% $Header: /cvsroot/latex-beamer/latex-beamer/solutions/generic-talks/generic-ornate-15min-45min.en.tex,v 1.4 2004/10/07 20:53:08 tantau Exp $
%\texhash
\documentclass{beamer}

% This file is a solution template for:

% - Giving a talk on some subject.
% - The talk is between 15min and 45min long.
% - Style is ornate.



% Copyright 2004 by Till Tantau <tantau@users.sourceforge.net>.
%
% In principle, this file can be redistributed and/or modified under
% the terms of the GNU Public License, version 2.
%
% However, this file is supposed to be a template to be modified
% for your own needs. For this reason, if you use this file as a
% template and not specifically distribute it as part of a another
% package/program, I grant the extra permission to freely copy and
% modify this file as you see fit and even to delete this copyright
% notice.


\mode<presentation>
{
  \usetheme{Warsaw}
  % or ...
  % or whatever (possibly just delete it)
  \setbeamercovered{invisible}
}

%\usetheme{Boadilla}


\usepackage[english]{babel}
% or whatever

\usepackage[latin1]{inputenc}
% or whatever
\usepackage[all]{xy}
\usepackage{xypic}
\usepackage{times}
\usepackage{ulem}
\usepackage[T1]{fontenc}
\usepackage{amsfonts,amsmath,amssymb}
\usepackage{hyperref}
\usepackage[all]{xy}

\usepackage{amssymb,amsthm,amsxtra}
%\usepackage[usenames]{color}

\usepackage{amscd}
\usepackage{amsthm}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{mathrsfs}


% Or whatever. Note that the encoding and the font should match. If T1
% does not look nice, try deleting the line with the fontenc.
%\newtheorem{theorem*}{Theorem}
%\newtheorem*{lemma}{Lemma}
\newtheorem*{proposition}{Proposition}
\newtheorem*{prop}{Proposition}
\newtheorem*{remark}{Remark}
\newtheorem*{rem}{Remark}
\newtheorem*{question}{Question}
%\newtheorem*{example}{Example}
%\newtheorem*{theorem}{Theorem}
%\newtheorem*{definition}{Definition}
\newtheorem*{notation}{Notation}
%\newtheorem*{result}{Result}
%\newtheorem*{property}{}
%\newtheorem*{corollary}{Corollary}
%\newtheorem*{construction}{Construction}
%\newtheorem*{case}{Case}
\newtheorem*{conjecture}{Conjecture}
\newtheorem*{setting}{Setting}
\newtheorem*{flowchart}{Flowchart}

%\newtheorem{cor}[lemma]{Corollary}
%\newtheorem{exm}[lemma]{Example}
%\newtheorem{exc}[lemma]{Exercise}
%\newtheorem{conj}[lemma]{Conjecture}
%\newtheorem{rem}[lemma]{Remark}
%\newtheorem{conc}[lemma]{Conclusion}



\newcommand{\F}{{\mathbb F}}
\newcommand{\R}{{\mathbb R}}
\newcommand{\C}{{\mathbb C}}
\newcommand{\Dist}{{\mathcal{D}}}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\Sc}{{\mathcal S}}
\newcommand{\Fre}{{Fr\'{e}chet \,}}


