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\begin{document}
\today

\bigskip
\title{Relevant and petite K-types for split groups}

\author{Dan Barbasch}
       \address[D. Barbasch]{Dept. of Mathematics\\
                Cornell University\\Ithaca, NY 14850}
        \email{barbasch@math.cornell.edu}

\maketitle

\bigskip

\section{Introduction}\label{sec:1} 

This paper discusses the role of the relevant and petite $K$-types in
the classification of the spherical unitary dual for split real
groups. 


\bigskip
Let $G$ be the rational points of a connected linear reductive group
over a local field $\bF$.  Assume $G$ is split, and denote a maximal
abelian torus by $\bA\cong (\bF^\times)^n.$ Let $K$ be a maximal
compact subgroup as in section \ref{sec:2}. A representation is called
spherical if it has nontrivial $K$-fixed vectors.  Any irreducible
spherical representation is realized in a canonical way as a
subquotient, denoted $L_{\bF}(\chi)$, of a principal series
$X_{\bF}(\chi)$ for a character $\chi\in\widehat{\bA}.$ (When the
context is clear, the subscript $\bF$ is dropped from the notation.)
Two such representations $L_{\bF}(\chi)$ and $L_{\bF}(\chi')$ are
equivalent if and only if there is an element $w$ of the Weyl group
$W$ such that $w\chi=\chi'.$ The module $L_\bF(\chi)$ admits a
nondegenerate hermitian form if and only if there is $w\in W$ such
that $w\chi=\ovl{\chi}^{-1}.$ The hermitian form is obtained from a
hermitian form on $X_{\bF}(\chi)$, and $L_{\bF}(\chi)$ is unitary if
and only if the form on $X_{\bF}(\chi)$ is positive semidefinite. For
each $K$-type $(E,\mu),$ the form on $X_{\bF}(\chi)$ restricts to a
hermitian form $A_{\bF}(\mu,\chi)$ on the corresponding $\mu$-isotypic
component.
Thus $L_{\bF}(\chi)$ is unitary if and only if
$A_{\bF}(\mu,\chi)$ is positive semidefinite for all $\mu.$

It is of interest to compare the cases when ${\bF}$ is real and p-adic.
To distinguish the latter notationally, $F$ will denote
any field of the form $\mathbb{Q}_p$.

In the p-adic case, the results in \cite{BM1} and
\cite{BM2} reduce the problem to a finite one, namely to show that
certain related Hermitian forms $a_F(\sig,\chi)$ 
(obtained from the $A_F(\mu,\chi)$) 
are positive semidefinite for all $\sig\in\widehat W$. 
These ideas are used in \cite{BM3} to obtain the spherical unitary
dual for split p-adic groups of classical type. The full unitary dual
for $GL(n)$ was obtained by Tadi\'{c}, and for $G_2$ by Mui\'{c}. 

\bigskip
Consider now the real case. Let $M$ be the centralizer of $\bA$ in $K$.
Then $(E^*_\mu)^M$ is a representation of the Weyl group. The form
$A_{\bR}(\mu,\chi)$ induces, by Frobenius reciprocity, a hermitian form
$a_{\bR}(\mu,\sigma)$ on $\sigma = (E^*_\mu)^M.$ 
In sections \ref{sec:3} and \ref{sec:4} we
define a set of $K$-types, which we call \textit{petite}. They have
the property that $A_{\bR}(\mu,\chi)$ only depends on $\sig$, and in fact
$a_{\bR}(\sig,\chi) = a_F(\sig,\chi)$, where the latter is
defined in the context of the p-adic group as mentioned in the previous
paragraph.

In \cite{B6}, for the classical cases, I introduced  a subset of
$\widehat W$ called \textbf{relevant}, which has the following
properties: 
\begin{itemize}
\item[(a)] A representation $L_F(\chi)$ is unitary if and only if
  $a_F(\sig,\chi)$ is positive semidefinite for all $\sig$ relevant.
\item[(b)] For every relevant $\sig$ there is a petite representation
$E_\mu$ of the maximal compact subgroup of the real group such that
$(E^*_\mu)^M\cong \sig.$
\end{itemize}
We will call the $K$-types in part (b) relevant as well.  The
existence of such a set of $W$-representations implies the following
result.
\begin{theorem}[1]\label{t:0}
The representation  $L_{\bR}(\chi)$ is unitary
\textbf{only if} $L_F(\chi)$ is unitary.
\end{theorem}


\bigskip
 
For split p-adic $F_4$, the spherical unitary dual (in fact the full
Iwahori spherical unitary dual) is obtained in \cite{C}. For p-adic
split type $E_6, E_7, E_8,$ the spherical unitary dual is obtained in
\cite{BC}.  In all these cases, a set satisfying part (a) is given. 
In this paper we show that these $W$-representations also satisfy part
(b). As a consequence, theorem (\ref{t:0}) holds in all cases. 

\medskip
\noindent It is natural to conjecture that the \textbf{only if} can be
strengthened to \textbf{if and only if}, in the theorem. This is done
for the classical groups in \cite{B6}. 

\medskip
The proof given below of the matchup between
petite $K$-types and relevant $W$-types in item (b) is
conceptually different from the one given in \cite{B6} for classical
groups. Instead of identifying the groups $M$ and $W$ explicitly and
using restrictions to Levi components, I use properties of fine
$K$-types, and tensor products. To decompose tensor products in the
exceptional cases, I used the packages \textit{GAP} and \textit{LiE}.
The results for $E_6-E_8$ contain matchups for a larger class of
petite $K$-types than needed for theorem (\ref{t:0}). The definitions
and background are in sections \ref{sec:2} and \ref{sec:3}. The
results are in section \ref{sec:4}, and the calculations are in
section \ref{sec:5}.

\medskip
Let us first consider the problem of restricting a $K$-type to $M$.
The standard restriction formulas are designed to deal with restricting to a
subgroup $H\subset K$ which shares a Cartan subgroup with $K.$ This is
not the case for $M.$ One strategy would be to compute the
restriction of the fundamental representations, and set up an
induction by realizing $K$-types as composition factors of tensor
products of fundamental representations. We can then exploit the fact
that there are good formulas for decomposing tensor products.

It is certainly feasible to obtain the restrictions of fundamental
representations to $M.$ Combined with the induction sketched above,
this gives an efficient way to restrict a $K$-type to $M.$ However the
restrictions of the fundamental representations to $M$ are not very
nice.  On the other hand, the fine $K$-types (section \ref{sec:3})
have very nice restrictions.  Precisely, each fine $K$-type decomposes
into the Weyl orbit of a single $M$-type, and every $M$-type occurs in
at least one fine $K$-type. So the general aim is to replace the
fundamental representations by the fine $K$-types.  This works very
well in the case of $GL(n,\bR)$ and $SO(n,n),$ and $Sp(n,\bR)$ where
all the fundamental representations are fine.  To carry out the
inductive strategy for other groups, it turns out that it is very
useful to consider fine representations of the double cover $\wti K$
(and their restrictions to $\wti M$) arising in the context of a
(nonalgebraic) double cover of the real group $G$.

For $SO(n+1,n),$ the
fine $K$-types do not generate the Grothendieck group of
representations of $K$. However we can establish a weaker form of part (b)
sufficient for theorem (1) by decomposing a tensor product of fine
$K$-types of a cover of $SO(n+1,n).$
It is also not difficult to restrict the fundamental representations
of $K$ to $M$ (this is implicit in \cite{B6}). But we need  a more explicit
knowledge of the structure of $M$. In any case, these ideas give  an efficient
method for restricting representations of $K$ to $M.$ 


For the real group of type $G_2$, the fine $K$-types do not generate the
Grothendieck group of representations of $K$. However we can use the
representation of $K$ on $\mathfrak s,$ from the Cartan decomposition
$\mathfrak g=\mathfrak k +\mathfrak s.$ Its restriction to $M$ is easy to
ascertain. In this case the full unitary is 
contained in the work of \cite{V1}. So the \textit{if and only if}
analogue of theorem (\ref{t:0}) is known to hold for spherical representations.

In the case of $F_4,$  the fine $K$-types do not generate the
Grothendieck group of representations of $K.$ We can again use the
representation of $\mathfrak s.$ 

In the cases of $E_6,\ E_7,$ and $E_8,$ the Grothendieck group is not
generated by the fine $K$-types either. In addition to using
$\mathfrak s,$ the calculations are simplified considerably in the
cases of $E_6$ and $E_8$ by the fact that $\wti M$ has a unique
genuine representation. Thus a genuine representation of $\wti K$
restricts to a multiple of the genuine $\wti M$-type, and the
multiplicity is determined from the dimension.

To establish part (b), we also need to determine the Weyl group
representation on the $M$-fixed vectors. 
This is done by using theorem \ref{t:3.3} together with
sections \ref{sec:5.1}-\ref{sec:5.2}.



\bigskip
I would like to thank David Vogan for generously sharing his ideas
about the relation between $K$-types, Weyl group representations and
signatures. They were the catalyst for this paper. Thanks are also due
to P. Trapa for carefully reading the manuscript, and making valuable
suggestions for improvements.

\medskip 
The paper is an outgrowth of a series of talks given at the conference
in  Dubrovnik in the summer of 2003. I would like to thank the
organizers for giving me  the opportunity of participating, and
express my gratitude for their work to make the conference a success.

\bigskip
\noindent This research was supported by NSF grants DMS-0070561 and DMS0300172.

\bigskip

\section{Notation and background}\label{sec:2}
\subsection{}\label{sec:2.1}
Let $\bF$ be a local field, either $\bR$ or a p-adic field. In the
latter case, let $\CR\supset\C P$ be the ring of integers and its
maximal prime ideal respectively. 
Let $G:=\mathbf G(\bF)$ be the rational points of a  connected
reductive group, $\bF$ real or p-adic. Assume $G$ is split, and fix a 
rational Borel subgroup $B=AN.$ In the real case, fix a Cartan
involution $\theta$, and let $K$ be the maximal compact group which is
its fixed points. In the p-adic case, let    
$K:=\mathbf G(\CR).$ A character $\chi\in\wht A$ is called
\textit{unramified}, if $\chi\big|_{A\cap K}$ is trivial. Since $G$ is
split, $A\cong (\bF^\times)^n$ where $n$ is the rank of $G.$ Thus an
unramified  character is of the form
\begin{equation}
  \label{eq:2.1.1}
  \chi=(\chi_1,\dots , \chi_n)
\end{equation}
where $\chi_i(t)=|t|^{\nu_i}.$ Let $\C L:=(0)$ if $\bF=\bR,$
$\C L:=(\frac{2i\pi}{\log q}\bZ)$ in the p-adic case. We can therefore
identify $\chi$ with an element in $(\bC/\C L)^n.$ 
As is well known, any irreducible spherical
admissible representation of $G$ is the unique spherical  subquotient
of a \textit{standard} induced module
\begin{equation}
  \label{eq:2.1.2}
  X_B(\chi):=Ind_B^G(\chi\otimes\delta_B^{-1/2}),
\end{equation}
where $\chi$ is an unramified character, and $\delta_B$ is the modulus
function of $B.$  Denote this subquotient by
$L(\chi).$  Two such quotients $L(\chi)$ and $L(\chi')$ are
isomorphic if and only if there is 
an element $w\in W:=N_K(A)/C_K(A)$ such that $w\chi=\chi'.$ 
\subsection{}\label{sec:2.2} Suppose $Im\chi\ne 0.$ Then $Im\chi$
defines a proper standard parabolic subgroup $P=MU,$ where 
\begin{equation}
  \label{eq:2.2.1}
  \Delta(M)=\{ \al\ :\ (\al,Im\chi)=0\},\qquad \Delta(U)=\{\al\ :\
  (\al,Im\chi)>0 \}.
\end{equation}
Let $L_M(\chi)$ be the spherical irreducible module of $M$
corresponding to $\chi.$ Then $L_M(\chi)=L_M(\chi_{r})\otimes \chi_i$
where $\chi_r$ is such that $Im\chi_r=0$ and $\chi_i$ is unitary.
\begin{theorem}\label{t:2.2}
$$
L(\chi)=Ind_M^G [L_M(\chi_r)\otimes\chi_i],
$$
and $L(\chi)$ is unitary if and only if $L_M(\chi_r)$ is unitary.  
\end{theorem}
\begin{proof}
For the real case, this is theorem 16.10 in \cite{K}. For the p-adic case
and I-spherical representations it follows from \cite{BM1} and
\cite{BM2}.  
\end{proof}
Because of this theorem, we assume that $\chi$ is \textbf{real}, \ie
$Im\chi=0.$ In this situation, the lattice $\C L$ plays no role, and
we can treat the real and p-adic case in the same way. 

\subsection{}\label{sec:2.3} 
Let $(\chi,\al)\ge 0$ for all roots $\al\in\Delta(\fk n,\fk
a).$ then $X(\chi)$ has a unique irreducible quotient, namely
$L(\chi)$. We call such a $\chi$ \textit{dominant}. If on the other
hand $(Re\chi,\al)\le 0$ for all 
$\al\in\Delta(\fk n,\fk a),$ then $X(\nu)$ has a unique irreducible
submodule which is again $L(\chi).$ 
We call such a $\chi$ \textit{antidominant}. 

