an analysis of the geometry of an Arthur parameter ψ. Write ψ_{0} for the restriction of ψ to W_{ℝ}, a tempered Langlands parameter. According to Proposition 5.6, ψ_{0} is determined by a pair

(y_{0}, λ_{0}),  y_{0} ∈ ^{∨}G^{Γ} −^{∨}G,  λ_{0} ∈ ^{∨}𝔤.  (22.8)(a)

Write ψ_{1} for the restriction of ψ to SL(2, ℂ), an algebraic homomorphism into ^{∨}G. We can define

y_{1} = ψ_{1} ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) ∈ ^{∨}G,  λ_{1} = dψ_{1} ( \begin{matrix}1/2 & 0 \\ 0 & −1/2 \end{matrix} ) ∈ ^{∨}𝔤.  (22.8)(b)

An elementary calculation from Definition 22.4 and Proposition 5.6 shows that the Langlands parameter ϕ_{ψ} is associated to the pair (y, λ), with

y = y_{0}y_{1},  λ = λ_{0} + λ_{1}.  (22.8)(c)

(Here and elsewhere in the following discussion, one should keep in mind that elements of ψ_{0}(W_{ℝ}) commute with elements of ψ_{1}(SL(2, ℂ).) Using λ and y, we can introduce all of the structure considered in section 6 (cf. (6.1), (6.10)). We also write θ_{y} for the conjugation action of y on ^{∨}G(λ), and

∨𝔤(λ) = 𝔨(y) + 𝔰(y)  (22.8)(c)

for the corresponding Cartan decomposition. Define

E_{ψ} = dψ_{1} ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) ∈ ^{∨}𝔤,  F_{ψ} = dψ_{1} ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ) ∈ ^{∨}𝔤;  (22.8)(d)

these are nilpotent elements. By calculation in 𝔰𝔩(2), E_{ψ} lies in the +1-eigenspace of ad(λ_{1}). Since λ_{0} and E_{ψ} commute, it follows that

E_{ψ} ∈ ^{∨}𝔤(λ)_{1} ⊂ 𝔫(λ).  (22.8)(e)

Similarly, we check that Ad(y_{1})(E_{ψ}) = −E_{ψ}, and deduce

E_{ψ} ∈ 𝔰(y).  (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.

Proposition 22.9. Suppose ^{∨}G^{Γ} is an E-group, and ψ ∈ Q(^{∨}G^{Γ}) is an Arthur parameter. Let x = p(ϕ_{ψ}) ∈ X(^{∨}G^{Γ}) (Proposition 6.17) be the corresponding point, and S = ^{∨}G · x the corresponding orbit. Fix a non-degenerate ^{∨}G^{Γ}-invariant symmetric bilinear form on ^{∨}𝔤, and use it as in Proposition 21.3 to identify the conormal space T_{S,x}^{∗}(X(^{∨}G^{Γ})) with 𝔫(λ)∩𝔰(y). Accordingly regard E_{ψ} as an element of this conormal space (cf. (22.8)(e),(f)):

E_{ψ} ∈ 𝔫(λ) ∩ 𝔰(y) ≃ T_{S,x}^{∗}(X(^{∨}G^{Γ})).

a) We have

[𝔭(λ) ∩ 𝔨(y), E_{ψ}] = 𝔫(λ) ∩ 𝔰(y).

b) The orbit (P (λ) ∩ K(y)) · E_{ψ} (with the natural action of the isotropy group on the conormal space) is Zariski dense in T_{S,x}^{∗}(X(^{∨}G^{Γ})).

c) The orbit ^{∨}G · E_{ψ} is Zariski dense in the full conormal bundle T_{S}^{∗}(X(^{∨}G^{Γ})).

d) The isotropy groups of E_{ψ} in the actions of (b) and (c) coincide; write ^{∨}G_{E_{ψ}} for this subgroup of ^{∨}G_{x}. Then ^{∨}G_{ψ} is a Levi subgroup of ^{∨}G_{E_{ψ}}.

e) The inclusion in (d) identifies the Arthur component group (Definition 22.4) naturally with the component group of ^{∨}G_{E_{ψ}}:

A_{ψ} ≃ ^{∨}G_{E_{ψ}}/(^{∨}G_{E_{ψ}})_{0}.

Similarly,

A_{ψ}^{alg} ≃ ^{∨}G_{E_{ψ}}^{alg}/(^{∨}G_{E_{ψ}}^{alg})_{0}.

Proof. By the representation theory of 𝔰𝔩(2), ad(λ_{1}) has half-integral eigenvalues. (For this discussion, we call any element of 1/2ℤ a half-integer.) By definition of 𝔤(λ), ad(λ) has integral eigenvalues on 𝔤(λ). It follows that λ_{0} = λ − λ_{1} has halfintegral eigenvalues on 𝔤(λ). In analogy with (6.1)(b), we define for half-integers r and s

∨𝔤_{r,s} = { μ ∈ ^{∨}𝔤 | [λ_{0}, μ] = rμ, [λ_{1}, μ] = sμ }.  (22.10)(a)

Then

∨𝔤(λ)_{n} = ∑_{r+s=n}^{∨}𝔤_{r,s}.  (22.10)(b)

Consulting the definitions (6.1), we deduce that

𝔫(λ) = ∑_{r+s∈ℕ−{0}}^{∨}𝔤_{r,s},  𝔭(λ) = ∑_{r+s∈ℕ}^{∨}𝔤_{r,s}.  (22.10)(c)

Now θ_{y}(λ_{0}) = θ_{y_{0}}(λ_{0}) commutes with λ_{0} and with λ_{1} (Proposition 5.6(c)) and so acts diagonalizably on each ^{∨}𝔤_{r,s}. Since ψ_{0} is tempered, λ_{0} + θ_{y_{0}}(λ_{0}) has purely imaginary eigenvalues (Definition 22.3(b)). It follows that θ_{y}(λ_{0}) acts by −r on ∨𝔤_{r,s}. On the other hand,

θ_{y}(λ_{1}) = Ad(y_{1})(λ_{1}) = λ_{1};

so θ_{y}(λ_{1}) acts by s on ^{∨}𝔤_{r,s}. Combining these two facts, we get

θ_{y} : ^{∨}𝔤_{r,s} → ^{∨}𝔤_{−r,s}.  (22.10)(d)

Combining the descriptions of θ_{y} and 𝔭(λ) in terms of the eigenspaces ^{∨}𝔤_{r,s}, we get

𝔨(y) ∩ 𝔭(λ) = { X + θ_{y}X | X ∈ ^{∨}𝔤_{r,s}, s ± r ∈ ℕ }  (22.10)(e)
𝔰(y) ∩ 𝔫(λ) = { X − θ_{y}X | X ∈ ^{∨}𝔤_{r,s}, s ± r ∈ ℕ − {0} }  (22.10)(f).

To study the adjoint action of E_{ψ}, we use the representation theory of 𝔰𝔩(2). We deduce that

ad(E_{ψ}) : ^{∨}𝔤_{r,s−1} → ^{∨}𝔤_{r,s};  (22.10)(g)

this map is surjective for s ≥ 1/2 and injective for s ≤ 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 − θ_{y}X belongs to the left side, with X ∈ ^{∨}𝔤_{r,s} and s ± r positive integers. By (22.10)(g), we can find Y ∈ ^{∨}𝔤_{r,s−1} with [Y, E_{ψ}] = X. Since θ_{y}E_{ψ} = −E_{ψ}, it follows that

[Y + θ_{y}Y, E_{ψ}] = X − θ_{y}X.

Furthermore (s−1)±r must be non-negative integers, so (by (22.10)(e)) Y +θ_{y}Y ∈ 𝔨(y) ∩ 𝔭(λ). This is (a).

For (b), recall that the conormal space T_{S,x}^{∗}(X(^{∨}G^{Γ})) has been identified with 𝔰(y) ∩ 𝔫(λ). This identification sends the isotropy action of ^{∨}G_{x} = K(y) ∩ P (λ) (Proposition 6.16) to the restriction of the adjoint action. It follows that the tangent space to the orbit (P (λ) ∩ K(y)) · E_{ψ} is just [𝔨(y) ∩ 𝔭(λ), E_{ψ}]. (Here we identify the tangent space to the vector space 𝔰(y) ∩ 𝔫(λ) at the point E_{ψ} with the vector space.) By (a), this tangent space is the full tangent space at E_{ψ}; so the orbit is open, as we wished to show.

For (c), the conormal bundle T_{S}^{∗}(X(^{∨}G^{Γ}) is an equivariant bundle over the homogeneous space S = ^{∨}G/^{∨}G_{x}. The orbits of such an action are in one-to-one correspondence with those of ^{∨}G_{x} on the fiber over x (cf. Lemma 6.15); this bijection preserves isotropy groups and codimension of orbits. Since (P (λ) ∩ K(y)) · E_{ψ} has codimension 0 in T_{S,x}^{∗}(X(^{∨}G^{Γ})), it follows that ^{∨}G · E_{ψ} has codimension 0 in T_{S}^{∗}(X(^{∨}G^{Γ})), as we wished to show.

We have already explained the first claim of (d). For the rest, we begin with the Levi decomposition

P (λ) ∩ K(y) = (L(λ) ∩ K(y))(N_{θ_{y}} ∩ K(y))  (22.11)(a)

of Proposition 6.17 and Lemma 6.18. We claim that this decomposition is inherited by the subgroup ^{∨}G_{E_{ψ}}:

∨G_{E_{ψ}} = (L(λ) ∩ K(y))_{E_{ψ}}(N_{θ_{y}} ∩ K(y))_{E_{ψ}}.  (22.11)(b)

So suppose that ln is the decomposition of an element of ^{∨}G_{E_{ψ}}; it is enough to show that l ∈ ^{∨}G_{E_{ψ}}. Using (22.10), one calculates easily that

𝔫_{θ_{y}} = ∑_{s≥|r|,s>0,s+r∈ℤ}^{∨}𝔤_{r,s}.  (22.11)(c)

It follows that

Ad(n)(E_{ψ}) = E_{ψ} + (terms in ^{∨}𝔤_{r,s} with s > 1).  (22.11)(d)

(Recall that E_{ψ} ∈ ^{∨}𝔤_{0,1}.) Applying Ad(l) to this, we get

E_{ψ} = Ad(ln)(E_{ψ}) = Ad(l)(E_{ψ}) + (terms in ^{∨}𝔤_{r,s} with s > 1).  (22.11)(e)

Projecting on the 1-weight space of λ_{1}, we find that Ad(l)(E_{ψ}) = E_{ψ}, proving (22.11)(b).

To complete the proof of (d), it suffices to show that

(L(λ) ∩ K(y))_{E_{ψ}} = ^{∨}G_{ψ}.  (22.12)(a)

That the left side contains the right is trivial from the definitions; so suppose l ∈ (L(λ) ∩ K(y))_{E_{ψ}}. We must show that l centralizes the image of ψ. It is enough to show that it fixes the images under dψ_{1} of the basis {E_{ψ}, 2λ_{1}, F_{ψ}} of 𝔰𝔩(2) (see (22.8)). Since l commutes with λ and y, it commutes also with θ_{y}λ. Now it follows from (22.10)(c) that the only eigenvalue of λ_{0} + θ_{y}λ_{0} on ^{∨}𝔤(λ) is 0; so it is central in ^{∨}𝔤(λ), and in particular is fixed by l. Hence

2λ_{1} = (λ + θ_{y}λ) − (λ_{0} + θ_{y}λ_{0})  (22.12)(b)

is also fixed by l. By the injectivity statement in (22.10)(g), there is at most one element F ∈ ^{∨}𝔤_{0,−1} with the property that

[E_{ψ}, F] = 2λ_{1}.  (22.12)(c)

By calculation in 𝔰𝔩(2), F_{ψ} satisfies (22.12)(c). Since l fixes E_{ψ} and 2λ_{1}, Ad(l)(F_{ψ}) is a second solution of (22.12)(d). By the uniqueness, l fixes F_{ψ}. Consequently l commutes with E_{ψ}, 2λ_{1}, and F_{ψ}, as we wished to show.

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

23. Local geometry of constructible sheaves.

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 = ⋃ ; H-orbits S S.  (23.1)(a)

Each orbit S is smooth and locally closed. If y ∈ S, then

S ≃ H/H_{y}  (23.1)(b)

Proposition 23.2. 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.

a) The rank of Q_{S}^{loc}(C) is equal to the dimension of the stalks of C along S.

b) Suppose that ξ = (S, V) is a complete geometric parameter for H acting on Y (Definition 7.1), and that μ(ξ) (the extension of ξ by zero — see (7.10(c)) is the corresponding irreducible H-equivariant constructible sheaf. Then Q_{S}^{loc}(μ(ξ)) = ξ, and Q_{S′}^{loc}(μ(ξ)) = 0 for S′ ≠ S.

c) Q_{S}^{loc} is an exact functor from C(Y, H) to H-equivariant local systems on S.

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

Corollary 23.3. In the setting (23.1), suppose C is an H-equivariant constructible sheaf on Y, and S ⊂ Y is an H-orbit. Attached to C there is a representation τ_{S}^{loc}(C) of the equivariant fundamental group A_{S}^{loc} (Definition 7.1).

a) The dimension of τ_{S}^{loc}(C) is equal to the dimension of the stalks of C on S.

b) Suppose that ξ = (S, τ) is a local complete geometric parameter for H acting on Y (with τ an irreducible representation of A_{S}^{loc}), and μ(ξ) is the corresponding irreducible constructible sheaf (see (7.10)(c)). Then τ_{S}^{loc}(μ(ξ)) = τ, and τ_{S′}^{loc}(μ(ξ)) = 0 for S ≠ S′.

c) The functor τ_{S}^{loc} from C(Y, H) to representations of A_{S}^{loc} is exact. In particular, it gives a well-defined map (also denoted τ_{S}^{loc}) from the Grothendieck group K(Y, H) (cf. (7.10)) to virtual representations of A_{S}^{loc}.

Definition 23.4. Suppose P ∈ 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^{i}P 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^{i}P)

on S (Proposition 23.2), and a representation

(τ_{S}^{loc})^{i}(P) = τ_{S}^{loc}(H^{i}P)

of the equivariant fundamental group A_{S}^{loc} (Corollary 23.3). We write

τ_{S}^{loc}(P) = ∑(−1)^{i}(τ_{S}^{loc})^{i}(P),

a virtual representation of A_{S}^{loc}.

Because of Lemma 7.8, this notation is consistent with that of Corollary 23.3(c). We have already seen the virtual representation τ_{S}^{loc}(P), at least for P irreducible, as the geometric character matrix. More precisely, suppose γ = (S_{γ}, τ_{γ}) and ξ = (S_{ξ}, τ_{ξ}) are complete geometric parameters (Definition 7.1), with τ_{γ} an irreducible representation of A_{S_{γ}}^{loc} and τ_{ξ} an irreducible representation of A_{S_{ξ}}^{loc}. Write P (γ) for the irreducible perverse sheaf parametrized by γ (cf. (7.10)). Then

multiplicity of τ_{ξ} in τ_{S_{ξ}}^{loc}(P (γ)) = (−1)^{dim S_{ξ}}c_{g}(ξ, γ)  (23.5)

(notation (7.11)).

