#  Wee Teck Gan (Talk 2) 

:::{.remark}
Correction from last time:
\[
\inp{v_2}{v_1} = \eps \inp{v_1}{v_2}^c
.\]

Notation from last time:

- $V$ Hermitian, $W$ skew-Hermitian
- An invariant $\disc(V) \da (-1)^m \det V\in F\units/\Norm(E\units)$ where $m\da {n\choose 2}$ and $n\da \dim V$
- $\disc(W) = \disc( \delta^{-n} V)$ where \( \delta\in E_0\units \).

:::

:::{.fact}
Over \(p\dash \)adic fields, $\disc(V)$ determines $V$.
By composing with a quadratic character $w_{E/F}$, we obtain
\[
\disc \circ w_{E/F} = (V \mapsvia{\disc} F\units/\Norm(E\units)) \mapsvia{w_{E/F}} \gens{\pm 1}
,\]
so there are exactly two classes of Hermitian vector spaces of a given dimension, which we'll denote $V^+, V^-$.
:::

:::{.remark}
Over a real field, this is not enough -- one also needs the signature $\signature(V) = (p, q)$ where $p+q=n$, in which case
\[
\disc(V_{p, q}) = (-1)^q (-1)^{p\choose 2}
.\]
For $E/K$ an extension of number fields, there is a local-global principle:
\[
\Herm\Vect\slice K &\injects \prod_{v\in \Places(K)} \Herm\Vect \slice {K_v} \\
V &\mapsto \ts{V\tensor_K K_v}_{v\in \Places(K)}
.\]

We'll call spaces in the image of this correspondence **coherent**.
:::

:::{.fact}
$V$ is coherent iff for almost every place $v$, \[
V\text{ is coherent} \iff 
\eps(V_v) = 1 \ae \text{ and }
\prod_v \eps(V_v) = 1
.\]
:::

:::{.example title="of classification"}
Let $k$ be a \(p\dash \)adic field.

In rank 1:

- $E_0\units / \Norm(E\units) = \ts{ \delta, \delta'}$,
- $W_1^+ = \gens{ \delta}$,
- $W_1^- = \gens{ \delta'}$.

In rank 2:

- $\HH = W_2^+ = E e_1 + E e_2$,
- $\inp{e_i}{e_i} = 0$ and $\inp{e_1}{e_2} = 1$, which yields  matrix $\matt 0 1 {-1} 0$,
- $W_2^-$ is described by a quaternionic division algebra.

In rank $2n$:

- $W_{2n}^+ = \HH\sumpower{n}$
- $W_{2n}^- = W_2^- \oplus \HH\sumpower{(n-1)}$

In rank $2n+1$:

- $W_{2n+1}^+ = \gens{ \delta} \oplus \HH\sumpower{n}$,
- $W_{2n+1}^- = \gens{ \delta' } \oplus \HH\sumpower{n}$.
:::

## Howe-PS: Counterexample to the Ramanujan-Petersson Conjecture

:::{.remark}
Let $\dim W = 3$, so $U(W) = \U_3$, then $\Res_{E/K}(W)\in \Vect^{\dim = 6}\slice k$.
The trace to $K$ yields a symplectic form:
\[
\omega(\wait, \wait) \da \Tr_{E/k}\inp{\wait}{\wait}_W
.\]

There is an embedding $U(W) \embeds \Sp(\Res_{E/k}(W))$, so $\U_3 \embeds \Sp_6$
There is a simple something:
\[
\Omega \subseteq \mca_2(\Sp(\wait))
,\]
which we'll call **theta functions**.
Note that $ZU(W) = E^1 \da \ts{x\in E\units \st \Norm(x) = 1}$.
Consider $i^* \Omega \subseteq \mca(U(W))$; there is a central character decomposition
\[
i^*( \Omega) = \bigoplus _{\chi} \Omega_{\chi}
,\]
where the sum is over automorphic characters of $E^1$.


:::{.claim}
$\Omega_\chi$ is an irreducible cuspidal representation, with at most one exception $\chi$, and this $\Omega_\chi$ produced a counterexample for the RP conjecture.
:::


:::

:::{.warnings}
A complication: the theta functions don't live on $\Sp_6$, but rather on a double cover, and this leads to many technicalities.
:::

:::{.remark}
Howe-PS produces a correspondence:
\[
\ts{\text{Automorphic characters on } E^1\cong \U_1} &\mapstofrom \ts{\text{Automorphic reps of } \U_3} \\
\chi &\mapsto \Omega_{\chi}
.\]
:::

:::{.question}
How can one produce an injective map
\[
\Irr(G) \injects \Irr(H)
?\]
:::

:::{.answer}
Recall that 
\[
\Irr(G\times H) = \ts{\pi \tensor \sigma \st \pi \in \Irr(G), \sigma \in \Irr(H)}
.\]
The idea to produce this map: find $(G\times H)\dash$reps $\Omega$ and produce a subset 
\[
\Sigma_{ \Omega} = \ts{(\pi, \sigma) \st \Hom_{G\times H}(\Omega, \pi\tensor\sigma) \neq 1} \subseteq \Irr(G) \times \Irr(H)
.\]
:::

:::{.question}
Is the correspondence $\Sigma_{\Omega}$ a graph?
:::

:::{.remark}
There is a decomposition
\[
\Omega_{G\times H}
&= \bigoplus _\pi \bigoplus _\sigma m(\pi, \sigma) \pi\tensor\sigma \\
&= \bigoplus _\pi \qty{\bigoplus _\sigma m(\pi, \sigma) \sigma }\tensor \pi \\
&\da \bigoplus _\pi \Theta(\pi) \tensor \pi 
.\]
Is $\Theta(\pi)$ an irreducible rep, or zero?
If so, this produces a map
\[
\Theta: \Irr(G) \to \Irr(H)\union\ts{0}
.\]
:::