Let $B'=AN'$ be another Borel subgroup. Then there is an intertwining
operator
\begin{equation}
  \label{eq:2.3.3}
  A(B,B',\chi):X_B(\chi)\longrightarrow X_{B'}(\chi)
\end{equation}
given by
\begin{equation}
  \label{eq:2.3.4}
  A(B,B',\chi)f(g):=\int_{N'/N\cap N'} f(gn')\ dn'.
\end{equation}
The integral is convergent for $\chi$ very dominant, and has a
meromorphic extension to all $\chi.$ It has no poles if $\chi$ is
dominant. Let $w\in W,$ and write $B':=w(B).$ There is an isomorphism
$R_w:X_{B'}(\chi)\longrightarrow X_{B}(w\chi)$ given by
\begin{equation}
  \label{eq:2.3.5}
R_wf(g):= f(gw^{-1}).  
\end{equation} 
We write 
\begin{equation}
  \label{eq:2.3.6}
A(w,\chi):=R_w\circ A(B,B',\chi):\ X_B(\chi)\longrightarrow X_B(w\chi).
\end{equation}

Recall that  $G=KB,$ and let  $(E,\mu)$ be a $K$-type. Then
the intertwining operator (\ref{eq:2.3.3}) gives rise to a map
\begin{equation}
  \label{eq:2.3.7}
  A(\mu,w,\chi):\Hom_K[E,X_B(\chi)]\longrightarrow\Hom_K[E,X_{B}(w\chi)].
\end{equation}
By Frobenius reciprocity, we get a map
\begin{equation}
  \label{eq:2.3.8}
  a(\mu,w,\chi): (E^*)^M\longrightarrow (E^*)^M.
\end{equation}
We normalize the intertwining operator (\ref{eq:2.3.3}) so that
$a(triv,w,\chi)=1$ by multiplying it with the appropriate meromorphic
function in $\chi.$  


\medskip
The hermitian dual of $X_B(\chi)$ is $X_B(\hchi),$ where
$\hchi:=(\ov{\chi})^{-1}$ ($\chi$ not necessarily rea). Indeed, 
if $v\in X_B(\chi)$ and $w\in X_B(\hchi),$ then $v\ovl{w}$
transforms according to $\delta_B$ under the right action of $B.$ 
Thus the integral of $v\ovl{w}$ over $G/B$ makes sense, 
and the hermitian pairing is
\begin{equation}
  \label{eq:2.3.9}
  \big( v,w\big) :=\int_{G/B} v(g)\ov{w(g)}\ dg.
\end{equation}
Thus the hermitian dual of $L(\chi)$ is $L(\hchi),$ and
$L(\chi)$ is hermitian if and only if there is $w\in W$ such that
$w\chi=\hchi.$ Suppose this is the case and $B':=w(B).$ 
If $\chi$ is dominant, then $\hchi$ is antidominant. So the image of
$A(w,\chi)$  is exactly $L(\chi)$. The hermitian form is given by
\begin{equation}
  \label{eq:2.3.10}
  \langle v_1,v_2\rangle:=\big( v_1,A(w,\chi)(v_2)\big),
\end{equation}
where $\big(\ , \ \big)$ is the pairing in (\ref{eq:2.3.9}). 
Fix a positive definite K-invariant hermitian form for each $K$-type
$(E,\mu).$ Then as in (\ref{eq:2.3.5})-(\ref{eq:2.3.6}), we get hermitian
symmetric maps $a(\mu,\chi).$ Then $L(\chi)$ is unitary if and only if
$a(\mu,\chi)$ is positive semidefinite for all $\mu.$ 


\section{Fine K-types}\label{sec:3}


\subsection{}\label{sec:3.1}
A reference for the results in this section is chapter 4 in \cite{V3}.

Let $(\mu_a,V_a)$ and $(\mu_b,V_b)$ be representations of $K.$ Then
$\Hom_M[V_a,V_b]$ is endowed with a representation of $N_K(M)$ via
\[
n\cdot f(v):=\mu_a(n)f(\mu_b(n^{-1})v).
\] 
Under this action, $M\subset N_K(M)$ acts trivially, so we get a
representation of $W.$ Note that
\[
\Hom_M[V_a,V_b]\cong \Hom_M[V_a\otimes V_b^*,Triv].
\]
In particular, if $V_a=E_\mu,$ and $V_b=Triv,$ we get the more
familiar action  of $W$ on $(E_\mu^*)^M.$ 

\subsection{}\label{sec:3.2}
Let $\al$ be a simple root and $P_\al=M_\al N$ be the standard
parabolic subgroup so that $M_\al$ is $\theta$-stable, and the Lie
algebra of $M_\al$ is isomorphic to 
the $sl(2,\bb R)$ generated by the root vectors $E_{\pm \al}.$ 

The map $\psi_\al: sl(2,\bb R)\longrightarrow \fk g$ determined by 
\[
\psi_\al\begin{bmatrix} 0&1\\0&0\end{bmatrix}=E_\al,\qquad 
\psi_\al\begin{bmatrix} 0&1\\0&0\end{bmatrix}=E_{-\al},
\]
gives a map
  \begin{equation}\label{eq:4.1.1}
    \Psi_\al\ :\ SL(2,\bb R)\longrightarrow G
  \end{equation}
with image $G_\al\subset M_\al,$ a connected group with Lie algebra
isomorphic to $sl(2,\bb R).$  We assume that $\theta E_\al=-E_{-\al}.$ Let
$D_\al=\sqrt{-1}(E_\al-E_{-\al})$, and $s_\al=e^{\sqrt{-1}\pi D_\al/2}.$ Then
$s_\al^2=m_\al$ is in $M\cap M_\al,$ while $s_\al$ itself gives rise
to the corresponding nontrivial element in the Weyl group.



\begin{definition}
  \label{d:3.2}

A $K$-type is called \textbf{fine} (Vogan), if $\mu(iD_\al)=0,\pm 1.$ 

More generally, a $K$-type is said to have level less than or equal to $r,$ if
$|\mu(iD_\al)|\le r.$ 



\end{definition}
The fine $K$-types are the lowest $K$-types of principal series. For every
$\delta,$ there is a fine $K$-type $\mu_\delta$ whose restriction to M
contains $\delta.$ The restriction of every fine $K$-type to $M$ is
the sum over the $W$-orbit of a single $\delta\in\widehat M.$  

Fix a representative $\delta$ for each $W$-orbit, and a fine $K$-type
$\mu_\delta.$
In the case of a linear group, $M$ is abelian, so $\wh M$ is formed of
characters.  

 Then 
\[
\mu_\delta\otimes \mu_{\delta'}^* \text{ is formed of level $\le 2$
  K-types only.} 
\]
\subsection{}\label{sec:3.3}
We will use the ideas in section \ref{sec:3.2} to determine the Weyl
group representation on $\mu_\delta\otimes\mu_{\delta'}^*.$
The following properties hold.
\begin{itemize}
\item $^\vee \Delta^\delta:=\{ \cha\mid \delta(m_\al)=1\}$ is a roots
  system.
\item The Weyl group generated by the roots in $^\vee \Delta^\delta$
  is a normal subgroup of $W_\delta.$
\item The quotient  $R_\delta:=W_\delta/W_\delta^0$ is is a product of
  $\bZ_2$'s.
\item $\wh{R_\delta}$ acts simply transitively on the fine $K$-types
  containing $\delta.$ 
\end{itemize}
Since $R_\delta$ is a quotient of $W_\delta,$ we can inflate any
$\tau\in\wh{R_\delta}$ to a representation of $W_\delta,$ trivial on
$W_\delta^0.$  Having fixed a $\mu_\delta,$ associate it
to the trivial representation of $W_\delta,$ and set up a 1-1 correspondence 
\[
\{\tau\in\wh{W_\delta}\mid \tau
|_{W_\delta^0}=triv\}\longleftrightarrow \{\mu_{\delta,\tau}\},\qquad
triv\longleftrightarrow \mu_\delta.
\]
\begin{theorem}\label{t:3.3}
As a  W-module,
\begin{equation*}
  \Hom_M[\mu_{\delta,1},\mu_{\delta,\tau}]\cong Ind_{W_\delta}^W[\tau].
\end{equation*}
\end{theorem}
\begin{proof}
Recall that a fine $K$-type has the property that its restriction to $M$ is
multiplicity free, and forms a single orbit under the action of $M.$ 
The proof follows from the properties listed before the theorem, and 
standard properties of representations of finite groups.
\end{proof}

\section{Relevant and Petite K-types}\label{sec:4}
\subsection{}\label{sec:4.1}
Recall the notation from section \ref{sec:3.2}.
Since the square of any element
in $M$ is in the center and $M$ normalizes the root vectors, $\Ad
m (D_\al) =\pm D_\al.$ Let $(E,\mu)$ be a $K$-type, and grade
$E^*=\oplus E^*_i$ according to the 
absolute values of the eigenvalues of $D_\al$ (which are
integers). Then $M$ preserves this grading, and 
\[
(E^*)^M=\bigoplus_{i\ even} (E^*_i)^M.
\]
Let $A_\al$ be the maps (\ref{eq:2.3.6}) for $M_\al.$
Recall that for $SL(2,\bR),$ $K$-types are parametrized by integers, and
only $K$-types parametrized by even integers occur in the spherical
principal series.

\begin{proposition}\label{p:4.1} On $(E^*_{2m}+E^*_{-2m})^M,$
$$
a(2m,s_\al,\nu)=\begin{cases} Id &\text{ if } m=0,\\
\prod_{0<j\le m} \frac{2j-1-<\nu,\cha>}{2j-1+<\nu,\cha>}\ Id\quad 
                &\text{ if } m\ne 0.
\end{cases}
$$  
\end{proposition}
\begin{proof}
The formula is well known for $SL(2,\bb R).$ The general assertion
follows from this and  the listed properties of intertwining operators.
\end{proof}
Let $w\in W.$ Then any reduced decomposition $w=s_1\cdots s_k$ gives
rise to a factoring
\begin{equation}
  \label{eq:4.1.6}
  A(w,\chi)=\prod A(s_i,s_{i+1}\cdot\ \dots\ \cdot s_k\chi).
\end{equation}
The resulting operator $A(w,\chi)$ does not depend on the particular
reduced decomposition of $w,$  and each $A(s_i,s_{i+1}\cdot\dots\cdot
s_k\chi)$ is induced from the corresponding $A_{\al_i}.$  
\begin{corollary}
  \label{c:4.1}
In particular, $A(w,\chi)$ is an isomorphism unless $\langle
\nu,\cha\rangle\in \bN$ for some root $\al.$
\end{corollary}
\begin{proof}
  This follows from the properties of the intertwining operator listed above.
\end{proof}
\begin{definition}\label{d:4.1} 
A $K$-type $\mu$ is called \textrm{petite} if each $A_{\al_i}$ satisfies
$$
a(\mu,s_i,\nu)=\begin{cases} Id &\text{ on the +1 eigenspace of } s_{\al_i},\\ 
             \frac{1-<\nu,\cha>}{1+<\nu,\cha>}\ Id
                &\text{ on the -1 eigenspace of } s_{\al_i}.
\end{cases}
$$
\end{definition}
In other words, the restriction of $\mu$ to $K_\al:=M_\al\cap K$ consists of
$K$-types with $2m=0,\pm 2$ only. The hermitian symmetric matrix
$a(\mu,\chi)$ depends only on the $W$-structure of $(E^*_\mu)^M.$
\begin{theorem}
  \label{t:4.1} Level $\le 3$ $K$-types are petite.
\end{theorem}
\begin{proof}
This follows from the above discussion.
\end{proof}
\subsection{Affine Hecke algebra}\label{sec:4.2}
We now show that the formulas in the previous section  coincide with
corresponding ones in the $p$-adic case. Recall from
\cite{BM3} that the induced module is $X(\chi):=\bH\otimes_\bA \bC_\chi$
where  $\bH=\bb C[W]\otimes\bA$ 
is the graded affine Hecke algebra. The abelian subalgebra $\bA$ is  
$S(\fk a)$ ($\fk a =Lie (A)_c$), 
and $\bb C[W]$ is generated by $\{t_\al\}_{\al\ simple}$
satisfying $t_\al^2=1.$ They are subject to the relations
\begin{equation}\label{eq:4.2.1}
\om t_\al=s_\al(\om)t_\al + c_\al<\om,\al>,\qquad \om\in \fk a.
\end{equation}
Because the group is assumed to be split, $c_\al=1.$ 
The intertwining operator  $A(w,\nu)$ is a product of operators
$A_{\al_i}$ according to a reduced decomposition of
$w=s_{\al_1}\cdot\dots\cdot s_{\al_k}.$ If $\al$ is a simple root, let
\begin{equation}\label{eq:4.2.2}
r_{\al}:=(t_\al\cha - c_\al)\frac{1}{\cha -c_\al},\qquad A_\al\ :\
x\otimes \one_\nu \mapsto xr_{\al}\otimes \one_{s_{\al}\nu}.
\end{equation}
The
$r_\al$ are being multiplied on the right, so we can replace $\cha$ with 
$\langle s_\al\chi,\cha\rangle$ in the formulas.
The $A(w,\chi)$ have the same properties as their counterparts in the
real case.  

Furthermore,  
\[
\bb C[W]=\sum_{\sig \in\widehat W} V_\sig\otimes V_\sig^*.
\]
Since $r_\al$ acts as multiplication on the right, it gives rise to an
operator  
\[
r_\sig(s_\al,\nu):V^*_\sig\longrightarrow V^*_\sig.
\]
\begin{theorem}\label{t:4.2}
Let $(E,\mu)$ be a petite $K$-type such that $V^*_\sig\cong (E_\mu^*)^M.$
Then $a(\mu, w,\chi)$ on $(E_\mu^*)^M$ coincide with
 $r_\sig(w,\chi)$ on  $V^*_\sig.$ 
\end{theorem}
\begin{proof}
The assertion is clear from definition (\ref{d:4.1})
and formula (\ref{eq:4.2.2}). 
\end{proof}
The algebra $\bH$ has a $*$, coming from the corresponding p-adic group,
given by
\begin{equation}
  \label{eq:4.2.3}
  \omega^*=-\ov{\om} +\sum_{\al\in\Delta(\fk n)} (\om,\cha)t_\al.
\end{equation}
So it makes sense to talk about its unitary dual.

\bigskip
We remark  that in the work of Lusztig and Kazhdan-Lusztig, the Hecke
algebra is defined in terms of the dual root system so that there is
no discrepancy between $\al$ and $\cha$ in the formulas. 
For p-adic groups the determination of the Iwahori spherical unitary
dual is equivalent to the determination of the unitary dual of $\bH.$
As explained in \cite{BM1} and \cite{BM2}, an infinitesimal character
is given by the conjugacy class of a semisimple  element $s\in \vG.$
In turn this element has a decomposition $s=s_es_h,$ where $s_e$ is
elliptic and $s_h$ is hyperbolic. The Iwahori spherical dual
consisting of representations with
infinitesimal character $s$ is in 1-1 correspondence with  the unitary
dual of an affine graded Hecke algebra $\bH(s_e)$ with real
infinitesimal character.  A representation is called \textit{spherical} if it
contains the trivial W-type. The classification of spherical (and hermitian)
representations with real infinitesimal character is the same as for
the real and p-adic cases. 

\subsection{Relevant K-types}\label{sec:4.3} 
A set of $K$-types is called \textbf{relevant}, if the $(E^*)^M$ form a
relevant set, \ie they satisfy part (a) of the
introduction. We list them for the simple root systems.  For the
classical groups they are already in \cite{B6}. 

\medskip
The representations of $W(A_{n-1})=S_n$ are parametrized by partitions
of $n,$ written as $(a_1,\dots, a_k), \ a_i\le a_{i+1}.$ 
The representations of $W(B_n)\cong W(C_n)$ are parametrized as in \cite{L1} by
pairs of partitions 
\begin{align}
  \label{eq:4.3.1}
  (a_1,\dots , a_k)&\times (b_1,\dots,b_l),\notag \\
  &a_i\le a_{i+1},\quad b_j\le b_{j+1},\quad \sum a_i +\sum b_j=n.
\end{align}
Precisely, the representation $\sig$ parametrized by (\ref{eq:4.3.1}) is as
follows. Let $k=\sum a_i,\ l=\sum b_j.$ Recall that $W\cong S_n\rtimes
\bZ_2^n.$ Let $\chi$ be the character of $\bZ_2^n$ which is trivial on
the first $k$ $\bZ_2$'s, sign on the rest. Its centralizer in $S_n$ is
$S_k\times S_l.$ Let $\sig_1$ and $\sig_2$ be the representations of
$S_k,\ S_l$ corresponding to the partitions $(a)$ and $(b).$ Then
$\sig,$ is
\[
Ind_{(S_k\times S_l)\times \bZ_2^n}^{W}\
[(\sig_1\otimes\sig_2)\otimes\chi].
\]
For  $W(D_n),$ the representations are parametrized as in
(\ref{eq:4.3.1}) except that $(a)\times (b)$ and $(b)\times (a)$
parametrize the same representation, and when $(a)=(b),$  there are two of them
$(a)\times(a)_{I,\ II}.$ This is because the restriction of
$(a)\times (b)$ to $W(D_n)$ is irreducible when $(a)\ne (b)$ and equal
to the restriction of $(b)\times (a),$ while the restriction of
$(a)\times (a)$ consists of two nonisomorphic irreducible
representations labelled $(a)\times(a)_{I,\ II}.$ These are usually easy to
deal with.


