Symmetry breaking operators for the reductive dual pair (U[formula omitted],U[formula omitted])

Abstract

We consider the dual pair in the symplectic group . Fix a Weil representation of the metaplectic group . Let and be the preimages of and in , and let be a genuine irreducible representation of . We study the Weyl symbol of the (unique up to a possibly zero constant) symmetry breaking operator (SBO) intertwining the Weil representation with . This SBO coincides with the orthogonal projection of the space of the Weil representation onto its -isotypic component and also with the orthogonal projection onto its -isotypic component. Hence can be computed in two different ways, one using and the other using . By matching the results, we recover Weyl’s theorem stating that occurs in the Weil representation with multiplicity at most one and we also recover the complete list of the representations occurring in Howe’s correspondence.