Weyl calculus and dual pairs

Abstract

We consider a dual pair $(G,G^{\prime })$, in the sense of Howe, with $G$ compact acting on $L^2({\mathbb{R}}^n)$ for an appropriate $n$ via the Weil Representation. Let $\tilde{G}$ be the preimage of $G$ in the metaplectic group. Given a genuine irreducible unitary representation $\Pi$ of $\tilde{G}$ we compute the Weyl symbol of orthogonal projection onto $L^2({\mathbb{R}}^n{)}_{\Pi }$, the $\Pi$-isotypic component. We apply the result to obtain an explicit formula for the character of the corresponding irreducible unitary representation ${\Pi }^{\prime }$ of $\widetilde{G^{\prime }}$ and to compute of the wave front set of ${\Pi }^{\prime }$ by elementary means.

Cited by in corpus (1)

• Characters of some unitary highest weight representations via the theta correspondence