Symmetry breaking operators for dual pairs with one member compact

Abstract

We consider a dual pair $(G,G^{\prime })$, 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 ${\Pi }^{\prime }$ be the corresponding irreducible unitary representation of $\widetilde{{\mathrm{G}}^{\prime }}$ 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 ${\mathrm{Hom}}_{\tilde{\mathrm{G}}\widetilde{{\mathrm{G}}^{\prime }}}({\mathcal{H}}_{\omega }^{\infty },{\mathcal{H}}_{\Pi }^{\infty }\otimes {\mathcal{H}}_{{\Pi }^{\prime }}^{\infty })$. 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 $\widetilde{{\mathrm{G}}^{\prime }}$. They also allow us to compute the wavefront set of ${\Pi }^{\prime }$ by elementary means.

arXiv commentThe present paper subsumes and extends sections 2,3,4 and 6 of the (unpublished) preprint arXiv:1405.2431. To appear in J. Math. Sci. Univ. Tokyo

References in corpus (1)

• Characters of some unitary highest weight representations via the theta correspondence