:::{.remark}
Upshot: one needs $\dim \Omega$ to be small.
Suppose $G\times H \to E$, take the smallest non-trivial representation $\Omega$ of $E$ and pull it back to $G\times H$.
If $G\times H \subseteq E$, this can be done by restriction.
:::

:::{.remark}
The **theta correspondence** is an instance of all of these ideas.
:::

## The Theta Correspondence

:::{.remark}
Let 

- $F\in\Field$ be a \(p\dash \)adic, 
- $E/F$ a quadratic extension, 
- $V$ Hermitian and $W$ skew-Hermitian so that $V \tensor_E W$ is skew-Hermitian under the symplectic form induced by the trace,

This yields a map of the form $G\times H\to E$:
\[
\U(V) \times \U(W) \to \Sp(V\tensor_E W)
.\]
What is $\Omega$?
To get small enough weights, one needs to pass to the **metaplectic cover** $\Mp$.
:::

## Metaplectic Groups and Weil Reps

:::{.remark}
For $\psi: F\to \CC\units$ a nontrivial character:

\begin{tikzcd}
	{S^1} && {\Mp(V\tensor_E W)} \\
	&&&& {\GL(S)} \\
	&& {\Sp(V\tensor_E W)}
	\arrow["{\Omega = \omega_\psi}", from=1-3, to=2-5]
	\arrow[two heads, from=1-3, to=3-3]
	\arrow[from=1-1, to=1-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJTXjEiXSxbMiwwLCJcXE1wKFZcXHRlbnNvcl9FIFcpIl0sWzIsMiwiXFxTcChWXFx0ZW5zb3JfRSBXKSJdLFs0LDEsIlxcR0woUykiXSxbMSwzLCJcXE9tZWdhID0gXFxvbWVnYV9cXHBzaSJdLFsxLDIsIiIsMix7InN0eWxlIjp7ImhlYWQiOnsibmFtZSI6ImVwaSJ9fX1dLFswLDFdXQ==)

Here $\ts{\omega_\psi}$ is the smallest infinite-dimensional representation of $\Mp$ and referred to as the **Weil representation**.
:::

:::{.remark}
On where this comes from: QM.
One looks at the Heisenberg group, uses the Stone-von-Neumann theorem, see 2.3 and 2.4 in the notes.
:::

:::{.remark}
One needs a lift of the following form:

\begin{tikzcd}
	&& {\Mp(V\tensor_E W)} \\
	\\
	{\U(V)\times \U(W)} && {\Sp(V\tensor_E W)}
	\arrow["i", from=3-1, to=3-3]
	\arrow["{\exists\, \tilde i?}", from=3-1, to=1-3]
	\arrow[from=1-3, to=3-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMiwwLCJcXE1wKFZcXHRlbnNvcl9FIFcpIl0sWzIsMiwiXFxTcChWXFx0ZW5zb3JfRSBXKSJdLFswLDIsIlxcVShWKVxcdGltZXMgXFxVKFcpIl0sWzIsMSwiaSJdLFsyLDAsIlxcZXhpc3RzXFwsIFxcdGlsZGUgaT8iXSxbMCwxXV0=)

By Xudle, $\tilde i$ exists and is determined by a pair of characters $(\chi_V, \chi_W)$ of $E\units$ such that

- $\ro{ \chi_V}{F\units} = \omega_{E/F}^{\dim V}$
- $\ro{\chi_W}{F\units} = \omega_{E/F}^{\dim W}$

Such a $\chi_V$ gives $\tilde \U(W) \to \Mp$, and similarly for $W$.
:::

:::{.remark}
Set 
\[
\Omega_{V, W, \chi_V, \chi_W, \psi} \da \tilde\iota_{\chi_V, \chi_W}^*(\omega_\psi)
,\]
which has properties described in the lecture notes.
:::

:::{.definition title="The big theta lift as a multiplicity space"}
For $\pi \in \Irr \U(V)$, define
\[
\Theta(\pi) \da \coinv_{\U(V)}(\Omega \tensor \pi \dual)
.\]
:::

:::{.remark}

Note that there is a $\U(W)$ action on both sides.
Moreover,
\[
\Hom( \coinv_G (\Omega\tensor \pi\dual), \CC) 
\cong \Hom_G(\Omega\tensor\pi\dual, \CC) 
\cong \Hom_G(\Omega, ?)
.\]
:::

:::{.theorem title="Howe-Kudla"}
\envlist

- $\Theta(\pi)$ has finite length as a $\U(W)$ rep, and thus has finitely many irreducible quotients.
- For any pair $(\pi, \sigma)$,
\[
\dim \Hom_{\U(V)\times \U(W)}(\Omega,\pi\tensor \sigma) < \infty
.\]
:::

:::{.definition title="Small theta lift"}
Define $\theta(\pi)$ to be the maximal semisimple quotient of $\Theta(\pi)$.
This is a finite length semisimple rep.
:::

:::{.theorem title="Howe Duality"}
\envlist

- $\theta(\pi)$ is irreducible if $\Theta(\pi) \neq 0$.
- Uniqueness: $\theta(\pi) \cong \theta(\pi') \implies \pi \cong \pi'$.
  Thus $\theta: \Irr \U(V) \to \Irr \U(W) \smz$ is injective on $\supp \theta$, those reps which are not sent to zero.

:::

:::{.question}
Is $\theta(\pi)$ zero or not?
:::