#  Wee Teck Gan (Talk 3) 

:::{.remark}
Last time: we describe the Howe-PS correspondence
\[
\text{Automorphic characters of } \U_1
&\mapstofrom
\text{Automorphic reps of } \U_3 \\
\chi &\mapsto \Omega_\chi
.\]
A correction: it's not true that $\Omega_\chi$ is cuspidal except for at most one $\chi$; instead if can be cuspidal for many $\chi$.
We defined $\Omega, \Theta(\pi)$ with a $\U(W)$ action, and Howe duality which took $\Theta(\pi)\neq 0$ to a unique irreducible quotient $\theta(\pi)$.
Thus $\Theta: \Irr \U(V)\injects \Irr \U(W) \disjoint \ts{0}$ is injective away from the zero locus.
:::

:::{.question}
When is $\Theta(\pi)\neq 0$?
:::

:::{.remark}
Let $\dim W$ be odd, and label $W_r^\eps = 2r+1$.
We know all skew-Hermitian spaces of a particular dimension, so we obtain towers:

\begin{tikzcd}
	{\U W_r^-} &&&& {\U W_r^+} \\
	\vdots && {\U(V), \pi} && \vdots \\
	{\U W_1^-} &&&& {\U W_1^+} \\
	{\U W_0^-} &&&& {\U W_0^+}
	\arrow["{\oplus \HH}", from=4-1, to=3-1]
	\arrow["{\oplus \HH}", from=3-1, to=2-1]
	\arrow[from=2-1, to=1-1]
	\arrow["{\oplus \HH}"', from=4-5, to=3-5]
	\arrow["{\oplus \HH}"', from=3-5, to=2-5]
	\arrow[from=2-5, to=1-5]
	\arrow["{\theta_1^+}", dashed, from=2-3, to=3-5]
	\arrow["{\theta_r^-}"', dashed, from=2-3, to=1-1]
\end{tikzcd}

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

Note that $W_{r+1}^+ = W_r^+ \oplus \HH$.
:::

:::{.question}
Which $\theta_r^\eps(\pi)$ are nonzero?
:::

:::{.theorem title="?"}
\envlist

- For $\pi \in \Irr \U(V)$ and a fixed $\eps = 1$, there is a smallest $r_0^\eps(\pi) \leq \dim V$ such that this is the first occurrence of $\pi$ in the $\eps$ tower, i.e. $\theta_{r_0^\eps(\pi)}^\eps (\pi) \neq 0$.
- For all $r>r_0, \Theta^\eps(\pi)\neq 0$,
- If $\pi$ is a supercuspidal rep, then by Kudla, $\Theta_r^\eps(\pi)$ is irreducible and is s.c. at the first occurrence but not after.

:::

:::{.remark}
\envlist

- Nonvanishing is reduced to determining $r_0^+(\pi)$ and $r_0^-(\pi)$.
- If $r\geq \dim V$, so $r$ is in the stable range and $\Theta_r^\eps(\pi)\neq 0$.

Thus reduces checking infinitely many nonzero conditions to just computing the values of these two numbers.
We can reduce this further to just checking *one* number by the following:
:::

:::{.theorem title="Conservation relation (B.Y. Sun, C.B. Zhu, Kudla-Rallis)"}
\[
\dim W_{r_0^+(\pi)}^+ + \dim W_{r_0^-(\pi)} = 2\dim V + 2
.\]
:::

:::{.corollary title="Dichotomy"}
If $\dim W^+ + \dim W^- = 2\dim V$, then for any $\pi\in \Irr \U(V)$, exactly one of $\Theta_{W^+}(\pi)$ or $\Theta_{W^-}(\pi)$ is nonzero.
:::

:::{.example title="?"}
Take $\U_1\times \U_1 = \U(V) \times \U(W_0)$ where $\U(V) = E^1$, and let $\chi\in \Irr E^1$.
Then
\[
\dim W_{r^+(\chi)}^+ + \dim W_{r^-(\chi)}^- = 4
,\]
These two dimensions are numbers in $\ts{1,3}$, and exactly one of $\theta_0^{\pm}(\chi)$ is nonzero, and for $r>0$ we have $\theta_r^\eps(\chi) \neq 0$.
Which $\theta_0^\eps(\chi)$ are nonzero?
:::

:::{.theorem title="Moen, Rogawski?, Hams-Kudla-Sweat"}
\[
\theta_{V, W_0, \psi}(\pi) \neq 0 \iff \eps(v) \eps(W_0) = \eps_E\qty{ {1\over 2}, \chi_E \chi_W\inv, \psi(\Trace_{E/F} (\delta-1)) }
\]
where $\eps_E$ is the local epsilon factor defined in Tate's thesis.
Here $\chi_E$ is the composite character $\chi_E(x) = \chi\qty{x\over x^?}$ defined by

\begin{tikzcd}
	{E\units/F\units} && {E^1} && \CC\units \\
	x && {{x\over x^c}}
	\arrow["\chi", from=1-3, to=1-5]
	\arrow["\cong", from=1-1, to=1-3]
	\arrow["{\chi_E}"', curve={height=-30pt}, dashed, from=1-1, to=1-5]
	\arrow[maps to, from=2-1, to=2-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMCwwLCJFXFx1bml0cy9GXFx1bml0cyJdLFsyLDAsIkVeMSJdLFs0LDAsIlxcQ0NcXHVuaXRzIl0sWzAsMSwieCJdLFsyLDEsInt4XFxvdmVyIHheY30iXSxbMSwyLCJcXGNoaSJdLFswLDEsIlxcY29uZyJdLFswLDIsIlxcY2hpX0UiLDIseyJjdXJ2ZSI6LTUsInN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFszLDQsIiIsMix7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Im1hcHMgdG8ifX19XV0=)