\subsection{Orthogonal groups}\label{sec:4.4}
Because we are dealing with the spherical case, we can use the
orthogonal group instead of its connected component. We follow
Weyl's conventions to parametrize the representations of $O(n).$ Embed
$O(a)\subset U(a)$ in the standard way. An irreducible representation
of $O(n)$ is parametrized by
\begin{equation}
  \label{eq:4.4.1}
  (a_1,\dots ,a_k,0,\dots ,0;\ep),\qquad a_i\ge a_{i+1},\ \ep=\pm 1.
\end{equation}
The $\ep$ is (sometimes) abbreviated as  $\pm.$
The parameter in (\ref{eq:4.4.1})  is the irreducible representation
generated by the highest weight 
vector of  the irreducible representation of $U(a)$ with
highest weight
\begin{equation}
  \label{eq:4.4.2}
  (a_1,\dots ,a_k,\unb{n-(1-\ep)k}{1,\dots ,1},0,\dots ,0).
\end{equation}

\bigskip
For $O(n,n)$ we have $K=O(n)\times O(n).$ The fine $K$-types (in the
sense of \cite{V4}), and their restrictions to $M$ are
\begin{equation}
  \label{eq:4.4.2.1}
  \begin{array}{ll}
K-type & M-type \\
&\\
(\unb{k}{1,\dots ,1},0,\dots,0)\otimes(0,\dots ,0)&\\
&\binom{n}{k}\ characters\  \delta_k\\ 
(0,\dots ,0)\otimes  (\unb{k}{1,\dots ,1},0,\dots ,0) ,
  \end{array}
\end{equation}
and we suppress the $\pm.$ The meaning of (\ref{eq:4.4.2.1}) is that
both representations of $K$ in the left column restrict to an orbit of
characters of $M$. The size of the orbit is $\binom{n}{k},$ and we fix a
representative denoted $\delta_k.$

\bigskip
The $K$-types
\begin{align}
&({{0,\dots ,0}};+)\otimes (\uset{r}{\ub{2,\dots ,2}},{{0,\dots ,0}};+)
\label{eq:4.4.3}\\
&(\uset{k}{\ub{1,\dots,1}},0,\dots ,0;\ep)\otimes 
(\uset{k}{\ub{1,\dots ,1}},0,\dots , 0;\ep)\qquad \ep=\pm\label{eq:4.4.4}
\end{align}
\textbf{form a relevant set}.
The representation of $W$ on $V^M$ is 
\begin{align}
&(r,n-r)\times (0)&&\longleftrightarrow (\ref{eq:4.4.3})\label{eq:4.4.5}\\
&(n-k)\otimes  (k),\quad k\le [n/2]&&\longleftrightarrow (\ref{eq:4.4.4}) 
\text{ with } +,\label{eq:4.4.6}\\
&(n-k)\otimes  (k),\quad k> [n/2]&&\longleftrightarrow (\ref{eq:4.4.4}) 
\text{ with } -.  \label{eq:4.4.7}
\end{align}

For $O(n+1,n),$ we have $K=O(n+1)\times O(n).$ The fine $K$-types
(\cite{V4}), and their restriction to $M$ are
\begin{equation}
  \label{eq:4.4.2.2}
  \begin{array}{ll}
K-type & M-type \\
&\\
(0,\dots ,0)\otimes  (\unb{k}{1,\dots ,1},0,\dots ,0) &
\binom{n}{k}\ characters\  \delta_k\\ 
  \end{array}
\end{equation}
The $K$-types
\begin{align}
&({{0,\dots ,0}};+)\otimes (\uset{r}{\ub{2,\dots ,2}},{{0,\dots ,0}};+)
\label{eq:4.4.8}\\
&(\uset{k}{\ub{1,\dots,1}},0,\dots ,0;+)\otimes 
(\uset{k}{\ub{1,\dots ,1}},0,\dots , 0;+)\qquad 
\text{ for } k\le [n/2]\label{eq:4.4.9}\\
&(\uset{n+1-k}{\ub{1,\dots,1}},0,\dots ,0;-)\otimes 
(\uset{n-k}{\ub{1,\dots ,1}},0,\dots , 0;-)\qquad 
\text{ for } k > [n/2]\label{eq:4.4.10}
\end{align}
\textbf{form a relevant set,} and the corresponding representations of
$W(B_n)$ on $V^M$ are 
\begin{align}
&(r,n-r)\times (0),\quad r\le n/2&&\longleftrightarrow
(\ref{eq:4.4.9})\label{eq:4.4.11}\\ 
&(n-k)\otimes  (k),\quad k\le [n/2]&&\longleftrightarrow (\ref{eq:4.4.10}) 
\text{ with } +,\label{eq:4.4.12}\\
&(n-k)\otimes  (k),\quad k> [n/2]&&\longleftrightarrow (\ref{eq:4.4.10}) 
\text{ with } -.  \label{eq:4.4.13}
\end{align}
\begin{theorem}[\cite{B6}]
  \label{t:4.4}
A spherical irreducible representation for $\bH$ is unitary if and only if the
hermitian form is positive definite on the W-types
$$
(r,n-r)\times (0),\qquad (n-k)\times (k).
$$
\end{theorem}
\begin{corollary}[\cite{B6}]
  \label{c:4.4}
A spherical representation $L_R(\chi)$ for a real split orthogonal group
is unitary only if the corresponding $L_F(\chi)$ for $\bH$ is unitary. 
\end{corollary}
In other words, the set of spherical unitary parameters for the real
split group is contained in the set of spherical unitary parameters
for $\bH.$ In fact in \cite{B6} the stronger result  is proved that the unitary
duals of $G(\bR)$ and $\bH$ coincide.

\subsection{Symplectic groups}\label{sec:4.5} The maximal compact
subgroup of $Sp(n,\bR)$ is $U(n).$ Its irreducible representations are
parametrized by
\begin{equation}
  \label{eq:4.5.1}
  (a_1,\dots ,a_n),\qquad a_i\ge a_{i+1},\ a_i\in\bZ.
\end{equation}
The fine $K$-types and their restrictions to $M$ are

\begin{equation}
  \label{eq:4.5.1.1}
  \begin{array}{ll}
K-type & M-type \\
&\\
\mu_+(k):=(\unb{k}{1,\dots ,1},0,\dots,0)&\\
&\binom{n}{k}\ characters\  \delta_k\\ 
\mu_-(k):=(0,\dots ,0,\unb{k}{-1,\dots ,-1}).
  \end{array}
\end{equation}


The $K$-types 
\begin{align}
&(\unb{k}{2,\dots ,2},0,\dots ,0)\longleftrightarrow (n-k)\times (k),
\label{eq:4.5.2}\\
&(0,\dots ,0,\unb{k}{-2,\dots,-2})\longleftrightarrow(k)\times(n-k),
\label{eq:4.5.2.1}\\
&(\unb{r}{1,\dots ,1},0,\dots ,0,\unb{r}{-1,\dots  ,-1})\longleftrightarrow
(r,n-r),\label{eq:4.5.3}
\end{align}
form a \textbf{relevant set}.
\begin{theorem}[\cite{B6}]
  \label{t:4.5}
A spherical representation for $\bH$ is unitary if and only if the
hermitian form is positive definite on the W-types
$$
(r,n-r)\times (0),\qquad (n-k)\times (k).
$$
\end{theorem}
\begin{corollary}
  \label{c:4.5}
A spherical representation $L_R(\chi)$ for a real split orthogonal group
is unitary only if the corresponding $L_F(\chi)$ for $\bH$ is unitary. 
\end{corollary}
In fact in \cite{B6} the stronger result  is proved that the unitary
duals of $G(\bR)$ and $\bH$ coincide.

\subsection{General linear group}\label{sec:4.6} The maximal compact
subgroup of $GL(n,\bR)$ is $O(n).$ The fine $K$-types and their
restrictions to $M$ are
\begin{equation}
  \label{eq:4.6.0}
  \begin{array}{ll}
K-type & M-type \\
&\\
(\unb{k}{1,\dots ,1},0,\dots,0)&\binom{n}{k}\ characters\  \delta_k\\ 
  \end{array}
\end{equation}
According to \cite{V2}, a set of relevant $K$-types is given by 
\begin{equation}
  \label{eq:4.6.1}
(\unb{k}{2,\dots ,2},0,\dots ,0).  
\end{equation}
The corresponding representation of $S_n$ on $V^M$ is, according to \cite{B6},
\begin{equation}
  \label{eq:4.6.2}
(\unb{k}{2,\dots ,2},0,\dots ,0)\longleftrightarrow (n-k,k)
\end{equation}
In the case $n$ even, we have
\begin{equation}
  \label{eq:4.6.3}
  (2,\dots ,2,\pm 2)\longleftrightarrow (n/2,n/2)_{I,II}.
\end{equation}

\bigskip
The theorems and corollaries analogous to the ones in sections
\ref{sec:4.4} and \ref{sec:4.5} follow from \cite{V2} and
\cite{T}. For a construction of the relevant $K$-types for $SL(n)$ 
starting from the Weyl group module on $E^M,$ the interested
reader may consult \cite{P}.
\subsection{G2}\label{sec:4.6.1}
We refer to \cite{V1} for more details about the structure of the split
group of type $G_2.$ 
The split real form which consists of the real points of the complex
simply connected linear group of type $G_2$ has 
\begin{equation}
  \label{eq:4.6.1.1}
  K:=[SU(2)_s\times SU(2)_l]/\{\pm Id\}.
\end{equation}
It is more convenient to work with the (nonlinear) double cover for which
\begin{equation}
  \label{eq:4.6.1.2}
  \wti{K}=SU(2)_s\times SU(2)_l.
\end{equation}
The Borel subgroup is $\wti{B}=\wti{M}AN$ where $\wti{M}$ is the group
of order 8 which contains $\{\pm Id\}$ in
(\ref{eq:4.6.1.1}) as a central sugroup, with quotient $\bZ_2\times \bZ_2.$
Let $T$ be the maximal compact subgroup. We identify the Lie algebra
$\fk t$ with $\{(a_1,a_2,a_3)\mid a_1+a_2+a_3=0\}$, and use the 
standard positive system of roots
\begin{equation}
  \label{eq:4.6.3a}
  \begin{aligned}
  \{ &(1,-1,0),\ (-1,2,-1),\ (0,1,-1),\ (1,0,-1),\\
       &(1,1,-2),\ (2,-1,-1)\}.    
  \end{aligned}
\end{equation}
A $K$-type is determined by its highest weight which we denote
\begin{equation}
  \label{eq:4.6.1.4}
  (p\mid q)\longleftrightarrow (q,\frac{p-q}{2},\frac{-p-q}{2}),\qquad
  p,\ q\in \bN.
\end{equation}
The positive roots in $\fk k$ are $(0,1,-1)$ and $(2,-1,-1).$
The fine $K$-types are
\begin{equation}
  \label{eq:4.6.1.5}
  \begin{aligned}
&\wti K-type &\wti M-type\\
&(0\mid 0) &\delta_0,\ \text{ the trivial character,}\\
&(1\mid 0) &\delta_2,\ \text{ a representation of
  dimension 2,}\\
&(2\mid 0) &\delta_3,\ \text{ three characters.}
  \end{aligned}  
\end{equation}
The one dimensional representations all factor to $M,$ while the two
dimensional one is a genuine representation of $\wti M.$ The Weyl
group is generated by $s_s,$ the reflection about the simple short
root, and $s_l,$ the reflection about the simple long root. The
representations of $W$  are parametrized as in \cite{L1},
\begin{equation}
  \label{eq:4.6.1.6}
  \begin{aligned}
    &1_1 & trivial\\ 
    &1_2 & s_s=1,\ s_l=-1,\\
    &1_3 & s_s=-1,\ s_l=+1,\\
    &1_4 & s_s=-1,\ s_l=-1,\\
    &2_1 & reflection,\\
    &2_2 &  2_1\otimes 1_2.
  \end{aligned}
\end{equation}
\begin{theorem}[\cite{C}]\label{t:g2} A spherical representation of $\bH$ is
unitary if and only if the $r_\sig$ are positive definite on the $W$-types
  \begin{equation*}
    1_1,\ 2_1,\ 2_2.
  \end{equation*}
  \end{theorem}
  \begin{proposition}The following list consists of petite $K$-types.
    \begin{equation*}
   \begin{aligned}
    &K-type &\qquad W-type \\
&(0\mid 0) &1_1,\\
&(3\mid 1) &2_1,\\
&(4\mid 0) &2_2.
  \end{aligned}     
    \end{equation*}
\end{proposition}
Thus the W-types in theorem \ref{t:g2} form a \textbf{relevant set}.
More detailed results along these lines can be found in \cite{C}.


\subsection{F4}\label{sec:4.7}
\begin{comment}
We use Bourbaki's realization of the root system. Precisely, the
Coxeter Dynkin diagram is
\begin{equation}
  \label{eq:4.7.1}
\begin{smallmatrix}
\beta&-&\al_1&-&\al_2&=&\al_3&-&\al_4
\end{smallmatrix} 
\end{equation}
where $\beta$ is the negative of the longest root.

\medskip 
The reflections $s_1,\ s_2\in W$ correspond to the long simple roots,
$s_3,\ s_4\in W$ to the short roots.

