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% - Giving a talk on some subject.
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% 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.
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% However, this file is supposed to be a template to be modified
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\title % (optional, use only with long paper titles)
{Derivatives for representations of $GL(n,{\mathbb{R}})$ and $GL
(n,{\mathbb{C}})$}
%
%%\subtitle
%%{$F$ a $p$-adic field} % (optional)
%
\author % (optional, use only with lots of authors)
{A. Aizenbud}
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%   affiliation.


\institute[Massachusetts Institute of Technology] % (optional, but mostly needed)
{Massachusetts Institute of Technology}
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\date[] % (optional)
%{August 14, 2008}

%\subject{}
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\begin{document}


\begin{frame}
\titlepage
\center{Joint with Dmitry Gourevitch and Siddhartha Sahi} \\$ $ \\ $ $ \\
\center{\url{http://math.mit.edu/~aizenr} }

\end{frame}






\begin{frame}
    \frametitle{The p-adic case}
\thispagestyle{empty}
%\begin{small}

\pause
\begin{definition}
$$P_n=\left\{\begin{pmatrix}* & \cdots & * & * \\
\vdots & \ddots & \vdots & \vdots \\
* & \cdots & * & * \\
0 & \cdots  & 0 & 1 \\
\end{pmatrix}\right\} \subset G_n:=GL_n$$
\end{definition}
\pause
\begin{theorem}
The category $\cM(P_n)$ of smooth $P_n$ representations is equivalent to the category of $G_{n-1}$ equivariant sheaves on $\mathbb{A}^{n-1}$
\end{theorem}
\pause
\begin{proof}
%Proof:
\begin{multline*}\cM(P_n)=\cM(\mathcal{H}(P_n))  =\cM(\mathcal{H}(G_{n-1}\ltimes \mathbb{A}^{n-1})) =
\\  =\cM(\mathcal{H}(G_{n-1})\otimes \mathcal{H}(\mathbb{A}^{n-1}))
 \cong\cM(\mathcal{H}(G_{n-1})\otimes \Sc(\mathbb{A}^{n-1}))
\end{multline*}
\end{proof}
%\end{small}
\end{frame}



\begin{frame}
    \frametitle{The p-adic case}
\pause
\begin{corollary}
We have a short exact sequence
$$0  \to \cM(P_{n-1})  \to  \cM(P_{n}) \to \cM(G_{n-1}) \to 0$$
\end{corollary}
\pause
\begin{definition} $ $
\begin{itemize}
\item $\Phi:\cM(P_{n})  \to \cM(P_{n-1})$ -- the restriction
\pause
\item $\Psi:\cM(P_{n})  \to \cM(G_{n-1})$ -- the fiber
\pause
\item $D^k=\Psi \circ \Phi^{k-1}$
\end{itemize}
\end{definition}
\end{frame}



\begin{frame}
    \frametitle{The Harish-Chandra category}
    \thispagestyle{empty}
\pause
Let $G$ be a real reductive group\pause, $\g$ be its complexified Lie algebra \pause and $K$ be its maximal compact subgroup.
\pause
\begin{definition}
A $(\g,K)$-module is a  $\g$-module $\pi$ with a locally finite action of $K$ such the two actions are compatible.
\pause

A finitely generated $(\g,K)$-module is called admissible if any representation of $K$ appears in it with finite
%(or zero)
 multiplicity.
\end{definition}
\pause
\begin{theorem}[Harish-Chandra, Osborne, Stafford, Wallach] \label{thm:EqAdm}
Let $\pi$ be a finitely generated $(\g,K)$-module. Then the following properties of $\pi$ are equivalent.
\pause
\begin{itemize}
\item \label{Eqit:Adm} $\pi$ is admissible.
\pause
\item \label{Eqit:FinLen} $\pi$ has finite length.
\pause
\item \label{Eqit:ZFin} $\pi$ is $Z_G$-finite.
\pause
\item \label{Eqit:NFinGen} $\pi$ is finitely generated over $\mathfrak{n}$.
\end{itemize}
\end{theorem}
\end{frame}



\begin{frame}
    \frametitle{The category of smooth admissible representations}
\pause
\begin{definition}
Denote by $\cM_{\infty}(G)$ the category of smooth admissible \Fre representations of $G$ of moderate growth \pause and by $\cM_{HC}(G)$ the category of admissible Harish-Chandra modules.\pause

