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\begin{document}

\title[Generalized \& degenerate Whittaker
models]{{\Large Generalized and degenerate  Whittaker
models\\  for  representations of reductive groups\\  }}
\author[Dmitry Gourevitch]{Dmitry Gourevitch  \\
Weizmann Institute of Science, Israel
\url{http://www.wisdom.weizmann.ac.il/~dimagur}\\
%EndAName
$\,$\\
Workshop: Automorphic forms, mock modular forms and string theory, SCGP, Stony Brooke, NY}
%\url{http://www.wisdom.weizmann.ac.il/~dimagur}
\date{September 2016}
\maketitle

%\section{Degenerate Whittaker functionals for real reductive groups}

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% \frametitle{Whittaker models}
%
% \begin{stepitemize}
% \item $\mathbb{F}$: local field of char.0, $G$: reductive group over $%
% \mathbb{F}$.
%
% \item $\mathcal{M}(G)$: smooth admissible f.g. representations (of moderate
% growth).
%
% \item Just for this slide assume $G$ is quasisplit and  fix a Borel $B=HU$,\\
% \item $\Psi^{\times}:=$ non-degenerate unitary  characters of $U$.
%
% %$\mathfrak{n}:=Lie_{\mathbb{F}}(N)$.
%
% % \begin{equation*}
% % \text{ Define\quad }\mathfrak{n}^{\prime }=\left[ \mathfrak{n},\mathfrak{n}%
% % \right] ,\mathfrak{v=n/n}^{\prime },\Psi =\mathfrak{v}^{\ast }\subset
% % \mathfrak{n}^{\ast }
% % \end{equation*}
%
%
%
% \item For $\psi \in \Psi^{\times}$  define $\cW_{\psi}:= \ind_{U}^G(\psi)$ .
% \item For $\pi \in \mathcal{M}(G)$  define
% $$\cW_{\psi}(\pi):= \Hom_G(\cW_{\psi}, \pi^*)=\Hom_U(\pi,\psi).$$
%
%
% %\item (Casselman) For any $\psi \in \Psi^{\times}, \quad \pi \mapsto
% %Wh_{\psi }^{\ast }(\pi )$ is an exact functor.
% \end{stepitemize}
%
% \pause
% \begin{theorem}[Jacquet,Gelfand-Kazhdan, Shalika, Rodier, Kostant, Vogan, Casselman-Hecht-Milicic, 70s-90s]
% \begin{stepitemize}
% \item For any $\psi \in \Psi^{\times}, \quad \pi \mapsto
% \cW_{\psi}(\pi)$ is an exact functor.
%
% \item For any $\pi \in Irr(G), \, \psi \in \Psi^{\times}$ we have $\dim \cW_{\psi}(\pi) \leq 1$.
%
% \item For any $\pi \in \cM(G),$
%   $$(\exists \psi \in \Psi^{\times} \text{ with }\cW_{\psi}(\pi)\neq 0)\, \Leftrightarrow\, \pi \text{ is large}.$$
%
% \end{stepitemize}
% \end{theorem}
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% \frametitle{Wave front set and wave-front cycle}
%
%
% \begin{theorem}[Howe, Harish-Chandra, Barbasch-Vogan, 70s]
% Near $e \in G$, the character distribution (asymptotically) equals to a
% linear combination of Fourier transforms of Haar measures of nilpotent
% coadjoint orbits.
% \begin{equation*}
% \exp^*(\chi_{\pi}) \approx \sum c_{{\mathcal{O}}} {\mathcal{F}}(\mu_{{\mathcal{O}}})
% \end{equation*}
% \end{theorem}
% \pause
% \begin{stepitemize}
% \item Let $\mathcal{N}\subset
% \mathfrak{g}^{\ast }$ denote the nilpotent cone.
% \item Define $\WF(\pi)=\cup \{\overline{{\mathcal{O}}} \, \vert \, c_{{%
% \mathcal{O}}} \neq 0\} \subset \mathcal{N}$.
%
% \item $\pi$ is called \emph{large} if $\dim\WF(\pi)=\dim\mathcal{N}$.
% \end{stepitemize}
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\frametitle{Degenerate Whittaker space}

\begin{stepitemize}
\item $\mathbb{F}$: local field of char.0, $G$: reductive group over $%
\mathbb{F}$, $\fg:=Lie(G)$.

