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 formula_1 be a local or a global field, not of characteristic formula_2. Let formula_3 be a symplectic vector space over formula_1, and formula_5 the symplectic group.
Fix a reductive dual pair formula_6 in formula_5. There is a classification of reductive dual pairs. 
Local theta correspondence.
formula_1 is now a local field. Fix a non-trivial additive character formula_9 of formula_1. There exists a Weil representation of the metaplectic group formula_11 associated to formula_9, which we write as formula_13.
Given the reductive dual pair formula_6 in formula_5, one obtains a pair of commuting subgroups formula_16 in formula_11 by pulling back the projection map from formula_11 to formula_5.
The local theta correspondence is a 1-1 correspondence between certain irreducible admissible representations of formula_20 and certain irreducible admissible representations of formula_21, obtained by restricting the Weil representation formula_13 of formula_11 to the subgroup formula_24. 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 formula_25 the set of irreducible admissible representations of formula_20, which can be realized as quotients of 
formula_13. Define formula_28 and formula_29, likewise.
The Howe duality conjecture asserts that formula_29 is the graph of a bijection between formula_25 and formula_28.
The Howe duality conjecture for archimedean local fields was proved by Roger Howe. For formula_33-adic local fields with formula_33 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.