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Weyl calculus and dual pairs

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 $Π$ of $\widetilde{G}$ we compute the Weyl symbol of orthogonal projection onto $L^2(\mathbb R^n)_Π$, the $Π$-isotypic component. We apply the result to obtain an explicit formula for the character of the corresponding irreducible unitary representation $Π'$ of $\widetilde{G'}$ and to compute of the wave front set of $Π'$ by elementary means.