# 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 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.

## Statement
### Setup

Let $F$ be a local or a global field, not of characteristic $2$. Let $W$ be a symplectic vector space over $F$, and $Sp(W)$ the symplectic group.
Fix a reductive dual pair $(G,H)$ in $Sp(W)$. There is a classification of reductive dual pairs.

### Local theta correspondence

$F$ is now a local field. Fix a non-trivial additive character $\psi$ of $F$. There exists a Weil representation of the metaplectic group $Mp(W)$ associated to $\psi$, which we write as $\omega _{\psi }$.
Given the reductive dual pair $(G,H)$ in $Sp(W)$, one obtains a pair of commuting subgroups $({\widetilde {G}},{\widetilde {H}})$ in $Mp(W)$ by pulling back the projection map from $Mp(W)$ to $Sp(W)$.
The local theta correspondence is a 1-1 correspondence between certain irreducible admissible representations of ${\widetilde {G}}$ and certain irreducible admissible representations of ${\widetilde {H}}$, obtained by restricting the Weil representation $\omega _{\psi }$ of $Mp(W)$ to the subgroup ${\widetilde {G}}\cdot {\widetilde {H}}$. 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 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 ${\mathcal {R}}({\widetilde {G}},\omega _{\psi })$ the set of irreducible admissible representations of ${\widetilde {G}}$, which can be realized as quotients of $\omega _{\psi }$. Define ${\mathcal {R}}({\widetilde {H}},\omega _{\psi })$ and ${\mathcal {R}}({\widetilde {G}}\cdot {\widetilde {H}},\omega _{\psi })$, likewise.
The Howe duality conjecture asserts that ${\mathcal {R}}({\widetilde {G}}\cdot {\widetilde {H}},\omega _{\psi })$ is the graph of a bijection between ${\mathcal {R}}({\widetilde {G}},\omega _{\psi })$ and ${\mathcal {R}}({\widetilde {H}},\omega _{\psi })$.
The Howe duality conjecture for archimedean local fields was proved by Roger Howe. For $p$-adic local fields with $p$ 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.