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

Abstract

We consider the dual pair (G,G′)=(Ul,Ul′) in the symplectic group Sp2ll′(R). Fix a Weil representation of the metaplectic group Sp˜2ll′(R). Let G˜ and G˜′ be the preimages of G and G′ in Sp˜2ll′(R), and let Π⊗Π′ be a genuine irreducible representation of G˜×G˜′. We study the Weyl symbol fΠ⊗Π′ of the (unique up to a possibly zero constant) symmetry breaking operator (SBO) intertwining the Weil representation with Π⊗Π′. This SBO coincides with the orthogonal projection of the space of the Weil representation onto its Π-isotypic component and also with the orthogonal projection onto its Π′-isotypic component. Hence fΠ⊗Π′ can be computed in two different ways, one using Π and the other using Π′. By matching the results, we recover Weyl’s theorem stating that Π⊗Π′ occurs in the Weil representation with multiplicity at most one and we also recover the complete list of the representations Π⊗Π′ occurring in Howe’s correspondence.