Weyl calculus and dual pairs

Abstract

We consider a dual pair $(G,G')$, in the sense of Howe, with $G$ compact acting on $L^2(\mathbb R^n)$ for an appropriate $n$ via the Weil Representation. Let $\widetilde{G}$ be the preimage of $G$ in the metaplectic group. Given a genuine irreducible unitary representation $\Pi$ of $\widetilde{G}$ 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 $\Pi'$ of $\widetilde{G'}$ and to compute of the wave front set of $\Pi'$ by elementary means.