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 {{harvtxt| Howe | 1979}}. Its name arose due to its origin in André Weil's representation theoretical formulation of the theory of theta series in {{harvtxt| Weil | 1964 }}. The Shimura correspondence as constructed by Jean-Loup Waldspurger in {{harvtxt | Waldspurger | 1980 }} and {{harvtxt | 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.{{sfn | Howe | 1979 }} {{sfn | Mœglin | Vignéras | Waldspurger | 1987 }}

=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 {{harvtxt|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 {{sfn|Kudla|1986}} and conservation relations concerning the first occurrence indices along Witt towers .{{sfn | Sun | Zhu | 2015 }}

=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. {{sfn|Rallis|1984}}

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.{{sfn | Howe | 1989 }} For $p$ -adic local fields with $p$ odd it was proved by Jean-Loup Waldspurger.{{sfn | Waldspurger | 1990 }} Alberto Mínguez later gave a proof for dual pairs of general linear groups, that works for arbitrary residue characteristic. {{sfn | Mínguez | 2008 }} For orthogonal-symplectic or unitary dual pairs, it was proved by Wee Teck Gan and Shuichiro Takeda. {{sfn | Gan | Takeda | 2016 }} The final case of quaternionic dual pairs was completed by Wee Teck Gan and Binyong Sun.{{sfn | Gan | Sun | 2017 }}

References

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Category:Langlands program

Category:Representation theory