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\begin{document}

\title[Fourier coefficients]{{\Large Fourier coefficients of automorphic forms \&\\ applications to minimal and next-to-minimal forms.}}
\author[Dmitry Gourevitch]{Dmitry Gourevitch\\
Weizmann Institute of Science, Israel
\url{http://www.wisdom.weizmann.ac.il/~dimagur}\\
%EndAName
$\,$\\
Automorphic Structures in String Theory, SCGP, Stony Brooke, NY$\,$\\
j.w. H. P. A. Gustafsson, A. Kleinschmidt, D. Persson, and S. Sahi\\$\,$\\
Following Piatetski-Shapiro--Shalika, Ginzburg--Rallis--Soudry, Moeglin-Waldspurger, Jiang--Liu--Savin, Gomez, Ahlen, Hundley--Sayag, Green-Miller-Vanhove, Kazhdan--Polishchuk, Bossard--Pioline}
%\url{http://www.wisdom.weizmann.ac.il/~dimagur}
\date{March 2019}
\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{Definitions}

\begin{stepitemize}
\item $\mathbb{K}$: number field, $\A:=\A_{\K}$, $\bf G$: reductive group over $%
\mathbb{K}$, $\Gamma:={\bf G}(\K)$, $G:={\bf G}(\A)$, $\fg:=Lie(\Gamma)$.
\item  Fix a semisimple $H\in \g$, and let $\g_i:=\g_i^{H}$ denote the eigenspaces of $ad(H)$.
Assume that all the eigenvalues $i$ lie in $\mathbb{Q}$.
\item Let $f \in \g_{-2}$. Call  $(H,f)\in \fg\times \fg$  a \emph{Whittaker pair}.
\item Define $\fn:=\fn_{H,f}:=(\fg_1\cap \fg^f)\oplus \bigoplus_{i> 1}\g_i, \, N:=\Exp(\fn)(\A).$
\item  Fix a non-trivial unitary additive  character $\psi:\K\backslash \A\to \C$ and define $\chi_{f}:N\to \C$ by $\chi_{f}(\Exp X):=\psi(\langle f, X \rangle).$
\item $C^{\infty}(\Gamma \backslash G):=$ functions on $\Gamma \backslash G$ smooth on $G_{\infty}$ and finite under $K_{\mathrm{fin}}$.
\item Let $[N]
:=(\Gamma \cap N)\backslash N$. For $\eta \in C^{\infty}(\Gamma \backslash G)$, define  \emph{Fourier coefficient} $$\Fou_{H,f}[\eta](g):=\int_{[N]}\eta(ng)\chi_f(n)^{-1}dn.$$


%  \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{Two central cases of Fourier coefficients}
$[H,f]=-2f,$
$\fn=(\fg_1\cap \fg^f)\oplus \bigoplus_{i> 1}\g_i, \, N=\Exp(\fn)(\A),$ $\eta \in C^{\infty}(\Gamma \backslash G)$,
$$\Fou_{H,f}[\eta](g):=\int_{[N]}\eta(ng)\chi_f(n)^{-1}dn.$$

\pause
%Two central special cases: %Some examples for $G=\GL_4(\F)$:
\begin{stepitemize}
\item Neutral Fourier coefficient, coming from $\fsl_2$-triple (e,H,f), {\it e.g.}:
$$
H=\left(
   \begin{array}{cccc}
     1 & 0 & 0 & 0 \\
     0 & -1 & 0 & 0 \\
     0 & 0 & 1 & 0 \\
     0 & 0 & 0 & -1 \\
   \end{array}
 \right) f=\left(
   \begin{array}{cccc}
     0 & 0 & 0 & 0 \\
     1 & 0 & 0 & 0 \\
     0 & 0 & 0 & 0 \\
     0 & 0 & 1 & 0 \\
   \end{array}
 \right)  \fn=\left(
   \begin{array}{cccc}
     0 & \underline{*} & 0 & * \\
     0 & 0 & 0 & 0 \\
     0 & * & 0 & \underline{*} \\
     0 & 0 & 0 & 0 \\
   \end{array}
 \right)$$

