Symmetry breaking operator for the reductive dual pair $(\mathrm{U}_l,\mathrm{U}_{l'})$

Abstract

We consider the dual pair $(G,G')=(\mathrm{U}_l,\mathrm{U}_{l'})$ in the symplectic group $\mathrm{Sp}_{2ll'}(\mathbb{R})$. Fix a Weil representation of the metaplectic group $\tilde{\mathrm{Sp}}_{2ll'}(\mathbb{R})$. Let $\tilde{G\,}$ and $\tilde{G'}$ be the preimages of $G$ and $G'$ under the metaplectic cover $\tilde{\mathrm{Sp}}_{2ll'}(\mathbb{R})\to \mathrm{Sp}_{2ll'}(\mathbb{R})$, and let $\Pi\otimes\Pi'$ be a genuine irreducible representation of $\tilde{G\,}\times\tilde{G'}$. We study the Weyl symbol $f_{\Pi\otimes\Pi'}$ of the (unique up to a possibly zero constant) symmetry breaking operator (SBO) intertwining the Weil representation with $\Pi\otimes\Pi'$. This SBO coincides with the orthogonal projection of the space of the Weil representation onto its $\Pi$-isotypic component and also with the orthogonal projection onto its $\Pi'$-isotypic component. Hence $f_{\Pi\otimes\Pi'}$ can be computed in two different ways, one using $\Pi$ and the other using $\Pi'$. By matching the results, we recover Weyl's theorem stating that $\Pi\otimes\Pi'$ occurs in the Weil representation with multiplicity at most one and we also recover the complete list of the representations $\Pi\otimes\Pi'$ occurring in Howe's correspondence.