Definition 23.6. Suppose F ∈ K(Y, H) (notation (7.10)), S is an H-orbit on Y, and σ ∈ A_{S}^{loc}. The local trace of σ on F is

χ_{S}^{loc}(F)(σ) = tr(τ_{S}^{loc}(F)(σ)).  (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 χ_{S}^{loc}(C)(σ) instead of χ_{S}^{loc}(F)(σ). Explicitly, fix y ∈ S, so that A_{S}^{loc} is the group of connected components of H_{y}, and choose a representative s ∈ H_{y} for σ. Then

χ_{S}^{loc}(C)(σ) = ∑(−1)^{i}tr(s on (H^{i}C)_{y}),  (23.6)(b)

the alternating sum of the traces of s on the stalks of the cohomology sheaves of C. The local multiplicity of F along S is

χ_{S}^{loc}(F) = χ_{S}^{loc}(F)(1)  (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.)

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

Proposition 23.7. Suppose we are in the setting (7.17).

a) Suppose C ∈ Ob C(X, G) is a G-equivariant constructible sheaf on X, so that ϵ^{∗}C is an H-equivariant constructible sheaf on Y (Proposition 7.18). Fix an orbit S of H on Y. Then the representation τ_{S}^{loc}(ϵ^{∗}C) (Corollary 23.3) is given by

τ_{S}^{loc}(ϵ^{∗}C) = A^{loc}(ϵ) ∘ τ_{Φ(ϵ)S}^{loc}(C)

(notation (7.17)).

b) Suppose ξ′ = (S′, τ′) ∈ Ξ(X, G) is a local complete geometric parameter for G acting on X, with τ′ an irreducible representation of A_{S′}^{loc} (Definition 7.1). Then

ϵ^{∗}μ(ξ′) = ∑_{{ξ=(S,τ)∈Ξ(Y,H) \atop S′=Φ(ϵ)S}} (multiplicity of τ in A^{loc}(ϵ) ∘ τ′)μ(ξ).

c) Suppose F ∈ K(X, G) (cf. (7.10)(a)). Fix an H-orbit S ⊂ Y, and an element σ ∈ A_{S}^{loc} (Definition 7.1). Define

S′ = Φ(ϵ)S ⊂ X, σ′ = A^{loc}(ϵ)(σ) ∈ A_{S′}^{loc}.

Then the local trace of σ on ϵ^{∗}F (Definition 23.6) is

χ_{S}^{loc}(ϵ^{∗}F)(σ) = χ_{S′}^{loc}(F)(σ′).

In particular, the local multiplicity of ϵ^{∗}F along S is equal to the local multiplicity of F along S′.

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 σ. The general case follows from the fact that both sides are homomorphisms from K(X, G) to ℂ. Q.E.D.

24. Microlocal geometry of perverse sheaves.

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 ^{∨}G_{ψ}^{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) = ⋃ ; H-orbits S T_{S}^{∗}(Y).  (24.1)(a)

A covector (λ, y) ∈ T_{S}^{∗}(Y) is called 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 ⊂ \overline{S′}.) We indicate non-degenerate conormal covectors with the subscript reg, so that

T_{H}^{∗}(Y)_{reg} = ⋃ ; H-orbits S T_{S}^{∗}(Y)_{reg}.  (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} ≃ H ×_{H_{y}} T_{S,y}^{∗}(Y)_{reg},  (24.1)(c)

a smooth cone bundle over S ≃ 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.

Example 24.2. Suppose Y is the projective line ℂℙ^{1}, and H is the additive group 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) ×_{H} Y — does occur as a geometric parameter space, in connection with representations of P GL(4, 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 ∈ Y and an orbit S ⊂ Y. Write N and N′ for the corresponding unipotent radicals, and T for a maximal torus in B ∩ B′. Then

B_{y} = B ∩ B′ = T (N ∩ N′),  T_{S,y}^{∗}(Y) ≃ 𝔫 ∩ 𝔫′  (24.2)(a)

(compare Proposition 20.14). Suppose now B′ is chosen so that N ∩ N′ is abelian. Then the unipotent radical N ∩ N′ of B ∩ B′ acts trivially on 𝔫 ∩ 𝔫′, so

orbits of B_{y} on T_{S,y}^{∗}(Y) ≃ orbits of T on 𝔫 ∩ 𝔫′.  (24.2)(b)

Now let B′ be obtained from B by permuting the first two and the last two coordinates. Then

𝔫 ∩ 𝔫′ = ( \begin{matrix}0 & A \\ 0 & 0 \end{matrix} );  (24.2)(c)

here we use two by two block matrices. Consequently 𝔫 ∩ 𝔫′ is abelian and fourdimensional. Since T is three-dimensional, there can be no open orbits of T on 𝔫 ∩ 𝔫′. By (24.2)(b), there are no open orbits of B_{y} on T_{S,y}^{∗}(Y).

Lemma 24.3. 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 ∈ X.

a) There is a variety G_{X} and a morphism f : G_{X} → X (a group scheme over X) with the property that the fiber over x is G_{x}.

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 ∈ U_{0}.

c) There is an open G-invariant subvariety U_{1} ⊂ 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 .

d) There is an open G-invariant subvariety U_{2} ⊂ 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 ∈ U_{2}). Then (G_{U_{2}})_{0} is (the set of closed points of) an open and closed subgroup scheme of G_{U_{2}}. e) There is a an étale group scheme A_{U_{2}} over U_{2}, with the property that the fiber

A_{x} over x ∈ U_{2} is isomorphic to the component group G_{x}/(G_{x})_{0}.

f) There is an open G-invariant subvariety U_{3} ⊂ U_{2} over which the étale 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}.

g) For each x ∈ U_{3}, the (algebraic) fundamental group π_{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 π_{1}(U_{3}; x).

Proof. The definition of the action provides a morphism a : G × X → X. We define G_{X} to be the subvariety of G × X

G_{X} = { (g, x) | a(g, x) = x }.

Obviously this is a closed group subscheme of G × 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 · 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 ℂ^{m}, and a holomorphic immersion γ : Z_{0} ×X_{0} → G×X, with the following properties: γ(z, x) ∈ G_{X}, f ∘γ(z, x) = x, and γ(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 γ is a neighborhood of (g_{0}, x_{0}) in G_{X}. (We will call γ a 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×X ⊂ G_{X}. The family of Lie algebras 𝔤_{x} may be regarded as a map from the smooth variety X to the Grassmanian variety of m-dimensional subspaces of 𝔤. In an analytic neighborhood X_{0} of any point x_{0} on X, we can find m holomorphic functions

s_{i} : X_{0} → 𝔤 (i = 1, …, m)

so that {s_{i}(x)} is a basis of 𝔤_{x}. For a sufficiently small neighborhood of zero Z_{0} in ℂ^{m}, the map

γ_{e} : X_{0} × Z_{0} → G_{X}, γ(x, z) = (exp(∑ z_{i}s_{i}(x)), x) ∈ G × X

is a local trivialization at (e, x_{0}); so f is smooth near e × 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

σ : X_{0} → C, σ(x_{0}) = (g_{0}, x_{0}).

Now the group scheme structure μ (that is, the multiplication in G in the first factor) gives a holomorphic map

γ : X_{0} × Z_{0} → V, γ(x, z) = μ(γ_{e}(x, z), σ(x)).

The map γ 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_{0}g′_{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 étale 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.

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 ±1; it has two connected components. Explicitly,

G_{X} = \{ (( \begin{matrix}a & b \\ 0 & a^{−1} \end{matrix} ) , ( \begin{matrix}x & y \\ 0 & −x \end{matrix} )) | 2xab − y(a^{2} − 1) = 0 \} ,  (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 (±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 ≠ 0 }.

Its image in X is the one-dimensional cone of non-zero nilpotent elements; so we take

U_{2} = \{ ( \begin{matrix}x & y \\ 0 & −x \end{matrix} ) | x ≠ 0 \} .

The group scheme A_{U_{2}} is trivial, so we can take U_{3} = U_{2}.

Definition 24.5. In the setting of Lemma 24.3, the finite group A_{x} attached to any x ∈ U_{3} is called the 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.

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.

Lemma 24.6. 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 ⊂ 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.

a) There is a locally constant family of natural homomorphisms

i_{x} : A_{x} → A (x ∈ U).

The image of these homomorphisms is independent of x.

b) If G has an open orbit on X, then the homomorphisms of (a) are surjective.

c) Suppose V is a G-equivariant local system of complex r-dimensional vector spaces on X. Then V defines a local system of r-dimensional representations of A_{x}.

Proof. The maps in (a) are just

i_{x} : G_{x}/(G_{x})_{0} → 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} · 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 · x = G_{0} · x, and i_{x} is surjective. For (c), we apply Lemma 7.3 to the restriction of V to the orbit G · x. That is, we take the isotropy action of G_{x} on the fiber V_{x}. Q.E.D.

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 ∈ 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} ⊂ T_{S,y}^{∗}(Y)_{reg} as in Lemma 24.3(f). Notice that

U_{S} = H ×_{H_{y}} U_{S,y}

is then an open set in T_{S}^{∗}(Y)_{reg}. Fix a point ν ∈ U_{S,y}. The (local) equivariant micro-fundamental group at (y, ν) is the pro-finite group

A_{y,ν}^{mic} = H_{y,ν}/(H_{y,ν})_{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,ν} : A_{y,ν}^{mic} → A_{y}^{loc},

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 equivariant micro-fundamental group for S is

A_{S}^{mic} = A_{y,ν}^{mic} ((y, ν) ∈ U_{S} ⊂ T_{S}^{∗}(Y)).

By Lemma 24.3, it is independent of the choice of (y, ν) up to automorphism; if H has an open orbit on T_{S}^{∗}(Y), then it is independent of choices up to inner automorphism. There is a natural homomorphism

i_{S} : A_{S}^{mic} → A_{S}^{loc},

which is surjective if H has an open orbit on T_{S}^{∗}(Y).

Theorem 24.8 ([KK], [GM]). 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)).

a) The rank of Q^{mic}(P) at any point (y, ν) of T_{S}^{∗}(Y)_{reg} is equal to the multiplicity χ_{S}^{mic}(P) of P in the characteristic cycle of P (Proposition 19.12).

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} → S.

c) Q^{mic} is an exact functor from P(Y, H) to H-equivariant local systems on T_{H}^{∗}(Y)_{reg}.

Corollary 24.9. In the setting (24.1), suppose P is an H-equivariant perverse sheaf on Y, and S ⊂ Y is an H-orbit. Attached to P there is a representation τ_{S}^{mic}(P) of the equivariant micro-fundamental group A_{S}^{mic} (Definition 24.7).

a) The dimension of τ_{S}^{mic}(P) is equal to the multiplicity χ_{S}^{mic}(P).

b) Suppose that ξ = (S, τ) is a local complete geometric parameter for H acting on Y (with τ an irreducible representation of A_{S}^{loc}), and P (ξ) is the corresponding irreducible perverse sheaf. Then τ_{S}^{mic}(P (ξ)) = τ ∘ i_{S} (Definition 24.7). c) The functor τ_{S}^{mic} from H-equivariant perverse sheaves on Y to representations

of A_{S}^{mic} is exact. In particular, it gives rise to a map (also denoted τ_{S}^{mic}) from the Grothendieck group K(Y, H) (cf. (7.10)) to virtual representations of A_{S}^{mic}.

This theorem is absolutely fundamental to “microlocal geometry,” and it is wellknown 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 D-module M corresponding to P under Theorem 7.9. By extension of scalars, they get a module M^{mic} for the sheaf E of microdifferential operators. The restriction of 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 E, but also the much larger sheaf of algebras E^{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 O_{T^{∗}Y} -module gr(M) (cf. (19.4)). We first consider what can be done easily. If the filtration of M is chosen to be H-invariant (as is possible on the geometric parameter space by Proposition 20.15) then it follows easily that gr(M) is annihilated by the defining ideal of T_{H}^{∗}(Y) (see the proof of Propostion 19.12(b)). We may therefore regard gr(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 ×_{H_{y}} N for some H_{y}-equivariant coherent O_{T_{S,y}^{∗}(Y)}-module N (cf. (24.1)(c)). There is an open H_{y}- invariant subvariety U of T_{S,y}^{∗}(Y) over which N is (the sheaf of sections of) an H_{y}-equivariant vector bundle V (compare [Shaf], Proposition VI.3.1). Essentially by definition, the rank of V is the multiplicity of T_{S}^{∗}(Y) in the characteristic cycle of M. If ν ∈ U is arbitrary, we get an isotropy representation τ_{y,ν} of H_{y,ν} on the stalk of V at ν. It is a consequence of Lemma 19.6 that the class of τ_{y,ν} in the Grothendieck group of virtual representations of H_{y,ν} is independent of the choice of good filtration on M. In order to prove at least Corollary 24.9 (which is all we will use) the main difficulty is to prove that τ_{y,ν} must be trivial on the identity component of H_{y,ν}. 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 gr(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, ν) ∈ 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) = ν. (The assumptions on f and ν 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 ϵ and δ ([GM], I.3.6). Write B_{δ}(y) for the closed ball of radius δ about y in Y, and define a pair of compact spaces

J = N ∩ B_{δ}(y) ∩ f^{−1}[−ϵ, ϵ] ⊃ K = N ∩ B_{δ}(y) ∩ f^{−1}(−ϵ)  (24.10)(a)

Then the stalk of the local system Q^{mic}(P) at the point (y, ν) is by definition

Q^{mic}(P)_{y,ν} = H^{− dim S}(J, K; P),  (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 χ_{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 χ_{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 ∩B_{δ}(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,ν} = H^{− dim S}({y}, ∅; P) = stalk at y of H^{− dim S}(P).

By Definition 23.4, this is the stalk of (Q_{S}^{loc})^{− 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  (P perverse and i ≠ − dim S).  (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.

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, ν) ∈ T_{S}^{∗}(Y)_{reg}) as in (24.10), and define J ⊃ K as in (24.10)(a) . Then the stalk of (Q^{mic})^{i}(C) at (y, ν) 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 (τ_{S}^{mic})^{i}(C) of A_{S}^{mic}. (When we wish to emphasize the base point, we will write (τ_{y,ν}^{mic})^{i}(C).) The virtual representation of A_{S}^{mic} in Corollary 24.9(c) is

τ_{S}^{mic}(C) = ∑_{i}(−1)^{i}(τ_{S}^{mic})^{i}(C).

Definition 24.12. Suppose F ∈ K(Y, H) (notation (7.10)), S is an H-orbit on Y, and σ ∈ A_{S}^{mic}. The microlocal trace of σ on F is

χ_{S}^{mic}(F)(σ) = tr(τ_{S}^{mic}(F)(σ)).  (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 χ_{S}^{mic}(C)(σ) instead of χ_{S}^{mic}(F)(σ). Explicitly, fix (y, ν) ∈ T_{S}^{∗}(Y)_{reg}, so that A_{S}^{mic} is the group of connected components of H_{y,ν}, and choose a representative s ∈ H_{y,ν} for σ. Then

χ_{S}^{mic}(C)(σ) = (−1)^{dim S} ∑(−1)^{i}tr(s 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) ≃ H^{i}(J′, K′; C) explained in [GM] to define the action of s.)

Definition 24.13. In the setting of Theorem 10.4, fix an equivalence class ϕ ∈ Φ^{z}(G/ℝ) of Langlands parameters (Definition 5.3), and write S = S_{ϕ} for the corresponding orbit of ^{∨}G on X(^{∨}G^{Γ}) (Definition 7.6). The micro-component group for ϕ is by definition the ^{∨}G^{alg}-equivariant micro-fundamental group A_{S}^{mic} (Definition 24.7):

A_{ϕ}^{mic,alg} = A_{S}^{mic}.

