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\begin{document}

\title[Endoscopic decomposition]%
{Endoscopic decomposition of characters of certain cuspidal
representations}

\author{David Kazhdan}
\address{Institute of Mathematics,
Hebrew University, Givat-Ram, Jerusalem,  91904 Israel}
\email{kazhdan@math.huji.ac.il}
\thanks{The work of the second author was supported by
the Israel Science Foundation (Grant No. 38/01-1)}

\author{Yakov Varshavsky}
\address{Institute of Mathematics,
Hebrew University, Givat-Ram, Jerusalem,  91904 Israel}
\email{vyakov@math.huji.ac.il }


\date{September 18, 2003 and, in revised form, January 19, 2004}

\keywords{Endoscopy, Deligne-Lusztig representations}

\subjclass[2000]{Primary 22E50; Secondary 22E35}


\commby{Svetlana Katok}

\begin{abstract}
We construct an endoscopic decomposition 
for local $L$-packets
associated to irreducible cuspidal Deligne-Lusztig representations. 
Moreover, the obtained decomposition is compatible with inner 
twistings.
\end{abstract}
\maketitle

\section{Introduction}

Let $E$ be a local non-Archimedean field, with ring of integers $\C{O}$ 
and residue field
$\fq$ of characteristic $p$. We denote by $\Gm\supset W\supset I$ the 
absolute Galois,
the Weil, and the inertia groups of $E$. Let $G$ be a reductive group over
$E$, ${}^LG=\wh{G}\rtimes W$ its complex Langlands dual group, and 
$\C{D}(G(E))$ the
space of invariant distributions on $G(E)$.

Every admissible homomorphism $\la:W\to {}^LG$ (see \cite[$\S$ 10]{Ko1})
gives rise to a finite group 
$S_{\la}:=\pi_0(Z_{\wh{G}}(\la)/Z(\wh{G})^{\Gm})$,
where $Z_{\wh{G}}(\la)$ is the centralizer of $\la(W)$ in $\wh{G}$.
Every conjugacy class $\ka$ of $S_{\la}$ defines an endoscopic
subspace $\C{D}_{\ka,\la}(G(E))\subset\C{D}(G(E))$. For simplicity, we 
will restrict
ourselves to the elliptic case, where $\la(W)$ does not lie in any 
proper Levi subgroup of ${}^LG$.

Langlands conjectured that every elliptic $\la$ corresponds to a finite 
set $\Pi_{\la}$,
called an $L$-packet, of cuspidal irreducible representations of $G(E)$.
Moreover, the subspace $\C{D}_{\la}(G(E))\subset\C{D}(G(E))$, generated 
by the characters
$\{\chi(\pi)\}_{\pi\in\Pi_{\la}}$, must have an endoscopic 
decomposition.
More precisely, it is expected (\cite[IV, 2]{La})
that there exists a basis $\{a_{\pi}\}_{\pi\in \Pi_{\la}}$ of the space 
of central functions on
$S_{\la}$ such that
$\chi_{\ka,\la}:=\sum_{\pi\in \Pi_{\la}}a_{\pi}(\ka) \chi(\pi)$ belongs 
to
$\C{D}_{\ka,\la}(G)$ for every conjugacy class $\ka$ of $S_{\la}$.

The goal of this paper is to construct the endoscopic decomposition of 
$\C{D}_{\la}(G(E))$ for
tamely ramified $\la$'s such that $Z_{\wh{G}}(\la(I))$ is a maximal 
torus. In this case, $G$ splits over
an unramified extension of $E$, and $\la$ factors through 
${}^LT\hra{}^LG$ for an elliptic
unramified maximal torus $T$ of $G$.
By the local Langlands correspondence for tori (\cite{La2}), a 
homomorphism
$\la:W\to {}^LT$ defines a tamely ramified homomorphism 
$\theta:T(E)\to\B{C}\m$.
Each $\ka\in S_{\la}=\wh{T}^{\Gm}/Z(\wh{G})^{\Gm}$ gives rise to an 
elliptic endoscopic
datum $\C{E}_{\ka,\la}$ of $G$, while the characters of $S_{\la}$ are in 
bijection with the
conjugacy classes of embeddings $T\hra G$, stably conjugate to the 
inclusion.
Therefore each character $a$ of $S_{\la}$ gives rise to an irreducible 
cuspidal
representation
$\pi_{a,\la}$ of $G(E)$ (denoted by $\pi_{a,\theta}$ in \rn{repr}).


Our main result asserts, for fields $E$ of sufficiently large residual 
characteristic,
that each $\chi_{\ka,\la}:=\sum_{a}a(\ka) \chi(\pi_{a,\la})$ is
$\C{E}_{\ka,\la}$-stable. Moreover, the resulting endoscopic 
decomposition of
$\C{D}_{\la}(G(E))$ is compatible with inner twistings.
For simplicity, we restrict ourselves to local fields of characteristic 
zero,
while the case of positive characteristic follows by approximation (see 
\cite{Ka3}, \cite{De}).


Our argument goes as follows. First we prove the stability
of the restriction of $\chi_{\ka,\la}$ to the subset of topologically 
unipotent elements of
$G(E)$. If $p$ is sufficiently large, this assertion reduces to
the analogous assertion about distributions on the Lie algebra.
Now the stability follows from a combination of a Springer hypothesis 
\cite{Ka1}
and a generalization of a theorem of Waldspurger \cite{Wa}.
To prove the result in general, we use the topological Jordan 
decomposition (\cite{Ka2}).


When this work was in the process of writing, we have heard that S. 
DeBacker and  M. Reeder
obtained similar results.

\subsection*{Notation and conventions}In addition to the notation 
introduced above, we use the following
conventions:

  For a reductive group $G$, always assumed to be connected, we denote 
by $Z(G)$, $G^{\ad}$, $G^{\der}$,
$G^{\ssc}$, $G_{\dt}$,
and $G^{\sr}$ the center of $G$, the adjoint group of $G$, the derived 
group of $G$,
the simply connected covering of $G^{\der}$, the centralizer of $\dt\in 
G$, and
the set of strongly regular semisimple elements of  $G$ (that is, the 
set of $\dt\in G$ such that
$G_{\dt}\subset G$ is a maximal torus), respectively.

  Denote by $\C{G}, \C{T},$ and $\C{L}$ the Lie algebras of the algebraic 
groups $G, T$, and
$L$.

