Weyl calculus and dual pairs

Abstract

We consider a dual pair (G, G ′ ), in the sense of Howe, with G compact acting on L 2 (R n ) for an appropriate n via the Weil Representation. Let G be the preimage of G in the metaplectic group. Given a genuine irreducible unitary representation Π of G we compute the Weyl symbol of orthogonal projection onto L 2 (R n ) Π , the Π-isotypic component. We apply the result to obtain an explicit formula for the character of the corresponding irreducible unitary representation Π ′ of G ′ and to compute of the wave front set of Π ′ by elementary means. Contents 1. Introduction. 1 2. The group G and the Weyl denominator. 7 3. A Theorem of G. W. Schwartz. 12 4. An almost semisimple orbital integral on the symplectic space. 13 5. Limits of orbital integrals. 28 6. Intertwining distributions. 40 7. The pair G = U l , G ′ = U l ′ , l ≤ l ′ . 59 8. Limits of orbital integrals in the stable range. 73 Appendix A: A few facts about nilpotent orbits 76 Appendix B: Pull-back of a distribution via a submersion 77 Appendix C: Some confluent hypergeometric polynomials 82 Appendix D: Wave front set of an asymptotically homogeneous distribution 90 Appendix E: A proof of a cocycle property 92 References 98 1. Introduction.Let W be a vector space of finite dimension 2n over R with a non-degenerate symplectic form •, • . Denote by Sp ⊆ GL(W) the symplectic group and let G, G ′ ⊆ Sp = Sp(W) be an irreducible dual pair. Denote by G, G ′ the preimages of G, G ′ in the metaplectic group