  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.