 \item Whittaker coefficient $\cW_f$, with $N$ maximal unipotent, {\it e.g.}:
$$
H=\left(
   \begin{array}{cccc}
     3 & 0 & 0 & 0 \\
     0 & 1 & 0 & 0 \\
     0 & 0 & -1 & 0 \\
     0 & 0 & 0 & -3 \\
   \end{array}
 \right)f=
\left(
   \begin{array}{cccc}
     0 & 0 & 0 & 0 \\
     1 & 0 & 0 & 0 \\
     0 & 0 & 0 & 0 \\
     0 & 0 & 1 & 0 \\
   \end{array}
 \right)
 \fn=\left(
   \begin{array}{cccc}
     0 & \underline{*} & * & * \\
     0 & 0 & * & * \\
     0 & 0 & 0 & \underline{*} \\
     0 & 0 & 0 & 0 \\
   \end{array}
 \right)$$
\end{stepitemize}
\pause
Coefficients that are both neutral and Whittaker  are Eulerian by local uniqueness of Whittaker models.
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\frametitle{Examples of Fourier coefficients}
$[H,f]=-2f,$
$\fn=(\fg_1\cap \fg^f)\oplus \bigoplus_{i> 1}\g_i, \, N=\Exp(\fn)(\A),$ $\eta \in C^{\infty}(\Gamma \backslash G)$,
$$\Fou_{H,f}[\eta](g):=\int_{[N]}\eta(ng)\chi_f(n)^{-1}dn.$$


 Comparison for $G=\GL_3(\K)$:
\begin{stepitemize}
\item Neutral Fourier coefficient:
$$
H=\left(
   \begin{array}{ccc}
     1 & 0 & 0  \\
     0 & 0 & 0  \\
     0 & 0 & -1  \\

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

 \item Whittaker  coefficient:
$$
H=\left(
   \begin{array}{ccc}
     1 & 0 & 0  \\
     0 & -3 & 0  \\
     0 & 0 & -1  \\

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

%In both cases $\fn=\fv.$
\end{stepitemize}
\end{frame}%

\begin{frame}
\frametitle{Root exchange}



\begin{stepitemize}
\item  $\fu:=\fg_1/(\fg_1\cap \fg^f)$. $\omega_f(X,Y):=\langle f, [X,Y]\rangle$- symplectic form.

\item $\forall$ isotropic subspace $\fii \subset \fu,$ let $I:=\Exp(\fii)(\A)$ and
$$\, \cF^I_{H,f}[\eta](g):=\int_{[I]} \cF_{H,f}[\eta](ug) \, du $$
\end{stepitemize}
\pause
% Let $\fl^{\bot}$ denote the orthogonal complement to $\fl$ in $\fu$ under the form $\omega_{\varphi}$ and let $L^\bot:=\Exp(\fl^\bot)$.
%  Then,
% \begin{align}\label{eq:FL-as-F}
%
%     \intertext{and}
%     \label{eq:F-as-FL}
%     \cF_{S,\varphi,\varphi'}[\eta](g) &=\sum_{\gamma \in (U/L^\bot)(\K)}\cF_{S,\varphi,\varphi'}^{L}[\eta](\gamma g).
% \end{align}

\begin{lemma}
\begin{enumerate}[(i)]
\item $\cF_{H,f}[\eta](g) =\sum_{\gamma \in (U/I^\bot)(\K)}\cF_{H,f}^{I}[\eta](\gamma g)$
\item For any isotropic subspace $\fj \subset \fu$ with $\dim \fj=\dim \fii$ and $\fj\cap \fii^{\bot}=\{0\}$,
$$\cF_{H,f}^{J}[\eta](g) =\int_{J(\A)} \cF_{H,f}^I[\eta](ug) \, du $$
\end{enumerate}
\end{lemma}

$$\text{For }
H=\left(
   \begin{array}{ccc}
     1 & 0 & 0  \\
     0 & 0 & 0  \\
     0 & 0 & -1  \\

   \end{array}
 \right), f=\left(
   \begin{array}{cccc}
     0 & 0 & 0  \\
     0 & 0 & 0  \\
     1 & 0 & 0  \\
   \end{array}
 \right): \quad \left(
   \begin{array}{cccc}
     0 & \fii & \underline{\fn}  \\
     0 & 0 & \fj  \\
     0 & 0 & 0  \\
   \end{array}
 \right)$$