By Definition 24.7, there is a natural homomorphism

i_{ϕ} : A_{ϕ}^{mic,alg} → A_{ϕ}^{loc,alg},

which is surjective if ^{∨}G has an open orbit on T_{S}^{∗}(X(^{∨}G^{Γ})). To each irreducible representation π ∈ Π^{z}(G/ℝ)_{ϕ}^{mic} (Definition 19.15), we associate a (possibly reducible) representation τ_{ϕ}^{mic}(π), as follows. Let P (π) be the irreducible perverse sheaf corresponding to π. Then

τ_{ϕ}^{mic}(π) = τ_{S}^{mic}(P (π))

(Corollary 24.9). This definition makes sense for any irreducible representation π (not necessarily in the micro-packet of ψ). Corollary 24.9(a) guarantees that τ_{ϕ}^{mic}(π) ≠ 0 if and only if π ∈ Π^{z}(G/ℝ)_{ϕ}^{mic}. Following Definition 24.12, we write for σ ∈ A_{ϕ}^{mic,alg}

χ_{ϕ}^{mic}(π)(σ) = tr(τ_{ϕ}^{mic}(π)(σ)).

Theorem 24.14. Suppose (G^{Γ}, W) is an extended group for G, and (^{∨}G^{Γ}, D) is an E-group for the corresponding inner class of real forms, with second invariant z (Definition 4.6). Fix an equivalence class ϕ ∈ Φ^{z}(G/ℝ) of Langlands parameters, and define the map τ_{ψ} (from irreducible representations in the micro-packet Π^{z}(G/ℝ)_{ϕ}^{mic} to representations of A_{ϕ}^{mic,alg}) as in Definition 24.13.

a) Suppose π belongs to the L-packet Π_{ϕ}^{z}, and τ_{ϕ}^{loc}(π) is the corresponding irreducible representation of A_{ϕ}^{loc,alg} (Definition 5.11 and Theorem 10.4). Then

τ_{ϕ}^{mic}(π) = τ_{ϕ}^{loc}(π) ∘ i_{ϕ}.

b) The dimension of τ_{ϕ}^{mic}(π) is the multiplicity χ_{ϕ}^{mic}(π) (Corollary 19.16).

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.

Definition 24.15. In the setting of Theorem 10.4, suppose ψ ∈ Ψ(^{∨}G^{Γ}) is an Arthur parameter (Definition 22.4), and π ∈ Π_{ψ}^{z} is an irreducible representation in the corresponding Arthur packet (Definition 22.6). We define a (possibly reducible) representation τ_{ψ}(π) of the universal component group A_{ψ}^{alg} as follows. Let x = p(ϕ_{ψ}) ∈ X(^{∨}G^{Γ}) be the geometric parameter corresponding to the Langlands parameter ϕ_{ψ}, and S ⊂ X(^{∨}G^{Γ}) the corresponding orbit. Define E_{ψ} ∈ T_{S,x}^{∗}(^{∨}G^{Γ}) as in Proposition 22.9, so that the orbit of E_{ψ} is open and dense in T_{S}^{∗}(^{∨}G^{Γ}). By Proposition 22.9 and Definition 24.7, A_{ψ}^{alg} is naturally identified with the equivariant micro-fundamental group A_{S}^{mic} for the action of ^{∨}G^{alg} on X(^{∨}G^{Γ}); that is, with A_{ϕ_{ψ}}^{mic,alg}. We can therefore define

τ_{ψ}(π) = τ_{ϕ_{ψ}}^{mic,alg}(π), χ_{ψ}(π)(σ) = tr τ_{ψ}(π)(σ)

(Definition 24.13).

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 χ_{S}^{mic}. The main point is the following observation of MacPherson taken from [Ch].

Proposition 24.16. Suppose X is a compact space with a finite Whitney stratification S = {S_{j}} having connected strata, and C is a complex of sheaves on X constructible with respect to S. Define F_{ij} to be the dimension of the stalks of the ith cohomology sheaf H^{i}C at points of S_{j}, and set

F_{j} = ∑_{i}(−1)^{i}F_{ij}.

Then the Euler characteristic of the hypercohomology

χ(X; C) = ∑(−1)^{i} dim H^{i}(X; C)

depends only on the integers F_{j}. More precisely, define

c_{j} = χ_{c}(S_{j}) = ∑_{i}(−1)^{i} dim H_{c}^{i}(S_{j}; ℂ),

the Euler characteristic with compact supports of the stratum S_{j}. Then

χ(X; C) = ∑_{j} c_{j}F_{j}.

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^{i}C have constant dimension on each S_{j}.

To prove Theorem 1.31, fix S and any point (y, ν) ∈ T_{S}^{∗}(Y)_{reg}. Define J ⊃ K as in (24.10). These compact sets have Whitney stratifications so that the various J ∩ S′ and K ∩ 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(τ_{S′}^{loc})^{i}(P) (Definition 23.4). The alternating sum of these ranks is χ_{S′}^{loc}(P) (Definition 23.6):

∑_{i}(−1)^{i} dim H^{i}(P)_{x} = χ_{S′}^{loc}(P)  (x ∈ J ∩ S′).  (24.17)(a)

Define

c(S, S′) = (−1)^{dim S}χ_{c}(J∩S′, K∩S′) = (−1)^{dim S} ∑_{i}(−1)^{i} dim H_{c}^{i}(J∩S′, K∩S′; ℂ).

(24.17)(b)

Finally, recall that we are defining

χ_{S}^{mic}(P) = (−1)^{dim S} ∑_{i} dim H^{i}(J, K; P).  (24.17)(c)

Then Theorem 1.31 is immediate from Proposition 24.16 and (24.17). Q.E.D.

25. A fixed point formula.

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

σ : G → G, σ : X → X  (25.1)(a)

(A little more formally, we could consider an action of some finite cyclic group ℤ/nℤ on everything, with σ the action of the distinguished generator 1 + nℤ.) The compatibility means for example that

(σ · g) · (σ · x) = σ · (g · x).  (25.1)(b)

We fix a subgroup H of G, with the property that

(G^{σ})_{0} ⊂ H ⊂ G^{σ}.  (25.1)(c)

We fix also a subvariety Y of X^{σ}, with the property that

H · Y = Y, and Y is open and closed in X^{σ}.  (25.1)(d)

We write

ϵ : Y → X,  ϵ : H → G  (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 σ of finite order, compatible with those of (25.1)(a):

σ : C → C  (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, ν) are given by

(Q^{mic})^{i}(C)_{x,ν} = H^{i−dim G·x}(J, K; C),  (25.1)(g)

with J ⊃ K as in (24.10). Accordingly we get an action of σ on (Q^{mic})^{i}(C):

σ : (Q^{mic})^{i}(C)_{x,ν} → (Q^{mic})^{i}(C)_{σ·(x,ν)}.  (25.1)(h)

Assume finally that we are given a point x_{0} ∈ Y ⊂ X, and a conormal covector

(x_{0}, ν_{0}) ∈ T_{G}^{∗}(X)_{reg},  σ · (x_{0}, ν_{0}) = (x_{0}, ν_{0}),  (25.1)(i)

so that σ defines an automorphism (denoted (τ_{x_{0},ν_{0}}^{mic})^{i}(C)(σ), in analogy with Definition 24.11) of finite order of each of the finite-dimensional vector spaces (Q^{mic})^{i}(C)_{x_{0},ν_{0}}. In analogy with Definition 24.12, we write

χ_{x_{0},ν_{0}}^{mic}(C)(σ) = ∑(−1)^{i}tr(τ_{x_{0},ν_{0}}^{mic})^{i}(C)(σ),  (25.1)(j)

the microlocal trace of σ on C at (x_{0}, ν_{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 σ. 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 ∈ G of finite order, and put

σ · g = sgs^{−1}  (g ∈ G).  (25.2)(a)

We make σ act on X and C as s does. In this case (τ_{x_{0},ν_{0}}^{mic})^{i}(C)(σ) is just the action of s on the stalk of the local system (Q^{mic})^{i}(C) at the point (x_{0}, ν_{0}). Suppose in addition that (x_{0}, ν_{0}) is sufficiently generic that

G_{x_{0},ν_{0}}/(G_{x_{0},ν_{0}})_{0} = A_{x_{0},ν_{0}}^{mic}  (25.2)(b)

(Definition 24.7). Then the element s (which by assumption (25.1)(i) belongs to G_{x_{0},ν_{0}}) defines a coset

\overline{s} = s(G_{x_{0},ν_{0}})_{0} ∈ A_{x_{0},ν_{0}}^{mic}  (25.2)(c)

(Definition 24.7). It follows that

(τ_{x_{0},ν_{0}}^{mic})^{i}(C)(σ) = (τ_{x_{0},ν_{0}}^{mic})^{i}(C)(\overline{s})  (25.2)(d)

(Definition 24.11). Taking the alternating sum of traces, we get

χ_{x_{0},ν_{0}}^{mic}(C)(σ) = χ_{x_{0},ν_{0}}^{mic}(C)(\overline{s}).  (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.

Lemma 25.3. Suppose Z is a smooth algebraic variety, and σ is an algebraic automorphism of Z of finite order. Write Z^{σ} for the subvariety of fixed points.

a) Z^{σ} is a closed smooth subvariety of Z.

b) The tangent space to Z^{σ} at a fixed point of σ may be identified with the fixed points of the differential of σ on the tangent space to Z:

T_{z}(Z^{σ}) = T_{z}(Z)^{dσ}.

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^{σ} belongs to an open affine set U, and therefore to a σ-invariant affine open set ⋂_{n∈ℤ} σ^{n} · U. (The intersection is finite since σ has finite order.) We may therefore assume Z = Spec A is affine. Then σ arises from an automorphism (also called σ) of A, also of finite order. The subvariety Z^{σ} is defined by the ideal I generated by elements f − σ · f, with f ∈ A. Since σ acts trivially on A/I and has finite order, we have

A/I ≃ A^{σ}/(A^{σ} ∩ I).  (25.4)(a)

Finally, write 𝔪 for the maximal ideal of the point z; necessarily I ⊂ 𝔪. The Zariski cotangent spaces to Z and Z^{σ} at z are

T_{z}^{∗}(Z) = 𝔪/𝔪^{2},  T_{z}^{∗}(Z^{σ}) = 𝔪/(𝔪^{2} + I) ≃ 𝔪^{σ}/(𝔪^{σ} ∩ (𝔪^{2} + I)). (25.4)(b)

(The last isomorphism uses (25.4)(a).) The (equivalent) cotangent version of (b) in the lemma therefore amounts to

𝔪^{σ}/(𝔪^{σ} ∩ 𝔪^{2}) ≃ 𝔪^{σ}/(𝔪^{σ} ∩ (𝔪^{2} + I)).  (25.4)(c)

Here the first space evidently maps surjectively to the second, so the isomorphism is equivalent to

(𝔪^{2} + I)^{σ} ⊂ 𝔪^{2}.

Since σ preserves both 𝔪^{2} and I, this in turn is equivalent to

I^{σ} ⊂ 𝔪^{2}  (25.4)(d)

(This says that a σ-invariant function vanishing on Z^{σ} must actually vanish to first order there.) To prove (25.4)(d), suppose f ∈ I^{σ}. By the definition of I, we can write f = ∑ f_{i}(g_{i} − σ · g_{i}). We can expand each f_{i} and g_{i} in eigenfunctions of σ. This gives

f = ∑ f_{j}(g_{j} − σ · g_{j}), σ · f_{j} = λ_{j}f_{j}, σ · g_{j} = μ_{j}g_{j}

for some roots of unity λ_{j} and μ_{j}. Consequently

f = ∑(1 − μ_{j})f_{j}g_{j}, σ · f = ∑(1 − μ_{j})λ_{j}μ_{j}(f_{j}g_{j}).

It follows that the terms in the expansion of f in eigenfunctions of σ may be found by restricting to the summands with λ_{j}μ_{j} fixed. Since f is fixed by σ, we may throw away all the terms with λ_{j} ≠ (μ_{j})^{−1}. Of course we may also throw away the terms with μ_{j} = 1. This leaves

f = ∑(1 − μ_{j})f_{j}g_{j},  σ · f_{j} = (μ_{j})^{−1}f_{j}, σ · g_{j} = μ_{j}g_{j},  μ_{j} ≠ 1.

Therefore

f = ∑(1 − μ_{j}^{−1})^{−1}(f_{j} − σ · f_{j})(g_{j} − σ · g_{j}),

which exhibits f as an element of I^{2} ⊂ 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^{σ}; for the dimension must jump up at a singular point. If z ∈ Z^{σ}, choose functions f_{1}, ⋯, f_{n} on Z so that (df_{1}, ⋯, df_{n}) is a basis of T_{z}^{∗}Z. Clearly we may assume that the f_{i} are eigenfunctions for σ; say σ · f_{i} = λ_{i}f_{i}. There is a Zariski open set U ⊂ 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^{σ}) = number of i such that λ_{i} = 1

for all y ∈ U ∩ Z^{σ}. Thus this dimension is constant on a neighborhood of z in Z^{σ}, 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.

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

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_{x}S ≃ 𝔤/𝔤_{x}.

Here G_{x} is the isotropy group of the action at x. If σ · x = x, then σ must preserve G_{x}, and so act on 𝔤_{x}. By Lemma 25.3(b),

T_{x}(S^{σ}) ≃ 𝔤^{σ}/𝔤_{x}^{σ}.

This is also the tangent space to the orbit G^{σ} · x. We conclude that the (smooth) orbit G^{σ} · x has the same dimension at x as the (smooth) fixed point set S^{σ}, and therefore that the orbit contains a neighborhood of x in S^{σ}. It follows that every orbit of G^{σ} on S^{σ} is open. As S^{σ} is the disjoint union of orbits of G^{σ}, every orbit must also be closed. Since an algebraic variety has only finitely many connected components, the lemma follows. Q.E.D.

Theorem 25.6. 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 ϵ : (Y, H) → (X, G) of (25.1) has the properties considered in (7.17). Fix a point y ∈ Y.

a) The conormal bundle to the H action on Y at y (cf. (19.1)) may be identified

naturally as

T_{H,y}^{∗}(Y) = (T_{G,y}^{∗}(X))^{σ} .

b) In the identification of (a), we have

(T_{G,y}^{∗}(X)_{reg})^{σ} ⊂ T_{H,y}^{∗}(Y)_{reg}.

c) Suppose y ∈ Y. Then there is a natural homomorphism of equivariant fundamental groups (Definition 7.1)

A^{loc}(ϵ) : A_{y}^{loc}(Y, H) → A_{y}^{loc}(X, G).

d) Suppose (y, ν) ∈ T_{H,y}^{∗}(Y); use (a) to identify (y, ν) with a point of T_{G,y}^{∗}(X). Assume that these points belong to the open sets U_{H·y} and U_{G·y} of Definition 24.7. Then there is a natural homomorphism of equivariant micro-fundamental groups

A^{mic}(ϵ) : A_{y,ν}^{mic}(Y, H) → A_{y,ν}^{mic}(X, G).

