Theta correspondence

Setup Let F be a local or a global field, not of characteristic 2. Let W be a symplectic vector space over F, and Sp(W) the symplectic group. Fix a reductive dual pair (G,H) in Sp(W). There is a classification of reductive dual pairs. Local theta correspondence F is now a local field. Fix a non-trivial additive character \psi of F. There exists a Weil representation of the metaplectic group Mp(W) associated to \psi, which we write as \omega_{\psi}. Given the reductive dual pair (G,H) in Sp(W), one obtains a pair of commuting subgroups (\widetilde{G}, \widetilde{H}) in Mp(W) by pulling back the projection map from Mp(W) to Sp(W). The local theta correspondence is a 1-1 correspondence between certain irreducible admissible representations of \widetilde{G} and certain irreducible admissible representations of \widetilde{H}, obtained by restricting the Weil representation \omega_{\psi} of Mp(W) to the subgroup \widetilde{G}\cdot\widetilde{H}. The correspondence was defined by Roger Howe in . The assertion that this is a 1-1 correspondence is called the Howe duality conjecture. Key properties of local theta correspondence include its compatibility with Bernstein-Zelevinsky induction and conservation relations concerning the first occurrence indices along Witt towers . Global theta correspondence Stephen Rallis showed a version of the global Howe duality conjecture for cuspidal automorphic representations over a global field, assuming the validity of the Howe duality conjecture for all local places. ==Howe duality conjecture==

Define \mathcal{R}(\widetilde{G},\omega_{\psi}) the set of irreducible admissible representations of \widetilde{G}, which can be realized as quotients of \omega_{\psi}. Define \mathcal{R}(\widetilde{H},\omega_{\psi}) and \mathcal{R}(\widetilde{G}\cdot\widetilde{H},\omega_{\psi}), likewise. The Howe duality conjecture asserts that \mathcal{R}(\widetilde{G}\cdot\widetilde{H},\omega_{\psi}) is the graph of a bijection between \mathcal{R}(\widetilde{G},\omega_{\psi}) and \mathcal{R}(\widetilde{H},\omega_{\psi}). The Howe duality conjecture for archimedean local fields was proved by Roger Howe. For p-adic local fields with p odd it was proved by Jean-Loup Waldspurger. Alberto Mínguez later gave a proof for dual pairs of general linear groups, that works for arbitrary residue characteristic. For orthogonal-symplectic or unitary dual pairs, it was proved by Wee Teck Gan and Shuichiro Takeda. The final case of quaternionic dual pairs was completed by Wee Teck Gan and Binyong Sun. ==See also==