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\frametitle{Relating different coefficients}
%\thispagestyle{empty}

\begin{stepitemize}
\item  $\WO(\eta):=\{\cO\in \cN(\fg) \, \vert \, \forall \text{ neutral } (h,f) \text{ with }f\in \cO, \, \cF_{h,f}(\eta)\not\equiv 0$\}.
\item Say $(H,f)\succ(S, f)$ if $[H,S]=0$ and $\fg^{f}\cap \fg^H_{\geq 1}\subseteq \fg^{S-H}_{\geq0}$.
\item $f$ is $\K$-distinguished if $\forall$ Levi $\fl\ni f$ defined over $\K$, $\fl=\fg$.
\item $(S,f)$ is called Levi-distinguished if $\exists$ parabolic $\fp=\fl\fu$\\ s.t. $f$ is $\K$-distinguished in $\fl$, and $\fn_{S,f}=\fl_{S,f}\oplus \fu$. \item Whittaker coefficients are Levi-distinguished.
\item For Whittaker pairs with the same $f$ and commuting $H$-s,\\
neutral $\succ$ any $\succ$ Levi-distinguished.
\end{stepitemize}
\pause
\begin{theorem}
Let $(H,f)\succ(S, f)$. Then
\begin{enumerate}[(i)]
\item $\cF_{H,f}[\eta]$ linearly determines  $\cF_{S,f}[\eta]$.
\item If $\Gamma f\in \WO^{\max}(\eta)$ and $\fg^H_{1}= \fg^{S}_{1}=0$ let  $\fv:= \fg^H_{> 1}\cap \fg^{S}_{<1}$. Then
%            \begin{equation}\label{=HfromS_easy}
 $$               \cF_{H,f}[\eta](g) =
                \int_{V(\A)} \cF_{S,f}[\eta](vg) \, dv 
$$
%            where .
\end{enumerate}
\end{theorem}
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%EndExpansion
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%\frametitle{Corollaries}
\thispagestyle{empty}
%\begin{theorem}
%Let $(H,f)\succ(S, f)$. Then $\cF_{H,f}[\eta]$ linearly determines  $\cF_{S,f}[\eta]$. \\
%%\begin{enumerate}[(i)]
%%\item
%%\item
%If in addition $\Gamma f\in \WO^{\max}(\eta)$ and $\fg^H_{1}= \fg^{S}_{1}=0$ then
%%            \begin{equation}\label{=HfromS_easy}
%$$                \cF_{H,f}[\eta](g) =
%                \int_{V(\A)} \cF_{S,f}[\eta](vg) \, dv \, ,$$
%%            \end{equation}
%            where $\fv:= \fg^H_{> 1}\cap \fg^{S}_{<1}$.
%%\end{enumerate}
%\end{theorem}
\begin{theorem}
Let $(H,f)\succ(S, f)$. Then
\begin{enumerate}[(i)]
\item $\cF_{H,f}[\eta]$ linearly determines  $\cF_{S,f}[\eta]$.
\item If $\Gamma f\in \WO^{\max}(\eta)$ and $\fg^H_{1}= \fg^{S}_{1}=0$ let  $\fv:= \fg^H_{> 1}\cap \fg^{S}_{<1}$. Then
%            \begin{equation}\label{=HfromS_easy}
 $$               \cF_{H,f}[\eta](g) =
                \int_{V(\A)} \cF_{S,f}[\eta](vg) \, dv
$$
%            where .
\end{enumerate}
\end{theorem}
\pause
\begin{corollary}
\begin{enumerate}[(i)]
\item If $\eta$ is cuspidal then any $\cO\in \WO^{\max}(\eta)$ is $\K$-distinguished.\\
In particular, $\cO$ is totally even for $G={\mathrm{Sp}}_{2n}$, totally odd for $G=SO(V)$, not minimal for $\mathrm{rk} G>1$, \\ and not next-to-minimal for $\mathrm{rk} G>2, \, G\neq F_4$.
\item If $f\notin \WO(\eta)$ then $\cF_{H,f}(\eta)=0$ for any $H$.
\item Let G be simply-laced, $H$ define a maximal parabolic, $f\in \WO^{\max}(\eta)$:\\
$f$ is minimal $\Rightarrow \cF_{H,f}(\eta)=\cW_f(\eta)$\\
$f$ is next-to-minimal $\Rightarrow \cF_{H,f}(\eta)=\int_{V(\A)}\cW_f(\eta)$.\\
The RHS is frequently Eulerian.
\end{enumerate}
\end{corollary}
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\frametitle{Example }
%\thispagestyle{empty}
%\begin{example}
$G:=\GL(4,\A)$, $f:=E_{21}+E_{43}$,
%, where $E_{21}$ and $E_{43}$ are elementary matrices. Let $h =\diag(1,-1,1,-1)$.
%\begin{example}\label{ex:GLSame}
$H:=\diag(3,1,-1,-3)$,
%\pause
 $h =\diag(1,-1,1,-1)$, $Z=H-h=\diag(2,2,-2,-2), H_t:=h+tZ$.\\
\pause
Then $\fn_{0}\subset \fn_{1/4}\oplus \fii \sim \fn_{1/4}\oplus \fj\subset \fn_{3/4}=\fn_1:$
$$
%\begin{multline*}
\left(
   \begin{array}{cccc}
     0 & - & 0 & - \\
     0 & 0 & 0 & 0 \\
     0 & - & 0 & - \\
     0 & 0 & 0 & 0 \\
   \end{array}
 \right) \subset \left(
   \begin{array}{cccc}
     0 & - & a & - \\
     0 & 0 & 0 & a \\
     0 & * & 0 & - \\
     0 & 0 & 0 & 0 \\
   \end{array}
 \right)  \sim
\left(
   \begin{array}{cccc}
     0 & - & * & - \\
     0 & 0 & 0 & * \\
     0 & 0 & 0 & - \\
     0 & 0 & 0 & 0 \\
   \end{array} \right)$$

