Theta correspondence

In mathematics, the theta correspondence or Howe correspondence is a mathematical relation between representations of two groups of a reductive dual pair. The local theta correspondence relates irreducible admissible representations over a local field, while the global theta correspondence relates irreducible automorphic representations over a global field. The theta correspondence was introduced by Roger Howe in Its name arose due to its origin in André Weil's representation theoretical formulation of the theory of theta series in The Shimura correspondence as constructed by Jean Loup Waldspurger in and may be viewed as an instance of the theta correspondence.

Let be a local or a global field, not of characteristic Let be a symplectic vector space over , and the symplectic group. Fix a reductive dual pair in There is a classification of reductive dual pairs.

is now a local field. Fix a non trivial additive character of There exists a Weil representation of the metaplectic group associated to , which we write as
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 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 .

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.