Multiplicity one theorems: The Archimedean case

Abstract

Let G be one of the classical Lie groups GL n+1(R), GL n+1(C), U (p, q + 1), O(p, q + 1), O n+1(C), SO(p, q + 1), SO n+1(C), and let G' be re-spectively the subgroup GL n(R), GL n(C), U(p, q), O(p, q), O n(C), SO(p, q), SO n(C), embedded in G in the standard way. We show that every irreducible Casselman-Wallach representation of G' occurs with multiplicity at most one in every irreducible Casselman-Wallach representation of G. Similar results are proved for the Jacobi groups GL n(R)⋉H 2n+1(R), GL n(C)⋉ H 2n+1(C), U(p, q)⋉H 2p+2q+1(R), Sp 2n(R)⋉H 2n+1(R), Sp 2n(C)⋉H 2n+1(C), with their respective subgroups GL n(R), GL n(C), U(p, q), Sp 2n(R), and Sp 2n(C).