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, ν) to H · y fails to be regular. By (24.1), (y, ν) is a limit of normal covectors to orbits in Y distinct from H · 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 · y. Consequently (y, ν) 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 ν 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

C(X, G; σ),  P(X, G; σ),  D(X, G; σ)  (25.7)(a)

in analogy with Definition 7.7; we include in the objects an automorphism σ of finite order, compatible with σ on (X, G). Thus for example an object of C(X, G; σ) is an G-equivariant constructible sheaf on X, endowed with an automorphism σ 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; σ).  (25.7)(b)

By Theorem 25.6 and Proposition 7.18, there are corresponding categories

C(Y, H; σ),  P(Y, H; σ),  D(Y, H; σ)  (25.7)(c)

and a homomorphism

ϵ^{∗} : K(X, G; σ) → K(Y, H; σ).  (25.7)(d)

Now the action of σ on (Y, H) is trivial. It follows that any object in one of these categories is a direct sum of “eigenobjects” on which σ acts by scalar multiplication (necessarily by a root of unity). For example, a constructible sheaf C ∈ Ob C(Y, H; σ) can be written

C = ∑_{λ∈ℂ^{×}} C_{λ}  (25.7)(e)

using the eigenspace decomposition of the action of σ on each stalk. This decomposition is inherited by K(Y, H; σ). (An unnecessarily fancy way to say this is that K(Y, H; σ) is the tensor product of K(Y, H) with the group algebra of the group of roots of unity.)

Theorem 25.8. Suppose we are in the setting (25.1). (Thus σ 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 σ at the σ-fixed point (x_{0}, ν_{0}) ∈ T_{G}^{∗}(X)_{reg} (cf. (25.1)(j)) may be computed along the fixed points of σ. More precisely, use Theorem 25.6(b) to identify (x_{0}, ν_{0}) as a point of T_{H}^{∗}(Y)_{reg}. Write ϵ^{∗}C for the restriction to Y of C, with its inherited action of σ (Proposition 7.18). Then

χ_{x_{0},ν_{0}}^{mic}(C)(σ) = (−1)^{dim G·x_{0}−dim H·x_{0}}χ_{x_{0},ν_{0}}^{mic}(ϵ^{∗}C)(σ)  (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

χ_{x_{0},ν_{0}}^{mic}(C)(\overline{s}) = (−1)^{dim G·x_{0}−dim H·x_{0}}χ_{x_{0},ν_{0}}^{mic}(ϵ^{∗}C)(A^{mic}(ϵ)(\overline{s}))  (25.9)(b)

(notation as in Theorem 25.6(d)).

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 ϵ^{∗}C need not be perverse, however, so the right side is only the value of a character of a virtual representation.

Proof. Recall from (24.10) the construction of the pair of spaces J ⊃ K attached to the regular conormal covector (x_{0}, ν_{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 σ; and in this case N will be preserved by σ. Of course we can choose the Riemannian metric to be preserved by σ. Since (x_{0}, ν_{0}) is assumed to be fixed by σ, we may replace the smooth function f by the average of its (finitely many) translates by σ without affecting the requirements f(x_{0}) = 0 and df(x_{0}) = ν_{0}. That is, we may assume that σ·f = f. When the choices are made in this way, the automorphism σ preserves J ⊃ K, and so we may interpret the microlocal trace directly as a Lefschetz number:

χ_{x_{0},ν_{0}}^{mic}(C)(σ) = (−1)^{dim G·x_{0}} ∑(−1)^{i}tr (σ on H^{i}(J, K; C)).  (25.9)(a)

By the long exact sequence for the pair, this is also the difference of the Lefschetz numbers of σ 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^{σ} is a smooth submanifold meeting H · x_{0} transversally in the single point x_{0}. The restriction of f to Y still has differential ν_{0}; so we see that the pair of spaces J^{σ} ⊃ K^{σ} may be taken as the ones constructed in Y using (x_{0}, ν_{0}) (cf. (24.10)). Consequently

χ_{x_{0},ν_{0}}^{mic}(ϵ^{∗}C)(σ) = (−1)^{dim H·x_{0}} ∑(−1)^{i}tr (σ on H^{i}(J^{σ}, K^{σ}; ϵ^{∗}C)). (25.9)(b)

Again this is a difference of Lefschetz numbers for J^{σ} and K^{σ} 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, distance to Y );

this is σ-invariant (since the Riemannian metric is) and therefore satisfies Goresky and MacPherson’s requirements

t^{−1}(0, 0) = Y,  t_{1}(σ · y) ≥ t_{1}(y),  t_{2}(σ · y) ≤ t_{1}(y).

Their local group A_{3}^{i} ([GMFP], Definition 4.4) for J is precisely

H^{i}(J^{σ}; ϵ^{∗}C),

and similarly for K. Their Theorem 1 ([GMFP], section 4.7) now says

∑(−1)^{i}tr (σ on H^{i}(J; C)) = ∑(−1)^{i}tr (σ on H^{i}(J^{σ}; ϵ^{∗}C)),  (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 naïve 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.

26. Endoscopic lifting.

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^{Γ}, W) and (H^{Γ}, 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 (^{∨}G^{Γ}, D) and (^{∨}H^{Γ}, D_{H}) are E-groups for these extended groups (Definition 4.6), say with second invariants z ∈ Z(^{∨}G)^{θ_{Z}} and z_{H} ∈ Z(^{∨}H)^{θ_{Z}} respectively. Suppose we are given an L-homomorphism

ϵ : ^{∨}H^{Γ} → ^{∨}G^{Γ}  (26.1)(a)

(Definition 5.1). As in Definition 10.10, fix a quotient Q of π_{1}(^{∨}G)^{alg}, and form the corresponding quotient

1 → Q → ^{∨}G^{Q} → ^{∨}G → 1  (26.1)(b)

of ^{∨}G^{alg}. As in (5.13)(b), we can pull this extension back by ϵ to

1 → Q → ^{∨}H^{Q} → ^{∨}H → 1

Define Q_{H} to be the intersection of Q with the identity component (^{∨}H^{Q})_{0}, so that

1 → Q_{H} → ^{∨}H^{Q_{H}} → ^{∨}H → 1  (26.1)(c)

is a connected pro-finite covering of ^{∨}H. It is easy to see that this is a quotient of the canonical cover of ^{∨}H, so Q_{H} is naturally a quotient of π_{1}(^{∨}H)^{alg}. Proposition 7.18 provides a natural homomorphism

ϵ^{∗} : KX(^{∨}G^{Γ})^{Q} → KX(^{∨}H^{Γ})^{Q_{H}}  (26.1)(d)

(cf. (7.19)(d)).

As in Theorem 10.11, we now define a subgroup J ⊂ Z(G)^{σ_{Z},fin} so that

Q̂ ≃ J,  (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

Π^{z}(G/ℝ)_{J} ↔ Ξ^{z}(G/ℝ)^{Q},  Π^{z}(H/ℝ)_{J_{H}} ↔ Ξ^{z}(H/ℝ)^{Q_{H}} .  (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.

Theorem 26.2. Suppose we are in the setting of Theorem 10.11. Write

\overline{K}_{ℂ}Π^{z}(G/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}_{ℂ}Π^{z}(G/ℝ)_{J} may be identified with the space of complex-valued linear functionals on the Grothendieck group KX(^{∨}G^{Γ})^{Q}:

\overline{K}_{ℂ}Π^{z}(G/ℝ)_{J} ≃ Hom_{ℤ}(KX(^{∨}G^{Γ})^{Q}, ℂ).

We will call elements of \overline{K}_{ℂ}Π^{z}(G/ℝ)_{J} formal complex virtual representations, or simply virtual representations if no confusion can arise. As in Definition 18.6, we write \overline{K}_{ℂ,f}Π^{z}(G/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).

Definition 26.3. Suppose we are in the setting (26.1). 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,

ϵ_{∗} : \overline{K}_{ℂ}Π^{z_{H}} (H/ℝ)^{Q_{H}} → \overline{K}_{ℂ}Π^{z}(G/ℝ)^{Q}.  (26.3)(a)

It is by definition the transpose of ϵ^{∗} (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

⟨ϵ_{∗}η_{H}, F_{G}⟩_{G} = ⟨η_{H}, ϵ^{∗}F_{G}⟩_{H}.  (26.3)(b)

Here η_{H} is a formal complex virtual representation for H, and F_{G} belongs to the Grothendieck group of equivariant constructible sheaves on X(^{∨}G^{Γ}).

We can calculate the Langlands functoriality map ϵ_{∗} 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 ϵ^{∗} does to irreducible constructible sheaves on X(^{∨}G). This is easy (Proposition 23.7). To see what it does to irreducible representations for H, we need to see what ϵ^{∗} does to irreducible perverse sheaves. This is much harder. Here is a precise statement.

Proposition 26.4. Suppose we are in the setting (26.1). Fix complete geometric parameters ξ_{H} = (S_{H}, τ_{H}) ∈ Ξ^{z_{H}} (H/ℝ)^{Q_{H}} and ξ_{G} = (S_{G}, τ_{G}) ∈ Ξ^{z}(G/ℝ)^{Q}. (Here S_{H} ∈ Φ(H/ℝ) is an orbit of ^{∨}H on X(^{∨}H^{Γ}), τ_{H} is an irreducible representation of the corresponding Q_{H}-component group A_{S_{H}}^{loc,Q_{H}}, and similarly for G.)

a) If Φ(ϵ)(S_{H}) ≠ S_{G} (cf. (7.19)(b)) then the standard representation M(ξ_{G}) does not occur in the expression of ϵ_{∗}M(ξ_{H}) in terms of standard representations.

b) Suppose Φ(ϵ)(S_{H}) = S_{G}; recall from (7.19)(c) the induced map A^{loc}(ϵ) : A_{S_{H}}^{loc,Q_{H}} → A_{S_{G}}^{loc,Q}. Then the multiplicity of M(ξ_{G}) in the expression of ϵ_{∗}M(ξ_{H}) in terms of standard representations is equal to the multiplicity of τ_{H} in the representation A^{loc}(ϵ) ∘ τ_{G} of A_{S_{H}}^{loc,Q_{H}}, multiplied by the quotient e(ξ_{G})/e(ξ_{H}) of the Kottwitz invariants (Definition 15.8).

c) If Φ(ϵ)(S_{H}) is not contained in the closure of S_{G}, then the irreducible representation π(ξ_{G}) does not occur in the expression of ϵ_{∗}π(ξ_{H}) in terms of irreducible representations. d) Suppose Φ(ϵ)(S_{H}) ⊂ \overline{S_{G}}. Recall from (7.10) the irreducible equivariant per-

verse sheaf P (ξ_{G}). Its restriction ϵ^{∗}P (ξ_{G}) is a ^{∨}H^{Q_{H}} -equivariant constructible complex on X(^{∨}H^{Γ}), and may therefore be written (in the Grothendieck group KX(^{∨}H^{Γ})) as a sum of irreducible equivariant perverse sheaves, with integer coefficents. Write m for the coefficient of P (ξ_{H}) in this sum. Then the multiplicity of π(ξ_{G}) in the expression of ϵ_{∗}π(ξ_{H}) in terms of irreducible representations is equal to

m(e(ξ_{G})(−1)^{dim S_{G}})/(e(ξ_{H})(−1)^{dim S_{H}}).

Proof. According to Definition 15.11, the multiplicity of M(ξ_{G}) in any formal virtual representation η_{G} is equal to e(ξ_{G})⟨η_{G}, μ(ξ_{G})⟩_{G}. Applying Definition 26.3(b), we find that the multiplicity of M(ξ_{G}) in ϵ_{∗}M(ξ_{H}) is equal to

e(ξ_{G})⟨ϵ_{∗}M(ξ_{H}), μ(ξ_{G})⟩_{G} = e(ξ_{G})⟨M(ξ_{H}), ϵ^{∗}μ(ξ_{G})⟩_{H}.

Applying Definition 15.11 for H, we find that the right side is e(ξ_{G})/e(ξ_{H}) times the multiplicity of μ(ξ_{H}) in ϵ^{∗}μ(ξ_{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 ^{p}H^{i}(ϵ^{∗}P (ξ_{G})) (see [BBD]) to represent the image of ϵ^{∗}P (ξ_{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} = multiplicity of P (ξ_{H}) in ^{p}H^{i}(ϵ^{∗}P (ξ_{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.

Definition 26.5. In the setting of Theorem 10.11, suppose ϕ ∈ P (^{∨}G^{Γ}) is a Langlands parameter, and x ∈ X(^{∨}G^{Γ}) is the corresponding point of the geometric parameter space. Write S for the orbit of x, so that the ^{∨}G^{Q}-equivariant fundamental group of S is isomorphic to the Q-component group for ϕ (Definition 7.6):

A_{S}^{loc,Q} = A_{x}^{loc,Q} ≃ A_{ϕ}^{loc,Q}

(Definition 7.6). Fix σ ∈ A_{ϕ}^{loc,Q}. The standard formal complex virtual representation attached to ϕ and σ is a sum of the standard final limit representations in the L-packet Π^{z}(G/ℝ)_{ϕ}^{Q}, with coefficients given by character values for the corresponding representation of A_{ϕ}^{loc,Q}. Specifically,

η_{ϕ}^{loc,Q}(σ) = ∑_{ξ=(ϕ,τ)∈Ξ^{z}(G/ℝ)^{Q}} e(ξ)tr τ(σ)M(ξ);

the sum is over complete Langlands parameters of type Q with ϕ fixed (Definition 5.11 and (5.13)). We may also write this as η_{S}^{loc,Q}(σ) or η_{x}^{loc,Q}(σ). When we wish to emphasize the group G, we will write η_{ϕ,G}^{loc,Q}(σ). Incorporating the definition of e(ξ) (Definition 15.8), we may rewrite the definition as

η_{ϕ}^{loc,Q}(σ) = ∑ ; τ∈(A_{ϕ}^{loc,Q})tr τ(σ\overline{z(ρ)})M(ϕ, τ),

a sum over the irreducible representations of the Q-component group.

This virtual representation is locally finite (Definition 18.6). If δ is a strong real form of G of type J (Definition 10.10) we can write

η_{ϕ}^{loc,Q}(σ)(δ) = e(G(ℝ, δ)) ∑_{{ξ=(ϕ,τ)∈Ξ^{z}(G/ℝ) \atop δ(ξ)=δ}} tr τ(σ)M(ξ)

for the part of η_{ϕ}^{loc}(σ) living on G(ℝ, δ). This is a finite combination of irreducible representations with complex coefficients, so it has a well-defined character

Θ(η_{ϕ}^{loc,Q}(σ), δ),

a generalized function on G(ℝ, δ)^{can}.

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

Lemma 26.6. In the setting of Definition 26.5, the isomorphism of Theorem 26.2 identifies the formal complex virtual representation η_{ϕ}^{loc,Q}(σ) with the map χ_{S}^{loc,Q}(·)(σ) (Definition 23.6) that assigns to a constructible equivariant sheaf C the trace of σ acting on the stalk C_{x}. In terms of irreducible representations, we therefore have

η_{ϕ}^{loc,Q}(σ) = ∑_{ξ∈Ξ^{z}(G/ℝ)^{Q}} e(ξ)(−1)^{d(ξ)}χ_{S}^{loc,Q}(P (ξ))(σ)π(ξ).

Here as usual P (ξ) is the irreducible perverse sheaf and π(ξ) the irreducible representation corresponding to the complete geometric parameter ξ.

This is proved in exactly the same way as Lemma 18.15; the \overline{z(ρ)} has disappeared because of its occurrence in Definition 15.11.