We choose the positive roots
\begin{equation}
  \label{eq:4.7.2}
  \{ \ep_i \pm \ep_j,\ i<j,\ 
\frac12(\ep_1\pm\ep_2\pm\ep_3\pm\ep_4),\ \ep_k\}.
\end{equation}
The  coroots and  weights corresponding to the simple roots are 

\begin{tabular}{ll}
$\al_1=\ep_1-\ep_2-\ep_3-\ep_4$&\quad$\om_1=\ep_1$\\
&\\
$\al_2=2\ep_4$&\quad
$\om_2=\frac32\ep_1+\frac12\ep_2+\frac12\ep_3+\frac12\ep_4$\\

&\\
$\al_3=\ep_3-\ep_4$&\quad$\om_3=2\ep_1+\ep_2+\ep_3$\\
&\\
$\al_4=\ep_2-\ep_3$  &\quad$\om_4=\ep_1+\ep_2$\\
\end{tabular}

\medskip
\end{comment}
The split real form which is the rational points of the complex linear
group of type $F_4$ has maximal compact subgroup
\begin{equation}
  \label{eq:4.7.3}
  K:=[Sp(1)\times Sp(3)]/\{\pm Id\}.
\end{equation}
It is more convenient to use the double cover which has
\begin{equation}
  \label{eq:4.7.4}
  \wti{K}=[Sp(1)\times Sp(3)]
\end{equation}
The Borel subgroup is $\wti B=\wti MAN,$ where $\wti M$ is a finite
nonabelian group of order $2^5.$ It has a central subgroup of order 2
(the $\pm Id$ in (\ref{eq:4.7.3})) such that the quotient is $\bZ_2^4.$

Let $T$ be a maximal compact Cartan subgroup. We use the standard
positive system and roots, 
\begin{equation}
  \label{eq:4.7.5}
\{2\ep_1,\  2\ep_k,\ \ep_{k}\pm\ep_\ell\}_{2\le k\le \ell\le 4}.
\end{equation}
The highest weight of a  $\wti K$-type will be denoted
\begin{equation}
  \label{eq:4.7.6}
(a_1\ \big|\ a_2,a_3,a_4,a_5),\qquad a_i\in \bN,\quad  a_2\ge a_3\ge a_4\ge 0.
\end{equation}

The \textit{fine} $K$-types (in the sense of \cite{V4}), and their
restrictions to $\wti M$ are
\begin{equation}
  \label{eq:4.7.7}
  \begin{aligned}
&\wti K-type &\wti M-type&\quad W_\delta\\
&(0\mid 0,0,0) &\delta_0,\ \text{ one character}&\quad F_4,\\
&(1\mid 0,0,0) &\delta_2,\ \text{ a representation of
  dimension 2}&\quad F_4\\
&(2\mid 0,0,0) &\delta_3,\ \text{ three characters}&\quad C_4,\\
&(0\mid 1,0,0) &\delta_6,\ \text{ three representations of
  dimension 2}&\quad B_4,\\
&(1\mid 1,0,0) &\delta_{12},\ \text{ twelve characters}&\quad B_3A_1
  \end{aligned}
\end{equation}
The 1-dimensional representations all factor to $M.$ 
The remaining  ones are genuine for $\wti M.$

\bigskip
The representations of the Weyl group of type $F_4$ are parametrized as in
\cite{L1}. 
\begin{theorem}[\cite{C}]
  \label{t:4.7}
A spherical representation of $\bH$ of type $F_4$ is unitary if and only
if the $r_\sig$ are positive semidefinite on the W-types
\[
1_1,\ 2_3,\ 8_1,\ 4_2,\ 9_1.
\]
\end{theorem}
The next result was obtained joint with D. Vogan.
\begin{proposition}
  \label{p:4.7}
The following list consists of petite $K$-types.
\begin{equation*}
  \begin{aligned}
    &K-type &\qquad W-type \\
&(0\mid 0,0,0) &1_1,\\
&(0\mid 1,1,0) &2_1,\\
&(4\mid 0,0,0) &2_3,\\
&(1\mid 2,1,0) &8_1,\\
&(1\mid 1,1,1) &4_2,\\
&(2\mid 2,0,0) &9_1.
  \end{aligned}
\end{equation*}
\end{proposition}
Thus the $W$-types in theorem \ref{t:4.7} form a \textbf{relevant set}.
\begin{corollary}
  \label{c:4.7}
A spherical representation for the real split group $G$ of type $F_4$ is
unitary only if the corresponding $L_F(\chi)$ for $\bH$ is unitary.
\end{corollary}
The results in \cite{C} provide an explicit list of the unitary dual
of $\bH.$ The unitarity of the unipotent representations and the
relevant irreducibility results for the real case are not (yet) available.

\subsection{E6}\label{sec:4.8} 

The maximal compact subgroup of the rational points of the simply
connected complex group of type $E_6$ is
\begin{equation}
  \label{eq:4.8.1}
  K=Sp(4)/\{\pm Id\}.
\end{equation}
Again it is more convenient to use the double cover which has
\begin{equation}
  \label{eq:4.8.2}
  \wti K=Sp(4).
\end{equation}
We use the same conventions as in section \ref{sec:4.7} for the positive roots
and highest weights of representations of $\wti K.$ The finite group
$\wti M$ has size $2^7,$ and its quotient by the central group $\{\pm
Id\}$ is a $\bZ_2^6.$ The fine $K$-types and their restrictions to $\wti
M$ are
\begin{equation}
  \label{eq:4.8.3}
  \begin{aligned}
&\wti K-type &\wti M-type&\qquad W_\delta^0\\
&(0, 0,0,0) &\delta_1,\ \text{ the trivial character,}&\qquad E_6,\\
&(1,0,0,0) &\delta_8,\ \text{ a representation of
  dimension 8,}&\qquad E_6,\\
&(1,1,0,0) &\delta_{27},\ \text{ twenty seven characters,}&\qquad D_5,\\
&(2,0,0,0) &\delta_{36},\ \text{ thirty six characters,}&\qquad A_5A_1.
  \end{aligned}
\end{equation}
The second representation is genuine for $\wti M,$ the others all
factor to $M.$  

The $W$-types are parametrized as in \cite{L1}.
\begin{theorem}[\cite{BC}]
  \label{t:4.8}
A spherical representation of $\bH$ of type $E_6$ is unitary if and only
if the $r_\sig$ are positive semidefinite on the W-types
\[
1_p,\ 6_p,\ 20_p,\ 30_p,\ 15_q.
\]
\end{theorem}
We denote by $\om_i$ the fundamental weights of $sp(4).$ In
coordinates they are $\om_i=\sum_{j\le i} \ep_j.$
\begin{proposition}
  \label{p:4.8}
The following list consists of petite $K$-types.
\begin{equation*}
  \begin{aligned}
    &K-type &\qquad W-type\\
&(0)=(0,0,0,0) &1_p,\\
&\om_4=(1,1,1,1) &6_p\\
&2\om_2=(2,2,0,0) &20_p,\\
&4\om_1=(4,0,0,0) &15_q,\\
&2\om_1+\om_4=(3,1,1,1) &30_p,\\
&\om_1+\om_2+\om_3=(3,2,1,0) &64_p,\\
&3\om_1+\om_3=(4,1,1,0) &60_p,\\
&2\om_3=(2,2,2,0) &15_p,\\
&2\om_1+2\om_2=(4,2,0,0) &81_p,\\
&3\om_2=(3,3,0,0) &24_p,\\
&6\om_1=(6,0,0,0) &24'_p.
  \end{aligned}
\end{equation*}
\end{proposition}
\begin{corollary}
  \label{c:4.8}
A spherical representation $L_R(\chi)$ for the real group $G$ of type $E_6$ is
unitary only if the corresponding $L_F(\chi)$ for $\bH$ is unitary.
\end{corollary}
Thus the W-types in theorem \ref{t:4.8} form a \textbf{relevant set}.


\subsection{E7}\label{sec:4.9}

The maximal compact subgroup of the split real form of the simply
connected  group of type $E_7$ is
\begin{equation}
  \label{eq:4.9.1}
  K=SU(8)/\{\pm Id\}.
\end{equation}
We work with the double cover for which
\begin{equation}
  \label{eq:4.9.2}
  \wti K=SU(8).
\end{equation}
The finite group $\wti M$ has size $2^8,$ and its quotient by the
center in (\ref{eq:4.9.1}) is a $\bZ_2^7.$ The fine $K$-types are
\begin{equation}
  \label{eq:4.9.3} 
   \begin{aligned}
&\wti K-type &\wti M-type&\qquad W^0_\delta\\
&(0)   &\delta_1,\ \text{ trivial representation,}&\qquad E_7\\
&\om_1 &\delta_8,\ \text{ eight dimensional representation,}&\qquad E_7\\
&\om_7 &\delta_8^*,\ \text{ eight dimensional representation,}&\qquad E_7\\
&\om_2,\ \om_6 &\delta_{28},\ \text{ twenty eight characters,}&\qquad E_6\\
&2\om_1,\ 2\om_7 &\delta_{36},\ \text{ thirty six characters,}&\qquad A_7\\
&\om_1+\om_7     &\delta_{63},\ \text{ sixty three characters,}&\qquad D_6A_1
  \end{aligned}
\end{equation}
In (\ref{eq:4.9.3}), the $\om$ refer to the fundamental weights of
$\wti K,$ usual labelling. 


The second and third are genuine representations of $\wti M,$ the others
are single orbits under the action of W.

The Weyl group representations are parametrized as in \cite{L1}.
\begin{theorem}[\cite{BC}]
  \label{t:4.9}
A  spherical representation of $\bH$ of type $E_7$ is unitary if and only
if $r_\sig$ is positive semidefinite for
\[
1_a,\ 7_a',\ 27_a,\ 56_a',\ 21_b',\ 35_b,\ 105_b.
\]
\end{theorem}
\begin{proposition}
  \label{p:4.9}
The following list gives the petite $K$-types with $M$-spherical
vectors, and the corresponding Weyl
group representations:
\begin{equation*}
  \begin{aligned}
&K-type &\qquad W-type \\
&(0) &1_a,\\
&\om_4 &7'_a\\
&2\om_2,\ 2\om_6 &21'_b,\\
&\om_2+\om_6 &27_a,\\
&2\om_1+2\om_7 &35_b,
\end{aligned}
\end{equation*}

\begin{equation*}
\begin{aligned}
&K-type &\qquad W-type \\
&4\om_1,\ 4\om_7   &15_a',\\
&\om_2+\om_3+\om_7,\ \om_1+\om_5+\om_6 &105_a',\\
&\om_1+\om_4+\om_7 &56_a',\\
&2\om_1+\om_3 +\om_7,\ \om_1+\om_5+2\om_7 &189_b',\\
&\om_1+\om_3+\om_6,\ \om_2+\om_5+\om_7 &168_a,\\
&3\om_1+\om_5,\ \om_3+3\om_7 &105_b,\\
&\om_3+\om_5 &21_a,\\
&\om_1+\om_2+\om_5,\ \om_3+\om_6+\om_7 &120_a,\\
&\om_1+\om_2+\om_6+\om_7,\ &168_a+210_a,\\
&3\om_1+3\om_7,\ &84_a+105_c,\\
&\om_1+2\om_2+\om_7,\om_1+\om_6+\om_7,\ & 189'_c\\
&3\om_1+\om_2+\om_7,\om_1+\om_6+3\om_7,\ & 216_a',\\
&3\om_1+\om_6+\om_7,\om_1+\om_2+3\om_7,\ &280_b,\\
&5\om_1+\om_7,\om_1+5\om_7, &84_a'.
  \end{aligned}
\end{equation*}
\end{proposition}

\begin{corollary}
  \label{c:4.9}
A spherical representation $L_R(\chi)$ for the real group $G$ of type $E_7$ is
unitary only if the corresponding $L_F(\chi)$ for $\bH$ is unitary.
\end{corollary}
Thus the W-types in theorem \ref{t:4.9} form a \textbf{relevant set}.

\subsection{E8}\label{sec:3a.10} 
The maximal compact subgroup of the split real form of the simply
connected complex group of type $E_8$ is
\begin{equation}
  \label{eq:3a.10.1}
  Spin(16)/\{Id,\om\},
\end{equation}
for $\om$ the appropriate element of order two in the center (the quotient is
\textbf{not} SO(16)). As before we work with the double cover which
has
\begin{equation}
  \label{eq:3a.10.2}
  \wti K=Spin(16).
\end{equation}
The group $\wti M$ has size $2^9,$ and its quotient by the center in
(\ref{eq:3a.10.1}) is $\bZ_2^8.$
The fine $K$-types are
\begin{equation}
  \label{eq:3a.10.3} 
   \begin{aligned}
&\wti K-type &\wti M-type\\
&(0)   &\delta_1,\ \text{ trivial representation,}\\
&\om_1 &\delta_{16},\ \text{ sixteen dimensional representation,}\\
&\om_2 &\delta_{120},\ \text{ one hundred and twenty characters,}\\
&2\om_1 &\delta_{135},\ \text{ one hundred thirty five characters,}\\
  \end{aligned}
 \end{equation}

Only the second representation is genuine.

The representations of the Weyl group are parametrized as in \cite{L1}.
\begin{theorem}
  \label{t:3a.10}
A  spherical representation of $\bH$ of type $E_8$ is unitary if and only
if $r_\sig$ is positive semidefinite for
\[
1_x,\ 8_z,\ 35_x,\ 50_x,\ 84_x,\ 112_z,\ 400_z,\ 300_x,\ 210_x.
\]

\end{theorem}
\begin{proposition}
  \label{p:3a.10}
The following list gives petite $K$-types and the corresponding Weyl
group representations on $(V^*)^M:$
\begin{equation*}
  \begin{aligned}
&K-type &\qquad\text{ W-type on } (V^*)^M\\
&(0) &1_x,\\
&\om_8 &8_z\\
&\om_4 &35_x,\\
&2\om_2 &84_x,\\
&\om_2+\om_8 &112_z,\\
&4\om_1 &50_x,\\
&3\om_1+\om_7 &400_z,\\
&\om_3+\om_7 &160_z,\\
&\om_6       &28_x,\\
&\om_1+\om_5 &210_x,\\
&\om_1+\om_2+\om_7 &560_x,\\
&\om_2+\om_4& 567_x, \\
&2\om_3     &300_x,\\
&2\om_1+\om_4 &700_x,\\
&3\om_1+\om_3 &1050_x,\\
&\om_1+\om_2+\om_3 &1344_x,\\
&3\om_2 &525_x,\\
&2\om_1+2\om_2 &972_x,\\
&4\om_1+\om_2  &700_{xx},\\
&6\om_1        &168_y.
  \end{aligned}
\end{equation*}
\end{proposition}
\begin{corollary}
  \label{c:3a}
A spherical representation $L_R(\chi)$ for the real group $G$ of type $E_8$ is
unitary only if the corresponding $L_F(\chi)$ for $\bH$ is unitary.
\end{corollary}
\section{Some proofs}\label{sec:5}

\subsection{}\label{sec:5.1} 
Let $\wti K$ be a compact group, $\wti M\subset\wti N$ finite subgroups
such that $\wti M$ is 
normal in $\wti N.$ Denote by $W$ the quotient $\wti N/\wti M.$ In the
applications, $\wti N$ is the normalizer of $\mathfrak a$ in $\wti K,$
which is simply connected.