$$
%\subset\\
% \subset
% \left(
%    \begin{array}{cccc}
%      0 & - & - & - \\
%      0 & 0 & 0 & - \\
%      0 & 0 & 0 & - \\
%      0 & 0 & 0 & 0 \\
%    \end{array} \right)
\subset
\left(
   \begin{array}{cccc}
     0 & - & - & - \\
     0 & 0 & * & - \\
     0 & 0 & 0 & - \\
     0 & 0 & 0 & 0 \\
    \end{array} \right)
   =
\left(
   \begin{array}{cccc}
     0 & - & - & - \\
     0 & 0 & - & - \\
     0 & 0 & 0 & - \\
     0 & 0 & 0 & 0 \\
   \end{array} \right)$$
%\end{multline*}
    Both $*$ and $-$ denote arbitrary elements. $-$ denotes the entries in $\fg^{H_t}_{>0}$ and $*$ those in $\fg^{H_t}_1$. $a$ denotes equal elements in $\fg^{H_t}_1\cap \fg^f$.
%    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}

\end{frame}%

\begin{frame}
\frametitle{Expressing forms through their Whittaker coefficients}
\begin{theorem}
Any $\cF_{H,f}$ is linearly determined by all Levi-distinguished Fourier coefficients $\cF_{S,F}$ with $\Gamma F\geq \Gamma f$.
\end{theorem}
\pause
\begin{corollary}
\begin{stepenumerate}[(i)]
\item Any $\eta\in C^{\infty}(\Gamma \backslash G)$ is linearly determined by all its Levi-distinguished Fourier coefficients.
\item If all $\cO\in \WO(\eta)$ admit Whittaker coefficients then $\eta$ is linearly determined by its Whittaker coefficients.
\item If $G$ is split and simply-laced, and $\eta$ is minimal or next-to-minimal then all Fourier coefficients of $\eta$ are linearly determined  by Whittaker coefficients of $\eta$.
\end{stepenumerate}
\end{corollary}
\end{frame}%


\begin{frame}
\frametitle{Parabolic minimal Fourier coeff. of next-to-minimal forms}
%\thispagestyle{empty}
\begin{stepitemize}
\item $\g$ split simply laced, $\fh\subset \fg$ Cartan, $\fb = \fh \oplus \fu$ Borel
\item $\alpha$ simple root.
$\fq_\alpha = \fl_\alpha \oplus \fn_\alpha=\fg^{S_{\alpha}}_{\geq 0}$ max. parabolic.
\item $I^{(\perp\alpha)} = \{\beta_1, \ldots, \beta_k \}$  Bourbaki enumeration of the simple roots orthogonal to $\alpha$.