  Let $E$ be a local non-Archimedean field of characteristic zero,  
$\ov{E}$
a fixed algebraic closure of $E$, and $E^{\nr}$ a maximal unramified 
extension of $E$ in $\ov{E}$.

  For a reductive group $G$ (resp. its Lie algebra $\C{G}$) over $E$, 
we denote
by  $\C{S}(G(E))$ (resp.  $\C{S}(\C{G}(E))$) the space of locally 
constant measures with compact support.
We denote by $\C{D}(G(E))$ (resp. $\C{D}(\C{G}(E))$) the space of  
invariant distributions on $G(E)$
(resp. $\C{G}(E)$), namely $G(E)$-invariant linear functionals on
$\C{S}(G(E))$ (resp.  $\C{S}(\C{G}(E))$), where  $G(E)$ acts by 
conjugation.
Whenever necessary we equip $G(E)$ and $\C{G}(E)$ with invariant 
measures, denoted by $\mu$, defined
by a translation-invariant top degree differential form on $G$.
We denote by $G(E)_{\tu}$ (resp. $\C{G}(E)_{\tn}$) the set of 
topologically unipotent
(resp. topologically nilpotent) elements of $G(E)$ (resp. $\C{G}(E)$).
Finally, we denote by $\rk(G)$ the rank of $G$ over $E$, and put 
$e(G):=(-1)^{\rk(G^{\ad})}$. Note that our sign
$e(G)$ differs from that defined by Kottwitz.


\section{Formulation of the main result}


\begin{Emp} \label{E:finite}
Let $L$ be a connected reductive group over $\fq$, and
$\ov{a}:\ov{T}\hra L$ an embedding of a maximal elliptic torus of $L$.
Following Deligne and Lusztig,  we associate to every
character $\ov{\theta}:\ov{T}(\fq)\to\B{C}\m$ in general position an 
irreducible cuspidal representation
$\rho_{\ov{a},\ov{\theta}}$ of $L(\fq)$ (see \cite[Prop. 7.4 and Thm. 
8.3]{DL}).
\end{Emp}

\begin{Emp}
There is an equivalence of categories $T\mapsto \ov{T}$ between tori 
over $E$ splitting over
$E^{\nr}$ and tori over $\fq$. Every such $T$ has a canonical 
$\C{O}$-structure.
\end{Emp}

\begin{Not} \label{N:repr}

a) Let $G$ be a reductive group over $E$, $T$ a torus over $E$ 
splitting over $E^{\nr}$, and
$a:T\hra G$ an embedding of a maximal elliptic torus of $G$. Then
$G$ splits over $E^{\nr}$, and $a(T(\C{O}))$ lies in a unique parahoric 
subgroup $G_{a}$ of $G(E)$.
Let $G_{a^+}$ be the pro-unipotent radical of $G_a$.
Then there exists a canonical reductive group $L_a$ over $\fq$
with an identification $L_a(\fq)=G_a/G_{a^+}$.
Moreover, $a:T\hra G$ induces an embedding $\ov{a}:\ov{T}\hra L_a$ of a 
maximal elliptic torus of $L_a$.

b) Let $\theta:T(E)\to\B{C}\m$ be a character in general position, 
trivial on
$\Ker[T(\C{O})\to\ov{T}(\fq)]$. Denote by 
$\ov{\theta}:\ov{T}(\fq)\to\B{C}\m$
the character of $\ov{T}(\fq)$ defined by $\theta$.
Then there exists a unique irreducible representation $\rho_{a,\theta}$ 
of $Z(G)(E)G_a$,
whose central character is the restriction of $\theta$, extending the 
inflation to $G_a$
of the cuspidal Deligne-Lusztig representation 
$\rho_{\ov{a},\ov{\theta}}$ of $L_a(\fq)$.
We denote by $\pi_{a,\theta}$ the induced cuspidal representation
$\Ind_{Z(G)(E)G_a}^{G(E)}\rho_{a,\theta}$ of $G(E)$.
\end{Not}
\begin{Emp} \label{E:coh}
Recall (see \cite[Thm 1.2]{Ko2}) that for every reductive group $G$ over
$E$, $H^1(E,G)$ is canonically isomorphic to the group  
$\pi_0(Z(\wh{G})^{\Gm})^D$
of characters of $\pi_0(Z(\wh{G})^{\Gm})$. If $T$ is a maximal torus of 
$G$,
we get a commutative diagram:
\[
\CD
H^1(E,T)@>{\sim}>> \pi_0(\wh{T}^{\Gm})^D\\
@VVV          @VVV\\
 H^1(E,G)@>{\sim}>> \pi_0(Z(\wh{G})^{\Gm})^D.
\endCD
\]
In particular, we have a canonical surjection
\[
\wh{T}^{\Gm}/Z(\wh{G})^{\Gm}\to 
\Coker[\pi_0(Z(\wh{G})^{\Gm})\to\pi_0(\wh{T}^{\Gm})]\overset{\thicksim}{\to}
(\Ker\,[H^1(E,T)\to H^1(E,G)])^D.
\]
\end{Emp}

\begin{Not} \label{N:char}
a) To every pair $a, a'$ of stably conjugate embeddings $T\hra G$, one 
associates the class
$\inv(a',a)\in\Ker\,[H^1(E,T)\to H^1(E,G)]$. This is the class of a 
cocycle
$c_{\sigma}=g^{-1}\sigma(g)$, where $g\in G(\ov{E})$ is such that 
$a'=gag^{-1}$
(compare \cite[4.1]{Ko2}).

b) To each  $\ka\in \wh{T}^{\Gm}/Z(\wh{G})^{\Gm}$, an embedding  
$a_0:T\hra G$, and
a character $\theta$ of $T(E)$ as in \rn{repr}, we associate the 
invariant distribution
\[
\chi_{a_0,\ka,\theta}:=e(G)\sum_{a}\lan \inv(a,a_0),\ka\ran 
\chi(\pi_{a,\theta}).
\]
Here $a$ runs over a set of representatives of conjugacy classes of 
embeddings which are
stably conjugate to $a_0$, and $\chi(\pi_{a,\theta})$ denotes the 
character of $\pi_{a,\theta}$.
\end{Not}

\begin{Not} \label{N:endoscopy}
Each pair $(a,\ka)$, where $a:T\hra G$ is an embedding of a maximal 
torus of
$G$ and $\ka$ is an element of $\wh{T}^{\Gm}$, gives rise to an 
isomorphism class
 $\C{E}_{(a,\ka)}$ of an endoscopic datum of $G$.
Furthermore,  $\C{E}_{(a,\ka)}$ is elliptic if $a(T)$ is an elliptic 
torus of $G$
(see \cite[\S 7]{Ko1} for the definitions of endoscopic data, and 
compare \cite[II, 4]{La}).