Let $V_a,\ V_b$ be
representations of $\wti K.$ In the nicer cases, the restrictions
of $V_a,\ V_b$ to $\wti M$ are multiples of the same representation
$V_\delta,$ which in turn  extends to a representation
of $\wti K.$ This is the case for $E_8$. The standard 16-dimensional
representation is genuine, and restricts to a single $\wti M$-type denoted
$\delta_{16}.$ We already defined the action of $W$ in the next
proposition in section \ref{sec:3.2}.
\begin{proposition}
  \label{p:4}
There is a natural action of $W$ on $\Hom_M[V_a,V_b].$
\end{proposition}
\begin{proof}
The action on $f\in\Hom[V_a,V_b]$ is given by 
\begin{equation}
  \label{eq:5.1.1}
  (n\cdot f)(v)=\mu_a(n)f(\mu_b(n^{-1})v).
\end{equation}
This is clearly an action of $\wti N.$
The fact that the action
does not depend on the right $\wti M$-coset is straightforward.
\end{proof}
This action is compatible with the canonical isomorphism
\begin{equation}
  \label{eq:5.1.2}
  \Hom_\wti M[V_a,V_b]\cong [V_a^*\otimes V_b]^\wti M,
\end{equation}
and is also compatible with the isomorphism
\begin{equation}
  \label{eq:5.1.3}
  \Hom_\wti M[V_a,V_b]^*\cong\Hom_\wti M [V_b^*,V_a^*].
\end{equation}
This action is a generalization of the usual one on $V^\wti M.$ 
\subsection{}\label{sec:5.2}
Now consider two genuine modules $V_a,\ V_b.$ Then there is a
homomorphism
\begin{equation}
  \label{eq:5.2.1}
 \Hom_\wti M[V_a,V_\delta]\otimes\Hom_\wti
 M[V_\delta,V_a]\longrightarrow\Hom_\wti M[V_a,V_b]. 
\end{equation}
This is compatible with the action of $W.$ When $V_a,\ V_b$ are
multiples of a single orbit of $V_\delta,$ this is an isomorphism. 
\subsection{Orthogonal groups}\label{sec:5.2.1}
In the case of $SO(n+1,n),$ the Weyl group is $W(B_n).$ The
centralizer $W^0_\delta=W_\delta$ corresponding to $(0)\otimes
(\unb{k}{1,\dots ,1},0,\dots ,0)$ is isomorphic to $W(C_k\times
C_{n-k}).$ Then 
\begin{equation}
  \label{eq:5.2.1.1}
  Ind_{W(C_k\times C_{n-k})}^{W(C_n}[triv]=\sum (n-k+\ell,k-\ell)\times (0).
\end{equation}
The corresponding tensor product is
\begin{equation}
  \label{eq:5.2.1.2}
  \sum (0)\otimes (\unb{a}{2,\dots ,2},1,\dots ,1,\unb{b}{0,\dots ,0}).
\end{equation}
These $K$-types are automatically level $\le 2.$ The formulas imply that
these  $W$-types are realized in the $M$-spherical vectors of petite
$K$-types. This is sufficient for the purpose of corollary \ref{c:4.4}.
For the more precise  results in
(\ref{eq:4.4.11}-\ref{eq:4.4.13}) one needs to show first that the 
factors occuring in (\ref{eq:5.2.1.2}) with $1$'s do not have any $M$
fixed vectors. This can be done by showing that such factors occur in
tensor products of distinct fine $K$-types which cannot have any $M$-spherical
vectors. To sort out the matchup of the remaining $K$-types with
the W-representations, one needs to compare the formulas
(\ref{eq:5.2.1.1}) and (\ref{eq:5.2.2.2}) for various values of $k.$

Tensor products of  fine $K$-types of $SO(n+1,n)$ are not sufficient to
realize the remaining relevant Weyl group representations in the $M$-spherical
vectors of petite $K$-types. To achieve this, we observe that
$Spin(n+1,n),$ the rational points of $Spin(2n+1,\bC)$ has another
pair of fine $K$-types 
\begin{equation}
  \label{eq:5.2.1.3}
  spin\otimes spin^\pm.
\end{equation}
In this case $W_\delta^0=S_n,$ while $W_\delta=S_n\times \bZ_2.$ Then 
\begin{equation}
  \label{eq:5.2.1.4}
  \begin{aligned}
    &Ind_{S_n\times \bZ_2}[triv\otimes triv]=\sum (n-k)\times
    (k),\qquad k\ even,\\
    &Ind_{S_n\times \bZ_2}[triv\otimes sgn]=\sum (n-k)\times
    (k),\qquad k\ odd.
  \end{aligned}
\end{equation}
We combine these formulas with
\begin{equation}
  \label{eq:5.2.1.5}
  \begin{aligned}
    &spin\otimes spin=\sum (\unb{k}{1,\dots ,1},0,\dots ,0),\quad
    \text{ for Spin(2n+1),}\\
    &spin^+\otimes spin^-=\sum_{k\ even } (\unb{n-k-1}{1,\dots
      ,1},0,\dots ,0),\\
    &spin^+\otimes spin^+=(1,\dots ,1)+\sum_{k>0\ even } 
      (\unb{n-k}{1,\dots ,1},0,\dots ,0),\\
    &spin^-\otimes spin^-=(1,\dots 1,-1)+\sum_{k>0\ even } 
      (\unb{n-k}{1,\dots ,1},0,\dots ,0).
  \end{aligned}
\end{equation}
This proves that the sum of petite representations in formulas
(\ref{eq:4.4.2.2}) contains the sum of all the relevant
W-representations. This is sufficient for corollary \ref{c:4.4}.

In the case $O(n,n),$ we do not need to use the covers. We can also
derive the matchup of  the petite $K$-types with the relevant
W-representations in (\ref{eq:4.4.5}-\ref{eq:4.4.7})  
in a manner similar to the symplectic group case, where we give a few
more details.   

\subsection{Symplectic groups}\label{sec:5.2.2}
The basic tool is the tensor product formulas
\begin{equation}
  \label{eq:5.2.2.1}
  \begin{aligned}
&\mu_+(k)\otimes\mu_-(k)=\sum (\unb{a}{1,\dots ,1},\unb{b}{0,\dots
  ,0},-1,\dots ,-1),\\
&\mu_+(k)\otimes\mu_+(k)=\sum (\unb{a}{2,\dots ,2},
\unb{b}{1,\dots ,1},{0,\dots ,0}), \qquad 2k\le n 
\end{aligned}
\end{equation}
with $2a+b=2k.$ 
These $K$-types are automatically level $\le 2,$ so petite. The
stabilizer $W_\delta^0$ for $\mu_\pm(k)$ is $W(D_k)\times W(C_{n-k}),$
while $W_\delta=W(C_k)\times W(C_{n-k}).$ Then
\begin{equation}
  \label{eq:5.2.2.2}
Ind_{W(D_k)}^{W(C_k)}[triv]= (k)\times(0) + (0)\times(k).  
\end{equation}
The  induced modules corresponding to (\ref{eq:5.2.2.1}) are
\begin{equation}
  \label{eq:5.2.2.3}
  \begin{aligned}
    &Ind_{W(C_k)\times W(C_{n-k})}^{W(C_n)}[(k)\times (0)\otimes
    (n-k)\times (0)]=\sum
    (n-k+\ell,k-\ell)\times (0),\\
   &Ind_{W(C_k)\times W(C_{n-k})}^{W(C_n)}[(0)\times (k)\otimes
   (n-k)\times (0) ]= (n-k)\times(k).
  \end{aligned}
\end{equation}
These results imply that any relevant Weyl group representation is
realized in $(E_\mu^+)^M$ for some petite $K$-type. To get the more
precise result of (\ref{eq:4.5.2}-\ref{eq:4.5.2.1}), one needs to
decompose $\mu_+(r)\otimes\mu_+(s)$ for all $r,\ s.$  
For $r\ne s,$ none of the factors have $M$-fixed vectors. Then one
makes a comparison of the remaining representations occuring in the
tensor products for $r=s$ with the  induced
representations of the corresponding Weyl groups.

\subsection{G2}\label{sec:5.2.3}
The stabilizer of $\delta_3$ is the subgroup $W(A_{1,s}\times A_{1,l}).$ 
The induced module 
\begin{equation}
  \label{eq:5.2.3.1}
  Ind_{W(A_{1,s}\times A_{1,l}}^{W(G2)}[triv]= 1_1+2_2
\end{equation}
corresponds to the tensor product
\begin{equation}
  \label{eq:5.2.3.2}
  (2\mid 0)\otimes (2\mid 0)= (4\mid 0) + (2\mid 0) + (0\mid 0).
\end{equation}
On the other hand,
\begin{equation}
  \label{eq:5.2.3.3}
  (1\mid 0)\otimes (1\mid 0) =(2\mid 0) +(0\mid 0).
\end{equation}
Since the $M$-fixed vectors in the tensor product have dimension 1, it
follows that $(2\mid 0)$ has no $M$-fixed vectors. Combining this with
the fact that the representation of $W$ on the $M$-fixed vectors of
$\fk s=(3\mid 1)$ is the reflection representation, we conclude
\begin{equation}
  \label{eq:5.2.3.4}
\begin{aligned}
  &(0\mid 0)\longleftrightarrow &1_1,\\
  &(4\mid 0)\longleftrightarrow  &2_2,\\
  &(3\mid 1)\longleftrightarrow &2_1.
\end{aligned}
\end{equation}
\subsection{F4}\label{sec:5.3}

We first compute some tensor products of the $\delta$'s. In the process we
get the restrictions of the $\wti K$-types in theorem \ref{t:4.7} to
$\wti M.$
\begin{itemize}
\item $\delta_2\otimes\delta_2=\delta_0 + \delta_3$ because it equals  
$$
(1\mid 0,0,0)\otimes (1\mid 0,0,0)=(2\mid 0,0,0) + (0\mid 0,0,0).
$$
\item $\delta_2\otimes \delta_3$ equals
$$
(1\mid 0,0,0)\otimes (2\mid 0,0,0)=(3\mid 0,0,0) + (1\mid
0,0,0)=2\delta_2 + \delta_2,$$
because there is no room for a $\delta_6.$ So we also get $(3\mid
0,0,0)=2\delta_2$.  
\item $\delta_2\otimes\delta_6$ equals 
$$
(1\mid 0,0,0)\otimes (0\mid 1,0,0)=(1\mid 1,0,0)=\delta_{12}.
$$
\item $\delta_2\otimes\delta_{12}=\delta_2\otimes\delta_2\otimes\delta_6$ 
equals 
$$
(\delta_1+\delta_3)\otimes\delta_6=\delta_6+\delta_3\otimes\delta_6.
$$
It also equals
$$
(1\mid 0,0,0)\otimes (1\mid 1,0,0)=(2\mid 1,0,0)+(0\mid 1,0,0)=
\delta_3\otimes\delta_6+\delta_6=4\delta_6.
$$
Then $\delta_3\otimes\delta_6=a\delta_2+b\delta_6.$ But
$$
\delta_2\otimes\delta_3\otimes\delta_6=3\delta_2\otimes\delta_6=3\delta_{12},
$$ 
so $a=0.$
So also $(2\mid 1,0,0)=\delta_3\otimes\delta_6=3\delta_6.$ 
\item $\delta_6\otimes\delta_6=3\delta_1+a\delta_3+b\delta_{12}.$
Because 
$$
\delta_3\otimes\delta_6\otimes\delta_6=3\delta_6\otimes\delta_6,
$$
$a=3$ and $b=2.$ Since also
\begin{equation*}
(0\mid 1,0,0)\otimes (0\mid 1,0,0)=(0\mid 2,0,0) + (0\mid 1,1,0)
+(0\mid 0,0,0),  
\end{equation*}
we conclude
\begin{equation*}
  \begin{aligned}
    &(0\mid 2,0,0)=3\delta_3 + \delta_{12},\\
    &(0\mid 1,1,0)=2\delta_1 + \delta_{12}.
  \end{aligned}
\end{equation*}
\end{itemize}

We now compute the induced modules
\begin{equation}
  \label{eq:5.3.1}
  \begin{aligned}
  &Ind_{W(B_4)}^{W(F_4)}[triv]=1_1+2_1,\\
  &Ind_{W(C_4)}^{W(F_4)}[triv]=1_1+2_3,\\
  &Ind_{W(B_3A_1)}^{W(F_4)}[triv]=1_1+2_1+9_1,
  \end{aligned}
\end{equation}
corresponding to the tensor products
\begin{equation}
  \label{eq:5.3.2}
  \begin{aligned}
    &(0\mid 1,0,0)\otimes (0\mid 1,0,0)=(0\mid 2,0,0) + (0\mid 1,1,0)
    +(0\mid 0,0,0),\\
    &(2\mid 0,0,0)\otimes (4\mid 0,0,0)=(2\mid 0,0,0) + (0\mid 0,0,0),\\
    &(1\mid 1,0,0)\otimes (1\mid 1,0,0)=(2\mid 2,0,0) + (2\mid 1,1,0)
    +(2\mid 0,0,0)+\\  
    & +(0\mid 2,0,0) + (0\mid 1,1,0)+(0\mid 0,0,0). 
\end{aligned}
\end{equation}
Noting also that
\begin{equation}
(2\mid 2,0,0)=(2\mid 0,0,0)\otimes (0\mid 2,0,0)=\delta_3\otimes
(3\delta_3+\delta_{12}),  
\end{equation}
we conclude that the $M$-fixed vectors of $(2\mid 0,0,0)$ are nine
dimensional. Therefore 
\begin{equation}
  \label{eq:5.3.4}
  \begin{aligned}
    &(0\mid 1,1,0)\longleftrightarrow 2_1,\\
    &(4\mid 0,0,0)\longleftrightarrow 2_3,\\
    &(2\mid 2,0,0)\longleftrightarrow 9_1.
  \end{aligned}
\end{equation}

\bigskip
Because $(1\mid 1,1,1)$ is the representation of $K$ on $\mathfrak s,$ the
Weyl group representation on the fixed vectors is the reflection
representation $4_2.$ 

\bigskip
Now consider
\begin{equation}
  \label{eq:5.3.5}
 \Hom_M[(1\mid 1,0,0),(0\mid 1,1,0)]=\Hom_M[\delta_{12},2\delta_1+\delta_{12}].
\end{equation}
It is therefore 12-dimensional. Because $\delta_{12}$ corresponds to
$B_3A_1,$ in (\ref{eq:4.7.7}), the $W$-module is induced from a
character of $W(B_3A_1).$  Thus the tensor product
\begin{equation}
  \label{eq:5.3.6}
(1\mid 1,0,0)\otimes (0\mid 1,1,0)=(1\mid 2,1,0)+(1\mid 1,1,1)+(1\mid 1,0,0)  
\end{equation}
has the property that the $W$-representation on the $M$-fixed vectors
is induced from this 1-dimensional character.
It follows that the $M$-fixed vectors of $(1\mid 2,1,0)$ are
8-dimensional. To determine the character, we note that the
restriction of $4_2$ to $W(B_3A_1)$ is
\begin{equation}
  \label{eq:5.3.7}
4_2\longrightarrow [(2\times 1)\otimes (2)]+ [(3\times 0)\otimes (1^2)].  
\end{equation}
The induced module from the character which occurs, is
\begin{equation}
  \label{eq:5.3.8}
[(3\times 0)\otimes (1^2)]\longrightarrow 4_2 + 8_1.  
\end{equation}
The proof of theorem \ref{t:4.7} follows.