% We say that $\alpha$ is an abelian or Heisenberg simple root in $\fg$ if $\fn_\alpha$ is an abelian or Heisenberg Lie algebra respectively, or, equivalently, if $[\fn_\alpha, \fn_\alpha]$ has dimension zero or one. If $\alpha$ is either abelian or Heisenberg we call it \emph{quasi-abelian}.
%
% Let $I = (\beta_1, \ldots, \beta_n)$ be an enumeration of the simple roots of $\g$ in some order, and let $\fl_i$ be the Levi subalgebra with simple roots $\{\beta_1, \ldots, \beta_i\}$. We will say that $I$ is \emph{abelian} if each $\beta_i$ is abelian in $\fl_i$, and that $I$ is \emph{quasi-abelian} if each $\beta_i$ is abelian or Heisenberg in $\fl_i$.

% TODO : Heisenberg is where affine connects

% From the table we see that, for a simple Lie algebra $\fg$, the Bourbaki enumeration is quasi-abelian if $\fg = E_8$ and abelian otherwise.

% TODO : simply laced?
% Let $G$ be a reductive group and let $I$ be a quasi-abelian enumeration of the simple roots of its Lie algebra. Given an automorphic function $\eta$ on $G(\mathbb{K})\backslash G(\mathbb{A})$ we define functions $A_i[\eta]$, $B_i[\eta]$ and $C_i[\eta]$ on $G(\mathbb{A})$ as follows.

% TODO : Recall notation for $\lie g^*_{-\beta_i}$ and $\lie g^\times_{-\beta_i}$
\item $\forall i \, G\supset L_{i}:=$ Levi given by roots $\beta_1,\dots,\beta_i$
\item  $L_i \supset S_{i}:=$ stabilizer of the root space $\fg_{-\beta_i}$,  $\Gamma_{i} := (L_{i}\cap \Gamma)  / (S_{i}\cap \Gamma)$.
 \item For $f \in \fg^{\times}_{-\alpha}$ and next-to-minimal
$\eta_{\mathrm{ntm}}\in C^{\infty}(\Gamma \backslash G)$ let
%\begin{equation}
 $$   A_i^f[\eta_{\mathrm{ntm}}](g) := \sum_{\gamma \in \Gamma_{i-1}} \sum_{\varphi \in \fg^\times_{-\beta_i}} \mathcal{W}_{\varphi +f}[\eta_{\mathrm{ntm}}](\gamma g)$$
%\end{equation}
\end{stepitemize}
\pause
\begin{theorem}
%\begin{equation}
 $$   \mathcal{F}_{S_\alpha, f}[\eta_\mathrm{ntm}] = \mathcal{W}_f[\eta_\mathrm{ntm}] + \sum_{i=1}^k A_i^f[\eta_\mathrm{ntm}]$$
%\end{equation}
\end{theorem}
\end{frame}%

\begin{frame}
\frametitle{Explanation for $\GL_n$ (PS-Shalika, Ahlen--Gustafsson-Liu- Kleinschmidt-Persson)}
Let $\eta\in C^{\infty}(\Gamma \backslash \GL_n(\A))$. Restrict to the last column and decompose to  Fourier series. All non-trivial characters are conjugate by $\GL_{n-1}(\K)$.
$$\left(
   \begin{array}{ccccc}
     0 & 0 & 0 & 0 & *\\
     0 & 0 & 0 & 0 &* \\
     0 & 0 & 0 & 0 & * \\
     0 & 0 & 0 & 0 & * \\
     0 & 0 & 0 & 0 & 0\\
   \end{array}
 \right)+\left(
   \begin{array}{ccccc}
     0 & 0 & 0 & 0 & *\\
     0 & 0 & 0 & 0 &* \\
     0 & 0 & 0 & 0 & * \\
     0 & 0 & 0 & 0 & \underline{*} \\
     0 & 0 & 0 & 0 & 0\\
   \end{array}
 \right) $$