Proposition 26.7. Suppose we are in the setting (26.1). Fix a Langlands parameter ϕ_{H} ∈ P (^{∨}H^{Γ}), so that ϕ_{G} = ϵ ∘ ϕ_{H} is a Langlands parameter for ^{∨}G^{Γ}. Fix an element σ_{H} ∈ A_{ϕ_{H}}^{loc,Q_{H}}, and write

σ_{G} = A^{loc}(ϵ)(σ_{H}) ∈ A_{ϕ_{G}}^{loc,Q}

(cf. (5.14)(d)). Then Langlands functoriality sends the virtual representation attached to ϕ_{H} and σ_{H} to the one attached to ϕ_{G} and σ_{G}:

ϵ_{∗}η_{ϕ_{H}}^{loc,Q_{H}} (σ_{H}) = η_{ϕ_{G}}^{loc,Q}(σ_{G}).

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.

Definition 26.8. In the setting of Theorem 10.11, suppose ϕ ∈ P (^{∨}G^{Γ}) is a Langlands parameter, and x ∈ X(^{∨}G^{Γ}) is the corresponding point of the geometric parameter space. Write S for the orbit of x, so that the equivariant microfundamental group of S is isomorphic to the micro-component group for ϕ (Definition 24.13):

A_{S}^{mic,Q} ≃ A_{ϕ}^{mic,Q}.

Fix σ ∈ A_{ϕ}^{mic,Q}. The formal complex virtual representation attached to ϕ and σ is a sum of the irreducible representations in the micro-packet Π^{z}(G/ℝ)_{J,ϕ}^{mic}, with coefficients given by character values for the corresponding representation of A_{ϕ}^{mic,Q}. Specifically,

η_{ϕ}^{mic,Q}(σ) = ∑_{(π,δ)∈Π^{z}(G/ℝ)_{J,ϕ}^{mic}} e(δ)(−1)^{d(π)−d(ϕ)} (χ_{ϕ}^{mic}(π)(σ)) π

(notation 24.13). As in Definition 26.5, we may replace the Kottwitz invariant e(δ) with an extra factor \overline{z(ρ)} next to σ. We may also separate the terms η_{ϕ}^{mic,Q}(σ)(δ) corresponding to each strong real form, and attach a generalized function to each of these.

Suppose ψ ∈ Q(^{∨}G^{Γ}) is an Arthur parameter (Definition 22.4). We identify the Q-component group A_{ψ}^{Q} with A_{ϕ_{ψ}}^{mic,Q} (Definition 24.15). This allows us to define

η_{ψ}^{Q}(σ) = η_{ϕ_{ψ}}^{mic,Q}(σ)

for any σ ∈ A_{ψ}^{Q}. Using the notation of Definitions 22.6 and 24.15, this amounts to

η_{ψ}^{Q}(σ) = ∑_{(π,δ)∈Π^{z}(G/ℝ)_{J,ψ}} e(δ)(−1)^{d(π)−d(ψ)} (χ_{ψ}(π)(σ)) π.

Lemma 26.9. In the setting of Definition 26.8, fix a sufficiently generic point (x, ν) ∈ T_{S}^{∗}(X(^{∨}G^{Γ}))_{reg} (Definition 24.7). The formal complex virtual representation η_{ϕ}^{mic,Q}(σ) corresponds in the isomorphism of Theorem 26.2 to the map (−1)^{d(ϕ)}χ_{(x,ν)}^{mic,Q}(·)(σ) (Definition 24.12) that assigns to a ^{∨}G^{Q}-equivariant perverse sheaf P the trace of σ acting on the stalk of Q^{mic}(P) at (x, ν), multiplied by (−1)^{d(ϕ)}. In terms of standard representations, we therefore have

η_{ϕ}^{mic,Q}(σ) = (−1)^{d(ϕ)} ∑_{ξ∈Ξ^{z}(G/ℝ)} e(ξ)χ_{S}^{mic}(μ(ξ))(σ)M(ξ).

Proof. We use Theorem 15.12 to compute the pairing of η_{ϕ}^{mic,Q}(σ) with the irreducible perverse sheaf P (π). The result is

e(δ)(−1)^{d(π)−d(ϕ)} (χ_{ϕ}^{mic}(π)(σ)) ⟨π, P (π)⟩ = (−1)^{d(ϕ)}χ_{ϕ}^{mic}(π)(σ).

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 parametrization 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 ϕ, or in describing the associated characters of A_{ϕ}^{loc,Q}. In this sense the virtual representations η_{ϕ}^{loc,Q}(σ) 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 Θ(η_{ϕ}^{loc,Q}(σ), δ) may be regarded as known. To describe η_{ϕ}^{loc,Q}(σ) 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 η_{ϕ}^{mic,Q}(σ) 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 η_{ϕ}^{mic,Q}(σ) 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 η_{ϕ}^{mic,Q}(σ). It is immediately clear that some additional hypotheses are required: there is no homomorphism of equivariant micro-fundamental groups A^{mic}(ϵ) 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.

Definition 26.10. Suppose we are in the setting (5.14). Recall from Proposition 4.4 the group Z(^{∨}G)^{θ_{Z}}; write Z(^{∨}G)^{θ_{Z},Q} for its preimage in ^{∨}G^{Q}, so that we have a short exact sequence

1 → Q → Z(^{∨}G)^{θ_{Z},Q} → Z(^{∨}G)^{θ_{Z}} → 1.

If ϕ is any Langlands parameter for ^{∨}G^{Γ}, then this group centralizes the image of ϕ:

Z(^{∨}G)^{θ_{Z},Q} ⊂ ^{∨}G_{ϕ}^{Q}  (26.10)(a)

(notation as in (5.14)). The universal Q-component group for ^{∨}G^{Γ} is

A_{univ}^{Q}(^{∨}G^{Γ}) = Z(^{∨}G)^{θ_{Z},Q}/(Z(^{∨}G)^{θ_{Z},Q})_{0}.  (26.10)(b)

The map of (26.10)(a) provides a natural homomorphism

i_{univ}^{loc} : A_{univ}^{Q}(^{∨}G^{Γ}) → A_{ϕ}^{loc,Q}  (26.10)(c)

for every Langlands parameter ϕ; 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 (π, δ) is an irreducible canonical projective representation of type z of a strong real form δ of type J. Theorem 10.10 associates to (π, δ) a complete Langlands parameter (ϕ(π, δ), τ(π, δ)), with τ(π, δ) an irreducible representation of A_{ϕ}^{loc,Q}. Define

τ_{univ}(π, δ) = τ(π, δ) ∘ i_{univ}^{loc} : A_{univ}^{Q}(^{∨}G^{Γ}) → ℂ^{×},  (26.10)(d)

a one-dimensional character of A_{univ}^{Q}. In terms of geometric parameters, τ_{univ}(π, δ) is the character by which Z(^{∨}G)^{θ_{Z},Q} acts on the stalks of the equivariant local system (or the equivariant perverse sheaf) corresponding to (π, δ).

We have already considered several cases of this definition. The element z(ρ) belongs to Z(^{∨}G)^{θ_{Z},Q}, and Definition 15.8 says that

τ_{univ}(π, δ)(\overline{z(ρ)}) = e(δ),  (26.11)(a)

the Kottwitz invariant of the real form. As another example, let z_{0} ∈ Z(G)^{σ_{Z},fin} be the second invariant of the extended group (G^{Γ}, W) (cf. (3.5)). Then Lemma 10.9(b) says (in the notation defined there)

τ_{univ}(π, δ)(\overline{z^{Q}}) = χ_{z_{0}δ^{2}}(z^{Q})  (z^{Q} ∈ Q).  (26.11)(b)

These examples are special cases of the following general fact.

Lemma 26.12. In the setting of Definition 26.10, the character τ_{univ}(π, δ) of Z(^{∨}G)^{θ_{Z},Q} depends only on the G-conjugacy class of δ, and not on the representation π. We may therefore write it as τ_{univ}(δ).

We have not found a particularly compelling proof of this result, but one can proceed along the following lines. The group Z(^{∨}G)^{θ_{Z},Q} must act by scalars on an irreducible ^{∨}G^{Q}-equivariant perverse sheaf on the geometric parameter space. By Theorem 15.12, it follows that τ_{univ}(π, δ) is unchanged if π is replaced by any composition factor of the standard limit representation having π as a quotient. In this way (using the subquotient theorem) we can reduce to the case of ordinary principal series representations of G(ℝ, δ)^{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 τ(π, δ) directly (see section 13) and verify the lemma. We leave the details to the reader.

Definition 26.13. Suppose we are in the setting of Definition 26.10, and σ ∈ A_{univ}^{Q}(^{∨}G^{Γ}). Using the notation of (18.3), we now define a function ζ_{σ} on G(ℝ, ∗)_{SR}^{can} by

ζ_{σ}(g, δ) = τ_{univ}(δ)(σ)  (26.13)(a)

The locally finite formal complex virtual representation η ∈ \overline{K}_{ℂ,f}Π^{z}(G/ℝ)_{J} is said to be strongly σ-stable if the function

Θ_{SR}(η, ∗)(ζ_{σ})^{−1}  (26.13)(b)

is constant on the fibers of the first projection p_{1} : G(ℝ, ∗)_{SR}^{Q} → G^{Q} (Definition 18.6). Define

\overline{K}_{ℂ,f}Π^{z}(G/ℝ)_{J}^{σ−st}  (26.13)(c)

to be the vector space of these virtual representations.

Theorem 26.14. Suppose we are in the setting of Theorem 10.11, and σ ∈ A_{univ}^{Q}(^{∨}G^{Γ}) (Definition 26.10).

a) The restriction of a strongly σ-stable formal complex virtual representation to single strong real form is stable (Definition 18.2).

b) Suppose δ_{0} is a strong real form of G of type J, and η_{0} is a stable virtual representation of G(ℝ, δ_{0})^{can} (Definition 18.2). Then there is a strongly σ-stable locally finite formal complex virtual representation η of G with the property that

Θ_{SR}(η, ∗)(g, δ_{0}) = Θ_{SR}(η_{0})(g)

for every strongly regular element g ∈ G(ℝ, δ_{0})^{can}. If δ_{0} is quasisplit, then η is unique.

c) The vector space \overline{K}_{ℂ,f}Π^{z}(G/ℝ)_{J}^{σ−st} has as a basis the set \{ η_{ϕ}^{loc,Q}(i_{univ}^{loc}(σ)) \} (Definitions 26.5 and 26.10), as ϕ varies over Φ(^{∨}G^{Γ}).

d) The vector space \overline{K}_{ℂ,f}Π^{z}(G/ℝ)_{J}^{σ−st} has as a basis the set \{ η_{ϕ}^{mic,Q}(i_{univ}^{mic}(σ)) \} (Definitions 26.8 and 26.10), as ϕ varies over Φ(^{∨}G^{Γ}).

Proof. Part(a) follows from Lemma 18.5. The function ζ_{σ} 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 ±1 in each diagonal entry. Q.E.D.

In the setting of Example 19.17, it is not difficult to see that the function ζ_{σ} is nothing but the character of the complex formal virtual representation η_{ϕ_{0}}^{mic,Q}(σ).

Definition 26.15. Suppose ^{∨}G^{Γ} is a weak E-group (Definition 4.3), and Q is any quotient of π_{1}(^{∨}G)^{alg} (Definition 10.10). A weak endoscopic datum for ^{∨}G^{Γ} is a pair (s^{Q},^{∨}H^{Γ}), subject to the following conditions.

i) s^{Q} ∈ ^{∨}G^{Q} is a semisimple element, and ^{∨}H^{Γ} ⊂ ^{∨}G^{Γ} is a weak E-group. Write s for the image of s^{Q} in ^{∨}G.

ii) ^{∨}H^{Γ} is open in the centralizer of s in ^{∨}G^{Γ}.

In (i), it is understood that the weak E-group structure ^{∨}H^{Γ} → Γ is the restriction of that for G; that is, that the inclusion

ϵ : ^{∨}H^{Γ} → ^{∨}G^{Γ}

is an L-homomorphism (Definition 5.1). An endoscopic datum is a weak endoscopic datum endowed with an E-group structure on ^{∨}H^{Γ} (Definitions 4.6, 4.14). That is, it is a triple (s^{Q},^{∨}H^{Γ}, D_{H}), subject to the conditions above and iii) D_{H} is a ^{∨}H-conjugacy class of elements of finite order in ^{∨}H^{Γ} −^{∨}H, each acting

by conjugation on ^{∨}H as a distinguished involution.

Endoscopic data (s^{Q},^{∨}H^{Γ}, D_{H}) and ((s^{Q})′, (^{∨}H^{Γ})′, (D_{H})′) are equivalent if there is an element g^{Q} ∈ ^{∨}G^{Q} with the property that

g(^{∨}H^{Γ})′g^{−1} = ^{∨}H^{Γ},  g(D_{H})′g^{−1} = D_{H},  g^{Q}(s^{Q})′(g^{Q})^{−1} ∈ s^{Q}(Z(^{∨}H)^{θ_{Z},Q})_{0}.

Here g is the image of g^{Q} in^{∨}G. The last condition says that s^{Q} and g^{Q}(s^{Q})′(g^{Q})^{−1} should have the same image in A_{univ}^{Q_{H}}(^{∨}H^{Γ}).

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^{Q}Z(^{∨}H^{Q})_{0} by sZ(^{∨}H^{Γ})_{0}Z(^{∨}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} ∈ Z(^{∨}G)^{θ_{Z},Q} (notation as in Proposition 4.4) defines an endoscopic datum (s^{Q},^{∨}G^{Γ}, D). Two of these are equivalent if and only if s^{Q} and (s^{Q})′ have the same image in A_{univ}^{Q} (Definition 26.10). In general, suppose s^{Q} is any semisimple element of ^{∨}G^{Q}; write s for its image in ^{∨}G. Define

S = centralizer of s in ^{∨}G^{Γ}, S_{1} = centralizer of s in ^{∨}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 ϕ ∈ P (^{∨}G^{Γ}) is a Langlands parameter, and s^{Q} ∈ ^{∨}G_{ϕ}^{Q} is a semisimple element (Definition 5.11). (One should think of s^{Q} as representing an element σ ∈ A_{ϕ}^{loc,Q}.) Let ∨H be the identity component of the centralizer of s in ^{∨}G. Then

∨H^{Γ} = (^{∨}H)ϕ(W_{ℝ})  (26.16)

is a weak E-group, and the pair (s^{Q},^{∨}H^{Γ}) 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^{Γ}, W) a corresponding E-group (^{∨}G^{Γ}, D) with second invariant z, and a quotient Q of π_{1}(^{∨}G)^{alg}. Fix a set of endoscopic data

(s^{Q},^{∨}H^{Γ}, D_{H})  (26.17)(a)

for ^{∨}G^{Γ} as in Definition 26.15, say with second invariant z_{H}. (We may also call this endoscopic data for G.) An extended endoscopic group attached to this set of data is an extended group

(H^{Γ}, W_{H})  (26.17)(b)

with E-group ^{∨}H^{Γ} (Definition 4.3) and second invariant 1 (Proposition 3.6). Recall from Definition 1.12 that the extended group structure W_{H} includes an equivalence class of quasisplit strong real forms of H. Fix one of these δ_{0}. In the terminology of [LS], H(ℝ, δ_{0}) is an endoscopic group attached to the endoscopic data. The L-homomorphism

ϵ : ^{∨}H^{Γ} ↪ ^{∨}G^{Γ}  (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

ϵ^{∗} : KX(^{∨}G^{Γ})^{Q} → KX(^{∨}H^{Γ})^{Q_{H}}  (26.17)(d)

and Langlands functoriality

ϵ_{∗} : \overline{K}_{ℂ}Π^{z_{H}} (H/ℝ)^{Q_{H}} → \overline{K}_{ℂ}Π^{z}(G/ℝ)^{Q}.  (26.17)(e)

With these definitions, it is easy to check that the element s^{Q} from the endoscopic datum belongs to Z(^{∨}H)^{θ_{Z},Q_{H}}, and therefore defines a class

σ_{H} ∈ A_{univ}^{Q_{H}}(^{∨}H^{Γ})  (26.17)(f)

(Definition 26.10).

