Irreducible representations of product of real reductive groups

Abstract

Let $G_1,G_2$ be real reductive groups and $(\pi,V)$ a smooth, irreducible, admissible representation of $G_1 \times G_2$. We prove that $(\pi,V)$ is the completed tensor product of $(\pi_i,V_i)$, $i=1,2$, where $(\pi_i,V_i)$ is a smooth,irreducible,admissible representation of $G_i$, $i=1,2$. We deduce this from the analogous theorem for Harish-Chandra modules, for which one direction was proven in [AG] and the other direction we prove here. As a corollary, we deduce that strong Gelfand property for a pair $H \subset G$ of real reductive groups is equivalent to the usual Gelfand property of the pair $\Delta H \subset G \times H$.