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theta correspondence

Easier in a “stable range”.

local theta correspondence

For \(\pi \in \operatorname{Irr}(\mathrm{U}(V))\), one considers the maximal \(\pi\)-isotypic quotient of \(\Omega\): \begin{align*} \Omega / \bigcap_{f \in \operatorname{Hom}_{\mathrm{U}(V)}(\Omega, \pi)} \operatorname{Ker}(f) .\end{align*} which is a \(U(V) \times U(W)\)-quotient of \(\Omega\) expressible in the form \begin{align*} \pi \otimes \Theta(\pi) \end{align*} 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: \begin{align*} \Theta(\pi)=\left(\Omega \otimes \pi^{\vee}\right)_{\mathrm{U}(V)}, \end{align*} 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 \begin{align*} \Omega \rightarrow \pi \otimes \Theta(\pi), \end{align*} which satisfies the “universal property” that for any smooth representation \(\sigma\) of \(\mathrm{U}(W)\), \begin{align*} \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). } \end{align*} 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.