---
date: 2022-02-10 22:37
modification date: Thursday 10th February 2022 23:18:51
title: local theta correspondence
aliases: [theta correspondence]

---

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# theta correspondence

Easier in a "stable range".

![](attachments/Pasted%20image%2020220210231902.png)

# local theta correspondence

For $\pi \in \operatorname{Irr}(\mathrm{U}(V))$, one considers the maximal $\pi$-isotypic quotient of $\Omega$:
\[
\Omega / \bigcap_{f \in \operatorname{Hom}_{\mathrm{U}(V)}(\Omega, \pi)} \operatorname{Ker}(f)
.\]
which is a $U(V) \times U(W)$-quotient of $\Omega$ expressible in the form
\[
\pi \otimes \Theta(\pi)
\]
for some smooth representation $\Theta(\pi)$ of $\mathrm{U}(W)$ (possibly zero, and possibly infinite length a priori). We call $\Theta(\pi)$ the big theta lift of $\pi$. An alternative way to define $\Theta(\pi)$ is:
\[
\Theta(\pi)=\left(\Omega \otimes \pi^{\vee}\right)_{\mathrm{U}(V)},
\]
the maximal $\mathrm{U}(V)$-invariant quotient of $\Omega \otimes \pi^{\vee}$. In any case, it follows from definition that there is a natural $\mathrm{U}(V) \times \mathrm{U}(W)$-equivariant map
\[
\Omega \rightarrow \pi \otimes \Theta(\pi),
\]
which satisfies the "universal property" that for any smooth representation $\sigma$ of $\mathrm{U}(W)$,
\[
\operatorname{Hom}_{\mathrm{U}(V) \times \mathrm{U}(W)}(\Omega, \pi \otimes \sigma) \cong \operatorname{Hom}_{\mathrm{U}(W)}(\Theta(\pi), \sigma) \quad \text { (functorially). }
\]
The local theta lifts of $\pi$ are then the irreducible quotients of $\Theta(\pi)$.

The goal of local theta correspondence is to determine the representation $\Theta(\pi)$ or rather its irreducible quotients. Recall that our hope is that $\Theta(\pi)$ is close to irreducible or at least not too big.