%Note that both $\cM_{\infty}(G)$ and $\cM_{HC}(G)$ are naturally subcategories %of $\cM(\g)$.
We denote by $HC:\cM_{\infty}(G) \to \cM_{HC}(G)$ the functor of $K$-finite vectors.
\end{definition}
\pause

\begin{thm}[Casselman-Wallach] \label{thm:CW}
The functor $HC:\cM_{\infty}(G) \to \cM_{HC}(G)$ is an equivalence of categories.
\end{thm}
\end{frame}



\begin{frame}
    \frametitle{Definitions}
\pause
\begin{definition}
Define a functor $\Phi: \cM(\fp_n) \to \cM(\fp_{n-1})$ by $\Phi(\pi):=\pi_{\fv_n,\psi} \otimes  |det|^{-1/2}$.
\end{definition}
\pause
\begin{definition}
For a $\fp_n$-module $\pi$ we have 3 notions of derivative:
\pause
\begin{itemize}
\item $D_1^k(\pi):=\Phi^{k-1}(\pi)\otimes |det|^{-1/2}=\pi_{\fu_{k-1},\psi_{k-1}}\otimes |det|^{-k/2}.$ Clearly it has a structure of a $\p_{n-k+1}$ - representation.
%Here $\p_{n-k+1}$ is the Lie algebra of the ${n-k+1}$-mirabolic.
\pause
\item$D^k(\pi)=D_2^k(\pi)= (D_1^k(\pi))_{gen, \fv_{n-k+1}}$. Here $\fv_{n-k+1}$  is the nil-radical of $\p_{n-k+1}$ and $\cdot_{gen, \fv_{n-k+1}}$ denotes the generalized co-invariants.
\pause
\item $D_3^k(\pi)= (D_1^k(\pi))_{\fv_{n-k+1}}$.
\pause
\item $depth(\pi)$ -- the largest part in the associated partition of $\pi$
\end{itemize}
\end{definition}



\end{frame}



\begin{frame}
    \frametitle{Examples}
\pause
\begin{itemize}
\item
$D_1^{1}(\pi )=\pi |_{G_{n-1}}$,
$$depth(\pi
)=1 \Longleftrightarrow \pi \text{ is  f.d.  }\Longleftrightarrow D_i^{k}(\pi
)=0\text{ for any }k>1.$$
\pause
\item $D_i^{n}=(\Phi) ^{n-1}$ is the Whittaker functor. $$depth(\pi )=n\Longleftrightarrow D_2^{n}(\pi )\neq 0$$ \end{itemize}

\end{frame}



\begin{frame}
%    \frametitle{Results}
\thispagestyle{empty}
\begin{thm}[A. - Gourevitch - Sahi]
\pause
%\label{thm:main} Let $\mathcal{M}(G_{n})$ denote the category of smooth

%admissible {Fr\'{e}chet} representations of moderate growth.
Let $\mathcal{M}^{d}_{\infty}(G_{n})$ denote the subcategory of representations of depth $\leq d$. Then
\pause
\begin{itemize}
\item \label{mainit:Adm} $D_2^d$ defines a functor $\mathcal{M}^{d}_{\infty}(G_n) \to
\mathcal{M}_{\infty}(G_{n-d})$.
\pause
\item \label{mainit:Exact} The functor $D_2^d : \mathcal{M}^{d}_{\infty}(G_n) \to
\mathcal{M}_{\infty}(G_{n-d})$ is exact.
\pause
\item \label{mainit:nilp} For any $\pi \in \mathcal{M}^{d}_{\infty}(G_n), \,
D_{2}^d(\pi)=D_{1}^d(\pi)$.
\pause
\item \label{mainit:Zero} $D_2^{k}|_{\mathcal{M}^{d}_{\infty}(G_{n})}=0$ for any $k>d$.
\pause
\item \label{mainit:Prod} Let $n=n_1+...+n_d$ \pause and let $\chi_i$ be characters of $G_{n_i}$. 
    \pause Let $\pi= \chi_1 \times ... \times \chi_d \in \cM^{d}_{\infty}(G_n)$  
denote the corresponding degenerate principal series representation.
\pause Then $depth(\pi)=d$
    and\\
    $D_1^d(\pi)=D_2^d(\pi)=D_3^d(\pi)\cong(\chi_1)|_{G_{n_1-1}} \times ... \times (\chi_d)|_{G_{n_d-1}}$ \pause 
   \item For a unitarizable representation $\pi$
$$D_1^d(\pi)=D_2^d(\pi)=D_3^d(\pi)=A(\pi)$$%\oplus \tau$$ 
\end{itemize}
\end{thm}
\end{frame}



\begin{frame}
    \frametitle{Steps in the proof}
\pause
\begin{enumerate}
\item We prove admissibility of $D_1^d(\pi)$ in the HC-category -- ${\mathcal{M}_{HC,d}(G)}$
\pause
\item We deduce $D_{2}^d|_{\mathcal{M}_{HC,d}(G)}=D_{1}^d|_{\mathcal{M}_{d}(G_{n})}$.
\pause
\item We deduce $D_i^{k}|_{\mathcal{M}_{{HC,d}}(G_{n})}=0$ for any $k>d$.
\pause
\item We prove exactness of $D_1^i$ and Hausdorffness of $D_1^i(\pi)$ in the smooth category
\pause
\item Using the Hausdorffness we deduce 1-3 in the smooth category
\pause
\item Using the exactness we prove the product formula in the smooth category
  \pause
   \item We deduce from the product formula that  for a unitarizable representation $\pi$
$$D_1^d(\pi)=D_2^d(\pi)=D_3^d(\pi)=A(\pi)$$%\oplus \tau$$ 
\end{enumerate}


