#  Wee Teck Gan (Talk 4) 

:::{.remark}
Take $V = \gens{1}$ and $W_r^\eps = W_0^\eps \oplus \HH^r$ which has dimension $2r+1$, and let $\chi \in \mca(\U(V))$.
We know $\Theta_r^\eps(\chi)\neq 0$ for all $r>0$, which is the stable range.
Note that $\Theta_r^\eps(\chi) \subseteq \mca_2( \U(W_r^\eps))$, i.e. these are square-integrable.
What happens when $r=0$?
:::

:::{.theorem title="?"}
$\Theta_{W_0^\eps}(\chi) \neq 0 \iff$ several conditions hold:

- For all $v$, $\Theta_{W_0}^{\eps_v}(\chi_v)\neq 0$, so it is controlled by local conditions,
- $L(1/2, \chi_E \chi_W\inv)\neq 0$, a global condition.
:::

:::{.remark}
Note that 
\[
\eps_v &= \eps(1/2, \chi_E \chi_{W, v}\inv, \phi(\Trace ?)) \\
1 = \prod_v \eps_v &= \eps(1/2, \chi_E \chi_W\inv)
.\]

:::

:::{.proof title="of theorem, sketch"}
For $\psi\in W_\phi$, we produce $\Theta(\phi)$ and obtain an integral
\[
\Theta(\phi, \chi)(g) = \int_{[\U(V)]} \phi(g,h) \chi(h)\inv \, dh \in \mca(\U(W_0))
.\]
Is this function nonzero for some $\phi$?
There isn't a good notion of Fourier expansion here, so one instead computes $\inp{\Theta(\phi, \chi)}{\Theta(\phi, \chi)}$.
Write $V^\square = V \oplus -V$, where $-V$ is $V$ with the form negated.

\begin{tikzcd}
	& {\U(V^\square)} && {\U(W_0)\times \U(W_0)} & \textcolor{rgb,255:red,92;green,92;blue,214}{\Theta(\phi,\chi) \overline{\Theta(\phi,\chi)}} \\
	\\
	\textcolor{rgb,255:red,92;green,92;blue,214}{(\chi, \chi\inv)} & {\U(V)\times \U(V^-)} && {\U(W_0)^\Delta} \\
	& {}
	\arrow[hook, from=3-4, to=1-4]
	\arrow[hook, from=3-2, to=1-2]
	\arrow[hook, from=3-2, to=1-4]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=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)

One can them map $\U(V^\square) \to \U(W_0)^\square$; this diagram is called the **doubling see-saw**.
Combining this with Siegel-Weil associates to the above inner product the doubling zeta integral $Z(0, \phi, \chi)$.
By Ellen's lectures, this reduces to computing the central value of an \(L\dash \)function, $cL(1/2, \chi_E \chi_W\inv)$, up to a fudge factor $c$.
The process is the **Rallis inner product formula**:
\[
\inp{\Theta(\phi,\chi)}{\Theta(\phi,\chi)} \leadsto Z(0,\phi,\chi)\leadsto L(1/2, \chi_E \chi_W\inv)
.\]
:::

## Howe-PS

:::{.remark}
Setup:

- $V = \gens{1}$ a 1-dim space
- $W = W_0 \oplus \HH$
- A nonzero irreducible theta lift $\Theta_W(1) \subseteq \mca_2(\U(W))$

We know that the local components are contained in non-tempered principal series, i.e. 
\[
\Theta_W(1)_v \injects \Ind_?^{\U(W_v)} \abs{\wait}_v^+ \tensor 1_v?
.\]
It only remains to check that happens when this is not cuspidal.
If it is not, then $\Theta_{W_0}(1) \neq 0$, so pick 2 places $v_1, v_2$ of $K$ and swap the signs on $W_{0, v_i}$ to produce $W_0'$, and run the above argument on $W' = W_0' \oplus \HH$.
:::

## Arthur's Conjecture

:::{.remark}
Goal: classify constituents of $\mca_2(G)$, i.e. describe this as a $G(\AA)\dash$module.
We'll make a basic hypothesis (global Langlands for $\GL_n$) that there exists a group $L_F$ (thought of as $\Gal(\bar F/F)$) such that there is a bijection
\[
\Irr\Rep^{\dim = n} L_F \mapstofrom \Rep_\cusp \GL_n
,\]
where for all $v$ there is a Weil-Deligne group $L_{F_v} \approx \Gal(\bar F_v/F_v)$ with a map $L_{F_v} \injects L_F$.
:::

:::{.definition title="Near equivalence"}
Two adelic representations $\pi = \Tensor_v \pi_v$ and $\pi' = \Tensor_v \pi_v'$ are **nearly equivalent** iff $\pi_v \cong \pi_v'$ for almost all places $v$.
:::

:::{.remark}
We can decompose into near-equivalence classes $\mca_w(G) = \bigoplus_\psi \mca_{\psi}$, where $\psi: L_F\times \SL_2 \to \langL{G}$ is a map to the Langlands $L$ group $\langL{G} = G\dual \semidirect \Gal(\bar F/F)$, such that

- A tempered condition: the image is big, $\psi(L_F)$ is bounded, and
- Centralizers are small: $Z_{G\dual}/Z_{G\dual}^{\Gamma_F}$ is finite.