\newcommand{\Matrix}[4]{ \left( \begin{array}{cc}  #1 & #2 \\  #3 & #4 \\ \end{array} \right) }
\newcommand{\norm}[1]{\left\Vert#1\right\Vert}
\newcommand{\abs}[1]{\left\vert#1\right\vert}
\newcommand{\set}[1]{\left\{#1\right\}}
\newcommand{\Cat}[1]{ {\mathfrak #1} }
\newcommand{\Lie}[1]{ {\mathfrak #1} }
\newcommand{\filt}[1]{ {\mathcal #1} }
\newcommand{\scheme}[1]{\underline{ \mathbf{#1}}}
\newcommand{\alg}[1]{\mathbf{#1}}
\newcommand{\mc}[1]{\mathcal #1}
\newcommand{\Hom}{ {\mbox{Hom}} }
\newcommand{\Ext}{ {\mbox{Ext}} }
\newcommand{\Ind}{ {\mbox{Ind}} }
% \newcommand{\proof}{ {\sc Proof:} }
% \newcommand{\qed}{\begin{flushright}Q.E.D \end{flushright}}
\newcommand{\nats}{\mathbb N}
\newcommand{\ints}{\mathbb Z}
\newcommand{\rats}{\mathbb Q}
\newcommand{\reals}{\mathbb R}
\newcommand{\complex}{\mathbb C}
\newcommand{\quats}{\mathbb H}
\newcommand{\octs}{\mathbb O}
\newcommand{\hecke}{\mathcal H}
\newcommand{\fourier}{\mathcal F}
\newcommand{\FF}{\mathbb F}
\newcommand{\Gp}{\mathbb G}
\newcommand{\adeles}{\mathbb A}
%\newcommand{\eps}{\varepsilon}
\newcommand{\To}{\longrightarrow}
\newcommand{\BX}{\mathbf{B}(X)}
\newcommand{\A}{\mathcal{A}}
\newcommand{\D}{\mathcal{D}}
\newcommand{\DOP}{\mathbb{D}}
\newcommand{\M}{\mathcal{M}}
\newcommand{\Schwarz}{\mathcal{S}}
\newcommand{\U}{\mathcal{U}}
\newcommand{\OO}{\mathcal{O}}
\newcommand{\ZU}{\mathcal{Z}}
\newcommand{\e}{\textbf{e}}
\newcommand{\Orb}{\mathcal{O}}
\newcommand{\Period}{\mathcal{P}}
\newcommand{\isom}{\cong}
\newcommand{\gK}{( \Lie{g}_\complex, K)}
\newcommand{\J}{\mathcal J}
\newcommand{\g}{{\mathfrak{g}}}




\newcommand{\speh}{\operatorname{Speh}}
\newcommand{\aaa}{\mathfrak{a}}
\newcommand{\res}{\operatorname{res}}
\newcommand{\mix}{\mathcal{M}}
\newcommand{\tr}{\operatorname{Tr}}
\newcommand{\www}{\mathfrak{W}}
\newcommand{\ob}{\operatorname{ ob}}
\newcommand{\bes}{\mathfrak{B}}
\newcommand{\eisen}{\mathcal{E}}
\newcommand{\rest}{\lvert}
\newcommand{\sg}{\mathfrak{S}}
\newcommand{\proj}{\operatorname{proj}}
\newcommand{\sgn}{\operatorname{sgn}}
\newcommand{\disc}{\Delta}
\newcommand{\modulus}{\delta}
\newcommand{\I}{\mathbb{I}}
\newcommand{\eps}{\varepsilon}
\newcommand{\vol}{\operatorname{vol}}
\newcommand{\Ht}{H}
\newcommand{\diff}{\mathfrak{d}}
\newcommand{\triv}{{\bf 1}}
\newcommand{\ind}{\operatorname{ind}}
\newcommand{\inv}{\operatorname{inv}}
%\newcommand{\Ind}{\operatorname{Ind}}
\renewcommand{\Im}{\operatorname{Im}}
\newcommand{\match}[1]{\mathop{\leftrightarrow}\limits^{#1}}
\newcommand{\der}[1]{\mathfrak{D}^{(#1)}}
%\newcommand{\abs}[1]{\left|{#1}\right|}
%\newcommand{\norm}[1]{\lVert#1\rVert}
\newcommand{\sprod}[2]{\left\langle#1,#2\right\rangle}


\newcommand{\ff}{\mathbb{F}}
\newcommand{\bbc}{\mathbb{C}}
\newcommand{\bc}{\operatorname{bc}}
\newcommand{\ad}{\operatorname{ad}}
\newcommand{\uu}{\mathcal{U}}
%\newcommand{\hecke}{\mathcal{H}}
\newcommand{\pp}{\mathcal{P}}
\newcommand{\oo}{\mathcal{O}}
\newcommand{\ch}{\operatorname{ch}}
\newcommand{\sss}{\mathcal{S}}
\newcommand{\ww}{\mathcal{W}}
\newcommand{\bb}{\mathcal{B}}
\newcommand{\ii}{\mathcal{I}}
\newcommand{\ccc}{\mathcal{C}}
\newcommand{\nm}{\operatorname{Nm}}
\newcommand{\bs}{\backslash}
%\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\re}{\operatorname{Re}}
\newcommand{\diag}{\operatorname{diag}}
\newcommand{\kk}{\mathfrak{K}}
\newcommand{\zz}{\mathbb{Z}}
\newcommand{\nn}{\mathbb{N}}
\newcommand{\cc}{\mathbb{C}}
\newcommand{\qq}{\mathbb{Q}}
\newcommand{\rr}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}