\subsection{E6}\label{sec:4.8a}
The proof will occupy several sections. We will exploit the fact that
genuine representations are all multiples of $\delta_8.$ 

The following induced modules,
\begin{equation}
  \label{eq:4.8a.1}
  \begin{aligned}
    &Ind_{W(D5)}^{W(E6)}[triv]=1_p+6_p +20_p,\\
    &Ind_{W(A5A1)}^{W(E6)}[triv]=1_p+15_q +20_p,\\
  \end{aligned}
\end{equation}
correspond to the tensor products
\begin{equation}
  \label{eq:4.8a.2}
  \begin{aligned}
&\om_2\otimes\om_2=(2\om_1)+(\om_1+\om_3)+(2\om_2) +(\om_4)+(\om_2)+(0),\\
&2\om_1\otimes2\om_1=(4\om_1)+(2\om_1+\om_2)+(2\om_1) +(2\om_2)+(\om_2)+(0).
  \end{aligned}
\end{equation}
We know that $\om_4\longleftrightarrow 6_p$ because this is the
representation of $\wti K$ on $\mathfrak s$ of  the Cartan
decomposition. We conclude that
\begin{equation}
  \label{eq:4.8a.3}
  \begin{aligned}
  &(2\om_1+\om_2)+(2\om_2)\longleftrightarrow 20_p,\\
  &(\om_1+\om_3)+(2\om_2)\longleftrightarrow 20_p.
  \end{aligned}
\end{equation}
But $(2\om_1)\otimes(\om_2)$ does not have any $M$-fixed vectors, so
$2\om_1+\om_2$ which is a factor does not have any either. We conclude
that
\begin{equation}
  \label{eq:4.8a.4}
  \begin{aligned}
&(0)\longleftrightarrow &1_p,\\
&(\om_4)\longleftrightarrow &6_p,\\
&(2\om_2)\longleftrightarrow &20_p,\\
&(4\om_1)\longleftrightarrow &15_q.
  \end{aligned}
\end{equation}
The rest of the representations considered so far do not have any
$M$-fixed vectors. Using tensoring with $\om_1$ we also find that
\begin{equation}
  \label{eq:4.8a.5}
  \begin{aligned}
&(\om_1)\longleftrightarrow &1_p,\\
&(\om_3)\longleftrightarrow &6_p,\\
&(\om_1+\om_2)\longleftrightarrow &20_p,\\
&(3\om_1)\longleftrightarrow &15_q.
  \end{aligned}
\end{equation}
In these matchups, the $W$ representations are on $\Hom_M[\om_1,V].$
\subsection{}\label{sec:4.8b}
The tensor product
\begin{equation}
  \label{eq:4.8b.1}
(\om_3)\otimes(\om_3)=(2\om_1)+(\om_1+\om_3)+(2\om_2)+(\om_2+\om_4)+(\om_2)+
(2\om_3)+(0) 
\end{equation}
corresponds to
\begin{equation}
  \label{eq:4.8b.2}
  6_p\otimes 6_p=1_p + 20_p +15_p.
\end{equation}
Since $\om_4=6\delta_1+\delta_{36},$ $(\om_2)\otimes(\om_4)$ does not have
any $M$-fixed vectors. Thus $(\om_2+\om_4)$ doesn't either. We conclude
\begin{equation}
  \label{eq:4.8b.3}
  (2\om_3)\longleftrightarrow 15_p.
\end{equation}
\subsection{}\label{sec:4.8c}
The representations $(3\om_1)$ and $\om_3$ are genuine, and
\begin{equation}
  \label{eq:4.8c.1}
(3\om_1)\otimes
(\om_3)=(3\om_1+\om_3)+(2\om_1+\om_2)+(2\om_1+\om_4)+(\om_1+\om_3) 
\end{equation}
corresponds to
\begin{equation}
  \label{eq:4.8c.2}
  6_p\otimes 15_q=30_p+60_p.
\end{equation}
Since 
\begin{equation}
  \label{eq:4.8c.3}
(\om_1)\otimes(\om_1+\om_4)=(2\om_1+\om_4)+(\om_1+\om_3)+(\om_2+\om_4)+(\om_4),
\end{equation}
and the dimension of $(\om_1+\om_4)$ is $8\cdot 36,$ we conclude that
\begin{equation}
  \label{eq:4.8c.4}
  \begin{aligned}
&(2\om_1+\om_4)\longleftrightarrow &30_p,\\
&(3\om_1+\om_3)\longleftrightarrow &60_p.    
  \end{aligned}
\end{equation}
This completes the proof of corollary \ref{c:4.8}.
\subsection{}\label{sec:4.8d}
The tensor product
\begin{equation}
  \label{eq:4.8d.1}
  \begin{aligned}
&(\om_1+\om_2)\otimes(\om_3)=(\om_1+\om_2+\om_3)+(2\om_1+\om_2)+(2\om_1+\om_4)+
\\
&(2\om_1)+2(\om_1+\om_3)+(2\om_2)+(\om_2+\om_4)+(\om_2)+(\om_4)    
  \end{aligned}
\end{equation}
corresponds to
\begin{equation}
  \label{eq:4.8d.2}
  6_p\otimes 20_p=6_p+30_p+20_p+64_p.
\end{equation}
We conclude that
\begin{equation}
  \label{eq:4.8d.4}
(\om_1+\om_2+\om_3)\longleftrightarrow 64_p.
\end{equation}
\subsection{}\label{sec:4.8e}
The tensor product
\begin{equation}
  \label{eq:4.8e.1}
  \begin{aligned}
&(3\om_1)\otimes(\om_1+\om_2)=(4\om_1+\om_2)+(4\om_1)+(3\om_1+\om_3)+\\
&(2\om_1+2\om_2)+ 
2(2\om_1+\om_2)+(2\om_1)+\\
&(\om_1+\om_2+\om_3)+(\om_1+\om_3)+(2\om_2)+(\om_2)     
  \end{aligned}
\end{equation}
corresponds to
\begin{equation}
  \label{eq:4.8e.2}
  15_q\otimes 20_p=15_q+20_p+60_p+60_s+64_p+81_p.
\end{equation}
We conclude that
\begin{equation}
  \label{eq:4.8e.4}
(4\om_1+\om_2)+(2\om_1+2\om_2)\longleftrightarrow 81_p+60_s.
\end{equation}
\subsection{}\label{sec:4.8f}
The tensor product
\begin{equation}
  \label{eq:4.8f.1}
  \begin{aligned}
&(\om_1+\om_2)\otimes(\om_1+\om_2)=\\
&(4\om_1)+(3\om_1+\om_3)+(2\om_1+2\om_2)+3(2\om_1+\om_2)+\\
&(2\om_1+\om_4)+2(2\om_1)+2(\om_1+\om_2+\om_3)+3(\om_1+\om_3)+
\\&(3\om_2)+2(2\om_2)+(\om_2+\om_4)+(2\om_2)+(2\om_3)+(\om_4)+(0) 
\end{aligned}
\end{equation}
corresponds to
\begin{equation}
  \label{eq:4.8f.2}
20_p\otimes 20_p=1_p+6_p+15_q+15_p+2\cdot20_p+24_p+30_p+60_p+2\cdot 64_p+81_p.
\end{equation}
We conclude that
\begin{equation}
  \label{eq:4.8f.4}
(2\om_1+2\om_2)+(3\om_2)\longleftrightarrow 81_p+24_p.
\end{equation}
\subsection{}\label{sec:4.8g}
The tensor product
\begin{equation}
  \label{eq:4.8g.1}
  \begin{aligned}
&(3\om_1)\otimes (3\om_1)=(6\om_1)+(4\om_1+\om_2)+(4\om_1)+\\
&(2\om_1+2\om_2)+2(2\om_1+\om_2)+(2\om_1)+\\
&(3\om_2)+(2\om_2)+(\om_1)+(0)     
  \end{aligned}
\end{equation}
corresponds to
\begin{equation}
  \label{eq:4.8g.2}
  15_q\otimes 15_q=1_p+15_q+20_p+24_p+24'_p+60_s+81_p.
\end{equation}
We conclude that
\begin{equation}
  \label{eq:4.8g.4}
(6\om_1)+(4\om_1+\om_2)+(2\om_1+2\om_2)+ (3\om_2)\longleftrightarrow
24_p+81_p+60_s+24'_p. 
\end{equation}
Combining this with (\ref{eq:4.8f.4}), and  (\ref{eq:4.8e.4}), we get
\begin{equation}
  \label{eq:4.8g.5}
  \begin{aligned}
&(3\om_2)\leftrightarrow 24_p,\\
&(4\om_1+\om_2)\leftrightarrow 60_s,\\
&(2\om_1+2\om_2)\leftrightarrow 81_p,\\
&(6\om_1)\leftrightarrow 24_p'.
  \end{aligned}
\end{equation}
\subsection{E7}\label{sec:4.10}
In this case, genuine representations restrict to combinations of
$\delta_8$ and $\delta_8^*,$ so the proof will be more involved.
We first note that
\begin{equation}
  \label{eq:4.10.1}
  (\om_4)\longleftrightarrow 7'_a,
\end{equation}
because it is the representation of $K$
on $\mathfrak s$ in the Cartan decomposition.

We record the following induced representations:
\begin{equation}
  \label{eq:4.10.2}
  \begin{aligned}
&Ind_{A7}^{E7}[triv]= 1_a+ 35_b +21'_b +15_a',\\   
&Ind_{E6}^{E7}[triv]= 1_a+ 27_a+ 7'_a +21'_b,\\ 
&Ind_{D6}^{E7}[triv]= 1_a+ 27_a + 35_b. 
  \end{aligned}
 \end{equation}
For the  first one, $W_\delta\ne W_\delta^0.$ It  corresponds to
the sum of 
\begin{equation}
  \label{eq:4.10.3}
  \begin{aligned}
&(2\om_1)\otimes (2\om_1)= (4\om_1) + (2\om_1+\om_2) +(2\om_2),\\
&\text{ and }\\
&(2\om_1)\otimes (2\om_7)=(2\om_1+2\om_7) +(\om_1+\om_7) +(0).
  \end{aligned}
\end{equation}
The first line is $21'_b+ 15_a,$ the second one $1_a +35_b.$ Since
\begin{equation}
  \label{eq:4.10.3a}
  \begin{aligned}
&(\om_1)\otimes(\om_7)=(\om_1+\om_7)+(0),\\
&(\om_1)\otimes(\om_1)=(2\om_1)+(\om_2),\\
&(\om_7)\otimes(\om_7)=(2\om_7)+(\om_6),    
  \end{aligned}
\end{equation}
we conclude that none of the representations on the RHS of these
representations  (except (0)) are $M$-spherical. We conclude that 
\begin{equation}
  \label{eq:4.10.3b}
  \begin{aligned}
&(2\om_1+2\om_7)\longleftrightarrow 35_b,\\
&(4\om_1)+(2\om_1+\om_2)+(2\om_2)\longleftrightarrow 21_b'+15_a.
  \end{aligned}
\end{equation}
For the second equation, $W_\delta\ne W_\delta^0$ as well. It
corresponds to the sum
\begin{equation}
  \label{eq:4.10.4}
  \begin{aligned}
&(\om_2)\otimes (\om_2)=(2\om_2)+(\om_1+\om_3) +(\om_4),\\
&\text{ and }\\
&(\om_2)\otimes(\om_6)=(\om_2+\om_6)+(\om_1+\om_7)+(0).
  \end{aligned}
\end{equation}
The first line is $7'_a+ 21'_b,$ the second one $1_a +27_a.$ 
The same argument as before implies
\begin{equation}
  \label{eq:4.10.4a}
  \begin{aligned}
&(\om_2+\om_6)\longleftrightarrow 27_a,\\
&(2\om_2)+(\om_1+\om_3)\longleftrightarrow 21_b'.
  \end{aligned}
\end{equation}
But
\begin{equation}
  \label{eq:4.10.8}
  (\om_1)\otimes (\om_3)=(\om_1+\om_3) + (\om_4),
\end{equation}
and the dimension of $\om_3$ is $8\cdot 7,$ So $\om_1+\om_3$  cannot
have a 21-dimensional space of $M$-fixed vectors. It follows that
\begin{equation}
  \label{eq:4.10.9}
  \begin{aligned}
&(2\om_2)\longleftrightarrow 21'_b,\\
&(\om_3)\longleftrightarrow  21'_b\qquad \text{ on } \delta_7^*.    
  \end{aligned}
\end{equation}
The dimension of $(\om_1+\om_2)$ is $8\cdot 21,$ and
\begin{equation}
  \label{eq:4.10.11}
  (\om_1+\om_2)\otimes(\om_1)=(2\om_1+\om_2) + (\om_1+\om_3) +(2\om_2).
\end{equation}
It follows that neither $(2\om_1+\om_2)$ nor $(\om_1+\om_3)$ can
be $M$-spherical. So
\begin{equation}
  \label{eq:4.10.12}
  (4\om_1)\longleftrightarrow 15_a'.
\end{equation}
For the third
equation, $W_\delta=W_\delta^0.$ It  corresponds to
\begin{equation}
  \label{eq:4.10.5}
  \begin{aligned}
(\om_1+\om_7)&\otimes(\om_1+\om_7)= (2\om_1+2\om_7)
  +(2\om_1+\om_6)+\\ &+2(\om_1+\om_7) +(\om_2+\om_6)+(\om_2+\om_7)+(0).
  \end{aligned}
\end{equation}
We conclude that neither of 
\begin{equation}
  \label{eq:4.10.6}
(2\om_1+\om_6),\quad 2(\om_1+\om_7),\quad (\om_2+\om_7)
\end{equation}
are $M$-spherical.
\subsection{}\label{sec:4.11} The following tensor products, 
\begin{equation}
  \label{eq:4.11.1}
  \begin{aligned}
&(\om_1+\om_6)\otimes(\om_1)=(2\om_1+\om_6)+(\om_1+\om_7)+(\om_2+\om_6),\\    
&(\om_1+2\om_7)\otimes(\om_1)=(2\om_1+2\om_7)+(\om_1+\om_7)+(\om_2+2\om_7),
\end{aligned}
\end{equation}
and the results in section \ref{sec:4.10} allow us to conclude that
\begin{equation}
  \label{eq:4.11.2}
  \begin{aligned}
&(\om_1)\longleftrightarrow 1_a,\\
&(\om_3)\longleftrightarrow 7'_a\\
&(3\om_1)\longleftrightarrow 15_a'\\
&(\om_1+\om_2)\longleftrightarrow 21'_b,\\
&(\om_1+\om_6)\longleftrightarrow 27_a,\\
&(\om_1+2\om_7)\longleftrightarrow 35_b,
  \end{aligned}
\end{equation}
on the $\delta_7$-isotypic component. From
\begin{equation}
  \label{eq:4.11.3}
  \begin{aligned}
&(\om_1+\om_6)\otimes(\om_1)=(2\om_1+\om_6)+(\om_1+\om_7)+(\om_2+\om_6)\\
&(3\om_1)\otimes(\om_7)=(3\om_1+\om_7) +(2\om_1)
  \end{aligned}
\end{equation}
we conclude that $(3\om_1+\om_7)$ and $(2\om_1+\om_6)$ are not
$M$-spherical. 
\subsection{}\label{sec:4.12}
Now consider
\begin{equation}
  \label{eq:4.12.1}
  7'_a\otimes 7'_a=1_a + 21_a + 27_a,
\end{equation}
and
\begin{equation}
  \label{eq:4.12.2}
  \begin{aligned}
    &(\om_3)\otimes(\om_3)=(2\om_3)
    +(\om_1+\om_5)+(\om_2+\om_4)+(\om_6),\\
    &(\om_3)\otimes(\om_5)=(\om_3+\om_5)+(\om_1+\om_7)+(\om_2+\om_6)+(0).
  \end{aligned}
\end{equation}
It follows that 
\begin{equation}
  \label{eq:4.12.3}
  (\om_3+\om_5)\longleftrightarrow 21_a,
\end{equation}
and aside from the already listed cases, none of the representations
occuring in (\ref{eq:4.12.2}) are $M$-spherical.