\end{frame}

\begin{frame}
\frametitle{Explanation for $\GL_n$ }
\thispagestyle{empty}
Let $\eta\in C^{\infty}(\Gamma \backslash \GL_n(\A))$. Restrict to the last column and decompose to  Fourier series. All non-trivial characters are conjugate by $\GL_{n-1}(\K)$.
Conjugate, restrict to the next column and continue
$$\left(
   \begin{array}{ccccc}
     0 & 0 & 0 & 0 & *\\
     0 & 0 & 0 & 0 &* \\
     0 & 0 & 0 & 0 & * \\
     0 & 0 & 0 & 0 & \underline{*} \\
     0 & 0 & 0 & 0 & 0\\
   \end{array}
 \right) = \left(
   \begin{array}{ccccc}
     0 & 0 & 0 & * & *\\
     0 & 0 & 0 & * &* \\
     0 & 0 & 0 & * & * \\
     0 & 0 & 0 & 0 & \underline{*} \\
     0 & 0 & 0 & 0 & 0\\
   \end{array}
 \right)+\left(
   \begin{array}{ccccc}
     0 & 0 & 0 & * & *\\
     0 & 0 & 0 & * &* \\
     0 & 0 & 0 & \underline{*} & * \\
     0 & 0 & 0 & 0 & \underline{*} \\
     0 & 0 & 0 & 0 & 0\\
   \end{array}
 \right)=\cdots$$
 \pause
 \begin{stepitemize}
\item For $\eta_{\mathrm{min}}$, only one column can have a non-trivial character.

\item For $\eta_{\mathrm{ntm}}$,   $\cF_{S_{\alpha},f}$ is a minimal automorphic function on $L_{\alpha}$.
\end{stepitemize}
\pause
$\,$
$$\left(
   \begin{array}{ccccc}
     0 & 0 & * & * & *\\
     0 & 0 & \underline{*} & * &* \\
     0 & 0 & 0 & 0 & 0 \\
     0 & 0 & 0 & 0 & 0 \\
     0 & 0 & 0 & 0 & 0\\
   \end{array}
 \right) =\left(
   \begin{array}{ccccc}
     0 & * & * & * & *\\
     0 & 0 & \underline{*} & * &* \\
     0 & 0 & 0 & * & * \\
     0 & 0 & 0 & 0 & * \\
     0 & 0 & 0 & 0 & 0\\
   \end{array}
 \right)+ \left(
   \begin{array}{ccccc}
     0 & * & * & * & *\\
     0 & 0 & \underline{*} & * &* \\
     0 & 0 & 0 & * & * \\
     0 & 0 & 0 & 0 & \underline{*} \\
     0 & 0 & 0 & 0 & 0\\
   \end{array}
 \right)$$

\end{frame}

\begin{frame}
%\thispagestyle{empty}
%\frametitle{Explanation for $\GL_n$ }
For $\eta_{\mathrm{ntm}}$,   $\cF_{S_{\alpha},f}$ is a minimal automorphic function on $L_{\alpha}$.
$$\left(
   \begin{array}{ccccc}
     0 & 0 & * & * & *\\
     0 & 0 & \underline{*} & * &* \\
     0 & 0 & 0 & 0 & 0 \\
     0 & 0 & 0 & 0 & 0 \\
     0 & 0 & 0 & 0 & 0\\
   \end{array}
 \right) = \left(
   \begin{array}{ccccc}
     0 & * & * & * & *\\
     0 & 0 & \underline{*} & * &* \\
     0 & 0 & 0 & * & * \\
     0 & 0 & 0 & 0 & * \\
     0 & 0 & 0 & 0 & 0\\
   \end{array}
 \right)+\left(
   \begin{array}{ccccc}
     0 & * & * & * & *\\
     0 & 0 & \underline{*} & * &* \\
     0 & 0 & 0 & * & * \\
     0 & 0 & 0 & 0 & \underline{*} \\
     0 & 0 & 0 & 0 & 0\\
   \end{array}
 \right)$$


 $$   \mathcal{F}_{S_\alpha, f}[\eta_\mathrm{ntm}] = \mathcal{W}_f[\eta_\mathrm{ntm}] + \sum_{i=1}^k A_i^f[\eta_\mathrm{ntm}]$$
where $f \in \fg^{\times}_{-\alpha}$ and
%\begin{equation}
 $$   A_i^f[\eta](g) := \sum_{\gamma \in \Gamma_{i-1}} \sum_{\varphi \in \fg^\times_{-\beta_i}} \mathcal{W}_{\varphi + f}[\eta](\gamma g)$$
\pause