\newcommand{\aut}{\mathcal{A}}
\newcommand{\Supp}{\mathrm{Supp}}
%\newcommand{\g}{{\mathfrak{g}}}
\newcommand{\proofend}{\hfill$\Box$\smallskip}
\newcommand{\cO}{\mathcal O}
\newcommand{\cP}{\mathcal P}


\definecolor{DarkGreen}{rgb}{0,0.5,0}

\newcommand{\Rami}[1]{{\color{cyan}{#1}}}
\newcommand{\Dima}[1]{{\color{magenta}{#1}}}
\newtheorem{observation}{Observation}
\newtheorem{conclution}{Conclusion}
\newtheorem{conclusion}{Conclusion}


\begin{document}

\title[Representation theory - from zero characteristic
to positive one]{Representation theory - how to get from zero
characteristic to positive one}
\author{Dmitry Gourevitch}
\begin{frame}
%\maketitle
\titlepage
\begin{center}Joint work with Avraham Aizenbud and Nir Avni  \end{center}
$ $\\
\begin{center}\url{http://www.math.ias.edu/~dimagur}  \end{center}

\end{frame}
%\newpage

\begin{frame}
    \frametitle{Non-Archimedean local fields}

%%\thispagestyle{empty}
%
\begin{definition}
%A local field is a
%
%field that is complete in a topology defined by a non-trivial
%absolute value.
We will discuss two types of local fields:
\begin{itemize}
\item zero characteristic \quad $ \,$ -- $\Q_p$ and its finite extensions.
\item positive characteristic -- $\F_p((t))$ and its finite extensions.
\end{itemize}
Those fields satisfy:
$$ |x+y| \leq \max(|x|,|y|).$$
\end{definition}
\pause

\begin{notation}
For a local field $F$ we denote
\begin{itemize}
\item $\mathcal{O}_F:=\{x \in F \, | \, |x| \leq 1\} $
\item $\mathcal{P}_F:=\{x \in F \, | \, |x| < 1\}$
\end{itemize}
\end{notation}


\end{frame}

%\newpage
\begin{frame}
   \frametitle{The ring of integers}

\begin{notation}
For a local field $F$ we denote
\begin{itemize}
\item $\mathcal{O}_F:=\{x \in F \, | \, |x| \leq 1\} $
\item $\mathcal{P}_F:=\{x \in F \, | \, |x| < 1\}$
\end{itemize}
\end{notation}

\begin{example}%$ $\\
\begin{itemize}
\item $\mathcal{O}_{\Q_p}=\Z_p.$
\item $\mathcal{P}_{\Q_p}=p\Z_p.$
\item $\mathcal{O}_{\F_p((t))}=\F_p[[t]].$
\item  $\mathcal{P}_{\F_p((t))}=t\F_p[[t]].$
\end{itemize}
\end{example}

\end{frame}

%\newpage
\begin{frame}
    \frametitle{
   Close local fields
    }
%\Dima{Maybe you will just say or write once that you discuss only non-Archimedean local fields?}
\begin{definition}
Two local fields $F$ and $F'$ are said to be $n$-close if
$$\mathcal{O}_F/\mathcal{P}_F^n \simeq
\mathcal{O}_{F'}/\mathcal{P}_{F'}^n.$$
%\Dima{I introduced notation $F \overset{n}{\sim} F'$}
\end{definition}
\pause
\begin{proposition}
Any local field can be approximated up to any order by a local
field of  characteristic $0$.
\end{proposition}
\pause
\begin{example}
  $\F_p((t))$ is $n$-close to $\Q_p(\sqrt[n]{p})$.
\end{example}


\end{frame}
%\newpage



\begin{frame}
\thispagestyle{empty}
%\frametitle{ Reductive groups and their congruence subgroups }
\begin{example}[of reductive groups]
$GL_n, \, O_n , \, Sp_n,$ semisimple groups.
\end{example}
\pause