\item $\mathcal{M}(G)$: smooth (Frechet, moderate
growth) representations.
\item Fix a semisimple $H\in \g$, and let $\g_i$ denote the eigenspaces of $ad(H)$.\\
Assume that all the eigenvalues $i$ lie in $\mathbb{Q}$.
\item Let $\varphi \in \g_{-2}^*$ and let $\fl\subset \g_1$ be a maximal isotropic subspace w.r.t
$\omega(X,Y):=\varphi([X,Y]).$

\item Define $\fv:=\bigoplus_{i> 1}\g_i, \, \n:=\fv \oplus \fl, \, N:=\Exp(\fn),$
\item  Fix a non-trivial unitary additive  character $\psi:\F\to \C$ and define $\chi_{\varphi}:N\to \C$ by $\chi_{\varphi}(\Exp X):=\psi(\varphi(X)).$
\item For $\pi\in \cM(G)$ let
$ \cW_{H,\varphi}(\pi)=\Hom_N(\pi,\chi_\varphi).$
\item We call $(H,\varphi)$ a \emph{Whittaker pair}, and $\cW_{H,\varphi}(\pi)$ a \emph{degenerate Whittaker space}. We call  them \emph{generalized} or \emph{neutral}  if $(H,\varphi)$ can be completed to an $\mathfrak{sl}_2$-triple.
The generalized Whittaker space depends only on the coadjoint orbit  of  $\varphi$.
\item $\mathrm{WS}(\pi):=$maximal orbits with $\cW_{\cO}(\pi)\neq 0$ (w.r.t. the closure order)

%  \begin{theorem}[Moeglin-Waldspurger, 87']
% Assume that $\F$ is non-archimedean, and $H$ is tangent to a 1-parameter subgroup. Let $\pi\in \cM(G)$. Then
% \begin{stepitemize}
% \item If $\cW_{H,\varphi}(\pi)\neq 0$ then $\varphi \in \WF(\pi).$
% \item For any $(H,\varphi)$ with $G\varphi$ open in $\WF(\pi),$ $$\dim %\cW_{H,\varphi}(\pi)=c_{\varphi}.$$
% \end{stepitemize}
%\end{theorem}

\end{stepitemize}
%\pause


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\frametitle{Examples}
 Some examples for $G=\GL_4(\F)$:
\begin{stepitemize}
\item
$$
\left(
   \begin{array}{cccc}
     1 & 0 & 0 & 0 \\
     0 & -1 & 0 & 0 \\
     0 & 0 & 1 & 0 \\
     0 & 0 & 0 & -1 \\
   \end{array}
 \right), \left(
   \begin{array}{cccc}
     0 & 0 & 0 & 0 \\
     1 & 0 & 0 & 0 \\
     0 & 0 & 0 & 0 \\
     0 & 0 & 1 & 0 \\
   \end{array}
 \right)  \fn=\left(
   \begin{array}{cccc}
     0 & \underline{*} & 0 & * \\
     0 & 0 & 0 & 0 \\
     0 & * & 0 & \underline{*} \\
     0 & 0 & 0 & 0 \\
   \end{array}
 \right)$$

 \item
$$
\left(
   \begin{array}{cccc}
     3 & 0 & 0 & 0 \\
     0 & 1 & 0 & 0 \\
     0 & 0 & -1 & 0 \\
     0 & 0 & 0 & -3 \\
   \end{array}
 \right),
\left(
   \begin{array}{cccc}
     0 & 0 & 0 & 0 \\
     1 & 0 & 0 & 0 \\
     0 & 0 & 0 & 0 \\
     0 & 0 & 1 & 0 \\
   \end{array}
 \right)
 \fn=\left(
   \begin{array}{cccc}
     0 & \underline{*} & * & * \\
     0 & 0 & * & * \\
     0 & 0 & 0 & \underline{*} \\
     0 & 0 & 0 & 0 \\
   \end{array}
 \right)$$

