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\begin{document}

\title{{Wave-front sets of distinguished representations}}
\author[Dmitry Gourevitch]{Dmitry Gourevitch\\
Weizmann Institute of Science, Israel
\url{http://www.wisdom.weizmann.ac.il/~dimagur}\\
%EndAName
%$\,$\\
Representation theory \& Algebraic Geometry seminar, WIS\\
j.w. Eitan Sayag\\
following Harris-Weich, Prasad, Sakellaridis, Vogan\\
arXiv:2001.11746, arXiv:2001.11750}
%\url{http://www.wisdom.weizmann.ac.il/~dimagur}
\date{July 2020}
\maketitle

\begin{frame}%

\begin{stepitemize}
\item $F$: local field, $\bf G$: reductive group over $F$, $\bf H\subset G$ algebraic subgroup,  $G:={\bf G}(F)$, $H:={\bf H}(F)$, $\fg:=Lie({\bf G})$, $\fg_0:=Lie(G)$, $\cN(\fg^*)$:=nilpotent cone, $\fh := Lie({\bf H})$, $\fh_0:=Lie(H)$.
\item  $\pi \in \Irr(G)$ is called \emph{H-distinguished} if $(\pi^*)^H\neq 0$.
\item $\Irr_H(G)$:=all smooth admissible $H$-distinguished irreps of $G$.

\item $WF^{o}(\pi)\subset \cN(\fg_0^*)$:=  union of top G-orbits in the wave-front set of $\pi$.
\end{stepitemize}
\pause
\begin{conj}
For any $\pi\in \Irr_H(G)$, $WF^{o}(\pi)|_{\fh_0}\ni 0$.
\end{conj}
\pause
\begin{theorem}[G.-Sayag 2020]
\begin{enumerate}[(i)]
\item If $F$ is Archimedean and $\bf H \subset G$ is spherical then
$${\bf G} (WF^{0}(\pi))|_{\fh}\ni 0.$$
\item If $F$ is p-adic then $WF^{o}(\pi)\subset \overline{G\fh_0^{\bot}}\subset \fg_0^*$
\end{enumerate}
\end{theorem}
$\bf H$ is called spherical if it has an open orbit on the flag variety $\bf G/B$.
\end{frame}

\begin{frame}%
%EndExpansion
%\thispagestyle{empty}
\frametitle{Uniform applications}

\begin{itemize}
\item $\Irr_H(G)$:=all smooth admissible $H$-distinguished irreps.

\item $WF^{o}(\pi)\subset \cN(\fg_0^*)$:=  union of top G-orbits in the wave-front set of $\pi$.
\end{itemize}
%\pause
\begin{theorem}[G.-Sayag 2020]
\begin{enumerate}[(i)]
\item If $F$ is Archimedean and $\bf H \subset G$ is spherical then
\vspace{-2mm}
$${\bf G} (WF^{o}(\pi))|_{\fh}\ni 0.$$
\vspace{-7mm}
\item If $F$ is p-adic then $WF^{o}(\pi)\subset \overline{G\fh_0^{\bot}}$
\end{enumerate}
\end{theorem}
Assume $\mathrm{char}\, F=0$.

\begin{cor}[following Prasad - Sakellaridis]
$\forall$  unimodular spherical $H\subset G$,\\
$\quad \quad \max_{\pi \in \Irr_H(G)}\dim \WF^o(\pi)=\dim \cN(\fg_0^*)\cap G\fh_0^{\bot}\quad\text{ and }$
\begin{enumerate}[(i)]

\item
$ \exists$ inf. dim. $\pi \in \Irr_H(G)\iff\exists$ unipotent $u\in G - H$.
\item for quasisplit $G$:
$\exists$ generic $\pi \in \Irr_H(G)\iff H\cap U =\{1\}$ for some maximal unipotent subgroup $U\subset G$.