Let $G$ be a reductive group defined over $\Z$ and let $F$ be a
local field.
\begin{definition}
$K_0 :=K_0(F):=K_0(G,F):=G(\mathcal{O}_F) \subset G(F)$
\end{definition}
\pause
\begin{definition}
$K_n :=K_n(F):=K_n(G,F)$ is defined by the following exact
sequence: $$1 \to K_n \to K_0 \to G(\mathcal{O}_F/\mathcal{P}_F^n)
\to 1$$
%\Dima{I replaced 0 by 1, since those groups are not commutative.}
\end{definition}
\pause
\begin{example}
$F=\Q_p$, $G=GL_n$, $K_0=GL_n(\Z_p)$, $K_n=Id + p^n Mat_n(\Z_p)$%, $K_0/K_n=GL_n(\Z / p^n\Z)$.
\end{example}
\pause
\begin{remark}
If $F$ and $F'$ are $n$-close then $K_0(F)/K_n(F) \simeq
K_0(F')/K_n(F')$.
\end{remark}

\end{frame}
%\newpage
\begin{frame}
\frametitle{Cartan decomposition}
Let $\pi \in \mathcal{P}_{F} - \mathcal{P}_{F}^2.$ For simplicity let $G=GL_n$. Let $$\Lambda(G) := \left \{ \begin{pmatrix}
  \pi^{i_1} & 0 &  0 \\
  0 & ... & 0\\
  0 & 0 & \pi^{i_n}
\end{pmatrix}
 | \, i_1 \geq ... \geq i_n \right \}.$$
%
\begin{theorem}
$K_0(F)\setminus G(F)/K_0(F) = \Lambda(G).$
\end{theorem}

\end{frame}
%\newpage
\begin{frame}
%\frametitle{ Smooth representations }
\thispagestyle{empty}
\begin{definition}
\begin{itemize}
\item A smooth representation of $G(F)$ is a linear action of $G(F)$ on a complex linear
vector space $V$ such that any $v\in V$ has an open stabilizer.
\item
We denote the category of smooth representations of $G(F)$ by
$\mathcal{M}(G(F))$. \pause
\item Let $K<G$.
We denote the category of representations of $G(F)$ which are
generated by $K$-invariant vectors by $\mathcal{M}_K(G(F))$.
%$K-$representations of $G(F)$
\end{itemize}
\end{definition}
\pause
%\begin{remark}
 $$\text{Clearly }\bigcup_{n=0}^\infty \mathcal{M}_{K_n}^f(G(F))=\mathcal{M}^f(G(F)).$$
%\end{remark}
\pause
%\Rami{
\begin{definition}
The Hecke algebra, denoted by $\mathcal{H}_K(G(F))$, is the
algebra of double $K$-invariant measures on $G(F)$. The algebra
structure on $\mathcal{H}_K(G(F))$ is defined by convolution.
\end{definition}
%}
\end{frame}

%\newpage
\begin{frame}
\frametitle{ Bernstein center } %\thispagestyle{empty}
%\begin{notation}
%The Hecke algebra, denoted by $\mathcal{H}_K(G(F))$, is the
%algebra of double $K$-invariant measures on $G(F)$. The algebra
%structure on $\mathcal{H}_K(G(F))$ is defined by convolution.
%\end{notation}
\begin{figure}[htp]
\includegraphics[height=25mm]{Bernstein.jpg}
\end{figure}

\begin{theorem}[Bernstein ~ 1980]
\begin{itemize}
\item $\mathcal{M}_{K_n}(G(F))$ is a direct summand of $\mathcal{M}(G(F)).$
\item $\mathcal{M}_{K_n}(G(F))$ is equivalent to the category of $\mathcal{H}_{K_n}(G(F))-$modules
\item The algebra $\mathcal{H}_{K_n}(G(F))$ is finite over its center which is finitely generated. Therefore, $\mathcal{H}_{K_n}(G(F))$ is Noetherian and finitely presented.
\end{itemize}
\end{theorem}
\end{frame}
%\newpage
%
\begin{frame}
   \frametitle{
   Kazhdan's Theorem
   }
%\thispagestyle{empty}