This has something to do with elliptic $A\dash$parameters.
:::

:::{.question}
Given $\psi$, how can we describe $\mca_\psi$?
:::

:::{.remark}
From $\psi$ we'll obtain 

- a global component group/centralizer $S_\psi = Z_{G\dual}/Z_{G\dual}^{\Gamma_F}$,
- local factors $\psi_v: L_{F_v}\times \SL_2 \injects L_f \times \SL_2 \mapsvia{\psi} \langL{G}$,
- Local component groups $\pi_0\qty{Z_{G\dual}(\psi_v) / Z(G\dual)^{\Gamma_F} }$ which are finite?
- $S_\psi \mapsvia{\Delta} \prod_v S_{\psi_v} \da S_{\psi/\Delta}$ which is compact
- Quadratic characters $\eps_\psi: S_\psi \to \gens{\pm 1}$.

:::

:::{.remark}
For all $v$, we should have a finite set of unitary reps of $G(F_v)$,
\[
\prod_{\psi_v} = \ts{\pi_{\eta_v} \st \eta_v \in \Irr S_{\psi_v}}
,\]
i.e. for almost all $v$, $\pi_{1_v}$ is irreducible unramified with Satake parameters
\[
\psi_v\qty{\Frob_v, \diag(q_v^{1\over 2} , q_v^{-{1\over 2} })} \in \langL{G}
.\]
:::

:::{.observation}
The key point: if $\psi(\SL_2) = 1$, then $\pi_{1_v}$ is tempered. 
If not, $\psi_{1_v}$ is non-tempered.

> This explains how rigidity obstructs the Ramanujan-Petersson conjecture?

:::

:::{.remark}
Set 
$\pi_\psi = \Tensor_v \pi_{\psi_v}$ and let
\[
\mca_\psi = \bigoplus_{\eta \in \Irr S_{\psi, ?}} m_\eta \pi_\eta, \qquad m_\eta = \dim \Hom_{S_\psi}(\eps_\psi, \eta)
.\]
To define $\eps_\psi$, define a map
\[
(L_F\times \Omega_2)\times S_{\psi} \mapsvia{\psi\times \id} \langL{G}/ Z(G\dual)^{\Gamma_F} \actsonl \Ad \lieg\dual
\]
where $\lieg\dual = \Lie(G\dual) = \bigoplus _{i\in I} \rho_i \tensor S_{r_i} \tensor \eta_i$ for some index set $I$, and $S_r$ are $r\dash$dimensional irreps of $\SL_2$.
Set $T \subseteq I$ to be the indices such that $r_i$ is even, $\eta_i$ is orthogonal, and $\rho_i$ is symplectic, and $\eps(1/2, \phi_i) = -1$.
Then define
\[
\eps_\psi: S_\psi &\to \gens{\pm 1} \\
s &\mapsto \prod_{i\in T} \eta_i(s)
.\]
:::

:::{.example title="?"}
For $\psi(\SL_2) = \ts{1}$, $\eps_\psi = 1$ and $T = \emptyset$.
:::

## Specializing to $\U_n$

:::{.remark}
Fix $G=\U_n$, let $E/F$ be an extension, and
\[
G\dual = \GL_n(\CC) \normal \langL{G} = \GL_n(\CC) \semidirect \Gal(E/F)
.\]
An $L\dash$parameter is a map
\[
\psi: L_F \times \SL_2 \to \langL{G}
\]
where the subset $L_E \times \SL_2$ maps to $G\dual$.
By an email comment of Benedict Gross, $\psi$ is determined by this restriction.
Not every such map extends, but conjugate self-dual reps of sign $(-1)^{n-1}$ will.
:::

:::{.example title="?"}
For $\U_3$, $\ro{\psi}{L_E}: L_E\times \SL_2 \to \GL_3(\CC)$ which decomposes as $\ro{\psi}{L_E} = \mu \oplus \chi \tensor S_2$ such that $\chi$ are not characters of $L_E$, but rather automorphic characters of $\dcosetl{E\units}{\AA_E\units}$ with $\ro{\mu}{\AA_F\units} = 1$ and $\ro{\chi}{\AA_F\units} = \omega_{E/F}$.
For $S_\psi = \mu_2 \mapsvia{\Delta} \prod_v S_{\psi_v}$, we have $S_{\psi_v} = \mu_2$ if $v$ is inert in $E$ and 1 otherwise.
Then $\eps_{\psi}: \mu_2 \to \gens{\pm 1}$ which is trivial when $\eps(1/2, \chi\mu\inv) = 1$ and nontrivial if this is $-1$.
So $\prod_{\psi_v} = \ts{\pi^+_v, \pi_v^-}$ if $v$ is inert, and just $\ts{\pi_v^+}$ otherwise, meaning
\[
m(\pi^{\eps})
=
\begin{cases}
1 & \prod_v \eps_v = \eps(1/2) 
\\
0 & \text{otherwise}.
\end{cases}
.\]
For almost every $v$, $\pi_{1_v} = \pi_v^+$.
Something about $\Ind_{B_v}^{U_?} \chi \abs{\wait}_v^{-{1\over 2}} \tensor \tilde \mu$.
Something about Howe-PS.
:::