\end{enumerate}
\end{cor}

\end{frame}

\begin{frame}%
%EndExpansion
%\thispagestyle{empty}
\frametitle{Archimedean setting \& applications to symmetric pairs}
From now on $F:=\R$.
\begin{theorem}[Rossmann 1995, using Borho-Brylinski + Joseph ]
$\forall \pi\in \Irr(G), \, \WF^o(\pi)$ lies in a unique $\bf G$-orbit  $\cO(\pi)\subset \cN(\fg^*)$.
\end{theorem}
\pause
Let $\bf H\subset G$ be symmetric sbgrp, and $G_H^{\R}$ the corresponding real form of $\bf G$.
Kostant-Sekiguchi: $\bf H$-orbits on $\fh^{\bot}\cong(\fg/\fh)^*\leftrightarrow$
real $G_H^{\R}$-orbits on $\fg^*$.

Prasad philosophy: $G_H^{\R}$ describes  $H$-distinguished $\mathrm{rep}^{\mathrm{ns}}$
of $G$.
\pause
\begin{cor}[of our thm]
$\forall \pi \in \Irr_H(G), \, \cO(\pi)$ includes a real $G_H^{\R}$-orbit.
\end{cor}
Real orbits are classified by Ohta and Djokovic.
\pause
\begin{conj}[Prasad]
For split $G,H$:
$\exists$ tempered $\pi\in \Irr_H(G)\Rightarrow \exists$ generic $\pi\in \Irr_H(G)$.
\end{conj}
\pause
Our theorem + result of Harris $\Rightarrow$ conjecture holds for most pairs.
\end{frame}

\begin{frame}%
%EndExpansion
%\thispagestyle{empty}
\frametitle{Applications to strongly spherical pairs}

\begin{cor}
Assume that $\bf H$ is reductive and $\bf \Delta H \subset G \times H$ spherical.\\ Let $\pi\in \Irr(G), \, \tau \in \Irr(H)$ with $\Hom_H(\pi|_H,\tau)\neq 0$. Then  $$\cO(\tau)\subset \cO(\pi)|_{\fh}.$$
\end{cor}
\pause
For classical $G$, $\cO(\pi)$ is described by its Jordan partition $\lambda(\pi)$.


\begin{cor}
Let $\bf G$ be $\GL_{n+1},O_{n+1}$ or $U_{n+1}$ and $\bf H$ be $\GL_{n},O_{n}$ or $U_{n}$ respectively.
Let $\pi\in \Irr(G), \, \tau \in \Irr(H)$ with $\Hom_H(\pi|_H,\tau)\neq 0$. Then  for any index $i$:
$$|\lambda(\pi)^t_i-\lambda(\tau)^t_i|\leq 1.$$
\end{cor}


\end{frame}

\begin{frame}%
%EndExpansion
\thispagestyle{empty}
\frametitle{Applications to Jacquet quotients}

\begin{cor}
Let $P=LU\subset G$ be parabolic subgroup, $\pi \in \Irr(G)$, and $\tau\in \Irr(L)$ be a quotient of the Jacquet module $r_U(\pi)$. Then
$\cO(\tau)\subset \cO(\pi)|_{\fp^*}$.
\end{cor}
\vspace{-4mm}
$$\fl=\fp/\fu, \quad \cO(\tau)\subset \fl^*\subset \fp^* \twoheadleftarrow \fg^*\supset \cO(\pi)$$
\pause
\vspace{-4mm}
\begin{proof}
Let $H:=\{(p,pU)\in G\times L\}\subset G\times L$. Then $\Hom_H( \pi\hat \otimes \tilde\tau,\C)\neq 0$.
\end{proof}
\pause
For a maximal parabolic of $\GL_n$:
$$\text{if}\begin{pmatrix}A & 0 \\
0 & B \\
\end{pmatrix}\in \cO(\tau) \text{ then }\begin{pmatrix}A & C \\
0 & B \\
\end{pmatrix}\in \cO(\pi) \text{ for some }C.$$
\pause
\vspace{-3mm}
\begin{prop}[Zhang, work in progress]
$\exists C$ s. t. $\begin{pmatrix}A & C \\
0 & B \\
\end{pmatrix}$ has given Jordan partition $\mu\iff c_{\lambda(A),\lambda(B)}^{\mu}\neq 0$
(Littlewood-Richardson coefficient)
\end{prop}