The $\delta\in E_0\units$ appears because a Hermitian space depends on a choice of a traceless element.
:::


:::{.example title="?"}
Applying Howe-PS to $\U_1\times \U_3$: let $V = \gens{1} = V_0^+$ and $\chi \in \Irr E^1 = \Irr \U(V)$.
Since $\dim W^\eps = 3$, $\Omega^\eps$ is semisimple and decomposes as 
\[
\Omega^\eps = \bigoplus _{\chi\in\Irr E^1} \chi \tensor \Theta^\eps(\chi)
.\]

- Since $\dim V = 1$, we're in the stable range and thus $\Theta^\eps(\chi)\neq 0$ for all $\chi$.
- $\Theta^\eps(\chi)$ is irreducible by Howe duality and s.c.
- If $\eps = \eps_E\qty{{1\over 2}, \cdots}$ as in the theorem, $\Theta^\eps(\chi)$ is non-supercuspidal and $\Theta^{-\eps}(\chi)$ is supercuspidal.

In fact, $\Theta^\eps(\chi) \embeds \Ind_B^{\U(W)}\qty{ \chi_v \abs{\wait}^{- {1\over 2} } \tensor \chi }$ where $B = \diag(a, b, (a^c)\inv) + N^+$ (upper triangular) with $a\in E\units$ and $b\in E^1$.
:::

## Global Setting


:::{.remark}
For $K\in \NF$, writing $\theta = \prod_v \theta_v$, one might hope for a map $\Irr \U(V)(\AA) \to \U(W)(\AA)$.
Instead, we'll want a map
\[
\theta:
\ts{\text{Automorphic reps of } \U(V)} 
&\to 
\ts{\text{Automorphic reps of } \U(W)} 
,\]
i.e. a concrete way to transfer functions from a space $X$ to a space $Y$.
If $K\in C(X\times Y)$, we can define
\[
T_K: C(X) &\to C(Y) \\
T_k(f)(y) &\da \int_X K(x, y) f(x) \dx
,\]
so $K$ acts like a matrix.
In our case, we'll want a lift

\begin{tikzcd}
	&&& {\Mp(V\tensor W)(\AA)} \\
	\\
	{(\U(V)\times \U(W))(\AA)} &&& {\Sp(V\tensor W)(\AA)}
	\arrow["\iota", from=3-1, to=3-4]
	\arrow["{\exists \tilde\iota\,?}"', from=3-1, to=1-4]
	\arrow[from=1-4, to=3-4]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwyLCIoXFxVKFYpXFx0aW1lcyBcXFUoVykpKFxcQUEpIl0sWzMsMCwiXFxNcChWXFx0ZW5zb3IgVykoXFxBQSkiXSxbMywyLCJcXFNwKFZcXHRlbnNvciBXKShcXEFBKSJdLFswLDIsIlxcaW90YSJdLFswLDEsIlxcZXhpc3RzIFxcdGlsZGVcXGlvdGFcXCw/IiwyXSxbMSwyXV0=)

Here $\Omega = \tilde{\iota}^* W_\psi$.
For $\pi \in \mca_\cusp(\U(V))$, we have a map

\begin{tikzcd}
	{W_\psi} &&&& {\mca_2(\Mp(\cdots))} \\
	\\
	&&&& {C([\U(V)\times \U(W)])}
	\arrow["{\tilde\iota^*}", from=1-5, to=3-5]
	\arrow[from=1-1, to=1-5]
	\arrow["\theta"', from=1-1, to=3-5]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwwLCJXX1xccHNpIl0sWzQsMCwiXFxtY2FfMihcXE1wKCkpIl0sWzQsMiwiQyhbXFxVKFYpXFx0aW1lcyBcXFUoVyldKSJdLFsxLDIsIlxcdGlsZGVcXGlvdGFeKiJdLFswLDFdLFswLDIsIlxcdGhldGEiLDJdXQ==)

This yields
\[
w_\psi \tensor \pi &\to \mca(\U(V)) \\
\phi \tensor f &\mapsto \theta(\phi, f), \qquad \theta(\phi, f)(g) \da \int_{[\U(V)]} \theta(\phi)(g, h) \bar{f(h)}\, dh
.\]
So define the **global theta lift of $\pi$** as
\[
\Theta(\pi) \da \gens{\theta(\phi, f) \st \phi\in w_\phi, f\in \pi} \subseteq \mca(\U(W))
.\]

:::

:::{.question}
\envlist

- Is $\Theta(\pi)$ nonzero?
- Does it land in $\mca_2$ or $\mca_\cusp$?
- What is the relation with the local picture?

:::


:::{.proposition title="?"}
If $\Theta(\pi) \subset \mca_2(\U(W))$ is a proper subset, then $\Theta(\pi)$ is either zero or isomorphic to $\Tensor_v \theta(\pi_v)$.

:::


:::{.theorem title="?"}
Let $\pi \subseteq \mca_\cusp(\U(V))$,

1. There exists a smallest $r_0 = r_0^\eps(\pi)$ such that $\Theta_{r_0}^\eps(\pi) \neq 0$.
  In this case, $\Theta_{r_0}^\eps(\pi) \subseteq \mca_\cusp(\U(W))$.

2. For all $r > r_0$,$\Theta_r^\eps(\pi)\neq 0$ and is noncuspidal, i.e. not contained in $\mca_\cusp(\U(W))$.

3. For all $r\geq \dim V$ in the stable range, $0\neq \Theta_r^\eps(\pi) \subseteq \mca_2(\U(W))$.
  Note that being nonzero follows from 1 and 2.

:::