Symmetry breaking operators for the reductive dual pair (Ul,Ul′)

Abstract

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