\end{frame}
\begin{frame}
\thispagestyle{empty}
\frametitle{Twisted induction of spherical pairs}

Let $P=LU\subset G$ parabolic subgroup,  $\bf S\subset L$ be a spherical subgroup. Let $H:=SU$.
Fix algebraic character $\psi:U\to \R$, and let $\chi:H\to \C^{\times}$ be a character s.t. $\chi(u):=\exp(i\psi(u))\, \forall u\in U$. Consider $d\psi\in \fu^*\subset \fh^*$.
$$\Irr_{H,\chi}(G):=\{\pi \in \Irr(G)\, : \Hom_{H}(\pi|_H,\chi)\neq 0\}$$

\begin{theorem}[G.-Sayag 2020]
For all $\pi\in \Irr(G)_{H,\chi}, \quad d\psi\in \cO(\pi)|_{\fh}$
\end{theorem}
\pause
%Examples for $\bf G=\GL_n$:
Shalika models: ${\bf G}=\GL_{2n}, {\bf L}=\GL_n\times \GL_n,\, {\bf S}=\Delta\GL_n, \,\fu=Mat_{n\times n}$,
$$H=\left\{\begin{pmatrix}g & A \\
0& g \\
\end{pmatrix} \right\}, \, \psi(A)=\tr(A).$$
\begin{cor}For all $\pi\in \Irr_{H,\chi}(G), \,\,\lambda(\pi)$   consists of even parts.
\end{cor}


\end{frame}

\begin{frame}
%\frametitle{Twisted version - induction of spherical pairs}
\thispagestyle{empty}
$P=LU\subset G$ parabolic ,  $\bf S\subset L$  spherical,  $H:=SU, \psi:U\to \R$,  $\chi:H\to \C^{\times}$  character s.t. $\chi(u):=\exp(i\psi(u))\, \forall u\in U$.
%$$\Irr_{H,\chi}(G):=\{\pi \in \Irr(G)\, : \Hom_{H}(\pi|_H,\chi)\neq 0\}$$

\begin{theorem}[G.-Sayag 2020]
For all $\pi\in \Irr(G)_{H,\chi}, \quad d\psi\in \cO(\pi)|_{\fh}$
\end{theorem}
%Examples for $\bf G=\GL_n$:
\begin{itemize}
\item Shalika models: ${\bf G}=\GL_{2n}, {\bf L}=\GL_n\times \GL_n,\, {\bf S}=\Delta\GL_n, \,\fu=Mat_{n\times n}$,
$$H=\left\{\begin{pmatrix}g & A \\
0& g \\
\end{pmatrix} \right\}, \, \psi(A)=\tr(A).$$
%\begin{cor}
Cor. For all $\pi\in \Irr_{H,\chi}(G), \,\,\lambda(\pi)$   consists of even parts.
\vspace{1mm}
%\end{cor}
\item Klyachko models: ${\bf G}=\GL_{n+k}, \, {\bf L}=\GL_{n}\times (\GL_1)^k, \, {\bf S}=\Sp_{n},$
$$H=\left\{\begin{pmatrix}g & A \\
0& B \\
\end{pmatrix}: g\in \Sp_{n},\, B \text{ upper-uni-triangular}\right\},\psi(B)=\sum B_{i,i+1}$$
\vspace{-4mm}
 \end{itemize}
 \pause
\begin{cor}
 $\forall\pi\in \Irr_{H,\chi}(G),\, \#\{i  :  \lambda(\pi)^t_i \text{ is odd}\}=k.$\\
 \end{cor}
\vspace{-1mm}
 G.-Offen-Sahi-Sayag: for unitary $\pi$, $$\pi\in \Irr_{H,\chi}(G)\iff \#\{i  :  \lambda(\pi)^t_i \text{ is odd}\}=k.$$
\
\end{frame}