Next observe that
\begin{equation}
  \label{eq:4.12.4}
  (\om_5)\otimes(\om_1+\om_2)=(\om_1+\om_2+\om_5)+(2\om_1+\om_6)+(\om_1+\om_7)+(\om_2+\om_6) 
\end{equation}
corresponds to
\begin{equation}
  \label{eq:4.12.5}
  7'_a\otimes 21'_b=27_a+ 120_a.
\end{equation}
Because $(2\om_1+\om_6)$ does not have $M$-fixed vectors, we conclude that
\begin{equation}
  \label{eq:4.12.6}
  (\om_1+\om_2+\om_5)\longleftrightarrow 120_a.
\end{equation}
The same argument using
\begin{equation}
  \label{eq:4.12.7}
(\om_3)\otimes (\om_1+\om_2)
=(\om_1+\om_2+\om_3)+(2\om_1+\om_4)+(\om_1+\om_5)+(\om_2+\om_4)  
\end{equation}
shows that none of the representations in the RHS of (\ref{eq:4.12.7})
can be $M$-spherical.
\subsection{}\label{sec:4.13}
Note that
\begin{equation}
  \label{eq:4.13.1}
  \begin{aligned}
(\om_3)&\otimes(\om_1+\om_6)=(\om_1+\om_3+\om_6)+
(2\om_1)+\\&+(\om_1+\om_2+\om_7)+(\om_2)+(\om_3+\om_7)+(\om_4+\om_6)\\
(\om_5)&\otimes(\om_1+\om_6)=(\om_1+\om_5+\om_6)+
(\om_1+\om_4+\om_7)+\\ &+(\om_1+\om_3)+(\om_4)+(\om_5+\om_7)+(2\om_6).
  \end{aligned}
\end{equation}
The representations in the first equation do not have $M$-fixed
vectors, while the ones in the second equation correspond to
\begin{equation}
  \label{eq:4.13.2}
  7'_a\otimes 27_a=7'_a+21'_b+56'_a+105'_a.
\end{equation}
Since
\begin{equation}
  \label{eq:4.13.3}
  (\om_7)\otimes (\om_1+\om_4)=(\om_1+\om_4+\om_7)+(\om_1+\om_3)+(\om_4),
\end{equation}
and the dimension of $(\om_1+\om_4)$ is $8\cdot 63,$ we conclude that
the dimension of the $M$-fixed vectors in $(\om_1+\om_4+\om_7)$ is 56.
So it follows that
\begin{equation}
\label{eq:4.13.4}
\begin{aligned}
  &(\om_1+\om_5+\om_6)\longleftrightarrow 105_a',\\
  &(\om_1+\om_4+\om_7)\longleftrightarrow 56'_a.
\end{aligned}
\end{equation}
We also conclude that
\begin{equation}
  \label{eq:4.13.5}
  (\om_1+\om_4)\longleftrightarrow 7'_a+ 56'_a.
\end{equation}
\subsection{}\label{sec:4.14}
Note that
\begin{equation}
  \label{eq:4.14.1}
  \begin{aligned}
    (\om_3)\otimes(\om_1+2\om_7)=&
(\om_1+\om_3+2\om_7)+(\om_1+\om_2+\om_7)+\\
&+(\om_3+\om_7)+(\om_4+2\om_7),\\
   (\om_3)\otimes(\om_1+2\om_7)=&
(\om_1+\om_3+2\om_7)+(\om_1+\om_2+\om_7)+\\
&+(\om_3+\om_7)+(\om_4+2\om_7).
  \end{aligned}
\end{equation}
The first equation consists of representations without $M$-fixed
vectors, the second one corresponds to
\begin{equation}
  \label{eq:4.14.2}
  7'_a\otimes 35_b=56'_a+189'_b.
\end{equation}
It follows that
\begin{equation}
  \label{eq:4.14.3}
  (\om_1+\om_5+2\om_7)\leftrightarrow 189'_b.
\end{equation}
\subsection{}\label{sec:4.15}
Note that
\begin{equation}
  \label{eq:4.15.1}
  \begin{aligned}
   & (\om_3)\otimes(3\om_1)=(3\om_1+\om_3)+(2\om_1+\om_4) ,\\
   & (\om_5)\otimes(3\om_1)=(3\om_1+\om_5)+(2\om_1+\om_6).\\
  \end{aligned}
\end{equation}
The first equation does not contain any representations with $M$-fixed
vectors, while the second one corresponds to
\begin{equation}
  \label{eq:4.15.2}
  7'_a\otimes 15_a'=105_b.
\end{equation}
So
\begin{equation}
  \label{eq:4.15.3}
  (3\om_1+\om_5)\longleftrightarrow 105_b.
\end{equation}
The proof of corollary \ref{c:4.9} is complete.
\subsection{}\label{sec:4.16}
The tensor products decompose 
\begin{equation}
  \label{eq:4.16.1}
  \begin{aligned}
  (\om_1+\om_2)&\otimes (\om_1+\om_2)=
(2\om_1+2\om_2)+(3\om_1+\om_3)+(2\om_1+\om_4)+\\
&+2(\om_1+\om_2+\om_3)+(3\om_2)+(\om_2+\om_4)+(2\om_3),\\
  (\om_1+\om_2)&\otimes (\om_6+\om_7)=
(\om_1+\om_2+\om_6+\om_7)+(2\om_1+\om_6)+\\
&+(2\om_1+2\om_7)+2(\om_1+\om_7)+(\om_2+\om_6)+\\
&+(\om_2+2\om_7)+(0).
  \end{aligned}
\end{equation}
The first equation does not have any representations with $M$-fixed
vectors, while the second one corresponds to
\begin{equation}
  \label{eq:4.16.2}
  21'_b\otimes 21'_b=1_a+ 27_a + 35_b + 168_a +210_a.
 \end{equation}
Thus
\begin{equation}
  \label{eq:4.16.3}
  (\om_1+\om_2+\om_6+\om_7)\longleftrightarrow   168_a+ 210_a.
\end{equation}
\subsection{}\label{sec:4.17}
We decompose the tensor products
%\begin{equation}

\begin{align}  \label{eq:4.17.1}
&(\om_1+\om_2)\otimes (\om_1+\om_6)=
(3\om_1+\om_7)+(2\om_1+\om_2+\om_6)+(2\om_1)+\notag\\
&2(\om_1+\om_2+\om_7)+(\om_1+\om_3+\om_6)+
(2\om_2+\om_6)+(\om_2)+(\om_3+\om_7),\notag\\
&\\
    &(\om_1+\om_2)\otimes (\om_2+\om_7)=
(\om_1+2\om_2+\om_7)+(2\om_1+\om_3+\om_7)+(2\om_1+\om_2)+\notag\\
&2(\om_1+\om_3)+(\om_1+\om_4+\om_7)+
(\om_1+\om_4+\om_7)+(2\om_2)+(\om_2+\om_3+\om_7).\notag
  \end{align}
%\end{equation}
Since also 
\begin{equation}
  \label{eq:4.17.2}
  21'_b\otimes 27_a=7'_a +21'_b+56'_a+105'_a+189_b'+189_c',
\end{equation}
and the first equation does not contain any representations with
$M$-fixed vectors, we conclude
\begin{equation}
  \label{eq:4.17.3}
  \begin{aligned}
    &(2\om_1+\om_3+\om_7)\longleftrightarrow 189_{b,c}',\\
   &(\om_1+2\om_2+\om_7)\longleftrightarrow 189_{b,c}'.
  \end{aligned}
\end{equation}
The ambiguity is resolved in section \ref{sec:4.22}.
\subsection{}\label{sec:4.18}
We decompose the tensor products
\begin{equation}
  \label{eq:4.18.1}
  \begin{aligned}
 &(3\om_1)\otimes(3\om_1)=(6\om_1)+(4\om_1+\om_2)+(2\om_1+2\om_2)+(3\om_2),\\
 &(3\om_1)\otimes(3\om_7)=(3\om_1+3\om_7)+(2\om_1+2\om_7)+(\om_1+\om_7)+(0).
 \end{aligned}
\end{equation}
The first equation does not contain any representations with $M$-fixed
vectors, the second one corresponds to
\begin{equation}
  \label{eq:4.18.2}
  15_a'\otimes 15_a'=1_a +35_b+84_a+105_c.
\end{equation}
It follows that
\begin{equation}
  \label{eq:4.18.3}
  (3\om_1+3\om_2)\longleftrightarrow 84_a+105_c.
\end{equation}
\begin{comment}
\subsection{}\label{sec:4.19}
We decompose the tensor products
\begin{equation}
  \label{eq:4.19.1}
  \begin{aligned}
&(\om_1+\om_6)\otimes(\om_1+\om_6)=
(2\om_1+\om_4)+(2\om_1+\om_5+\om_7)+\\
&(2\om_1+2\om_6)+2(\om_1+\om_5)+2(\om_1+\om_6+\om_7)+
(\om_2+\om_4)+\\
&(\om_2+\om_5+\om_7)+(\om_2+2\om_6)+(\om_6)+(2\om_7),\\
&(\om_1+\om_6)\otimes(\om_2+\om_7)=
(\om_1+\om_2+\om_6+\om_7)+(2\om_1+2\om_7)+\\
&(\om_1+\om_2+\om_5)+2(\om_1+\om_7)+2(\om_2+\om_6)+
(\om_1+\om_2+\om_5)+\\
&(\om_2+2\om_7)+(\om_3+\om_5)+(\om_3+\om_6+\om_7)+(0).
  \end{aligned}
\end{equation}
The first equation does not contain any $M$-spherical representations,
while the second on corresponds to
\begin{equation}
  \label{eq:4.19.2}
  27_a\otimes 27_a=1_a+21_a +2\cdot 27_a+35_b+2\cdot 120_{a}+168_{a}+210_a.
\end{equation}
These equations don't give anything new; the same conclusion as
(\ref{eq:4.16.3}) follows.
\end{comment} 
\subsection{}\label{sec:4.20}
We decompose the tensor products
\begin{equation}
  \label{eq:4.20.1}
  \begin{aligned}
&(3\om_1)\otimes(\om_1+\om_6)=(4\om_1+\om_6)+(3\om_1+\om_7)+\\
&(2\om_1+\om_2+\om_6)+(\om_1+\om_2+\om_7),\\
&(3\om_1)\otimes(\om_2+\om_7)=(3\om_1+\om_2+\om_7)+(2\om_1+\om_2)+\\
&(2\om_1+\om_3+\om_7)+(\om_1+\om_3).
  \end{aligned}
\end{equation}
The first equation does not contain any $M$-spherical representations,
while the second one corresponds to
\begin{equation}
  \label{eq:4.20.2}
 15_a'\otimes 27_a=189'_b +216_a'.
\end{equation}
We conclude that
\begin{equation}
  \label{eq:4.20.3}
(3\om_1+\om_2+\om_7)\longleftrightarrow 216_a'.  
\end{equation}
\subsection{}\label{sec:4.21}
We decompose the tensor products
\begin{equation}
  \label{eq:4.21.1}
  \begin{aligned}
&(3\om_1)\otimes(\om_1+\om_2)=(4\om_1+\om_2)+(3\om_1+\om_3)+
(2\om_1+2\om_2)+\\
&(\om_1+\om_2+\om_3),\\
&(3\om_1)\otimes(\om_6+\om_7)=(3\om_1+\om_6+\om_7)+(2\om_1+\om_6)+\\
&(2\om_1+2\om_7)+(\om_1+\om_7).
  \end{aligned}
\end{equation}
The first equation does not contain any $M$-spherical representations,
while the second on corresponds to
\begin{equation}
  \label{eq:4.21.2}
 15'_a\otimes 21'_b=35_b+ 280_b
\end{equation}
We conclude that
\begin{equation}
  \label{eq:4.21.3}
(3\om_1+\om_6+\om_7)\longleftrightarrow 280_b.  
\end{equation}
\subsection{}\label{sec:4.22}
We decompose the tensor products
\begin{equation}\label{eq:4.22.1}
\begin{aligned}
&(3\om_1)\otimes(\om_1+2\om_7)=(4\om_1+2\om_2)+(3\om_1+\om_7)+\\
&(2\om_1+\om_2+2\om_7)+(2\om_1)+(\om_1+\om_2+\om_7)+(\om_2),\\
&(3\om_1)\otimes(2\om_1+\om_7)=(5\om_1+\om_7)+(4\om_1)+\\
&(3\om_1+\om_2+\om_7)+(2\om_1+\om_2)+(\om_1+2\om_2+\om_7)+(2\om_2).
\end{aligned}
\end{equation}
The first equation does not contain any $M$-spherical representations,
while the second on corresponds to
\begin{equation}
  \label{eq:4.22.2}
 15'_a\otimes 35_b=15'_ a+21'_b+84_a'+189_c' +216_a'.
\end{equation}
Combining this with section \ref{sec:4.17}, we conclude that
\begin{equation}
  \label{eq:4.22.3}
  \begin{aligned}
&(5\om_1+\om_7)\longleftrightarrow 84_a',\\ 
&(\om_1+2\om_2+\om_7)\longleftrightarrow 189_c',\\
&(2\om_1+\om_3+\om_7)\longleftrightarrow 189_b'  
  \end{aligned}
\end{equation}