\begin{figure}[htp]
\includegraphics[height=25mm]{kazhdan.jpg}
\end{figure}

\begin{theorem}[Kazhdan - 1984]
For any natural $n$ there exists $N$ s.t. if $F$ and $F'$ are $N-$close then $$\mathcal{H}_{K_n(F)}(G(F)) \simeq \mathcal{H}_{K_n(F')}(G(F')).$$
\end{theorem}
\end{frame}
%\newpage
%
%
\begin{frame}
%\thispagestyle{empty}
\frametitle{Ingredients of the proof}
\begin{itemize}
\item A construction of a linear isomorphism $$\phi: \mathcal{H}_{K_n(F)}(G(F)) \to \mathcal{H}_{K_n(F')}(G(F'))$$ whenever $F$ and $F'$ are $n-$close, using Cartan decomposition.
 \pause
\begin{multline*}
K_n(G,F)\setminus G(F)/K_n(G,F) =\\= K_n(G,F)\setminus K_0(G,F) \Lambda(G) K_0(G,F)/K_n(G,F) =\\= G(\cO_F/\cP_F^n) \Lambda(G) G(\cO_F/\cP_F^n)
\end{multline*}
%
%  \pause
%  \item A proof that for any $x,y \in \mathcal{H}_{K_n}(G(F))$ there exists $N$ s.t. if $F$ and $F'$ are $N-$close then $$\phi(xy)=\phi(x)\phi(y).$$
%  \pause
% %
% \item  The algebra $\mathcal{H}_{K_n}(G(F))$ is finitely presented.
\end{itemize}
\end{frame}

\begin{frame}
%\thispagestyle{empty}
\frametitle{Ingredients of the proof}
\begin{itemize}
\item A construction of a linear isomorphism $$\phi: \mathcal{H}_{K_n(F)}(G(F)) \to \mathcal{H}_{K_n(F')}(G(F'))$$ whenever $F$ and $F'$ are $n-$close, using Cartan decomposition.
%  \pause
% \begin{multline*}
% K_n(G,F)\setminus G(F)/K_n(G,F) =\\= (K_n(G,F)\setminus K_0(G,F)) \Lambda(G) (K_0(G,F)/K_n(G,F)) =\\= G(\cO_F/\cP_F)^n) \Lambda(G) G(\cO_F/\cP_F^n)  = G(\cO_{F'}/\cP_{F'}^n) \Lambda(G) G(\cO_{F'}/\cP_{F'}^n)=\\ = K_n(G,F')\setminus G(F')/K_n(G,F')
% \end{multline*}
%
 \item A proof that for any $x,y \in \mathcal{H}_{K_n}(G(F))$ there exists $N$ s.t. if $F$ and $F'$ are $N-$close then $$\phi(xy)=\phi(x)\phi(y).$$
 \pause
%
\item  The algebra $\mathcal{H}_{K_n}(G(F))$ is finitely presented.
\end{itemize}
\end{frame}

% %\newpage
%
\begin{frame}

\frametitle{
Relative representation theory -- Harmonic analysis over spherical varieties
}
\thispagestyle{empty}
\begin{observation}
\center{Representation theory of $G$\\
%$\Leftrightarrow$
$\Updownarrow$
\\
Harmonic analysis on $G$ w.r.t. the two sided action of $G \times G$}
\end{observation}
\pause
\begin{conclution}
Let $H\subset G$ be a spherical pair (i.e. the Borel subgroup of $G$ acts on $G/H$ with finite number of orbits). One can consider harmonic analysis over $G/H$ as a generalization of
representation theory.
\end{conclution}
%\pause
%\begin{example}
%Schur's lemma is analogous to the Gelfand property of
%%the pair
%$(G,H)$:\\
%%\center{
%\quad \quad \quad \quad \quad \quad $\forall \pi \in irr(G): \dim
%\Hom_H (\pi|_H,\C)\leq 1$
%\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad or equivalently \\
%\quad \quad \quad \quad \quad \quad  $\forall \pi \in irr(G): \dim \Hom (\Sc(G/H),\pi)\leq 1.$ \\
%%}
%%$$\text{For any irreducible representation of } G \text{ we have } \dim \Hom (\Sc(G/H),\pi)\leq 1$$
%\end{example}
%\pause
%\begin{example}
%The trace formula is analogous to the relative trace formula.
%\end{example}
\end{frame}