More precisely, each embedding $\eta:\wh{T}\hra\wh{G}$, whose conjugacy 
class
corresponds to the stable conjugacy class of $a$, defines an endoscopic 
datum
$\C{E}_{(a,\ka,\eta)}=(s,\rho)$, consisting of a semisimple element
$s=\eta(\ka)$ of $\wh{G}$ and a homomorphism
$\rho:\Gm\overset{\rho_T}{\lra}\Norm_{\Aut\,\wh{G}}(\eta(\wh{T}))_s/%
\eta(\wh{T})\overset{\rho'}{\lra}
\Out(\wh{G}_s^0)$. Here $\rho_T$ is induced by the $E$-structure of 
$T$, and $\rho'$ is induced
by the inclusion $\Norm_{\Aut\,\wh{G}}(\eta(\wh{T}))_s\subset 
\Norm_{\Aut\,\wh{G}}(\wh{G}^0_s)$.
Moreover, the isomorphism class of $\C{E}_{(a,\ka,\eta)}$, denoted by  
$\C{E}_{(a,\ka)}$,
does not depend on $\eta$.
\end{Not}

\begin{Not} \label{N:end}
For each $\gm\in G^{\sr}(E)$ and $\xi\in\wh{G_{\gm}}^{\Gm}$,
\begin{itemize}
\item[(i)] put $\C{E}_{(\gm,\xi)}:=\C{E}_{(a_{\gm},\xi)}$, where
$a_{\gm}:G_{\gm}\hra G$ is the inclusion map;
\item[(ii)] fix an
invariant measure $dg_{\gm}$ on $G_{\gm}(E)$, and put
\[
O_{\gm}(\phi):=\int_{G(E)/G_{\gm}(E)}f(g\gm g^{-1})\frac{dg}{dg_{\gm}}
\]
for each $\phi=fdg\in\C{S}(G(E))$;
\item[(iii)] denote by $\ov{\xi}\in 
\pi_0(\wh{G_{\gm}}^{\Gm}/Z(\wh{G})^{\Gm})$ the class of $\xi$;
\item[(iv)] denote by $O^{\ov{\xi}}_{\gm}\in \C{D}(G(E))$ the sum
$\sum_{\gm'}\lan\inv(\gm',\gm),\ov{\xi}\ran O_{\gm'}$,
taken over a set of representatives of the conjugacy classes stably 
conjugate to $\gm$, where each
$dg_{\gm'}$ is compatible with $dg_{\gm}$.
\end{itemize}
\end{Not}

\begin{Def} \label{D:stab}
Let $\C{E}$ be an endoscopic datum of $G$.
\begin{itemize}
\item[(i)] A measure $\phi\in\C{S}(G(E))$ is called {\em 
$\C{E}$-unstable} if
$O^{\ov{\xi}}_{\gm}(\phi)=0$ for all pairs $(\gm,\xi)$ as in \rn{end} 
for which
$\C{E}_{(\gm,\xi)}$ is isomorphic to $\C{E}$.
\item[(ii)] A distribution $F\in\C{D}(G(E))$ is called {\em 
$\C{E}$-stable} if $F(\phi)=0$ for all
$\C{E}$-unstable $\phi\in\C{S}(G(E))$.
\end{itemize}
\end{Def}



\begin{Thm} \label{T:main}
Assume that $p>\dim\,G^{\der}$. Then for each triple  
$(a_0,\ka,\theta)$, the distribution
$\chi_{a_0,\ka,\theta}$ is $\C{E}_{(a_0,\ka)}$-stable.
\end{Thm}

\begin{Not} \label{N: }
For an endoscopic datum $\C{E}=(s,\rho)$, choose a representative 
$\wt{s}\in\wh{G}^{\ssc}$ of $s$, and
let $Z(\C{E})$ be the set of $z\in Z(\wh{G}^{\ssc})^{\Gm}$ for which 
there exists $g\in\wh{G}_s$ commuting with
$\rho:\Gm\to\Out(\wh{G}^0_s)$ such that $g\wt{s}g^{-1}=z\wt{s}$.
Then $Z(\C{E})$ is a subgroup of $Z(\wh{G}^{\ssc})^{\Gm}$, depending 
only on the isomorphism class of $\C{E}$.
\end{Not}

\begin{Def} \label{D:adm}
Let $\C{E}$ be an endoscopic datum of $G$. An inner twisting 
$\varphi:G\to G'$ is called
{\em $\C{E}$-admissible} if the corresponding class
$\inv(G',G)\in H^1(E,G^{\ad})\cong (Z(\wh{G}^{\ssc})^{\Gm})^D$ is 
orthogonal to
$Z(\C{E})\subset Z(\wh{G}^{\ssc})^{\Gm}$.
\end{Def}

\begin{Def} \label{D:def2}
Let $G$ be a reductive group over $E$, $\C{E}=(s,\rho)$ an elliptic 
endoscopic datum of $G$,
and $\varphi:G\to G'$ an $\C{E}$-admissible inner twisting. Fix a 
triple $(a,a';\ka)$, consisting of
a pair $a:T\hra G$ and  $a':T\hra G'$ of stably conjugate embeddings of 
maximal tori, and an element
$\ka\in\wh{T}^{\Gm}$ such that $\C{E}_{(a,\ka)}\cong\C{E}$.