\begin{figure}[t!]
\centering
\begin{tikzpicture}[font=\small, scale=0.75]
\def\nodedist{1}
    \tikzstyle{dot}=[circle, draw, thick, fill=white, inner sep=2pt]
    \tikzstyle{fdot}=[circle, draw, thick, fill=black, inner sep=2pt]
    \tikzstyle{line}=[thick, shorten >=-2pt, shorten <=-2pt]
    \node (n1) at (0, 0) [dot, label=below:1] {};
    \node (n2) at (1*\nodedist, 0) [dot, label=below:3] {};
    \node (n3) at (2*\nodedist, 0) [dot, label=below:4] {};
    \node (n4) at (2*\nodedist, 1*\nodedist) [dot, label=above:2] {};
    \node (n5) at (3*\nodedist, 0) [dot, label=below:5] {};
     \node (n7) at (5*\nodedist, 0) [dot, label=below:$n-1$] {};
    \node (n8) at (6*\nodedist, 0) [dot, label=below:$n$] {};

    \begin{scope}%[on background layer]
        \draw [line] (n1) -- (n2) -- (n3) -- (n5);
        \draw [line] (n3) -- (n4);
        \draw [dashed, thick] (n5) -- (n7);
        \draw [line] (n7) -- (n8);
    \end{scope}

%     \node at (-1.5*\nodedist,0){(c)};
%     \node at (10*\nodedist,0) {\begin{tabular}{cc}$E_{n-1}\subset E_{n}$\\decompactification %limit\end{tabular}};

% \def\yshift{6}
%
%     \node (n1) at (0, \yshift) [fdot, label=below:1] {};
%     \node (n2) at (1*\nodedist, \yshift) [dot, label=below:3] {};
%     \node (n3) at (2*\nodedist, \yshift) [dot, label=below:4] {};
%     \node (n4) at (2*\nodedist, 1*\nodedist+\yshift) [dot, label=above:2] {};
%     \node (n5) at (3*\nodedist, \yshift) [dot, label=below:5] {};
%      \node (n7) at (5*\nodedist, \yshift) [dot, label=below:$n-1$] {};
%     \node (n8) at (6*\nodedist, \yshift) [dot, label=below:$n$] {};
%
%     \begin{scope}[on background layer]
%         \draw [line] (n1) -- (n2) -- (n3) -- (n5);
%         \draw [line] (n3) -- (n4);
%         \draw [dashed, thick] (n5) -- (n7);
%         \draw [line] (n7) -- (n8);
%     \end{scope}
%
%     \node at (-1.5*\nodedist,\yshift){(a)};
%     \node at (10*\nodedist,\yshift) {\begin{tabular}{cc}$\SO_{n-1,n-1}\subset %E_{n}$\\string perturbation limit\end{tabular}};

% \def\yshift{3}
%
%     \node (n1) at (0, \yshift) [dot, label=below:1] {};
%     \node (n2) at (1*\nodedist, \yshift) [dot, label=below:3] {};
%     \node (n3) at (2*\nodedist, \yshift) [dot, label=below:4] {};
%     \node (n4) at (2*\nodedist, 1*\nodedist+\yshift) [fdot, label=above:2] {};
%     \node (n5) at (3*\nodedist, \yshift) [dot, label=below:5] {};
%      \node (n7) at (5*\nodedist, \yshift) [dot, label=below:$n-1$] {};
%     \node (n8) at (6*\nodedist, \yshift) [dot, label=below:$n$] {};
%
%     \begin{scope}[on background layer]
%         \draw [line] (n1) -- (n2) -- (n3) -- (n5);
%         \draw [line] (n3) -- (n4);
%         \draw [dashed, thick] (n5) -- (n7);
%         \draw [line] (n7) -- (n8);
%     \end{scope}

%    \node at (-1.5*\nodedist,\yshift){(b)};
%    \node at (10*\nodedist,\yshift) {\begin{tabular}{cc}$\SL_n\subset E_{n}$\\M-theory %limit\end{tabular}};


\end{tikzpicture}
% \caption{\label{fig:limits}The various string theory limits associated with different maximal parabolic subgroups $P_\alpha$. Roots are labeled in the Bourbaki ordering.}
\end{figure}

\end{frame}



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\end{document}

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