In both cases $\fn=\fv.$
\end{stepitemize}
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% \frametitle{Some results in the real case}
%
% In this slide $\mathbb{F}=\mathbb{R.}$
%
% \begin{theorem}[Matumoto 87',92']
% \begin{itemize}
% \item If $\cW_{H,\varphi}(\pi)\neq 0$ then $\varphi \in G_{\C}\cdot\WF(\pi).$
% \item If $\varphi$ is regular nilpotent then $\cW_{H,\varphi}(\pi)\neq 0 \Leftrightarrow \varphi \in \WF(\pi). $
% \end{itemize}
% \end{theorem}
%
%
%
% \pause
% \begin{theorem}[G.-Sahi, 2013]
% \label{thm:OurMat} Let $G$ be quasi-split and $(H,\varphi)$ be \textbf{principal} degenerate Whittaker pair.
% $$\cW_{H,\varphi}(\pi)\neq 0 \Rightarrow \varphi \in\WF(\pi)\Rightarrow \exists g\in F_G \text{ s.t. }  \cW_{H,g\cdot\varphi}(\pi)\neq 0, $$
% where $F_G=\{1\}$ if $G=GL_{n}\left( {\mathbb{R}}\right)$ or if $G$ is a complex
% group,
%
%  and  $F_G=$ finite abelian group $Norm_{G_{\mathbb{C}}}\left( G\right) /\left(
% Z_{G_{\mathbb{C}}}\cdot G\right) $ otherwise.
%
% \pause
% Moreover, if $G$ is complex classical and $\pi \in Irr(G)$ then the set of  principal degenerate Whittaker pairs $(H,\varphi)$ with $\cW_{H,\varphi}(\pi)\neq 0$ determines $\WF(\pi)$.
% \end{theorem}
%
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%\frametitle{Our uniform results}

\begin{theorem}[Gomez-G.-Sahi, 2016 following Moeglin-Waldspurger]
Let $(H,\varphi)$ be a Whittaker pair, and $\pi\in\cM(G)$.
\begin{stepitemize}
\item  We have a natural functorial embedding $\nu:\cW_{H,\Phi}(\pi) \hookrightarrow\cW_\varphi(\pi) $.

\item If $G\varphi \in \mathrm{WS}(\pi)$ then $\nu\neq 0$.
\item If  $G\varphi \in \mathrm{WS}(\pi)$ and $F$ is $p$-adic then $\nu$ is an isomorphism.
\end{stepitemize}
\end{theorem}

\pause

We also have a global (adelic) analogue.
\pause
\begin{corollary}
Let $F$ be $p$-adic, $\pi$ be cuspidal, $\cO\in \WS(\pi)$ and $L\subset G$ a Levi subgroup with $\cO\cap \fl^*\neq \emptyset$. Then $L=G$.
\end{corollary}
\pause
\begin{proof}
Let $(e,h,f)\in \fl$ be an $\Sl_2$-triple such that $f\in \cO$. Let $Z\in \fg$ be a (rational) semi-simple element s.t. $\fl=\fg^Z$. Let $T>>0\in \bZ$ and let $H:=h+TZ$. Then $\cW_{H,\varphi}(\pi)$ embeds into the dual of  the Jacquet module $r_L(\pi)$. Since $\cW_{H,\varphi}(\pi)\neq 0$ and $\pi$ is cuspidal, $L=G$.
\end{proof}


