Weyl calculus and dual pairs

TL;DR

This paper computes Weyl symbols and characters for representations arising from dual pairs in the Weil representation, providing explicit formulas and elementary methods to analyze their wave front sets.

Contribution

It introduces explicit formulas for Weyl symbols and characters of representations associated with dual pairs, advancing the understanding of their structure and wave front sets.

Findings

Explicit Weyl symbol formulas for isotypic components

Character formulas for irreducible unitary representations

Elementary computation of wave front sets

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.