Definition 26.18. Suppose we are in the setting (26.17). Endoscopic lifting is the restriction of Langlands functoriality to σ_{H}-stable virtual representations. That is,

Lift : \overline{K}_{ℂ,f}Π^{z_{H}} (H/ℝ)_{J_{H}}^{σ_{H}−st} → \overline{K}_{ℂ,f}Π^{z}(G/ℝ)_{J}.

Fix a quasisplit strong real form δ_{0} of H from the distinguished class given by W_{H}, and write

\overline{K}_{ℂ,f}Π^{z_{H}} (H(ℝ, δ_{0}))^{st}

for the Grothendieck group of stable complex virtual representations (projective of type z_{H}) of the group H(ℝ, δ_{0}). By Theorem 26.14 (which relies on Shelstad’s Theorem 18.7), restriction to H(ℝ, δ_{0}) defines an isomorphism

\overline{K}_{ℂ,f}Π^{z_{H}} (H/ℝ)_{J_{H}}^{σ_{H}−st} ≃ K_{ℂ}Π^{z_{H}} (H(ℝ, δ_{0}))^{st}.

Using this identification, we may regard lifting as defined on stable virtual (canonical projective) representations of the endoscopic group H(ℝ, δ_{0}):

Lift_{0} : K_{ℂ}Π^{z_{H}} (H(ℝ, δ_{0}))^{st} → \overline{K}_{ℂ,f}Π^{z}(G/ℝ)_{J}.

If we fix a strong real form δ of G of type J (that is, satisfying δ^{2} ∈ Jz_{0}, with z_{0} the second invariant of the extended group G^{Γ}), then we may project the image of lift on the complex virtual (projective of type z) representations of G(ℝ, δ). This gives a map

Lift_{0}(δ) : K_{ℂ}Π^{z_{H}} (H(ℝ, δ_{0}))^{st} → K_{ℂ}Π^{z}(G(ℝ, δ)).

It is essentially this restricted map that Langlands and Shelstad use.

Lemma 26.19. Endoscopic lifting depends only on the equivalence class of endoscopic data. More precisely, suppose (s^{Q},^{∨}H^{Γ}, D_{H}) and ((s^{Q})′, (^{∨}H^{Γ})′, (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^{Γ}, W_{H}) and ((H^{Γ})′, W′_{H}) (cf. (26.17)). Fix an equivalence ^{∨}j from (s^{Q},^{∨}H^{Γ}, D_{H}) to ((s^{Q})′, (^{∨}H^{Γ})′, (D_{H})′), implemented by conjugation by an element g^{Q} ∈ ^{∨}G^{Q} (Definition 26.15). Finally, choose distinguished (quasisplit) strong real forms δ_{0} and δ′_{0} for H and H′.

a) The isomorphism ^{∨}j carries σ_{H} to σ_{H′} (cf. (26.17)(f)).

b) The isomorphism ^{∨}j induces an isomorphism

j : (H^{Γ}, W_{H}) →^{∼} ((H^{Γ})′, W′_{H})

unique up to an inner automorphism from H.

c) The isomorphism j allows us to identify equivalence classes of irreducible canonical projective representations of strong real forms of H and H′:

j : Π^{z_{H}} (H/ℝ)_{J_{H}} →^{∼} Π^{z_{H′}} (H′/ℝ)_{J_{H′}} .

d) The isomorphism j induces an isomorphism

j_{0} : H(ℝ, δ_{0}) →^{∼} H′(ℝ, δ′_{0})

unique up to an inner automorphism from the normalizer of H′(ℝ, δ_{0}) in H.

e) The isomorphism j_{0} allows us to identify stable virtual representations of H(ℝ, δ_{0}) and H′(ℝ, δ′_{0}):

j_{0} : K_{ℂ}Π^{z_{H}} (H(ℝ, δ_{0}))^{st} →^{∼} K_{ℂ}Π^{z_{H′}} (H′(ℝ, δ′_{0}))^{st}.

This isomorphism depends only on ^{∨}j.

f) If η_{H} is a σ_{H}-stable formal complex virtual (canonical projective of type z_{H}) representation of strong real forms of H of type J_{H}, then

Lift(η_{H}) = Lift(j(η_{H})).

g) If (η_{0})_{H} is a complex virtual (canonical projective of type z_{H}) representation of

H(ℝ, δ_{0}), then

Lift_{0}((η_{0})_{H}) = Lift_{0}(j_{0}((η_{0})_{H})).

Proof. Part (a) is immediate from the definition of equivalence and the definition of A_{univ}^{Q_{H}} (Definition 26.10). Part (b) is Proposition 3.6, and then (c) is immediate. For (d), j(δ_{0}) must be a distinguished strong real form coming from W_{H′}. Since these strong real forms constitute a single conjugacy class, conjugation by an element of H′ carries j(δ_{0}) to δ′_{0}. The composition of this inner automorphism with j gives j_{0}. For (e), we need only know that the normalizer of H′(ℝ, δ′_{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 δ in ours). To do this, notice that both kinds of lifting can be computed explicitly on the basis elements η_{ϕ_{H}}^{loc,Q_{H}} (σ_{H})(δ_{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 Π^{z}(G(ℝ, δ)_{ϕ_{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.

Lemma 26.20. Suppose G is a (possibly disconnected) complex algebraic group, and a_{0} ∈ 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).

a) The set of elements of finite order is Zariski dense in Z(H)_{s}.

b) The set

U = { a ∈ Z(H)_{s} | dim G_{a} = dim H }

is Zariski open in Z(H)_{s} and contains a_{0}.

c) There is an element a′ ∈ a_{0}(Z(H)_{s})_{0} of finite order such that H is open in the centralizer of a′ in G.

Proof. The identity component (Z(H)_{s})_{0} is a product of copies of ℂ^{×}. The elements of finite order are therefore dense in it. We need only show that they meet every other component. Suppose b ∈ Z(H)_{s}. Since the group of connected components of Z(H) is finite, b^{n} ∈ (Z(H)_{s})_{0} for some positive integer n. Since ℂ^{×} is a divisible group, there is an element b_{0} ∈ (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} ⊃ H for every a ∈ Z(H)_{s}. Consequently

U = { a ∈ Z(H)_{s} | det(Ad(a)|_{𝔤/𝔥}) ≠ 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′ ∈ sZ(^{∨}H^{Γ})_{0} so that s′ has finite order, and

∨H^{Γ} is open in the centralizer of s′ in ^{∨}G^{Γ}.  (26.21)(a)

Now Z(^{∨}H^{Γ})_{0} is just Z(^{∨}H)_{0}^{θ_{Z}}. We may therefore find a preimage

(s^{Q})′ ∈ s^{Q}(Z(^{∨}H)^{θ_{Z},Q_{H}})_{0} ⊂ ^{∨}G^{Q}  (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 ^{∨}H^{Q}, and this is all we need to apply the results from section 25. (For example, any particular ∨G^{Q}-equivariant constructible sheaf on X(^{∨}G^{Γ}) will actually be equivariant for an algebraic quotient of ^{∨}G^{Q}.) By Definition 26.15,

((s^{Q})′,^{∨}H^{Γ}, D_{H})  (26.21)(c)

is again an endoscopic datum, equivalent to (s^{Q},^{∨}H^{Γ}, D_{H}) by the identity map on ∨H^{Γ}. In this setting, we will write σ for the action of s′ on X(^{∨}G^{Γ}) or on ^{∨}G (by conjugation); these are compatible automorphisms of finite order. By the choice of s′,

(^{∨}G^{σ})_{0} ⊂ ^{∨}H ⊂ ^{∨}G^{σ}.  (26.21)(d)

Lemma 26.22 In the setting (26.21), X(^{∨}H^{Γ}) is open and closed in X(^{∨}G^{Γ})^{σ}.

Proof. It is immediate from the definition of the geometric parameter space that

X(^{∨}G^{Γ})^{σ} = { (y, Λ) | y ∈ ^{∨}G^{σ}, Ad(s′)Λ = Λ }.

Because a canonical flat Λ 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(^{∨}G^{Γ})^{σ} = { (y, Λ) | y ∈ ^{∨}G^{σ}, Λ ∩ 𝔥 ≠ 0 }.

On the other hand, the image of the closed immersion X(ϵ) (Corollary 6.21) is

{ (y, Λ) | y ∈ ^{∨}H, Λ ∩ 𝔥 ≠ 0 }.

Since ^{∨}H is open and closed in ^{∨}G^{σ}, the lemma follows. Q.E.D.

Proposition 26.23. In the setting (26.17), suppose x ∈ X(^{∨}H^{Γ}); write ϵ(x) for the corresponding point of X(^{∨}G^{Γ}).

a) The element s ∈ ^{∨}G from the endoscopic datum defines an automorphism σ of the conormal space T_{^{∨}G,ϵ(x)}^{∗}(X(^{∨}G^{Γ})) to the ^{∨}G-orbit of ϵ(x).

b) There is a natural isomorphism

ϵ : T_{^{∨}H,x}^{∗}(X(^{∨}H^{Γ})) →^{∼} (T_{^{∨}G,ϵ(x)}^{∗}(X(^{∨}G^{Γ})))^{σ} .

c) In the isomorphism of (b),

(T_{^{∨}G,ϵ(x)}^{∗}(X(^{∨}G^{Γ}))_{reg})^{σ} ⊂ ϵ (T_{^{∨}H,x}^{∗}(X(^{∨}H^{Γ}))_{reg}) .

d) Suppose (x, ν) ∈ T_{^{∨}H,x}^{∗}(X(^{∨}H^{Γ})), so that (ϵ(x), ϵ(ν)) ∈ T_{^{∨}G,ϵ(x)}^{∗}(X(^{∨}G^{Γ})). With notation as in Definition 24.7, assume that

(x, ν) ∈ U_{^{∨}H·x}, (ϵ(x), ϵ(ν)) ∈ U_{^{∨}G·ϵ(x)}.

(This is automatic if the orbit ^{∨}G · (ϵ(x), ϵ(ν)) is open in the conormal bundle.) Then there is a natural homomorphism of micro-component groups

A^{mic}(ϵ) : A_{x,ν}^{mic,Q_{H}} → A_{ϵ(x),ϵ(ν)}^{mic,Q}.

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 σ, 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.

Theorem 26.24. Suppose we are in the setting (26.17). Fix a Langlands parameter ϕ_{H} ∈ P (^{∨}H^{Γ}), so that ϕ_{G} = ϵ ∘ ϕ_{H} (that is, ϕ_{H} regarded as a map into the larger group ^{∨}G^{Γ}) is a Langlands parameter for G. Write x_{0} ∈ X(^{∨}H^{Γ}) for the point corresponding to ϕ_{H}, so that ϵ(x_{0}) ∈ X(^{∨}G^{Γ}) corresponds to ϕ_{G}. Fix a generic conormal covector ν_{0} ∈ U_{^{∨}H·x_{0}} (Definition 24.7), so that the microcomponent group for ϕ_{H} may be computed at (x_{0}, ν_{0}):

A_{ϕ_{H}}^{mic,Q_{H}} = A_{(x_{0},ν_{0})}^{mic,Q_{H}}

(Definition 24.13). Using Proposition 26.23, regard (ϵ(x_{0}), ϵ(ν_{0})) as a conormal covector to ^{∨}G · ϵ(x_{0}). Assume that this covector is also generic, so that

A_{ϕ_{G}}^{mic,Q} = A_{(}ϵ(x_{0}), ϵ(ν_{0}))^{mic,Q},

and Proposition 26.23 provides a homomorphism

A^{mic}(ϵ) : A_{ϕ_{H}}^{mic,Q_{H}} → A_{ϕ_{G}}^{mic,Q}.

Define σ_{H} ∈ A_{univ}^{Q_{H}}(^{∨}H^{Γ}) as in (26.17)(f). We write also

σ_{ϕ_{H}}^{mic} = i_{univ}^{mic}(σ_{H}) ∈ A_{ϕ_{H}}^{mic,Q_{H}} , σ_{ϕ_{G}}^{mic} = A^{mic}(ϵ)(σ_{ϕ_{H}}^{mic}) ∈ A_{ϕ_{G}}^{mic,Q}.

Then endoscopic lifting (Definition 26.18) sends the σ_{H}-stable formal complex virtual representation attached to ϕ_{H} and σ_{ϕ_{H}}^{mic} to the one attached to ϕ_{G} and σ_{ϕ_{G}}^{mic}:

Lift(η_{ϕ_{H}}^{mic,Q_{H}} (σ_{ϕ_{H}}^{mic})) = η_{ϕ_{G}}^{mic,Q}(σ_{ϕ_{G}}^{mic}).

In terms of the restricted map of Langlands and Shelstad, we have

Lift_{0}(δ)(η_{ϕ_{H}}^{mic,Q_{H}} (δ_{0})) = e(δ) ∑_{π∈Π^{z}(G(ℝ,δ))_{ϕ_{G}}^{mic}}(−1)^{d(π)−d(ϕ_{G})}(χ_{ϕ_{G}}^{mic,Q}(π)(σ_{ϕ_{G}}^{mic}))π

for each strong real form δ of type J.

Proof. By Lemma 26.19, we may replace the endoscopic datum by an equivalent one without changing Lift. By (26.21), we may therefore assume that s has finite order. Now regard Lift(η_{ϕ_{H}}^{mic,Q_{H}} (σ_{ϕ_{H}}^{mic})) as a linear functional on constructible^{∨}G^{Q}- equivariant sheaves (Theorem 26.2), and evaluate it on such a sheaf C. By Lemma 26.9, the result is (−1)^{d(ϕ_{H})} times the alternating sum of the traces of σ_{ϕ_{H}}^{mic} on the on the stalks at (x_{0}, ν_{0}) of (Q^{mic})^{i}(ϵ^{∗}C). By Theorem 25.8, this is equal to (−1)^{d(ϕ_{G})} times the alternating sum of the traces of σ_{ϕ_{G}}^{mic} on the stalks at (ϵ(x_{0}), ϵ(ν_{0})) of (Q^{mic})^{i}(C). By Lemma 26.9 again, this alternating sum is η_{ϕ_{G}}^{mic,Q}(σ_{ϕ_{G}}^{mic}) evaluated at C, as we wished to show. Q.E.D.

Here is the solution to Problem D from the introduction.