a) Consider $\phi\in\C{S}(G(E))$ and $\phi'\in\C{S}(G'(E))$. They are 
called
{\em $(a,a';\ka)$-in\-dis\-tin\-guish\-able} if they satisfy the following 
conditions.
\begin{itemize}
\item[(A)]
For every $\gm\in G^{\sr}(E)$ and  $\xi\in
\wh{G_{\gm}}^{\Gm}$
such that $\C{E}_{(\gm,\xi)}\cong\C{E}$ and
$O^{\ov{\xi}}_{\gm}(\phi)\neq 0$,
\begin{itemize}
\item[(i)] there exists $\gm'\in G'(E)$ stably conjugate to $\gm$;
\item[(ii)] we have $O^{\ov{\xi}}_{\gm'}(\phi')=\lan 
\frac{\gm',\gm;\xi}{a',a;\ka}\ran O^{\ov{\xi}}_{\gm}(\phi)$.
\end{itemize}
Here $\lan \frac{\gm',\gm;\xi}{a',a;\ka}\ran\in\B{C}\m$ is the 
invariant $\lan \frac{a_{\gamma'},a_{\gamma};\xi}{a',a;\ka}\ran$
defined in the Appendix for embeddings $a_{\gamma}:G_{\gm}\hra G$ and 
$a_{\gamma'}:G_{\gm}\hra G'$  such that
$a_{\gamma}(\gm)=\gm$ and $a_{\gamma'}(\gm)=\gm'$.
\item[(B)] Condition (A) holds if $G$, $a_0$, $\gm$, $\phi$ are 
interchanged with $G'$, $a'_0$, $\gm'$, $\phi'$.
\end{itemize}

b)  The distributions $F\in\C{D}(G(E))$ and $F'\in\C{D}(G'(E))$ are 
called
{\em $(a,a';\ka)$-equivalent} if $F(\phi)=F'(\phi')$ for every two  
$(a,a';\ka)$-indistinguishable
measures $\phi$ and $\phi'$.
\end{Def}

\begin{Rem}\label{R:end}
If $\phi$ is $\C{E}_{(a_0,\ka)}$-unstable, then $\phi$ and $\phi'\!=0$ are
$(a_0,a'_0;\ka)$-indistin\-guish\-able. Therefore every two 
$(a_0,a'_0;\ka)$-equivalent
distributions $F$ and $F'$ are $\C{E}_{(a_0,\ka)}$-stable.
\end{Rem}

\begin{Main} 
Assume that  $p>\dim\,G^{\der}$.
Let $\varphi:G\to G'$ be an $\C{E}_{(a_0,\ka)}$-admissible inner 
twisting.
Let $a'_0:T\hra G'$ be an embedding which is  stably conjugate to $a_0$.
Then the distributions $\chi_{a_0,\ka,\theta}$ on $G(E)$ and  
$\chi_{a'_0,\ka,\theta}$ on $G'(E)$ are
$(a_0,a'_0;\ka)$-equivalent.
\end{Main}

\begin{Rem}
a) By \rr{end}, \rt{main} follows from the Main Theorem.

b) We believe that a much smaller bound on $p$ would suffice.
\end{Rem}

\section{Basic ingredients of the argument}
\subsection{A generalization of a theorem of Waldspurger} \label{SS:Wa}

Suppose that we are in the situation of \rd{def2}. Then $\varphi$ 
induces an inner twisting
$\C{G}\to \C{G}'$. As in \rd{def2}, one can define 
$(a,a';\ka)$-equivalence of
$F\in\C{D}(\C{G}(E))$ and $F'\in\C{D}(\C{G}'(E))$.

Fix a nontrivial character $\psi: E\to \B{C}\m$, a nondegenerate 
$G$-invariant pairing
$\lan\cdot,\cdot\ran$ on $\C{G}$, and $\varphi$-compatible invariant 
measures on $\C{G}(E)$ and
$\C{G}'(E)$.
Then $\varphi$ defines a  nondegenerate $G'$-invariant pairing 
$\lan\cdot,\cdot\ran'$
on $\C{G}'$. These data determine the Fourier transforms $F\mapsto \C{F}(F)$ 
on $\C{G}(E)$ and $\C{G}'(E)$.


\begin{Thm} \label{T:Wa}
The distributions $F\in\C{D}(\C{G}(E))$ and $F'\in\C{D}(\C{G}'(E))$ are 
 $(a,a';\ka)$-equivalent
if and only if $e(G)\C{F}(F)$ and $e(G')\C{F}(F')$ are  
$(a,a';\ka)$-equivalent.
\end{Thm}

The proof is a generalization of that of Waldspurger \cite{Wa},
who treated the case $\phi'=0$ (compare also \cite[Thm. 2.7.1]{KP}, 
where the stable case is considered).

\subsection{Springer hypothesis}

In the notation of \re{finite}, assume that  
$\ov{a}(\ov{\C{T}})(\fq)\subset\C{L}(\fq)$
contains an $L$-regular element $\ov{t}$ [and that $p$ is so large that
the logarithm defines an isomorphism $\log:L_{\un}\isom \C{L}_{\nil}$ 
between unipotent elements
of $L$ and nilpotent elements of $\C{L}$].
Let $\dt_{\ov{t}}$ be the characteristic function of the 
$\Ad(L(\fq))$-orbit of $\ov{t}$, and let
$\C{F}(\dt_{\ov{t}})$ be its Fourier transform.
We need the following result of \cite{Ka1}.

\begin{Thm} \label{T:Spr}
For every $u\in L_{\un}(\fq)$, we have
\[
\Tr\,\rho_{\ov{a},\ov{\theta}}(u)=q^{-(\dim\,L-\dim\,\ov{T})/2}\C{F}(%
\dt_{\ov{t}})(\log(u)).
\]
\end{Thm}

\subsection{Topological Jordan decomposition}
We will call an element $\gm\in G(E)$ {\em compact} if it generates a 
relatively compact subgroup of $G(E)$.
We will call an element $\gm\in G(E)$ {\em topologically unipotent} if 
the sequence $\{\gm^{p^n}\}_n$
converges to $1$.
Every topologically unipotent element is compact.
The following result is a rather straightforward generalization of 
\cite[Lem. 2, p. 226]{Ka2}.
\begin{Lem} \label{L:jor}
For every compact element $\gm\in G(E)$ there exists a unique 
decomposition $\gm=\dt u$ such that
$\dt$ and $u$ commute, $\dt$ is of finite order prime to $p$, and $u$ 
is topologically unipotent.
In particular, this decomposition is compatible with conjugation and 
field extensions.
\end{Lem}