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% \frametitle{Basic lemma}
%
% Define anti-symmetric form $\omega$ on $\fg$  by $\omega(X,Y):=\varphi([X,Y])$.
%
% \begin{lemma}[{\small Following} Ginzburg-Soudry-Rallis, Jiang-Liu, Lapid-Mao]
% Let $\fn,\fm\subset \fg$ be nilpotent subalgebras such that $[\fn,\fm]\subset \fn\cap\fm$, $\omega|_{\fn}=0$, $\omega|_{\fm}=0$  and   the radical of $\omega|_{\fn+\fm}$ is $\fn\cap \fm$. Then $\fn+\fm$ is a nilpotent Lie algebra and
% $$\ind^{\Exp(\fn+\fm)}_{\Exp(\fn)}\varphi\simeq \ind^{\Exp(\fn+\fm)}_{\Exp(\fm)}\varphi.$$
% \end{lemma}
% \pause
% \begin{proof}
%  $\fk:=\fn\cap\fm\cap \Ker(\varphi)$. Then $\Exp(\fn+\fm)/\Exp(\fk)$ is  Heisenberg group corresponding to   $(\fn+\fm)/(\fn\cap\fm)$. The subspaces $\fm/(\fn\cap\fm),$  $\fn/(\fn\cap\fm)$ are Lagrangian, thus$$\ind^{\Exp(\fn+\fm)/\Exp(\fk)}_{\Exp(\fn)/\Exp(\fk)}\varphi\simeq \ind^{\Exp(\fn+\fm)/\Exp(\fk)}_{\Exp(\fm)/\Exp(\fk)}\varphi,$$ since both  $\simeq$  oscillator representation  of $\Exp(\fn+\fm)/\Exp(\fk)$ with  central character  $\varphi$.
% \end{proof}
% % \begin{remark}\label{rk:inttransform}
% % By induction by stages we obtain $\ind^{G}_{\Exp(\fl)}\chi_\varphi\simeq \ind^{G}_{\Exp(\fr)}\chi_\varphi.$ Observe that this isomorphism can be realized explicitly as an integral transform: given $f\in  \ind^{G}_{\Exp(\fl)}\chi_\varphi$ we can define $\check{f}\in  \ind^{G}_{\Exp(\fr)}\chi_\varphi$ simply by setting
% % \[
% % \check{f}(g)=\int_{\Exp(\fl\cap\fr) \backslash \Exp(\fr)} f(ng)\, dn.
% % \]
% % Then the previous results imply that the map $f\mapsto \check{f}$ defines an isomorphism between these two spaces.
% % \end{remark}
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\frametitle{Example for our theorem}
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%\begin{example}
Let $G:=\GL(4,\F)$ and define $\varphi$ by $\varphi(X):=\tr(X(E_{21}+E_{43}))$.\\
%, where $E_{21}$ and $E_{43}$ are elementary matrices. Let $h =\diag(1,-1,1,-1)$.
%\begin{example}\label{ex:GLSame}
Let $\Phi:=\varphi$, $H:=\diag(3,1,-1,-3)$,
%\pause
 $h =\diag(1,-1,1,-1)$, $Z=H-h=\diag(2,2,-2,-2), H_t:=h+tZ$.\\
\pause
Then $\fn_{0}\subset \fn_{1/4}\sim \fn'_{1/4}\subset \fn_{3/4}=\fn_1:$
$$
%\begin{multline*}
\left(
   \begin{array}{cccc}
     0 & - & 0 & - \\
     0 & 0 & 0 & 0 \\
     0 & - & 0 & - \\
     0 & 0 & 0 & 0 \\
   \end{array}
 \right) \subset \left(
   \begin{array}{cccc}
     0 & - & a & - \\
     0 & 0 & 0 & a \\
     0 & * & 0 & - \\
     0 & 0 & 0 & 0 \\
   \end{array}
 \right)  \sim
\left(
   \begin{array}{cccc}
     0 & - & * & - \\
     0 & 0 & 0 & * \\
     0 & 0 & 0 & - \\
     0 & 0 & 0 & 0 \\
   \end{array} \right)$$

$$
%\subset\\
% \subset
% \left(
%    \begin{array}{cccc}
%      0 & - & - & - \\
%      0 & 0 & 0 & - \\
%      0 & 0 & 0 & - \\
%      0 & 0 & 0 & 0 \\
%    \end{array} \right)
\subset
\left(
   \begin{array}{cccc}
     0 & - & - & - \\
     0 & 0 & * & - \\
     0 & 0 & 0 & - \\
     0 & 0 & 0 & 0 \\
    \end{array} \right)
   =
\left(
   \begin{array}{cccc}
     0 & - & - & - \\
     0 & 0 & - & - \\
     0 & 0 & 0 & - \\
     0 & 0 & 0 & 0 \\
   \end{array} \right)$$
%\end{multline*}
    Both $*$ and $-$ denote arbitrary elements. $-$ denotes the entries in $\fv_t$ and $*$ those in $\fl_t$. $a$ denotes equal elements.
%    The  passage from $\fl_{1/4}$ to $\fr_{1/4}$ is denoted by $\sim$. At  $3/4$ we have $\fl_{3/4}=\fr_{3/4}. $
%    %In the second matrix there was a choice involved, and there were no choices elsewhere.
%\end{example}

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\frametitle{Adelic setting}



\begin{stepitemize}
\item $K$ - number field,  $\mathbf{G}$ defined over $K$, $G:=\mathbf{G}(\A_K)$, $\Gamma:=\mathbf{G}(K)$
\item $(H,\varphi)\in \fg(K)\times \fg^*(K)$ - Whittaker pair, $N\subset G$ defined as before.
\item For any automorphic function $f\in C^{\infty}(G/\Gamma)$, let a new function $\cW\cF_{H,\varphi}(f)$ be
%
$$ \cW\cF_{H,\varphi}(f)(x):=\int_{N(\A)/N(K)}\chi_\varphi(n)^{-1}f(xn)dn.$$
\end{stepitemize}
\pause
 \begin{theorem}%[Gomez-G.-Sahi]
\begin{enumerate}[(i)]
\item $\cW\cF_{H,\varphi}(f)$ is obtained from $\cW\cF_{\varphi}(f)$ by an integral transform.
%In particular,  for any automorphic representation $\pi$ we have
%$$\cW\cF_{(H,\Phi)}(\pi)\neq 0\Rightarrow \cW\cF_{\varphi}^{gen}(\pi)\neq %0.$$
\item If $\varphi\in \WS(f)$ then $\cW\cF_{\varphi}$ is obtained from $\cW\cF_{H,\varphi}$ by another integral transform.
\end{enumerate}
\end{theorem}

