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. Statement

Setup
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. Local theta correspondence
 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 . 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  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. For -adic local fields with  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
Reductive dual pair
Metaplectic group

References

Bibliography

Langlands program
Representation theory