\section{A sketch of the proof of the main theorem}

\subsection{Reformulation of the problem}

\begin{Not}
To each $a:T\hra G$ and $\theta:T(E)\to\B{C}\m$ as in \rn{repr} we 
associate a function $t_{a,\theta}$ on $G(E)$
supported on  $Z(G)(E) G_a$ and equal to $\Tr\,\rho_{a,\theta}$ there.
Since $t_{a,\theta}$ is cuspidal, the integral
\[
F_{a,\theta}(\gm):=\frac{1}{\mu((G^{\ad})_a)}
\int_{G(E)/Z(G)(E)}t_{a,\theta}(g\gm g^{-1})dg
\]
stabilizes for every $\gm\in G^{\sr}(E)$ (see \cite[Lem. 23]{HC}), thus 
providing us with a locally constant
invariant function $F_{a,\theta}$ on $G^{\sr}(E)$.
\end{Not}

\begin{Lem} \label{L:HCH}
For each $a$ and $\theta$, $F_{a,\theta}$ is a locally $L^1$-function 
on $G(E)$.
Moreover, the corresponding distribution equals $\chi({\pi_{a,\theta}})$.
\end{Lem}
\begin{proof}
The assertion follows from Harish-Chandra's theorem \cite[Thm. 16]{HC}.
\end{proof}

\begin{Not} \label{N:sigma}
For every $\gm_0\in G^{\sr}$ and
$\ov{\xi}\in\pi_0(\wh{G_{\gm_0}}^{\Gm}/Z(\wh{G})^{\Gm})$ we define
\begin{equation} \label{E:redst}
\Sigma_{G;\gm_0,\ov{\xi};a_0,\ka}:=e(G)\sum_{a}\sum_{\gm}
\lan \inv(a,a_0),\ka\ran \lan \inv(\gm,\gm_0),\ov{\xi}\ran 
^{-1}F_{a,\theta}(\gm),
\end{equation}
where $a$ and $\gm$ run over sets of representatives of the conjugacy 
classes within the
stable conjugacy classes of $a_0$ and $\gm_0$, respectively.
\end{Not}

\begin{Thm} \label{T:main'}
For all $\gm_0\in G^{\sr}$ and 
$\ov{\xi}\in\pi_0(\wh{G_{\gm_0}}^{\Gm}/Z(\wh{G})^{\Gm})$
such that $\Sigma_{G;\gm_0,\ov{\xi};a_0,\ka}\!\neq 0$,
\begin{itemize}
\item[$(i)$] there exists a representative $\xi\in 
\wh{G_{\gm_0}}^{\Gm}$ of $\ov{\xi}$ such that
$\C{E}_{\gm_0,\xi}\cong\C{E}_{(a_0,\ka)};$
\item[$(ii)$] if $\varphi:G\to G'$ is
$(\C{E},a_{\gm_0},\ov{\xi})$-admissible (see \rd{app}),
then for every $\xi$ as in  $(i)$ and every stably conjugate $\gm'_0\in
G'(E)$ of $\gm_0$ we have
\[\Sigma_{G';\gm'_0,\ov{\xi};a'_0,\ka}= \lan
\frac{\gm'_0,\gm_0;\xi}{a',a;\ka}\ran\Sigma_{G;\gm_0,\ov{\xi};a_0,\ka}.\]
\end{itemize}
\end{Thm}

\begin{Emp}
It follows from \rl{HCH} that \rt{main'} is equivalent to the Main Theorem.
Moreover, by standard arguments, \rt{main'} reduces to the case when 
the derived group of
$G$ is simply connected.
\end{Emp}

\begin{Emp} \label{E:red}
From now on we will assume that $G^{\der}=G^{\ssc}$.
In particular, the centralizer of each semisimple element of $G$ is
connected, and each $G_a$ is a maximal compact subgroup of $G(E)$.
We fix $(\gm_0,\ov{\xi})$ such that 
$\Sigma_{G;\gm_0,\ov{\xi};a_0,\ka}\neq 0$.
Since $\Sigma_{G;z\gm_0,\ov{\xi};a_0,\ka}=\theta(z)\Sigma_{G;\gm_0,\ov{%
\xi};a_0,\ka}$
for each  $z\in Z(G)(E)$ and since the support of each $t_{a,\theta}$ 
consists of elements compact modulo center,
we can assume that $\gm_0$ is compact with topological Jordan 
decomposition  $\gm_0=\dt_0 u_0$.
Moreover, we can assume that either $\gm_0$ is topologically unipotent, 
or $\dt_0\notin Z(G)(E)$.
\end{Emp}

\subsection{The topologically unipotent case} \label{SS:topun}
\begin{Emp}
Since $p$ does not divide the order of $Z(G^{\der})$, the canonical map
$G^{\der}(E)_{\tu}\times Z(G)(E)_{\tu}\to G(E)_{\tu}$ is an isomorphism.
Therefore to prove \rt{main'} for topologically unipotent $\gm_0$, we 
can assume that
$G$ is semisimple and simply connected.
\end{Emp}

\begin{Not}
Denote by $\Phi_{G}:G\to\C{G}$ the composition map
\[
G\overset{\Ad}{\lra} GL(\C{G})\overset{\log_{(p)}}{\lra}\End(\C{G})%
\overset{\pr}{\lra}\C{G},
\]
where $\log_{(p)}(1-A)=-\sum_{i=1}^{p-1} \frac{A^i}{i}$,
and $\pr$ is the canonical projection, defined by the standard pairing
$(A,B)\mapsto\Tr AB$ on 
$\End(\C{G})$.
\end{Not}

\begin{Lem} \label{L:bij}
The map $\Phi_G$ defines a $G(E)$-equivariant homeomorphism
\[G(E)_{\tu}\isom \C{G}(E)_{\tn}\,,\]
where $G(E)$ acts by conjugation.
Moreover, for every parahoric subgroup $G_x$ of $G(E)$,
$\Phi_G$ induces a bijection $(\Phi_G)_x:(G_x)_{\tu}\isom 
(\C{G}_x)_{\tn}$, which in turn
induces the logarithm map $\log:(L_x)_{\un}(\fq)\isom 
(\C{L}_x)_{\nil}(\fq)$.
\end{Lem}

\begin{Not} 
a) By our assumption on $p$, there exists $t\in\C{T}(\C{O})$ whose 
reduction
$\ov{t}\in \ov{\C{T}}(\fq)$ is not fixed by any nontrivial element of the
Weyl group of $G$.

