Symmetry breaking operators for dual pairs with one member compact

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 $ω$. Let $\tilde{\mathrm{G}}$ be the preimage of G in the metaplectic group. Given a genuine irreducible unitary representation $Π$ of $\tilde{\mathrm{G}}$, let $Π'$ be the corresponding irreducible unitary representation of $\tilde{\mathrm{G}'}$ in the Howe duality. The orthogonal projection onto $L^2(\mathbb{R}^n)_Π$, the $Π$-isotypic component, is the essentially unique symmetry breaking operator in $\mathrm{Hom}_{\tilde{\mathrm{G}}\tilde{\mathrm{G}'}}(\mathcal{H}_ω^{\infty}, \mathcal{H}_Π^{\infty}\otimes \mathcal{H}_{Π'}^{\infty})$. 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 $Π'$ by elementary means.