Theorem 26.25. Suppose we are in the setting (26.17). Fix an Arthur parameter ψ_{H} ∈ Q(^{∨}H^{Γ}) (Definition 22.4), so that ψ_{G} = ϵ ∘ ψ_{H} (that is, ψ_{H} regarded as a map into the larger group ^{∨}G^{Γ}) is an Arthur parameter for G. Write

A^{Arthur}(ϵ) : A_{ψ_{H}}^{Q_{H}} → A_{ψ_{G}}^{Q}

for the induced map on component groups. Define

σ_{ψ_{H}} = i_{univ}^{Arthur}(σ_{H}) ∈ A_{ψ_{H}}^{Q_{H}} , σ_{ψ_{G}} = A^{Arthur}(ϵ)(σ_{ψ_{H}}) ∈ A_{ψ_{G}}^{Q}.

Then endoscopic lifting satisfies

Lift(η_{ψ_{H}} (σ_{ψ_{H}})) = η_{ψ_{G}}(σ_{ψ_{G}}).

In terms of the restricted lifting map of Langlands and Shelstad,

Lift_{0}(δ)(η_{ψ_{H}} (σ_{ψ_{H}})(δ_{0})) = e(δ) ∑_{π∈Π^{z}(G(ℝ,δ))_{ψ_{G}}}(−1)^{d(π)−d(ψ_{G})}(χ_{ψ_{G}}(π)(σ_{ψ_{G}}))π

for each strong real form δ of G of type J.

In light of the discussion at Definition 24.15, this is just a special case of Theorem 26.24.

27. Special unipotent representations.

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 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].

Definition 27.1. Suppose ^{∨}G^{Γ} is a weak E-group. An Arthur parameter ψ (Definition 22.4) is said to be unipotent if its restriction to the identity component ℂ^{×} of W_{ℝ} is trivial. Write Q_{unip}(^{∨}G^{Γ}) for the set of unipotent Arthur parameters, and Ψ_{unip}(^{∨}G^{Γ}) for the corresponding set of equivalence classes. Following Definition 22.4, write

P_{unip}(^{∨}G^{Γ}) ⊂ P_{Arthur}(^{∨}G^{Γ}) ⊂ P (^{∨}G^{Γ})

for the corresponding Langlands parameters ϕ_{ψ}, and

Φ_{unip}(^{∨}G^{Γ}) ⊂ Φ_{Arthur}(^{∨}G^{Γ}) ⊂ Φ(^{∨}G^{Γ})

for their equivalence classes.

Proposition 27.2. Suppose ^{∨}G^{Γ} is a weak E-group. The set of unipotent Arthur parameters for ^{∨}G^{Γ} (Definition 27.1) may be identified naturally with the set of Lhomomorphisms (Definition 5.1) from the L-group of P GL(2) to ^{∨}G^{Γ}. Suppose that ψ corresponds to ϵ. Then the Langlands parameter associated to ψ (Definition 22.4) is the lift by ϵ of the Langlands parameter of the trivial representation of P GL(2, ℝ) (Proposition 5.4).

Proof. The dual group of P GL(2) is SL(2). Since P GL(2, 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 Γ (section 4):

L-group of P GL(2) = SL(2) × Γ.

The L-homomorphisms in question are therefore certain homomorphisms from SL(2)× Γ to ^{∨}G^{Γ}. On the other hand, the Weil group W_{ℝ} modulo its identity component is just Γ, so it is immediate from Definitions 22.4 and 27.1 that unipotent parameters are certain homomorphisms from Γ × SL(2) to ^{∨}G^{Γ}. 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 γ : W_{ℝ} → Γ for the natural quotient map (Definition 5.2). The Langlands parameter of the trivial representation of P GL(2, ℝ) is represented by the homomorphism

ϕ : W_{ℝ} → SL(2) × Γ, ϕ(w) = (( \begin{matrix}|w|^{1/2} & 0 \\ 0 & |w|^{−1/2} \end{matrix} ) , γ(w)) .

(This is well-known: in addition to the definitions of sections 11–14, one must understand exactly which standard representation of P GL(2, ℝ) contains the trivial representation.) Comparing this with the definition of ϕ_{ψ} in Definition 22.4 gives the last assertion. Q.E.D.

Corollary 27.3. Suppose ^{∨}G^{Γ} is a weak E-group. Fix an algebraic homomorphism

ψ_{1} : SL(2, ℂ) → ^{∨}G,

and define

S = centralizer of ψ_{1}(SL(2, ℂ)) in ^{∨}G^{Γ}
S_{0} = S ∩^{∨}G,  S_{1} = S − S_{0},  S_{1}^{(2)} = { y_{0} ∈ S_{1} | y_{0}^{2} = 1 }.

Then the set of unipotent Arthur parameters ψ restricting to ψ_{1} on SL(2, 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 ψ corresponds to y_{0} ∈ S_{1}^{(2)}, then the Arthur component group for ψ is the component group of the centralizer S_{0}(y_{0}) of y_{0} in S_{0}:

A_{ψ} = S_{0}(y_{0})/(S_{0}(y_{0}))_{0}, A_{ψ}^{alg} = S_{0}(y_{0})^{alg}/(S_{0}(y_{0})^{alg})_{0}

(Definition 22.4).

This is immediate from the definitions.

Corollary 27.4. Suppose ^{∨}G^{Γ} is a weak E-group. Then the set Ψ_{unip}(^{∨}G^{Γ}) of equivalence classes of unipotent Arthur parameters is finite.

Proof. According to a classical result of Dynkin, the number of H-conjugacy classes of algebraic homomorphisms of SL(2, ℂ) into a complex reductive algebraic group H is finite. Applying this to ^{∨}G, we find that there are only finitely many possible ψ_{1} up to equivalence. Now Corollary 27.3 completes the proof. Q.E.D.

Example 27.5. Suppose G = GL(n), endowed with the inner class of real forms including GL(n, ℝ). Then the L-group is

∨G^{Γ} = GL(n, ℂ) × Γ.  (27.5)(a)

A homomorphism ψ_{1} of SL(2) into ^{∨}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} > ⋯ > p_{r}; say the representation of dimension p_{i} occurs m_{i} times, so that n = ∑ m_{i}p_{i}. It follows from Schur’s lemma that the centralizer in GL(n) of the image of ψ_{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})×⋯×GL(m_{r}),  S = S_{0}×Γ,  S_{1}^{(2)} = (elements of order 2)×{σ};
(27.5)(b)

here σ is the non-trivial element of Γ. The equivalence classes of ψ associated to ψ_{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) ⋯ (m_{r} + 1) different unipotent Arthur parameters associated to ψ_{1}.

It turns out that the Arthur packet of each of these parameters contains exactly one representation of GL(n, 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}} × ⋯ × GL(p_{r})^{m_{r}}.  (27.5)(c)

For each i, fix an integer k_{i} between 0 and m_{i}. Consider the one-dimensional character π_{i} of GL(p_{i})^{m_{i}} defined by

π_{i}(g_{1}, …, g_{m_{i}}) = sgn(det(g_{1}g_{2} … g_{k_{i}})).  (27.5)(d)

Set

π = Ind_{P (ℝ)}^{GL(n,ℝ)}(π_{1} ⊗ ⋯ ⊗ π_{r})  (27.5)(e)

Then π is a unipotent representation attached to the Arthur parameter specified by ψ_{1} and the various k_{i}. (This can be verified by computing ϕ_{ψ}; what is also true, but not so immediately clear, is that there are no other representations of GL(n, ℝ) in the Arthur packet of ψ.)

Example 27.6. Suppose G = G_{1} × G_{1}, endowed with the inner class of real forms including G_{1} (regarded as a real group). The L-group of G is

∨G^{Γ} = (^{∨}G_{1} ×^{∨}G_{1}) ⋊ {1, δ},  (27.6)(a)

with δ acting on the first factor by interchanging the two factors ^{∨}G_{1}. A homomorphism ψ_{1} of SL(2) into ^{∨}G is specified by a pair ψ_{1}^{L}, ψ_{1}^{R} of such homomorphisms into ^{∨}G_{1}; the centralizer S_{0} of its image is a product S_{0}^{L} × S_{0}^{R}. An element (g^{L}, g^{R})δ ∈ ^{∨}G^{Γ} −^{∨}G conjugates (ψ_{1}^{L}, ψ_{1}^{R}) to (Ad(g^{L})ψ_{1}^{R}, Ad(g^{R})ψ_{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 ψ_{1}^{L} is conjugate to ψ_{1}^{R}. So we assume that they are conjugate; after replacing ψ_{1} by a conjugate, we may even assume

ψ_{1}^{L} = ψ_{1}^{R},  S_{0}^{L} = S_{0}^{R}.  (27.6)(b)

An elementary calculation now shows that

S_{1}^{(2)} = { (x, x^{−1})δ | x ∈ S_{0}^{L} },  (27.6)(c)

and that this set is precisely the S_{0} conjugacy class of δ. There is therefore exactly one equivalence class of unipotent Arthur parameters associated to ψ_{1}. The corresponding representations are (as we will see) the ones considered in [BVunip].

Fix now an algebraic homomorphism

ψ_{1} : SL(2, ℂ) → ^{∨}G.  (27.7)(a)

Define λ = λ_{1} as in (22.8). Write Λ for the canonical flat through λ (Definition 1.7 and (6.4)), and O for the orbit of λ in ^{∨}𝔤. Representations in Arthur packets attached to unipotent Arthur parameters extending ψ_{1} must have infinitesimal character 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(Λ) = ψ_{1}(−I).  (27.7)(b)

The automorphism Ad e(λ) acts by +1 on the integer eigenspaces of ad(λ), and by −1 on the half-integer eigenspaces. The map ψ_{1} is called even if e(λ) is central; that is, if ad(λ) has no half-integral eigenspaces. Define

E = dψ_{1} ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) ∈ 𝔫(Λ) ⊂ ^{∨}𝔤(Λ)  (27.7)(c)

as in (22.8). As in Proposition 22.9, we can use an invariant bilinear form to identify (^{∨}𝔤(Λ)/𝔭(Λ))^{∗} with 𝔫(Λ), and so regard E as an element of T_{eP (Λ)}^{∗}(P(Λ)^{0}) or of N_{P(Λ)^{0}}^{∗} (cf. (21.5)).

Lemma 27.8. In the setting (27.7), the element E is P(Λ)^{0}-regular (Definition 20.9); that is, it belongs to the Richardson class Z_{P(Λ)^{0}} in ^{∨}𝔤(Λ) associated to Y (Proposition 20.4). The moment mapping μ has degree 1; equivalently, the only conjugate of 𝔭(Λ) to which E belongs is 𝔭(Λ) itself.

Proof. This is very well-known, but we sketch an argument. We begin with a direct construction of the parabolic subalgebra 𝔭(Λ) of ^{∨}𝔤(Λ) from E. Now 𝔭(Λ) is the sum of the non-negative eigenspaces of λ on ^{∨}𝔤(Λ). Suppose (π, W) is any

finite-dimensional representation of SL(2), with E = dπ ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ). Then the sum

of the non-negative weight spaces of W is

W^{E} + W^{E^{2}} ∩ E · W + W^{E^{3}} ∩ E^{2} · W + ⋯  (27.9)(a)

(It suffices to check this in an irreducible representation W, where it is immediate.) Consequently

𝔭(Λ) = ^{∨}𝔤(Λ)^{Ad E} +^{∨}𝔤(Λ)^{(Ad E)^{2}} ∩ (Ad E)^{∨}𝔤(Λ) + ⋯  (27.9)(b)

The first consequence of this description is that the centralizer of E in ^{∨}G(Λ) must normalize 𝔭(Λ), and consequently must belong to P (Λ):

∨G(Λ)^{E} ⊂ P (Λ).  (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 𝔫(Λ) of 𝔭(Λ), so

ad(E)𝔭(Λ) = 𝔫(Λ).  (27.9)(d)

This says that the P (Λ) orbit of E is open in 𝔫(Λ), which certainly implies that E is in the Richardson class for P (Λ). Proposition 20.4 now says that the containment (27.9)(c) is equivalent to the assertion about the degree of μ. 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 ϕ_{ψ} = ϕ(y, λ) (cf. (5.8) and (22.8)) the difference is this. In Corollary 27.3, we fix ψ_{1} (and therefore λ_{1} = λ 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 λ.

Now the equivalence classes of unipotent Arthur parameters ψ of infinitesimal character O are in one-to-one correspondence with the equivalence classes of associated Langlands parameters ϕ_{ψ} (Definition 22.4). These in turn correspond to certain ^{∨}G orbits on X(O,^{∨}G^{Γ}) (Proposition 1.10) and therefore to certain K_{j} orbits on P(Λ)^{0} (Proposition 6.24). It is these latter orbits that we wish to identify.

Theorem 27.10. Suppose we are in the setting (21.1), and that the orbit O arises from a homomorphism of SL(2, ℂ) as in (27.7). The following sets are in one-to-one correspondence.

1) Equivalence classes of unipotent Arthur parameters supported on X_{j}(O,^{∨}G^{Γ}).

2) K_{j}-conjugacy classes of parabolic subgroups P′ ∈ P(Λ)^{0} with the following two properties: a) θ_{j}P′ = P′; and

b) 𝔫′ ∩ 𝔰_{j} ∩ Z_{P(Λ)^{0}} ≠ ∅. Here 𝔫′ is the nil radical of the Lie algebra of P′.

3) K_{j} orbits on 𝔰_{j} ∩ Z_{P(Λ)^{0}}. 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 𝔰_{j} ∩ Z_{P(Λ)^{0}}.

Notice that the K_{j}-conjugacy classes in (2) are precisely the “regular orbits” of K_{j} on P(Λ)^{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.

Proof. Suppose S is an orbit on X_{j}(O,^{∨}G^{Γ}) corresponding to a unipotent Arthur parameter. According to Proposition 6.24, we may choose a point (y_{j}, Λ′) in S, with y_{j} one of the representatives chosen in (21.1)(g) (uniquely determined) and P′ = P (Λ′) ∈ P(Λ)^{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 ψ′ for which the corresponding elements y′, λ′, and E′ = E_{ψ′} (cf. (22.8)(c)) satisfy

y′ = y_{j},  λ′ ∈ Λ′.  (27.11)(a)

Since E′ is in the +1 eigenspace of ad λ′, it follows that E′ ∈ 𝔫(λ′). Since Ad(y′)E′ = −E′, E′ must also be in the −1 eigenspace 𝔰_{j} of θ_{j}. By Lemma 27.8, E′ ∈ Z_{P(Λ)^{0}}. This gives (2)(b). Since y′ and λ′ commute, θ_{j} fixes P (λ′) = 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 𝔰_{j} ∩ Z_{P}(Λ)^{0} that is dense in 𝔰_{j} ∩ Z_{P}(Λ)^{0}.

Finally, suppose we are given an orbit of K_{j} on 𝔰_{j} ∩ Z_{P(Λ)^{0}}. Fix a point E′ of this orbit, and construct a homomorphism ψ′_{1} from SL(2) to ^{∨}G(λ) satisfying

dψ′_{1} ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) = E′,  ψ′_{1}(θ_{SL(2)}x) = θ_{j}ψ′_{1}(x).  (27.11)(b)

(Here θ_{SL(2)} is the involutive automorphism conjugation by ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ).) That this

is possible is proved in [KR]. From [KR] one can also deduce that ψ′_{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 ^{∨}G(λ) to our original E, it follows that the element λ′ attached to ψ′_{1} must be conjugate to λ. Because e(λ) is central in ^{∨}G(λ), it follows that

ψ′_{1}(−I) = e(λ).  (27.11)(c)

Define

y_{0} = y_{j}ψ′_{1} ( \begin{matrix}−i & 0 \\ 0 & i \end{matrix} ) .  (27.11)(d)

Because of (27.11)(b), y_{0} commutes with the image of ψ′_{1}. Because of (27.11)(c), y_{0}^{2} = 1. By Corollary 27.3, the pair (y_{0}, ψ′_{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.