\begin{frame}
\frametitle{Twisted induction of strongly spherical pairs}
\thispagestyle{empty}
$P=LU\subset G$ parabolic,  $\bf S\subset L$  with $\Delta {\bf S}\subset {\bf S}\times {\bf L}$ spherical,  $H:=SU, \psi:U\to \R$,  $\chi:H\to \C^{\times}$  char s.t. $\chi(u):=\exp(i\psi(u))\, \forall u\in U$.
\begin{cor}[G.-Sayag 2020]
Let $\pi\in \Irr(G), \tau\in \Irr(S)$ with $\Hom_{H}(\pi|_H,\tau \otimes \chi)\neq 0$. Then
$$\cO(\tau)+d\psi\subset \cO(\pi)|_{\fh}$$
\end{cor}
%Examples for $\bf G=\GL_n$:
\pause
Rankin-Selberg models: ${\bf G}=\GL_{n+k}, \, {\bf L}=\GL_{n+1}\times (\GL_1)^{k-1}, \, {\bf S}=\GL_{n},$
$$H=\left\{ \begin{pmatrix}g & 0 & * & \dots & *\\
0 & 1 & \underline{*} & \dots & * \\
\vdots & \ddots & \ddots\ & \ddots & \vdots \\

0 & \dots & 0 & 1 & \underline{*} \\
0 & \dots & \dots & 0 & 1 \\
\end{pmatrix}\right\}, \quad \psi= \text{ sum of underlined entries}$$
\begin{cor}
If $\Hom_{H}(\pi|_H,\tau \otimes \chi)\neq 0$ then $|\lambda(\pi)^t_i-\lambda(\tau)^t_i|\leq 1$ for any index $i$.
\end{cor}

\end{frame}

\begin{frame}
%\frametitle{Twisted induction of strongly spherical pairs}
\thispagestyle{empty}
$P=LU\subset G$ parabolic,  $\bf S\subset L$  with $\Delta {\bf S}\subset {\bf S}\times {\bf L}$ spherical,  $H:=SU, \psi:U\to \R$,  $\chi:H\to \C^{\times}$  char s.t. $\chi(u):=\exp(i\psi(u))\, \forall u\in U$.
\begin{cor}[G.-Sayag 2020]
Let $\pi\in \Irr(G), \tau\in \Irr(S)$ with $\Hom_{H}(\pi|_H,\tau \otimes \chi)\neq 0$. Then
\vspace{-3mm}
$$\cO(\tau)+d\psi\subset \cO(\pi)|_{\fh}$$
\end{cor}
%Examples for $\bf G=\GL_n$:
\begin{itemize}
\item Rankin-Selberg: ${\bf G}=\GL_{n+k}, \, {\bf L}=\GL_{n+1}\times (\GL_1)^{k-1}, \, {\bf S}=\GL_{n},$
$$H=\left\{ \begin{pmatrix}g & 0 & * & \dots & *\\
0 & 1 & \underline{*} & \dots & * \\
\vdots & \ddots & \ddots\ & \ddots & \vdots \\

0 & \dots & 0 & 1 & \underline{*} \\
0 & \dots & \dots & 0 & 1 \\
\end{pmatrix}\right\}, \quad \psi= \text{ sum of underlined entries}$$