b) For every $a:T\hra G$ as in \rn{repr}, we denote by 
$\Om_{a,t}\subset \C{L}_a(\fq)$
the $\Ad(L_a(\fq))$-orbit of $\ov{a}(\ov{t})$, by 
$\wt{\Om}_{a,t}\subset \C{G}_{a}\subset \C{G}$
the preimage of $\Om_{a,t}$, and let $\dt_{a,t}$ be the characteristic 
function of $\wt{\Om}_{a,t}$.

c) As  the centralizer $G_y$ of each $y\in\wt{\Om}_{a,t}$ is 
$G_a$-conjugate to $a(T)$, the integral
\[
\Dt_{a,t}(x):=\frac{1}{\mu(G_a)}\int_{G(E)}\dt_{a,t}(\Ad(g)x) dg
\]
converges absolutely for each $x\in\C{G}(E)$. Thus it defines an 
element of $\C{D}(\C{G}(E))$.
Similarly to \rn{char}, we consider
${\Dt}_{a_0,\ka,t}:=e(G)\sum_{a}\lan \inv(a,a_0),\ka\ran \Dt_{a,t}\in 
\C{D}(\C{G}(E))$.
\end{Not}

\begin{Lem} \label{L:four}
Let $\C{I}^+\subset\C{G}(E)$ be a maximal topologically nilpotent 
subalgebra.
Assume that $\psi:E\to\B{C}\m$ is trivial on the maximal ideal 
$M\subset\C{O}$ and
induces a nontrivial character of $\fq$. Then for each $u\in 
G(E)_{\tu}$ we have
\[t_{a,\theta}(u)=\mu(\C{I}^+)^{-1} \C{F}(\dt_{a,t})(\Phi_G(u)).\]
\end{Lem}
\begin{proof}
The assumption on $\psi$ implies that $\C{G}_{a^+}$ is the orthogonal 
complement of
$\C{G}_{a}$ with respect to the pairing $(x,y)\mapsto\psi(\lan 
x,y\ran)$. Therefore our lemma
is an immediate consequence of the definition of the Fourier transform 
(over $E$ and $\fq$),
 \rt{Spr}, \rl{bij}, and the equality 
$q^{(\dim\,L_a-\dim\,\ov{T})/2}\mu(\C{G}_{a^+})=\mu(\C{I}^+)$.
\end{proof}

\begin{Emp}
Now we are ready to show that $(\chi_{a_0,\ka,\theta})_{|G(E)_{\tu}}\!$ 
and
$(\chi_{a'_0,\ka,\theta})_{|G'(E)_{\tu}}$ are \linebreak
$(a_0,a'_0;\ka)$-equivalent. First of all, by
direct calculations, $e(G){\Dt}_{a_0,\ka,t}$ is 
$(a_0,a'_0;\ka)$-equivalent to $e(G'){\Dt}_{a'_0,\ka,t}$.
Hence, by \rt{Wa}, 
$\C{F}({\Dt}_{a_0,\ka,t})$ is $(a_0,a'_0;\ka)$-equiv\-a\-lent 
to
$\C{F}({\Dt}_{a'_0,\ka,t})$. Using Lemmas \ref{L:HCH}, \ref{L:bij}, and 
\ref{L:four} we see that
$\chi_{a_0,\ka,\theta}$ has the same restriction to $G(E)_{\tu}$ as
$\mu(\C{I}^+)^{-1}\Phi_G^*(\C{F}({\Dt}_{a_0,\ka,t}))$, and similarly 
for $\chi_{a'_0,\ka,\theta}$ and
$\mu(\C{I'}^+)^{-1}\Phi_{G'}^*(\C{F}({\Dt}_{a'_0,\ka,t}))$.
Since $\Phi_G$ is a $G^{\ad}$-invariant algebraic morphism defined over
$E$, the assertion follows from the equality $\mu(\C{I}^+)=\mu(\C{I'}^+
)$.
\end{Emp}

\subsection{The general case}

It remains to prove \rt{main'} for $\dt_0\notin Z(G)(E)$ (see \re{red}).
We are going to deduce the assertion from that for $G_{\dt_0}$.

\begin{Prop} \label{P:reduction}
For every embedding $a:T\hra G$ and a compact element $\gm$ in $G(E)$ 
with topological Jordan decomposition
$\gm=\dt u$, we have
\[
e(G)F_{a,\theta}(\gm)=e(G_{\dt})\sum_{b}\theta(b^{-1}({\dt}))F_{b,%
\theta}(u).
\]
Here $b$ runs over the set of conjugacy classes of embeddings $b:T\hra 
G_{\dt}$
whose composition with the inclusion $G_{\dt}\subset G$ is conjugate to 
$a$.
\end{Prop}
\begin{proof}
The proposition follows by direct calculation from the recursive formula
(\cite[Thm. 4.2]{DL}) for characters of Deligne-Lusztig representations.
\end{proof}

\begin{Not}
a) We say that $t\in T(E)$ is {\em $(G,a_0,\gm_0)$-relevant} if there 
exists an embedding
$b_0:T\hra  G_{\dt_0}\subset G$ stably conjugate to $a_0$ such that 
$b(t)=\dt_0$.

b) Assume that  $t\in T(E)$ is $(G,a_0,\gm_0)$-relevant.
Since $a_0(T)\subset G$ is elliptic, for each $\dt\in G(E)$ stably 
conjugate to $\dt_0$
there exists an embedding $b_{t,\dt}:T\hra G_{\dt}\subset G$ stably 
conjugate to $a_0$ such that
$b_{t,\dt}(t)=\dt$. Further, $b_{t,\dt}$ is unique up to stable 
conjugacy, and
the endoscopic datum $\C{E}_{t,\ka}:=\C{E}_{(b_{{\dt},t},\ka)}$ of 
$G_{\dt_0}$ is independent of $\dt$.