\end{frame}



\begin{frame}
    \frametitle{Applications}

\pause
\begin{itemize}
\item Uniqueness of degenerate Whittaker
functionals for unitary representations.
\pause
\begin{multline*}
Wh_{(n_1, \dots, n_k)}(\tau)  = D_{3}^{n_k}(\cdots(D_{3}^{n_1}(\tau))\cdots)\twoheadleftarrow
D_{1}^{n_k}(\cdots(D_{1}^{n_1}(\tau))\cdots)\\ 
\twoheadleftarrow  D_{1}^{n_k}(\cdots(D_{1}^{n_1}(\chi_1 \times \cdots \times \chi_d))\cdots)
\end{multline*}
\pause
\item Computation of  adduced representations  of Speh complementary series
\pause
$$\chi_1 \times \chi_2 \times \chi_3\times \chi_4 \twoheadrightarrow \Delta_{4m}$$
\pause
\begin{multline*}\Delta_{4m-4} \twoheadleftarrow \chi_1|_{G_{m-1}} \times \chi_2|_{G_{m-1}} \times \chi_3|_{G_{m-1}}\times \chi_4|_{G_{m-1}} = \\ =D_1^4(\chi_1 \times \chi_2 \times \chi_3\times \chi_4 )\twoheadrightarrow D_1^4(\Delta_{4m}) \twoheadrightarrow A(\Delta_{4m})
\end{multline*}
\end{itemize}


\end{frame}



\begin{frame}
    \frametitle{Admissibility}
\pause
We need -- $D_1^d(\pi)$ is finitely generated over $\mathfrak{n}_{n-d}$
\pause

We know -- $D_1^d(\pi)$ is finitely generated over $\mathfrak{n}_{n-d+1}$
\pause

We use
\pause

\begin{itemize}
\item  Annihilator variety -- $\cV(\pi)$
\pause
\item Associated variety --  $AV(\pi)$
\pause
\item  $AV(\pi) \subset \cV(\pi) $

\end{itemize}

\pause
\begin{multline*}depth(\pi)=d 
 \Rightarrow \text{constrains on } \cV_{\g}(\pi)
 \Rightarrow \\ \Rightarrow   AV_{\mathfrak{n}_{n-d+1}}(D_{1}^{d}(\pi ))\subset \mathfrak{n}_{n-d}^{\ast }  \Rightarrow D_1^d(\pi) \text{ is f.g. over }\mathfrak{n}_{n-d}
\end{multline*}

\end{frame}



\begin{frame}
    \frametitle{Exactness and Hausdorffness}
\pause
\begin{itemize}
\item Strategy 1 -- $\Phi$ is equivalent to a restriction functor $\Rightarrow$ has to be exact
\pause

 {\color{red}{Problem -- we do not have the language}}
\pause
\item Strategy 2 -- [CHM] method: reduction to  acyclicity of principal series and proof orbit by orbit.

\pause
 {\color{red} Problems}
 \begin{enumerate}
 {\color{red}
\pause
 \item     Unlike [CHM]  there are  $\infty$ orbits
 \pause
    \item  Unlike [CHM]  there are bad orbits}
\end{enumerate}

\end{itemize}
\pause
  Solution -- to introduce a class of ``good" $\p_n$ representations
\end{frame}



\begin{frame}
    \frametitle{Good $\p_n$ representations}
\pause


\begin{example}
$$\Sc(P_n/Q)$$
\end{example}
\pause
\begin{keylemma}
\begin{itemize}
\item $L^i \Phi(\Sc(P_n/Q))=0$ for $ i>0$
\pause
\item $\Phi(\Sc(P_n/Q))=\Sc(Z_0)$ for suitable $Z_0 \subset Z:=P_n/(QV_{n})$
\end{itemize}
\end{keylemma}

\end{frame}



\begin{frame}
    \frametitle{The product formula}
\pause
The BZ product formula:
$$D^k(\pi \times \tau)\sim\sum D^{l}(\pi)\times D^{k-l}( \tau)$$
\pause
{\color{red}
Problems}
\pause
\begin{itemize}
{\color{red}\item Not true for $D_{1,2}^k$
\pause
\item might be true for  $D_{3}^k$ but without exactness we can't prove it.
\pause
\item we do not have  appropriate language of $\infty$ dimensional bundles.
}
\end{itemize}

\pause
Compromise -- prove it only for the highest derivatives and only for characters.

\pause
Method --  exactness, key lemma, induction
\end{frame}




\end{document}