\item Bessel models: analogues for $O(p,q), U(p,q), O_n(\C)$.
\end{itemize}
\begin{cor}[holds for both kinds of models: Rankin-Selberg and Bessel]
If $\Hom_{H}(\pi|_H,\tau \otimes \chi)\neq 0$ then $|\lambda(\pi)^t_i-\lambda(\tau)^t_i|\leq 1$ for any index $i$.
\end{cor}
Related to the non-tempered Gan-Gross-Prasad conjecture.
\end{frame}

\begin{frame}
\frametitle{Proof ingredients}

\begin{theorem}[Wen-Wei Li 2019]
For any nilpotent orbit $\cO$, $2\dim \cO\cap \fh^{\bot}\leq \dim \cO$.
\end{theorem}
\pause
\begin{stepitemize}
\item $\cU_n(\fg)$ - PBW filtration on universal enveloping algebra.
\item  $\gr\cU(\fg)\cong S(\fg)\cong \Pol(\fg^*)$.
\item For a $\fg$-module $M$, $Ann(M)\subset \cU(\fg)$ - annihilator, and\\ $\Anv(M)\subset \fg^*$ - annihilator variety$:=$ zero set of symbols of $Ann(M)$
\end{stepitemize}
\pause
\begin{thm}[Rossmann 1995]
For all $\pi \in \Irr(G), \,\Anv(\pi)=\overline{\cO(\pi)}$.
\end{thm}
\pause
For finitely-generated $M$, define $\Asv(M)$ to be the set of common zeros of symbols of annihilators of the generators.
\pause
\begin{thm}[Gabber-Joseph]
$2\dim \Asv(M)\geq\dim \Anv(M)$.
\end{thm}
% \begin{lemma}
% If $M$ is generated by an $\fh$-finite vector then $\Asv(M)\subset \Anv(M)\cap \fh^{\bot}$.
% \end{lemma}
\end{frame}

\begin{frame}
%\frametitle{Proof ingredients}
\thispagestyle{empty}
\begin{itemize}
\item For any nilpotent orbit $\cO$, $2\dim \cO\cap \fh^{\bot}\leq \dim \cO$.
\item $\Anv(\pi)=\overline{\cO(\pi)}$
\item For f.g. $M$, $2\dim \Asv(M)\geq \dim \Anv(M)$.
\item If $M$ is generated by an $\fh$-finite vector then $\Asv(M)\subset \Anv(M)\cap \fh^{\bot}$.
\end{itemize}
\pause
\begin{theorem}[G. - Sayag 2020]
For any $\pi \in \Irr(G)_H, \, \cO(\pi)\cap \fh^{\bot}\neq \emptyset$.
\end{theorem}
\pause
\begin{proof}
Let $\eta\neq 0\in (\pi^*)^H$ and $M:=\cU(\fg)\eta\subset \pi^*$.
$M$ is non-degenerately paired with $\pi$, thus $Ann(M)=Ann(\pi)$ and $\Anv(M)=\Anv(\pi)$.
From
\pause
\vspace{-1mm}
\begin{align}
2\dim \Asv(M)\geq \dim \Anv(M)&=\dim \Anv(\pi)=\dim \cO(\pi),\\
\Asv(M)\subset \Anv(M)&\cap \fh^{\bot}, \text{ and }\\
\text{intersection with }\fh^{\bot} \text{ reduces }&\text{dimension times }2,
\end{align}
\vspace{-1mm}
we get that $\cO(\pi)$ intersects $\fh^{\bot}$.
\end{proof}
\pause
Twisted version  uses Kazhdan filtration + other ideas from $W$-algebras.
\end{frame}




\begin{frame}

\frametitle{Definition of the wave front set}
Assume $\mathrm{char}\, F=0$.
%
\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{itemize}
\item Define $\WF(\pi)=\{{\mathcal{O}} \, : \, c_{{\mathcal{O}}} \neq 0\}$.
%
\item Say $\cO<\cO'$ if $\cO\subset \overline{\cO'}$.
\item $\WF^o(\pi):=$ union of maximal elements of $\WF(\pi)$.
%
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Evidence for the $p$-adic case, for ${\bf G}=\GL_n$}