Theorem 27.12. Suppose we are in the setting (21.1), and that the orbit O arises from a homomorphism of SL(2, ℂ) as in (27.7). Fix an equivalence class of unipotent Arthur parameters supported on X_{j}(O,^{∨}G^{Γ}), say corresponding to a ^{∨}G orbit S on X_{j}(O,^{∨}G^{Γ}). Write S_{K} for the corresponding orbit of K_{j} on Y, and Z_{S} for the corresponding orbit of K_{j} on 𝔰_{j} ∩ Z_{P(Λ)^{0}} (Theorem 27.10). Then the bijections of Theorem 21.2 identify the following sets:

1) irreducible (^{∨}𝔤(Λ), K_{j}^{alg})-modules M annihilated by I_{P(Λ)^{0}}, with the property that Z_{S} ⊂ Ch(M) (notation (20.5));

2) irreducible K_{j}^{alg}-equivariant D_{P(Λ)^{0}}-modules M, with the property that T_{S_{K}}^{∗} (P(Λ)^{0}) ⊂ Ch(M) (Definition 19.9); and

3) irreducible ^{∨}G^{alg}-equivariant D-modules N on X_{j}(O,^{∨}G^{Γ}), with the property that T_{S}^{∗}(X_{j}) ⊂ Ch(N). Suppose in addition we are in the setting of Theorem 10.4. Then each of these

sets is in bijection with

4) Π^{z}(G/ℝ)_{ψ}; that is, equivalence classes of irreducible canonical projective representations of type z of strong real forms of G, belonging to the Arthur packet for ψ. In cases (1)–(3), the bijections identify the corresponding multiplicities in char-

acteristic cycles. Their common value (a positive integer) is equal to the weight χ_{ψ}(π) of Theorem 22.7 for the representation in (4).

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.

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

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 π to be minimal, or to requiring the associated variety of the Harish-Chandra module of π to be contained in the closure of a certain complex nilpotent orbit in 𝔤^{∗} (namely the associated variety of J_{max}(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.

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, ℝ). The L-group is

∨G^{Γ} = SO(5, ℂ) × Γ.  (27.14)(a)

Henceforth we will drop the 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

ψ_{1} : SL(2) → SO(5) ≃ ^{∨}G.  (27.14)(b)

The centralizer in SO(5) of the image of ψ_{1} is evidently

S_{0} = \{( \begin{matrix}ϵI_{3} & 0 \\ 0 & A \end{matrix} ) | A ∈ O(2), det A = ϵ\} ≃ O(2).  (27.14)(c)

Here I_{3} is the 3 by 3 identity matrix, and ϵ = ±1. Just as in Example 27.5, we find that the equivalence classes of unipotent Arthur parameters ψ associated to ψ_{1} correspond to conjugacy classes of elements of order 2 in S_{0}. There are exactly three such classes, represented by the elements

s_{++} = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) ,  s_{+−} = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) ,  s_{−−} = ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ) (27.14)(d)

in O(2) ≃ S_{0}. Write ψ_{++}, etc., for the corresponding Arthur parameters. The corresponding centralizers are just the centralizers in O(2) of these elements, namely

S_{ψ_{++}} = O(2),  S_{ψ_{+−}} = O(1) × O(1),  S_{ψ_{−−}} = O(2).  (27.14)(e)

The component groups are therefore

A_{ψ_{++}} = ℤ/2ℤ,  A_{ψ_{+−}} = ℤ/2ℤ × ℤ/2ℤ,  A_{ψ_{−−}} = ℤ/2ℤ.  (27.14)(f)

We also want to calculate the various A_{ψ}^{alg}. The algebraic universal cover ^{∨}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 ψ_{1}^{alg} : SL(2) → Sp(4). This turns out to be the diagonal map of Sp(2) (which is isomorphic to SL(2)) into Sp(2) × Sp(2), followed by the obvious embedding of Sp(2) × Sp(2) into Sp(4). From this one can deduce that S_{0}^{alg} ≃ O(2); the covering map

O(2) ≃ S_{0}^{alg} → S_{0} ≃ O(2)  (27.15)(a)

is the double cover of SO(2) on the identity component. It follows that

S_{ψ_{++}}^{alg} = O(2),  S_{ψ_{+−}}^{alg} = quaternion group of order 8,  S_{ψ_{−−}}^{alg} = O(2)

(27.15)(b)

A_{ψ_{++}}^{alg} = A_{ψ_{++}}, A_{ψ_{+−}}^{alg} = quaternion group of order 8, A_{ψ_{−−}}^{alg} = A_{ψ_{−−}}.

(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 δ_{s}, δ_{(1,1)}, δ_{(2,0)}, and δ_{(0,2)}. The last two of these represent the compact real form, which can have no representations of the (singular) infinitesimal character associated to ψ_{1}. The other two represent the inequivalent real forms Sp(4, ℝ) and Sp(1, 1). It turns out that δ_{(1,1)}^{2} = −I,

so representations of Sp(1, 1) correspond to ^{∨}G^{alg}-equivariant perverse sheaves on which π_{1}(^{∨}G)^{alg} acts non-trivially (Lemma 10.9(b)). It follows from (27.15)(c) that the Arthur packets for ψ_{++} and ψ_{−−} can contain no representations of Sp(1, 1). Furthermore, the representation τ_{ψ_{+−}}(π) of A_{ψ_{+−}}^{alg} attached to any irreducible representation π ∈ Π(G/ℝ)_{ψ_{+−}} (Definition 24.15) must satisfy

τ_{ψ_{+−}}(π)(−I) = \{ \begin{matrix}1, & if π is a representation of Sp(4, ℝ); \\ −1, & if π is a representation of Sp(1, 1). \end{matrix}  (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, ℝ) has exactly eight unipotent representations attached to ψ_{1}. They are the three irreducible constituents of Ind_{GL(2,ℝ)}^{Sp(4,ℝ)}(| det |); the three irre-

ducible constituents of Ind_{GL(2,ℝ)}^{Sp(4,ℝ)}(det); and the two irreducible constituents of

Ind_{GL(1,ℝ)×Sp(2,ℝ)}^{Sp(4,ℝ)}(sgn(det) ⊗ 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 π_{s}(m, n).

An even simpler calculation shows that Sp(1, 1) has a unique irreducible unipotent representation attached to ψ_{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 π_{(1,1)}(0).

Here are the Arthur packets.

Π(G/ℝ)_{ψ_{++}} = {π_{s}(0, 0), π_{s}(1, −1)}  (27.17)(a)
Π(G/ℝ)_{ψ_{−−}} = {π_{s}(1, 0), π_{s}(0, −1)}  (27.17)(b)
Π(G/ℝ)_{ψ_{+−}} = {π_{s}(1, 1), π_{s}(−1, −1), π_{s}(2, 2), π_{s}(−2, −2), π_{(1,1)}(0)} (27.17)(c)

In each case the map π ↦ τ_{ψ}(π) is a bijection from the packet onto the set of equivalence classes of irreducible representations of A_{ψ}^{alg}; we have listed the representations so that the one with τ_{ψ}(π) trivial is first. In particular, τ_{ψ_{+−}}(π_{(1,1)}(0)) is the irreducible two-dimensional representation of the quaternion group (cf. (27.16)).

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 π ↦ τ_{ψ}(π). (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.)

Theorem 27.18. Suppose (G^{Γ}, W) is an extended group, and (^{∨}G^{Γ}, D) is an L-group for G^{Γ}. Fix a principal three-dimensional subgroup

ψ_{1} : SL(2) → ^{∨}G.

a) The centralizer S_{0} of ψ_{1} in ^{∨}G is Z(^{∨}G).

b) The set of equivalence classes of unipotent Arthur parameters attached to ψ_{1} may be identified with

{ z ∈ Z(^{∨}G) | zθ_{Z}(z) = 1 }/{ wθ_{Z}^{−1}(w) | w ∈ Z(^{∨}G) } = H^{1}(Γ, Z(^{∨}G)).

c) If ψ is any unipotent Arthur parameter attached to ψ_{1}, then

A_{ψ}^{alg} = Z(^{∨}G)^{θ_{Z},alg}/(Z(^{∨}G)^{θ_{Z},alg})_{0}.

This is precisely the universal component group A_{univ}^{alg}(^{∨}G^{Γ}) of Definition 26.10.

d) The unipotent representations (of some real form G(ℝ, δ)) attached to ψ_{1} are precisely the representations trivial on the identity component G(ℝ, δ)_{0}.

e) Suppose δ is any strong real form of G^{Γ}. Then there is a natural surjection

H^{1}(Γ, Z(^{∨}G)) ↠ Hom(G(ℝ, δ)/G(ℝ, δ)_{0}, ℂ^{×}).

If G(ℝ, δ) is quasisplit, this is an isomorphism.

Suppose ψ is a unipotent Arthur parameter attached to ψ_{1}, and δ is a strong real form of G^{Γ}. Write π(ψ, δ) for the character of G(ℝ, δ)/G(ℝ, δ)_{0} attached to ψ by composing the bijection of (b) with the surjection of (e).

f) Suppose π is an irreducible unipotent representation (attached to ψ_{1}) of a strong real form G(ℝ, δ). Then the integer χ_{ψ}(π) of Theorem 22.7 is given by

χ_{ψ}(π) = \{ \begin{matrix}1 & if π ≃ π(ψ, δ) \\ 0 & otherwise. \end{matrix}

The character τ_{ψ}(π) of A_{ψ}^{alg} ≃ A_{univ}^{alg}(^{∨}G^{Γ}) (see (c) above) is

τ_{ψ}(π, δ) = χ_{ψ}(π)τ_{univ}(δ).

If we fix the unipotent parameter ψ attached to ψ_{1}, we therefore find exactly one unipotent representation for each strong real form. The correspondence π ↦ τ_{ψ}(π) therefore fails to be injective as soon as the number of strong real forms exceeds the cardinality of A_{ψ}^{alg} = Z(^{∨}G)^{θ_{Z},alg}/(Z(^{∨}G)^{θ_{Z},alg})_{0}. The simplest example of this is perhaps for P GL(n) endowed with the class of inner forms including the projective unitary groups P U(p, q). There are [(n + 2)/2] real forms. The dual group is SL(n), with θ_{Z} acting on the center by inversion. Consequently A_{ψ}^{alg} consists of the elements of order 2 in the cyclic group of order n; injectivity fails for n ≥ 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_{ψ}^{alg} has order at most four.

Proof of Theorem 27.18. Define E, F, and λ for ψ_{1} as in (22.8). Because ψ_{1} is principal, λ is a regular element with integral eigenvalues. It follows that P (λ) is a Borel subgroup of ^{∨}G; write it as ^{d}B, and write ^{d}T for the unique maximal torus containing λ in its Lie algebra. Then the centralizer of λ in ^{∨}G is ^{d}T, so S_{0} ⊂ ^{d}T. The element E is the sum of certain root vectors {X_{α}} 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(^{∨}G). This proves (a). For (b), write S for the centralizer of ψ_{1} in ^{∨}G^{Γ}. By Definition 4.6, we can find an element ^{∨}δ ∈ D so that

∨δ ∈ Aut(^{∨}G,^{d}B,^{d}T , {X_{α}})  (27.19)(a)

(see Proposition 2.11). In particular, ^{∨}δ centralizes E. Now λ belongs to [^{∨}𝔤,^{∨}𝔤], and α(λ) = 1 for each simple root α. These properties characterize λ, and they are preserved by ^{∨}δ. Consequently ^{∨}δ also centralizes λ. Finally, it follows from the representation theory of SL(2) that F is characterized by the property [E, F] = 2λ, so ^{∨}δ also centralizes F. Therefore

∨δ ∈ S,  S = Z(^{∨}G) ∪ Z(^{∨}G)^{∨}δ  (27.19)(b)

Since ^{∨}G^{Γ} is an L-group, ^{∨}δ^{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 λ belongs to the derived algebra and takes the value 1 on each simple root. The corresponding infinitesimal character is therefore ρ, 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(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(^{∨}G) ≃ Hom(π_{1}(G), ℂ^{×})  (27.20)(a)

of Lemma 10.2(a). Along the lines of Lemma 10.2(d), we deduce that

H^{1}(Γ, Z(^{∨}G)) ≃ Hom(H^{1}(Γ, π_{1}(G)), ℂ^{×})  (27.20)(b)

To prove (e), it is therefore enough to exhibit an injection

G(ℝ, δ)/G(ℝ, δ)_{0} ↪ H^{1}(Γ, π_{1}(G)).  (27.20)(c)

For this, we begin with the short exact sequence of groups with Γ actions

1 → π_{1}(G) → G^{sc} → G → 1.  (27.20)(d)

(Here we make complex conjugation act by δ on G, and pull this automorphism back to G^{sc}.) The corresponding exact sequence in cohomology includes

H^{0}(Γ, G^{sc}) → H^{0}(Γ, G) → H^{1}(Γ, π_{1}(G)).  (27.20)(e)

Now H^{0} is just the Γ-invariant subgroup; that is (for G and G^{sc}) the group of real points. Hence

G^{sc}(ℝ, δ) → G(ℝ, δ) → H^{1}(Γ, π_{1}(G)).  (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(ℝ, δ)_{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(ℝ, δ) is quasisplit, it suffices to find |H^{1}(Γ, Z(^{∨}G))| linearly independent functions on G(ℝ, δ)/G(ℝ, δ)_{0}. The functions we use are the stable characters Θ(η_{ψ}(δ)) (notation as in Theorem 22.7 and (18.1)), with ψ a unipotent Arthur parameter attached to ψ_{1}. The strongly stable characters η_{ψ} are part of the basis { η_{ϕ}^{mic} } of strongly stable virtual characters (Corollary 19.16), and are therefore linearly independent. By Shelstad’s theorem 18.7, the stable characters Θ(η_{ψ}(δ)) are also linearly independent. By (d), these characters may be regarded as functions on G(ℝ, δ)/G(ℝ, δ)_{0}. By (b), there are |H^{1}(Γ, Z(^{∨}G))| such functions, as we wished to show.

For (f), suppose first that G(ℝ, δ) is quasisplit. The L-packets attached by the Langlands classification to the parameters ϕ_{ψ} (with ψ unipotent attached to ψ_{1}) are disjoint, and each is a non-empty set of representations of the component group of G(ℝ, δ). It follows from (e) that the L-packet Π_{ϕ_{ψ}} 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 π(ψ, δ). Because the integer d(π) (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(ℝ, δ) in the Arthur packet Π_{ψ} is π(ψ, δ). 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

η′_{ψ} = ∑_{δ} π(ψ, δ)

is strongly stable. By the preceding paragraph, it agrees with the Arthur character η_{ψ} on each quasisplit real form. By Theorem 18.7, η′_{ψ} = η_{ψ}. 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 ψ_{1} may be empty; if not, then it is a principal homogeneous space for H^{1}(Γ, Z(^{∨}G)).

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Jeffrey Adams Department of Mathematics University of Maryland College Park, MD 20742

Dan Barbasch Department of Mathematics Cornell University Ithaca, NY 14853

David A. Vogan, Jr. Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139