  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 $\omega$. Let $\tilde{\mathrm{G}}$ be the preimage of G in the
metaplectic group. Given a genuine irreducible unitary representation $\Pi$ of
$\tilde{\mathrm{G}}$, let $\Pi'$ be the corresponding irreducible unitary
representation of $\tilde{\mathrm{G}'}$ in the Howe duality. The orthogonal
projection onto $L^2(\mathbb{R}^n)_\Pi$, the $\Pi$-isotypic component, is the
essentially unique symmetry breaking operator in
$\mathrm{Hom}_{\tilde{\mathrm{G}}\tilde{\mathrm{G}'}}(\mathcal{H}_\omega^{\infty},
\mathcal{H}_\Pi^{\infty}\otimes \mathcal{H}_{\Pi'}^{\infty})$. We study this
operator by computing its Weyl symbol. Our results allow us to recover the
known list of highest weights of irreducible representations of
$\tilde{\mathrm{G}}$ occurring in Howe's correspondence when the rank of
$\tilde{\mathrm{G}}$ is strictly bigger than the rank of $\tilde{\mathrm{G'}}$.
They also allow us to compute the wavefront set of $\Pi'$ by elementary means.