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\frametitle{Wave front set and wave-front cycle}
Let $\pi\in \cM(G)$ be admissible and finitely generated.

\begin{theorem}[Howe, Harish-Chandra, Barbasch-Vogan, 70s]
Near $e \in G$, the character distribution (asymptotically) equals to a
linear combination of Fourier transforms of Haar measures of nilpotent
coadjoint orbits.
\begin{equation*}
\exp^*(\chi_{\pi}) \approx \sum c_{{\mathcal{O}}} {\mathcal{F}}(\mu_{{\mathcal{O}}})
\end{equation*}
\end{theorem}
\pause
\begin{stepitemize}
\item Let $\mathcal{N}\subset
\mathfrak{g}^{\ast }$ denote the nilpotent cone.
\item  $\WF(\pi):=\cup \{\overline{{\mathcal{O}}} \, \vert \, c_{{%
\mathcal{O}}} \neq 0\} \subset \mathcal{N}$.
\item $\WF^{\max}(\pi):= $ union of maximal orbits in $\WF(\pi)$.
%\item $\pi$ is called \emph{large} if $\dim\WF(\pi)=\dim\mathcal{N}$.
\end{stepitemize}
\pause
 \begin{theorem}[Moeglin-Waldspurger, 87']
Let $\F$ be $p$-adic and let $(H,\varphi)$ be a Whittaker pair.
\begin{stepitemize}
\item If $\cW_{H,\varphi}(\pi)\neq 0$ then $\varphi \in \WF(\pi).$
\item If $\varphi \in \WF^{\max}(\pi)$ then $\dim \cW_{H,\varphi}(\pi)=c_{\varphi}.$
\end{stepitemize}
\end{theorem}


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% {\Huge {\center{\hspace{20pt} \color{green}{Thank you  for your attention!}}}}

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\frametitle{Some extensions of the Moeglin-Waldspurger theorem}
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\begin{theorem}[Matumoto 87',92']
\begin{itemize}
\item If $\cW_{H,\varphi}(\pi)\neq 0$ then $\varphi$ lies in the Zariski closure of $\WF(\pi).$
\item If $\varphi$ is regular nilpotent then $\cW_{H,\varphi}(\pi)\neq 0 \Leftrightarrow \varphi \in \WF(\pi). $
\end{itemize}
\end{theorem}
\pause

\begin{theorem}[Gomez-G.-Sahi 2016]
Let $G=\GL(n,\F)$. Then
\begin{enumerate}[(i)]
\item $ \varphi \in \WF(\pi) \Leftrightarrow \cW_{\varphi}(\pi)\neq 0$
\item If $\varphi \in \WF^{max}(\pi)$ and $\pi$ is irreducible \& unitary then $\dim \cW_{H,\varphi}(\pi)=1.$
\end{enumerate}
\end{theorem}
\pause
\begin{theorem}[G.-Sahi-Sayag 2016]
Let $P\subset G$ be a parabolic subgroup and $\fp$ be its Lie algebra.
Let $(H,\varphi)$ be a Whittaker pair with $\varphi \in \fp^{\bot}$. Let $\sigma\in \cM(P)$ with $\dim \sigma <\infty$ and let $\pi:=Ind_P^G(\sigma)$. Then $\cW_{H,\varphi}(\pi)\neq 0$.\\
\pause
In particular, $\WS(\pi)$ consists of the Richardson orbits of $\fp$.
\end{theorem}


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\frametitle{Prederivatives}

\begin{definition}
$$P_n=\left\{\begin{pmatrix}* & \cdots & * & * \\
\vdots & \ddots & \vdots & \vdots \\
* & \cdots & * & * \\
0 & \cdots  & 0 & 1 \\
\end{pmatrix}\right\} \subset G_n:=GL_n(F)$$
\end{definition}

\begin{stepitemize}
\item Over p-adic $\F$, the category $\Rep^{\infty}(P_n)$ of smooth $P_n$-rep-ns is equivalent to the category of $G_{n-1}$-equivariant sheaves on $V_n^*:=(F^{n-1})^*$.
\item Define $\Phi:\Rep^{\infty}(P_n)\to \Rep^{\infty}(P_{n-1})$ by $\Phi(\pi):=\pi_{V_n,\psi}$, and $E^k:\cM(G_n)\to \Rep^{\infty}(G_{n-k})$ by $E^k(\pi):=\Phi^{k-1}(\pi|_{P_n})|_{G_{n-k}}$.
\item For $\pi\in \cM(G_n), \, \text{depth}(\pi):=$ size of max. Jordan block in $\WF(\pi)$.

