Given the reductive dual pair in , one obtains a pair of commuting subgroups in by pulling back the projection map from to .
The local theta correspondence is a 1-1 correspondence between certain irreducible admissible representations of and certain irreducible admissible representations of , obtained by restricting the Weil representation of to the subgroup . The correspondence was defined by Roger Howe in Howe (1979). 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 [3] and conservation relations concerning the first occurrence indices along Witt towers .[4]
Define the set of irreducible admissible representations of , which can be realized as quotients of
. Define and , likewise.
The Howe duality conjecture asserts that is the graph of a bijection between and .
The Howe duality conjecture for archimedean local fields was proved by Roger Howe.[6] For -adic local fields with odd it was proved by Jean-Loup Waldspurger.[7] Alberto Mínguez later gave a proof for dual pairs of general linear groups, that works for arbitrary residue characteristic. [8] For orthogonal-symplectic or unitary dual pairs, it was proved by Wee Teck Gan and Shuichiro Takeda. [9] The final case of quaternionic dual pairs was completed by Wee Teck Gan and Binyong Sun.[10]
Mínguez, Alberto (2008), "Correspondance de Howe explicite: paires duales de type II", Ann. Sci. Éc. Norm. Supér., 4, 41 (5): 717–741, doi:10.24033/asens.2080
Waldspurger, Jean-Loup (1980), "Correspondance de Shimura", J. Math. Pures Appl., 59 (9): 1–132
Waldspurger, Jean-Loup (1990), "Démonstration d'une conjecture de dualité de Howe dans le cas p-adique, p ≠ 2", Festschrift in Honor of I. I. Piatetski-Shapiro on the Occasion of His Sixtieth Birthday, Part I, Israel Math. Conf. Proc., 2: 267–324