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 Template:Harvtxt. Its name arose due to its origin in André Weil's representation theoretical formulation of the theory of theta series in Template:Harvtxt. The Shimura correspondence as constructed by Jean-Loup Waldspurger in Template:Harvtxt and Template:Harvtxt 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.Template:Sfn Template:Sfn

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 $(\tilde{G},\tilde{H})$ 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 $\tilde{G}$ and certain irreducible admissible representations of $\tilde{H}$, obtained by restricting the Weil representation ${\omega }_{\psi }$ of $Mp(W)$ to the subgroup $\tilde{G}\cdot \tilde{H}$. The correspondence was defined by Roger Howe in Template:Harvtxt. 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 Template:Sfn and conservation relations concerning the first occurrence indices along Witt towers .Template:Sfn

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. Template:Sfn

Howe duality conjecture

Define $ℛ(\tilde{G},{\omega }_{\psi })$ the set of irreducible admissible representations of $\tilde{G}$, which can be realized as quotients of ${\omega }_{\psi }$. Define $ℛ(\tilde{H},{\omega }_{\psi })$ and $ℛ(\tilde{G}\cdot \tilde{H},{\omega }_{\psi })$, likewise.

The Howe duality conjecture asserts that $ℛ(\tilde{G}\cdot \tilde{H},{\omega }_{\psi })$ is the graph of a bijection between $ℛ(\tilde{G},{\omega }_{\psi })$ and $ℛ(\tilde{H},{\omega }_{\psi })$.

The Howe duality conjecture for archimedean local fields was proved by Roger Howe.Template:Sfn For $p$-adic local fields with $p$ odd it was proved by Jean-Loup Waldspurger.Template:Sfn Alberto Mínguez later gave a proof for dual pairs of general linear groups, that works for arbitrary residue characteristic. Template:Sfn For orthogonal-symplectic or unitary dual pairs, it was proved by Wee Teck Gan and Shuichiro Takeda. Template:Sfn The final case of quaternionic dual pairs was completed by Wee Teck Gan and Binyong Sun.Template:Sfn

See also

• Reductive dual pair
• Metaplectic group

Bibliography

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