c) We will write $\dt_1\sim_{\C{E}_{t,\ka}}\dt$ (resp. 
$\dt'\sim_{\C{E}_{t,\ka}}\dt$) if
$\dt,\dt_1\in G(E)$ (resp. $\dt\in G(E)$ and $\dt'\in G'(E)$) are 
stably conjugate to $\dt_0$,
and $G_{\dt_1}$ (resp.  $G'_{\dt'}$) is an  $\C{E}_{t,\ka}$-admissible 
inner form of $G_{\dt}$
 (see \rd{adm}).
\end{Not}

\begin{Emp}
Using \rp{reduction}, we see that
\begin{equation} \label{E:for2}
\Sigma_{G;\gm_0,\ov{\xi};a_0,\ka}=\sum_{t}\theta(t)\sum_{{\dt}}I_{t,\dt},
\end{equation}
where
\begin{itemize}
\item[(i)] $t$ runs over the set of $(G,a_0,\gm_0)$-relevant elements 
of $T(E)$;
\item[(ii)] ${\dt}$ runs over a set of representatives of the conjugacy 
classes within the stable conjugacy class
of ${\dt}_0$;
\item[(iii)] $I_{t,\dt}$ vanishes unless there exists an element  
$\gm\in G(E)$ stably conjugate to $\gm_0$
with topological Jordan decomposition $\gm={\dt}u$, in which case we get
\begin{equation} \label{E:for4}
I_{t,\dt}=\lan \inv(b_{t,\dt},a_0),\ka\ran \lan 
\inv(\gm,\gm_0),\ov{\xi}\ran ^{-1}
\Si_{G_{\dt};u,\ov{\xi};b_{t,\dt},\ka}.
\end{equation}
\end{itemize}
\end{Emp}

\begin{Emp}
For simplicity of the exposition, we will restrict
     ourselves to the case when $\gamma_0\in G(E)$ is elliptic.
Choose $t$ which has a nonzero contribution to (\ref{E:for2}).
Replacing $\dt_0$ by a stably conjugate element we can assume that
$\sum_{\dt\sim_{\C{E}_{t,\ka}}\dt_0}I_{t,\dt}\neq 0$ and 
$I_{t,\dt_0}\neq 0$.
So $\Si_{G_{\dt_0};u_0,\ov{\xi};b_{t,\dt},\ka}\neq 0$.
Hence by \rt{main'} for $G_{\dt_0}$ there exists a representative 
$\xi\in \wh{G_{\gm_0}}^{\Gm}$ of
$\ov{\xi}$ such that the endoscopic datum $\C{E}_{(u_0,\xi)}$ of 
$G_{\dt_0}$ is isomorphic to $\C{E}_{t,\ka}$.
Therefore there exist embeddings 
$\eta_1:\wh{G_{\gm_0}}\hra\wh{G_{\dt_0}}$ and
$\eta_2:\wh{T}\hra\wh{G_{\dt_0}}$ such that
$\C{E}_{(\gm_0,\xi,\eta_1)}=\C{E}_{(b_{t,\dt_0},\ka,\eta_2)}$ (compare 
\rn{endoscopy}) and
$\eta_2(\ka)=z\eta_1(\xi)$ for a certain $z\in Z(\wh{G_{\dt_0}})^{\Gm}$.
Moreover, $z$ is defined up to multiplication by an element of 
$Z(\C{E}_{t,\ka})$. Therefore
for all $\dt\sim_{\C{E}_{t,\ka}}\dt_0$, the expression $\lan 
\inv(\dt,\dt_0),z\ran$ is
independent of the choice of the $\eta_i$'s.
\end{Emp}

\begin{Cl} \label{C:eq}
For each $\dt\sim_{\C{E}_{t,\ka}}\dt_0$ we have
$I_{t,\dt}=\lan \inv(\dt,\dt_0),z\ran I_{t,\dt_0}$.
\end{Cl}
\begin{proof}
Since $\Si_{G_{\dt_0};u_0,\ov{\xi};b_{t,\dt},\ka}\neq 0$, \rt{main'} 
for inner forms $G_{\dt}$ and $G_{\dt_0}$
implies that for every stably conjugate $u\in G_{\dt}(E)$ of 
$u_0\in G_{\dt_0}(E)$,  we have
\[
\Sigma_{G_{\dt};u,\ov{\xi};b_{t,\dt},\ka}=\lan
\frac{u,u_0;\xi}{b_{t,\dt} ,b_{t,\dt_0};\ka}\ran
\Sigma_{G_{\dt_0};u_0,\ov{\xi};b_{t,\dt_0},\ka}\,.
\]
Then $\gm:=\dt u\in G(E)$ is stably conjugate to $\gm_0$, and the 
assertion follows by
direct calculation from (\ref{E:for4}).
\end{proof}

\begin{Emp}
Now we are ready to show the validity of (i), (ii) of \rt{main'}.

\smallskip

(i) As $\sum_{\dt\sim_{\C{E}_{t,\ka}}\dt_0}I_{t,\dt}\neq 0$, we 
get from \rcl{eq} that
$\sum_{\dt\sim_{\C{E}_{t,\ka}}\dt_0}\lan 
\inv(\dt,\dt_0),z\ran\neq 0$. By the definition of
$\C{E}_{t,\ka}$-equivalence, this implies that $z$ belongs to 
$Z(\C{E}_{t,\ka})Z(\wh{G})^{\Gm}$.
Thus changing $\eta_1$ (or $\eta_2$), we can assume that $z\in 
Z(\wh{G})^{\Gm}$.
Since  $(\gm_0,\xi)$ and $(b_{t,\dt_0},\ka)$ define isomorphic 
endoscopic data of $G_{\dt_0}$,
we therefore conclude that $\C{E}_{(\gm_0,\xi)}\cong\C{E}_{(a_0,\ka)}$.

\smallskip

(ii) Since $T$ is elliptic, an element $t\in T(E)$ is 
$(G,a_0,\gm_0)$-relevant if and only if it is
 $(G',a'_0,\gm'_0)$-relevant. Thus it will suffice to show that for 
every such $t$ we have
$\sum_{\dt'}I_{t,\dt'}=\lan\frac{\gm'_0,\gm_0;\xi}{a'_0,a_0;\ka}\ran 
\sum_{\dt}I_{t,\dt}$.
For every stably conjugate $\dt\in G(E)$ of $\dt_0$, there exists
a stably conjugate $\dt'\in G'(E)$ of $\dt'_0$ such that  
$\dt'\sim_{\C{E}_{t,\ka}}\dt$.
Therefore it will suffice to show that for every such pair 
$\dt'\sim_{\C{E}_{t,\ka}}\dt$, we have
$I_{t,\dt'}=\lan\frac{\gm'_0,\gm_0;\xi}{a'_0,a_0;\ka}\ran I_{t,\dt}$.
The latter equality can be proved by the same arguments as \rcl{eq}.
\end{Emp}

