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 Howe (1979).
Its name arose due to its origin in André Weil's representation theoretical formulation of the theory of theta series in Weil (1964).
The Shimura correspondence as constructed by Jean-Loup Waldspurger in Waldspurger (1980) and Waldspurger (1991) may be viewed as an instance of the theta correspondence.
be a local or a global field, not of characteristic
be a symplectic vector space over
Fix a reductive dual pair
There is a classification of reductive dual pairs.
is now a local field.
Fix a non-trivial additive character
There exists a Weil representation of the metaplectic group
Given the reductive dual pair
, one obtains a pair of commuting subgroups
by pulling back the projection map from
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
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] 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.
the set of irreducible admissible representations of
The Howe duality conjecture asserts that
The Howe duality conjecture for archimedean local fields was proved by Roger Howe.
-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.