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\newcommand{\fw}{{\mathfrak w}} \newcommand{\fx}{{\mathfrak x}} \newcommand{\fy}{{\mathfrak y}} \newcommand{\fz}{{\mathfrak z}} \newcommand{\ti}{\tilde} \newcommand{\va}{\vartriangleleft} \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} \begin{thebibliography}{Ka3} \bibitem[De]{De} P. 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