\end{stepitemize}



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%    \frametitle{Results}
%\thispagestyle{empty}
\begin{theorem}[Aizenbud - G. - Sahi, 2012]

%\label{thm:main} Let $\mathcal{M}(G_{n})$ denote the category of smooth

%admissible {Fr\'{e}chet} representations of moderate growth.
Let $\mathcal{M}^{d}(G_{n})\subset \mathcal{M}(G_{n})$ be the subcategory of rep-s of depth $\leq d$. Then

\begin{stepitemize}
\item \label{mainit:Adm} $E^d$ defines a non-zero exact functor $\mathcal{M}^{d}(G_n) \to
\mathcal{M}(G_{n-d})$.

\item \label{mainit:Zero} $E^{k}|_{\mathcal{M}^{d}(G_{n})}=0$ for any $k>d$.

\item If $\WF(\pi)=\overline{G_nJ_{(d,d_2,\dots,d_k)}}$ then  $\WF(E^{d}(\pi))=\overline{G_nJ_{(d_2,\dots,d_k)}}$

\item \label{mainit:Prod} Let $n=n_1+...+n_d$  and let $\chi_i$ be characters of $G_{n_i}$.
 Let $\pi= \chi_1 \times ... \times \chi_d \in \cM^{d}(G_n)$
denote the corresponding degenerate principal series representation.
 Then $depth(\pi)=d$
    and\\
    $E^d(\pi)\cong(\chi_1)|_{G_{n_1-1}} \times ... \times (\chi_d)|_{G_{n_d-1}}$. %\pause
\item For a unitarizable irrep $\pi$ of depth $d$,
$E^d(\pi)=A(\pi)$.%\oplus \tau$$
\end{stepitemize}
\end{theorem}
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\begin{theorem}[Gomez - G. - Sahi, 2015]
Let $\lambda=(n_1,\dots,n_k)$ be a partition of $n$ and $J_{\lambda}$ be the corresponding nilpotent Jordan matrix. Then $\cW^{gen}_{J_{\lambda}}(\pi)\cong E^{n_k}(\cdots(E^{n_1}(\pi))\cdots)^*.$
\end{theorem}

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\frametitle{Example for $\lambda=(3,2,1)$}


$$\left\{\begin{pmatrix}
0 &0 & * & 0 & * & * \\
0 &0 & \underline{*} & 0 & * & * \\
0 &0 & 0 & 0 & * & * \\
0 &0 & 0 & 0 & \underline{*} & * \\
0 &0 & 0 & 0 & 0 & \underline{*} \\
0 &0 & 0 & 0 & 0 & 0
\end{pmatrix}\right\} $$

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\frametitle{Adelic setting}



\begin{stepitemize}
\item $K$ - number field,  $\mathbf{G}$ defined over $K$, $G:=\mathbf{G}(\A_K)$.
\item $(H,\Phi)$ - Whittaker pair, $N$ defined as before.
\item For any automorphic function $f$, let a new function $\cW\cF_{H,\varphi}(f)$ be
%
$$ \cW\cF_{H,\varphi}(f)(x):=\int_{N(\A)/N(K)}\Phi(n)^{-1}f(xn)dn.$$
\end{stepitemize}
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 \begin{theorem}%[Gomez-G.-Sahi]
%\begin{enumerate}[(i)]
Let $\varphi\in \overline{G\Phi}$ and suppose that either $G=\GL_n(\A)$ or $\varphi\in \overline{G_H \Phi}$.
Then $\cW\cF_{H,\Phi}$ is obtained from $\cW\cF^{gen}_{\varphi}$ by an integral transform.
In particular,  for any automorphic representation $\pi$ we have
$$\cW\cF_{(H,\Phi)}(\pi)\neq 0\Rightarrow \cW\cF_{\varphi}^{gen}(\pi)\neq 0.$$
%\end{enumerate}
\end{theorem}

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{\Huge {\center{\hspace{20pt} \color{green}{Thank you  for your attention!}}}}

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