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We consider the dual pair $(G,G')=(\mathrm{U}_l,\mathrm{U}_{l'})$ in thesymplectic group $\mathrm{Sp}_{2ll'}(\mathbb{R})$. Fix a Weil representation ofthe metaplectic group $\tilde{\mathrm{Sp}}_{2ll'}(\mathbb{R})$. Let$\tilde{G\,}$ and $\tilde{G'}$ be the preimages of $G$ and $G'$ under themetaplectic cover $\tilde{\mathrm{Sp}}_{2ll'}(\mathbb{R})\to\mathrm{Sp}_{2ll'}(\mathbb{R})$, and let $\Pi\otimes\Pi'$ be a genuineirreducible representation of $\tilde{G\,}\times\tilde{G'}$. We study the Weylsymbol $f_{\Pi\otimes\Pi'}$ 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 thespace of the Weil representation onto its $\Pi$-isotypic component and alsowith the orthogonal projection onto its $\Pi'$-isotypic component. Hence$f_{\Pi\otimes\Pi'}$ can be computed in two different ways, one using $\Pi$ andthe other using $\Pi'$. By matching the results, we recover Weyl's theoremstating that $\Pi\otimes\Pi'$ occurs in the Weil representation withmultiplicity at most one and we also recover the complete list of therepresentations $\Pi\otimes\Pi'$ occurring in Howe's correspondence.

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