\appendix
\section{}

Let $G$ be a reductive group over $E$, $\C{E}=(s,\rho)$ an elliptic 
endoscopic datum of $G$,
and $\varphi:G\to G'$ an $\C{E}$-admissible inner twisting. For every 
two triples
$(a'_i,a_i;\ka_i)$, $i=1,2$, where $a_i:T_i\hra G$ and  $a'_i:T_i\hra G'$
are stably conjugate embeddings of maximal tori, and $\ka_i$ is an 
element of $\wh{T_i}^{\Gm}$
such that $\C{E}_{(a_i,\ka_i)}$ is isomorphic to $\C{E}$, we are going 
to define an invariant
$\lan \frac{a'_1,a_1;\ka_1}{a'_2,a_2;\ka_2}\ran\in\B{C}\m$.

\begin{step}
Replacing $G$, $G'$, $T_i$, $\ka_i$, and $\C{E}$ by 
$G^{\ssc}$, $G'^{\ssc}$,
$T^{\ssc}_i:=a_i^{-1}(G^{\ssc})=a'^{-1}_i(G'^{\ssc})$, the image of 
$\ka_i$ in
$\wh{T^{\ssc}_i}^{\Gm}$, and the corresponding endoscopic datum of
$G^{\ssc}$, respectively, we can assume that $G$ is semisimple and 
simply connected.
Let $T_{1,2}$ be the quotient of the product
$T_1\times T_2$ by the subgroup $\{(z, z^{-1})| z\in Z(G)=Z(G')\}$.
\end{step}

\begin{step} Choose elements $g_1,g_2$, and 
$\{\wt{c}_{\si}\}_{\si\in\Gm}$ of $G(\ov{E})$ such that
$a'_i=\varphi(g_i a_i g_i^{-1})$ and each $\wt{c}_{\si}$ is a 
representative of
$\varphi^{-1}{}^{\si}\varphi\in G^{\ad}(\ov{E})$. Then  each
$g_i^{-1}\wt{c}_{\si}{}^{\si}g_i\in G(\ov{E})$ belongs to 
$a_i(T_i(\ov{E}))$, and the images of
$(a_1^{-1}(g_1^{-1}\wt{c}_{\si}{}^{\si}g_1),a_2^{-1}((g_2^{-1}\wt{c}_{%
\si}{}^{\si}g_2)^{-1}))$ in
$T_{1,2}(\ov{E})$ form a cocycle, whose cohomology class
$\inv(\frac{a'_1,a_1}{a'_2,a_2})\in  H^1(E,T_{1,2})$ is independent of 
the choices.
\end{step}

\begin{step}
Choose embeddings $\eta_i:\wh{T_i}\hra\wh{G}$ such that
$\C{E}_{(a_i,\ka_i,\eta_i)}=(s,\rho)$ and a representative 
$\wt{s}\in\wh{G}^{\ssc}=\wh{G^{\ad}}$ of $s$.
Put $T^{\ad}_i:=T_i/a_i^{-1}(Z(G))$. Each $\eta_i$ defines
an embedding $\wt{\eta}_i:\wh{T^{\ad}_i}\hra\wh{G^{\ad}}$, hence an
element $\wt{\ka}_i=\kappa(\widetilde{s},\eta_i):=\wt{\eta}_i^{-1}(\wt{s})\in\wh{T^{\ad}_i}$.
Then the image of $(\wt{\ka}_1,\wt{\ka}_2)$ in
$\wh{T^{\ad}_1}\times \wh{T^{\ad}_2}/Z(\wh{G^{\ad}})\cong\wh{T_{1,2}}$, 
denoted by $\ka_{1,2}$,
is $\Gm$-invariant. Moreover, as $\varphi:G\to G'$ is 
$\C{E}$-admissible, the expression
$\lan \frac{a'_1,a_1;\ka_1}{a'_2,a_2;\ka_2}\ran:=\lan\inv(%
\frac{a'_1,a_1}{a'_2,a_2}),\ka_{1,2}\ran\in\B{C}\m$
is independent of the choices.
\end{step}

\begin{Def} \label{D:app}
Let $\C{E}=(s,\rho)$ be an endoscopic datum of $G$, $\varphi:G\to
G'$ an inner twisting, $a:T\hra G$ an embedding of a maximal
torus, and $\ka$ an element of $\wh{T}^{\Gm}$ such that
$\C{E}_{(a,\ka)}\cong\C{E}$. We say that $\varphi:G\to G'$ is {\em
$(\C{E},a,\ov{\ka})$-admissible}, if for all representatives
$\ka'\in\wh{T}^{\Gm}$ of
$\ov{\ka}\in\pi_0(\wh{T}^{\Gm}/Z(\wh{G})^{\Gm})$ satisfying
$\C{E}_{(a,\ka')}\cong\C{E}$, all embeddings
$\eta,\eta':\wt{T}\hra\wh{G}$ such that
$\C{E}_{(a,\ka,\eta)}=\C{E}_{(a,\ka',\eta')}=(s,\rho)$, and all
representatives
 $\wt{s}\in\wh{G^{\ad}}$ of $s$,
the difference $\ka(\wt{s},\eta')-\ka(\wt{s},\eta)\in
Z(\wh{G^{\ad}})^{\Gm}$ is orthogonal to $\inv(G',G)\in
H^1(E,G^{\ad})$.
\end{Def}

\begin{Rem} \label{R:app}
a) Every $(\C{E},a,\ov{\ka})$-admissible inner twisting is
$\C{E}$-admissible.

b) If $a(T)\subset G$ is elliptic, then every $\C{E}$-admissible
inner twisting  is $(\C{E},a,\ov{\ka})$-admissible.

c) $\varphi:G\to G'$ is $(\C{E},a_1,\ov{\ka}_1)$-admissible if and
only if $\lan \frac{a'_1,a_1;\ka_1}{a'_2,a_2;\ka_2}\ran=
\lan\frac{a'_1,a_1;\ka'_1}{a'_2,a_2;\ka_2}\ran$  for all
representatives $\ka'_1\in\wh{T}_1^{\Gm}$ of
$\ov{\ka}_1\in\pi_0(\wh{T}_1^{\Gm}/Z(\wh{G})^{\Gm})$ satisfying
$\C{E}_{(a_1,\ka'_1)}\cong\C{E}$.
\end{Rem}

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\end{document}