\begin{frame}
%\frametitle{Schur's lemma is analogous to Gelfand property}
\thispagestyle{empty}
\begin{lemma}[Schur]
Let $\pi$ and $\tau$ be irreducible representations of $G$. Then
$\dim \Hom_G(\pi,\tau) \leq 1$.
\end{lemma}
\pause Equivalent formulations:
\begin{enumerate}
\item $\forall \pi, \tau \quad \dim \Hom_{\Delta G}(\pi \otimes \widetilde{\tau},\C) \leq 1$
\pause
\item $\forall$ irreducible representation $\Pi$ of $G \times G,
\, \Hom_{\Delta G}(\Pi, \C) \leq 1$
\end{enumerate}
\pause
\begin{definition}
$(G,H)$ is called a Gelfand pair if
%\center{
%\quad \quad \quad \quad \quad \quad
$\forall \pi \in Irr(G): \dim \Hom_H (\pi|_H,\C)\leq 1$
%\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad
%}
%$$\text{For any irreducible representation of } G \text{ we have } \dim \Hom (\Sc(G/H),\pi)\leq 1$$
\end{definition}
\pause
\begin{proposition}
$G(F),H(F)$ is a Gelfand pair if and only if $\forall \pi \in
Irr(G): \dim \Hom_G (\Sc(G/H),\widetilde{\pi})\leq 1.$
\end{proposition}
\pause
\begin{proof}
$\Hom_{H(F)}(\pi|_{H(F)},\C)\cong  \Hom_{G(F)}(\pi,C^\infty(G(F)/
H(F))) \cong$ \\ $\Hom_{G(F)}(\Sc(G(F)/ H(F)),\tilde \pi)$

\end{proof}% %%\newpage
\end{frame}
% %
\begin{frame}
\frametitle{
Relative version of Kazhdan's theorem
}
%\thispagestyle{empty}
\begin{theorem}[Aizenbud, Avni, Gourevitch - 2009]
Under certain conditions on the pair $(G,H),$
for any natural $n$ the exist $N$ s.t. if $F$ and $F'$ are $N-$close then $$\Sc(G(F)/H(F))^{K_n(F)} \simeq \Sc(G(F')/H(F'))^{K_n(F')} $$
as a
$$\mathcal{H}_{K_n(F)}(G(F)) \simeq \mathcal{H}_{K_n(F')}(G(F'))-\text{module}$$
\end{theorem}
\end{frame}
%
\begin{frame}
\frametitle{
Ingredients in the proof
}
%\thispagestyle{empty}
\begin{itemize}
\item A construction of a linear isomorphism $$\psi: \Sc(G(F)/H(F))^{K_n(F)}\to \Sc(G(F')/H(F'))^{K_n(F')}$$ whenever $F$ and $F'$ are $n-$close.
\pause
\item A proof that for any $x \in \mathcal{H}_{K_n}(G(F))$ and $y \in \Sc(G(F)/H(F))^{K_n(F)}$ there exist $N$ s.t. if $F$ and $F'$ are $N-$close then $$\psi(xy)=\phi(x)\psi(y).$$
\pause
\item  The module $\Sc(G(F)/H(F))^{K_n(F)}$ is finitely generated over the algebra $\mathcal{H}_{K_n}(G(F)).$
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{ Proof of the 3-rd ingredient }
%\thispagestyle{empty}
\begin{observation}
If the module $\Sc(G(F)/H(F))^{K_n(F)}$ is finitely generated over the
algebra $\mathcal{H}_{K_n}(G(F))$ for all $n$ then
$$\forall \pi \in irr(G): \dim \Hom_H (\pi|_H,\C) < \infty.$$
\end{observation}
\pause
\begin{conclution}
We have to impose $\forall \pi \in irr(G): \dim \Hom_H
(\pi|_H,\C)< \infty.$
\end{conclution}
\pause
\begin{theorem}
The following are equivalent
\begin{itemize}
\item $\forall \pi \in irr(G): \dim \Hom_H (\pi|_H,\C)< \infty$
\item The module $\Sc(G(F)/H(F))^{K_n(F)}$ is finitely generated over the algebra
$\mathcal{H}_{K_n}(G(F))$ for all $n$.
\end{itemize}
\end{theorem}
\end{frame}

\begin{frame}
\frametitle{
Proof of the 3-rd ingredient
}
\thispagestyle{empty}
\begin{theorem}
The following are equivalent
\begin{enumerate}
\item $\forall \pi \in irr(G): \dim \Hom_H (\pi|_H,\C)< \infty$
\item $\forall M \text{ and a cuspidal } \rho \in irr(M): \dim \Hom_{H_M} (\rho|_{H_M},\C)< \infty$
\item The module $\Sc(G(F)/H(F))^{K_n(F)}$ is finitely generated over the algebra
$\mathcal{H}_{K_n}(G(F))$ for all $n$.
\end{enumerate}
\end{theorem}
\pause
\begin{proof}
$(2) \Rightarrow (1),(3) \Rightarrow (1)$ - easy,

\pause
$(2) \Rightarrow (3)$ - follows from the theory of Bernstein center.
\pause