\begin{comment}
\subsection*{Remark}
Here's another way to match the 21-dimensional Weyl group 
representations. The restriction of $2\om_2$ to the maximal compact
subgroup of the Levi component of type $D_6$ contains
\begin{equation}\label{eq:4.9.4}
    (1,1,1\ |\ 1,1,1) + (1,0,0\ |\ 1,0,0) + (2,0,0\ |\ 0,0,0),
  \end{equation}
and representations which do not occur in the spherical principal series.
Meanwhile the restriction of $\om_3+\om_5$ does not contain the second
two factors. The restrictions of the Weyl group representations are
\begin{equation*}
  \begin{aligned}
    &21_a :\ 5\times 1 + 51\times 0 + 3\times3_-,\\
    &21_b':  5\times 1 + 4\times 11.
  \end{aligned}
\end{equation*}
Next we observe that
\begin{equation*}
  \begin{aligned}
&\om_2\otimes\om_2=(2\om_2)+(\om_1+\om_3) + (\om_4),\\
&\om_3\otimes\om_5=(\om_3+\om_5) + (\om_2+\om_6) +(0).    
  \end{aligned}
\end{equation*}
Recalling that $\om_2=\delta_{28},$  
$\om_3=7\delta_8,\ \om_5=7\delta_8^*,$ and the fact that
$\om_4=8\delta_1+ \delta_{63}$ implies that the multiplicity of
$\delta_1$  in $(2\om_2)$ and $(\om_3+\om_5)$ is 21. The representations
of $SO(6)\times SO(6)$ in (\ref{eq:4.9.4}) match the restriction of
$21_a.$ The restriction of $(\om_3+\om_5)$ contains
\begin{equation}
  \label{eq:4.9.5}
  (1,0,0\ |\ 1,0,0)
\end{equation}
exactly once, and none of the other ones in (\ref{eq:4.9.4}). 
Thus it cannot contain $1_a,$ $7'_a$ or $21_a,$ so it must be $21_b'.$  
\end{comment}
\subsection{E8}\label{sec:5.3.1.1} Let $(\mu,V)$ be the fine $K$-type
$(\om_2).$ Then 
\begin{equation}
  \label{eq:5.3.1.1}
  \begin{aligned}
&\Hom_M[V,V]\cong Ind_{W(E7A1)}^{W(E8)}[triv]=1_x+ 35_x+ 84_x,\\  
&(\om_2)\otimes(\om_2)=
(2\om_2)+(\om_1+\om_3)+(2\om_1)+(\om_2)+(\om_4)+(0).
  \end{aligned}
\end{equation}
This is because the stabilizer of $\delta_{120}$ in $W(E8)$ is
$W(E7A1).$ 

Similarly, if $(\mu,V)$ is the fine $K$-type $(2\om_1),$ 
\begin{equation}
  \label{eq:5.3.1.2}
  \begin{aligned}
&\Hom_M[V,V]\cong Ind_{W(D8)}^{W(E8)}[triv]=1_x+ 84_x+ 50_x, \\
&(2\om_1)\otimes(2\om_1)=(4\om_1)+(2\om_1+\om_2)+(2\om_1)+(2\om_2)+(\om_2)
+(0).    
  \end{aligned}
\end{equation}
Then  $(\om_3)$  has dimension $35\cdot 16,$ so $\delta_{16}$
occurs 35 times. The tensor product is
\begin{equation}
  \label{eq:5.3.1.4}
  (\om_1)\otimes(\om_3)=(\om_1+\om_3) +(\om_2) +(\om_4),
\end{equation}
and thus the multiplicity of $\delta_1$ in $(\om_1+\om_3)+(\om_4)$ is
35, and (\ref{eq:5.3.1.1}) implies that the representation on the $M$
fixed vectors is $35_x.$ 
On the other hand, $\dim\om_4=1820,$ so the multiplicity of
$\delta_1$  in $(\om_4)$ is nonzero. This is because 1820 is not
divisible by 3, but the contribution of the other
characters occuring must be divisible by 3. Using (\ref{eq:5.3.1.1})
again, we conclude that
\begin{equation}
  \label{eq:5.3.1.5}
  (\om_4)\longleftrightarrow 35_x,\quad (2\om_2)\longleftrightarrow 84_x.
\end{equation}
We also conclude that the multiplicity of $\delta_1$ in
$(\om_1+\om_3)$ is zero, and 
\begin{equation}
  \label{eq:5.3.1.7}
 (\om_3)\longleftrightarrow 35_x 
\end{equation}
(on the $\delta_{16}$ isotypic component). 

Consider $(\om_1+\om_2)$ which has dimension $84\cdot16,$ so
$\delta_{16}$ occurs $84$ times. Then
\begin{equation}
  \label{eq:5.3.1.6}
  (\om_1+\om_2)\otimes(\om_1)=
(2\om_1+\om_2)+(2\om_1)+(\om_1+\om_3)+(2\om_2)+(\om_2).
\end{equation}
Thus only $(2\om_2)$ contains $\delta_1.$ Combined with
(\ref{eq:5.3.1.2}) we get 
\begin{equation}
  \label{eq:5.3.1.8}
 (4\om_1)\longleftrightarrow 50_x.
\end{equation}



\subsection{}\label{sec:5.4}
The representation $\om_7$ equals $8\delta_{16}.$ Also,
$\om_8=8\delta_1+\delta_{120},$ and the W-representation on $(\om_8)^M$ is
$8_x,$ because this is the representation on $\mathfrak s.$ Then 
\begin{equation}
  \label{eq:5.4.1}
  (\om_1)\otimes(\om_7)=(\om_1+\om_7)+(\om_8).
\end{equation}
So $\delta_1$ does not occur in $(\om_1+\om_7)$, and it occurs 8 times
in $(\om_8).$ This implies that  
$\Hom_M[\om_1,\om_7]$ is $8_x.$ We combine this with 
\begin{equation}
  \label{eq:5.4.2}
  \begin{aligned}
&  35_x\otimes 8_x=160_z + 112_x + 8_x,\\
&(\om_3)\otimes(\om_7)=(\om_3+\om_7)+(\om_2+\om_8)+(\om_1+\om_7)+ 
(\om_8).     
  \end{aligned}
\end{equation}
We have to distribute the other two representations among
$(\om_3+\om_7)$ and $(\om_2+\om_8)$. We have
\begin{equation}
  \label{eq:5.4.3}
  (\om_2)\otimes(\om_8)=(\om_2+\om_8)+(\om_2)+(\om_8),
\end{equation}
and $\delta_1$ occurs 120 times. Thus $(\om_2+\om_8)$ contains
$\delta_1$ 112 times, and combined with (\ref{eq:5.4.2}) we get
\begin{equation}
  \label{eq:5.4.4}
  (\om_2+\om_8)\longleftrightarrow 112_x,\qquad
  (\om_3+\om_7)\longleftrightarrow 160_z.
\end{equation}
\subsection{}\label{sec:5.5}
We apply the previous reasoning to
\begin{equation}\label{eq:5.5.1}
  \begin{aligned}
    35_x\otimes 35_x=&210_x+84_x+567_x+300_x+35_x+28_x+1_x,\\
    (\om_3)\otimes (\om_3)=&
(2\om_3)+(\om_1+\om_3)+(\om_1+\om_5)+(2\om_1)+\\
&+(2\om_2)+(\om_2+\om_4)+(\om_2)+
(\om_4)+(\om_6)+(0).
  \end{aligned}
\end{equation}
Then 
\begin{equation}
  \label{eq:5.5.2}
  (\om_1)\otimes(\om_5)=(\om_1+\om_5)+(\om_6) +(\om_4),
\end{equation}
and the dimension of $\om_5$ being $273\cdot 16,$ we get that the
multiplicity of $\delta_1$ in $(\om_6)+(\om_1+\om_5)$ is 238. The
tensor product decomposes
\begin{equation}
  \label{eq:5.5.3}
(\om_2)\otimes(\om_4)=(\om_2+\om_4)+(\om_1+\om_3)+(\om_1+\om_5)+(\om_2)+
(\om_4)+ (\om_6), 
\end{equation}
Using the fact that $\om_4=35\delta_1 +7\delta_{120}+7\delta_{135},$ 
we conclude that $\delta_1$ must occur $567$ times in $(\om_2+\om_4).$
The formula for $(\om_4)$ is a consequence of (\ref{eq:5.3.7})
which yields the system 
\begin{equation}
  \label{eq:5.5.4}
  \begin{aligned}
    &1820=35+120a_1 +135b_1,\\
    &7020=   120a_2 +135b_2,\\
    &a_1+a_2=34,\\
    &b_1+b_2=35.
  \end{aligned}
\end{equation}
The numbers 1820,\ 7020 are the dimensions of $\om_4$ and
$(\om_1+\om_3),$ and the two last equations come from
\begin{equation}
  \label{eq:5.5.5}
  \begin{aligned}
 &(\om_1)\otimes(\om_1)=(2\om_1)+(\om_2)+(0)\\
 &\delta_{16}\otimes\delta_{16}=\delta_{135}+\delta_{120}+\delta_1.
  \end{aligned}
\end{equation}
Furthermore,
\begin{equation}
  \label{eq:5.5.6}
  (\om_8)\otimes(\om_8)=(2\om_8)+(\om_6)+(\om_4)+(\om_2)+(0)
\end{equation}
and $\delta_1$ occurs 184 times. Since the  dimension of $(\om_6)$ is
$8008=a+120b+135c,$ it follows that $a>0,$ and $a$ cannot be divisible
by 5. We conclude that the only choice from (\ref{eq:5.5.3}) given
that $84_x,\ 567_x$ are not available, is
\begin{equation}
  \label{eq:5.5.7}
  \begin{aligned}
 & (\om_1+\om_5)\longleftrightarrow 210_x,\\
&(\om_6)\longleftrightarrow 28_x,\\
 &(\om_2+\om_4)\longleftrightarrow 567_x,\\
&(2\om_3)\longleftrightarrow 300x,\\
&(\om_5)\longleftrightarrow 28_x+35_x+210_x.
  \end{aligned}
 \end{equation}
\subsection{}\label{sec:5.6}
The previous arguments also imply that
\begin{equation}\label{eq:5.6.1}
(3\om_1)\longleftrightarrow 50_x.  
\end{equation}
Then 
\begin{equation}
  \label{eq:5.6.2}
  \begin{aligned}
    &50_x\otimes 8_x=400_x,\\
&(3\om_1)\otimes(\om_7)=(3\om_1+\om_7)+(2\om_1+\om_8).
  \end{aligned}
\end{equation}
But
\begin{equation}
  \label{eq:5.6.3}
  (\om_1+\om_8)\otimes(\om_1)=(2\om_1+\om_8)+(\om_1+\om_7)+(\om_2+\om_8)+
(\om_8).
\end{equation}
Since $\dim (\om_1+\om_8)=120\cdot 16,$ the previous results show that
$\delta_1$ cannot occur in $(2\om_1+\om_8)$, and we conclude
\begin{equation}
  \label{eq:5.6.4}
  (3\om_1+\om_7)\longleftrightarrow 400_x.
\end{equation}
\subsection{}\label{sec:5.7}
Recall again that
\begin{equation}
  \label{eq:5.7.1}
 (3\om_1)\longleftrightarrow 50_x,\qquad (\om_3)\longleftrightarrow 35_x.
\end{equation}
The tensor product decomposes
\begin{equation}
  \label{eq:5.7.2}
  50_x\otimes 35_x=1050_x + 700_x.
\end{equation}
On the other hand,
%\begin{equation}
   \begin{align} \label{eq:5.7.3}
(3\om_1)&\otimes (\om_3)=(3\om_1+\om_3)
  +(2\om_1+\om_2)+(2\om_1+\om_4)+(\om_1+\om_3),\notag\\
(\om_1)&\otimes(\om_1+\om_4)=\\
&(2\om_1+\om_4)+(\om_2+\om_4)+(\om_1+\om_5)+(\om_1+\om_3)+
(\om_4).\notag
\end{align}
%\end{equation}
In the first equation $2(\om_1+\om_4)+(3\om_1+\om_3)$ contains all the
$M$-spherical vectors, and their dimensions add up to $1750.$ 
Counting dimensions of $M$-fixed vectors in the second equation, we
find that the dimension of the $M$-fixed vectors in $(2\om_1+\om_4)$
is 700. Thus $(2\om_1+\om_4)$ must contain $700_x$ and
$(3\om_1+\om_3)$ must contain $1050_x.$ 

This completes the proof of corollary \ref{c:3a}.

\subsection{}\label{sec:5.8}
Recall that $50_x\longleftrightarrow (3\om_1).$ The tensor product
decomposes
\begin{equation}
  \label{eq:5.8.1}
  50_x\otimes 50_x=1_x+ 50_x+ 84_x+ 168_y+525_x+700_{xx} + 972_x.
\end{equation}
It corresponds to
\begin{equation}
  \label{eq:5.8.2}
  \begin{aligned}
&(3\om_1)\otimes (3\om_1)=\\
&(6\om_1)+(4\om_1+\om_2)+(4\om_1)+(2\om_1+2\om_2)+(2\om_1+\om_2)+\\
&(2\om_1)+(3\om_2)+(2\om_2)+(\om_2)+(0).
  \end{aligned}
\end{equation}
Since 
\begin{equation}
  \label{eq:5.8.3}
  \begin{aligned}
&(\om_1+\om_2)\otimes\om_1=(2\om_1+\om_2)
+(2\om_2)+(\om_1+\om_3)+(2\om_1) +(\om_2),\\
&(\om_1)\otimes(\om_3)=(\om_4)+(\om_2)+(\om_1+\om_3),
  \end{aligned}
\end{equation}
and taking the earlier results into account, we find that
$(\om_1+\om_3)$ and $(2\om_1+\om_2)$ cannot have  $M$-fixed vectors. 
This shows that the sum of Weyl group representations in (\ref{eq:5.8.1})
corresponds to the $M$-fixed vectors in (\ref{eq:5.8.2}). We omit the
details of how they match the irreducible modules. Finally, recall
that $(\om_1+\om_2)$ corresponds to $84_x,$ while $\om_3$ corresponds
to $35_x.$ Then 
\begin{equation}
  \label{eq:5.8.4}
  84_x\otimes 35_x=1344_x+ 700_x+ 210_x+567_x+ 84_x+ 35_x,
\end{equation}
while
\begin{equation}
  \label{eq:5.8.5}
  \begin{aligned}
&(\om_1+\om_2)\otimes (\om_3)=\\
&(2\om_1+\om_2)+(2\om_1+\om_4)+(2\om_1)+(\om_1+\om_2+\om_3)+
2(\om_1+\om_3)+\\
&+(\om_1+\om_5)+(2\om_2)+(\om_2+\om_4)+(2\om_2)+(\om_2)+(\om_4). 
\end{aligned}
\end{equation}

The fact that the representation of $W(E_8)$ on the $M$-fixed vectors
of $(\om_1+\om_2+\om_3)$ is $1344_x$ follows from these two equations.   





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