Theta correspondence


In mathematics, the theta correspondence or Howe correspondence is a correspondence between automorphic forms associated to the two groups of a dual reductive pair, introduced by as a generalisation of the Shimura correspondence. It is a conjectural correspondence between certain representations on the metaplectic group and those on the special orthogonal group. The case was constructed by Jean-Loup Waldspurger in and.

Statement

Setup

Let be a non-archimedean local field of characteristic not, with its quotient field of characteristic. Let be a quadratic extension over. Let be an -dimensional Hermitian space over. We assume further to be the isometry group of . There exists a Weil representation associated to a non-trivial additive character of for the pair, which we write as. Let be an irreducible admissible representation of. Here, we only consider the case or. We can find a certain representation of, which is in fact a certain quotient of the Weil representation by.

Local theta correspondence

Let be the set of all irreducible admissible representations of . Let be the map, which associates every irreducible admissible representation of the irreducible admissible representation of. We call the local theta correspondence for the pair.

Global theta correspondence

The global theta lift can be defined on the cuspidal automorphic representations of as well.

Howe duality conjecture

The Howe duality conjecture states that:
is irreducible or ;
Let be two irreducible admissible representations of, such that. Then,.
The Howe duality conjecture for with odd residue characteristic was proved by Jean-Loup Waldspurger in 1990. Wee Teck Gan and Shuichiro Takeda gave a proof in 2014 that works for any residue characteristic.

Etymology

Let be the theta correspondence between and. According to, one can associate to a function, which can be proved to be a modular function of half integer weight, that is to say, is a theta function.