The
hard part, $(1) \Rightarrow (2)$, follows from estimation of
co-homologies.
\end{proof}

\end{frame}
% %\pause
%
%
% %\newpage
% %
\begin{frame}
\frametitle{ Applications } \thispagestyle{empty} \pause Set
$G:=GL_{m+k}$ and
$H= GL_{m}\times GL_{k}$%\pause
\pause\\
\begin{theorem}[Jacquet, Rallis - 1996]
%\textbf{Theorem (Jacquet, Rallis - 1996)}\\
Let $F$ be a local field of characteristic 0. Then the pair
$(G(F),H(F))$ is a  Gelfand pair i.e. for any irreducible
representation $\pi$ of $G(F)$  we have $$\dim
\Hom_{H(F)}(\pi|_{H(F)},\C)\leq 1.$$
\end{theorem}
\pause
\begin{corollary}
Let $F$ be a local field of positive characteristic. Then the pair
$(G(F),H(F))$ is a  Gelfand pair.
\end{corollary}
\pause
\begin{proof}
%Assume $ \pi $ have a $K_n-$fixed vector.
%\pause
%\begin{multline*}
%$$
$\Hom_{H(F)}(\pi|_{H(F)},\C)%\cong  \Hom_{G(F)}(\pi,C^\infty(G(F)/ H(F)))
\cong \Hom_{G(F)}(\Sc(G(F)/ H(F)),\tilde \pi) \pause
\cong\Hom_{\mathcal{H}_{K_{n}}(G(F))}(\Sc(G(F)/
H(F))^{{K_{n}}},\tilde \pi^{{K_{n}}}).$
%$$
%\end{multline*}

\end{proof}

\end{frame}
% %\newpage
%
\begin{frame}
\frametitle{ Applications }
%\thispagestyle{empty}
 \begin{theorem}[Aizenbud, Gourevitch, Rallis, Schiffmann - 2007]
%\textbf{Theorem (Aizenbud, Gourevitch, Rallis, Schiffmann - 2007)}\\
Let $F$ be a local field of characteristic 0. Then the pair
$(GL_{n+1}(F),GL_{n}(F))$ is a strong Gelfand pair i.e. for any
irreducible representations $\pi$ of $GL_{n+1}(F)$ and $\tau$ of
$GL_{n}(F)$ we have $$\dim
\Hom_{GL_{n}(F)}(\pi|_{GL_{n}(F)},\tau)\leq 1.$$
\end{theorem}
\pause
\begin{corollary}
Let $F$ be a local field of positive characteristic. Then the pair
$(GL_{n+1}(F),GL_{n}(F))$ is a strong Gelfand pair.
\end{corollary}
\pause
\begin{proof}
 The fact that $(G,H)$ is a strong Gelfand pair is equivalent to the fact that $(G\times H,\Delta H)$ is a Gelfand pair.
\pause
\\
%\begin{multline*}
$ \Hom_{H}(\pi|_{H},\tau)\cong \Hom_{H}(\pi|_{H},\tilde {\tilde
\tau})\cong \Hom_{H}(\pi|_{H},{\tilde  \tau}^*)\cong \Hom_{\Delta
H}(\pi \otimes \tilde \tau,\C) $
%\end{multline*}
\end{proof}
\end{frame}


\begin{frame}
\frametitle{
Open questions
}
%\thispagestyle{empty}
Let $(G,H)$ be a spherical pair.
\begin{itemize}
\item To what generality our theorem extends?\\
\pause
Conjecture: It holds for all symmetric pairs defined over $\Z$. %?? quasi - split?
\pause
 \item Is $\Hom_{H}(\pi|_{H},\C)$ finite for any irreducible smooth representation $\pi$ of $G$?\\
\pause
Conjecture: Yes.\\ Proven for large classes of spherical pairs, including symmetric pairs, by Delorme and by Sakellaridis-Venkatesh.
\item If $\Hom_{H}(\pi|_{H},\C)$ is finite, is it bounded as a function of $\pi$?\\
\pause Our theorem implies that it is locally bounded, i.e.
bounded on $\mathcal{M}_{K}(G(F))$ for every $K$. \pause
\item What spherical pairs are Gelfand pairs?
\pause
Conjecture: $(G,H)$ is a Gelfand pair iff $\dim \Hom_{H}(\pi|_{H},\C) \leq 1$ for any irreducible representation $\pi$ of principle series.\\
\end{itemize}

\end{frame}


\end{document}