\begin{stepitemize}
\item Klyachko models: Offen-Sayag
\item Jacquet quotients of ladder representations: Lapid - Minguez
\item Restriction from $\GL_{n+1}$ to $\GL_n$ for representations of Arthur type: Maxim Gurevich, Kei Yuen Chan
\end{stepitemize}
\end{frame}

\begin{frame}
\frametitle{$p$-adic theorem}
$\forall$ nilpotent $\varphi\in \fg^*$, choose $\mathrm{sl}_2$-triple $f,h,e\in \fg$ using Killing form + Jacobson-Morozov; let $\fv_{\varphi}:=\fg^h_{\geq 2}\subset \fg$, $V_{\varphi}:=\Exp(\fv_{\varphi})\subset G$.
\begin{definition}
For any rep-n $\Pi$, let $\WO(\Pi):=\{\varphi\in \fg^* \, : \Pi_{V_{\varphi},\varphi}\neq 0\}$.
\end{definition}
\pause
\begin{thm}[Moeglin-Waldspurger 1987]
If $F$ is $p$-adic then $\forall\pi\in \Irr(G)$, $\overline{\WO(\pi)}=\overline{\WF^o(\pi)}.$
\end{thm}
\pause
\begin{thm}[G.-Sayag 2020]
$G\fh^{\bot}\cap \cN\subset \WO(\Sc(G/H)) \subset \overline{G\fh^{\bot}}\cap \cN$.
\end{thm}
\end{frame}


\begin{frame}
\frametitle{Ingredients and sketch of proof}
\thispagestyle{empty}
$\forall$ nilpotent $\varphi\in \fg^*$, choose $\mathrm{sl}_2$-triple $f,h,e\in \fg$ using Killing form + Jacobson-Morozov; let $\fv_{\varphi}:=\fg^h_{\geq 2}\subset \fg$, $V_{\varphi}:=\Exp(\fv_{\varphi})\subset G$.
\begin{definition}
For any rep-n $\Pi$, let $\WO(\Pi):=\{\varphi\in \fg^* \, : \Pi_{V_{\varphi},\varphi}\neq 0\}$.
\end{definition}
\begin{stepitemize}

% \item Moment map $\mu:T^*(G/H) \to \fg^*$ given by $\mu_x=da_x^*$, where $a_x:G \to G/H$ action map. Image: $G\fh^{\bot}$.

\item Kazhdan action of $F^{\times}$ on $\fg^*$:   $t\cdot \psi:=t^2 \exp(th)\psi$. Fixes $\varphi$.

\item $\Sc(G/H)_{V_{\varphi},\varphi}\neq 0\iff \varphi(\fv_{\varphi}\cap g(\fh))=\{0\}$ for some $g\in G$.
\end{stepitemize}
\pause
\begin{thm}[G.-Sayag 2020]
$G\fh^{\bot}\cap \cN\subset \WO(\Sc(G/H)) \subset \overline{G\fh^{\bot}}\cap \cN$.
\end{thm}
\pause
\begin{proof}
%\begin{enumerate}[(i)]
%\item
(i) $\varphi \in g(\fh^{\bot}) \Rightarrow\varphi(\fv_{\varphi}\cap g(\fh))=\{0\} \Rightarrow \Sc(G/H)_{V_{\varphi},\varphi}\neq 0 $.\\
\pause
(ii) $\Sc(G/H)_{V_{\varphi},\varphi}\neq 0 \Rightarrow \varphi(\fv_{\varphi}\cap g(\fh))=\{0\}\Rightarrow \varphi\in (\fg^*)^h_{> -2}+g(\fh^{\bot})$.
Kazhdan action preserves $\varphi$ and$\,G\fh^{\bot}$,$\,$contracts $(\fg^*)^h_{> -2}\,$thus$\,\varphi \in \overline{G\fh^{\bot}}$
%\end{enumerate}
\end{proof}

\end{frame}



\end{document}



