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Journal of Lie Theory 23 (2013), No. 4, 1005--1010
Copyright Heldermann Verlag 2013

Irreducible Representations of a Product of Real Reductive Groups

Dmitry Gourevitch
Faculty of Mathematics and Computer Science, Weizmann Institute of Science, POB 26, Rehovot 76100, Israel

Alexander Kemarsky
Mathematics Department, Technion, Israel Institute of Technology, Haifa 32000, Israel


\def\R{{\Bbb R}} Let $G_1,G_2$ be real reductive groups and $(\pi,V)$ be a smooth admissible representation of $G_1 \times G_2$. We prove that $(\pi,V)$ is irreducible if and only if it 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 moderate growth of $G_i$, $i=1,2$. We deduce this from the analogous theorem for Harish-Chandra modules, for which one direction was proved by A. Aizenbud and D. Gourevitch [``Multiplicity one theorem for $(GL_{n+1}(\R), GL_n(\R))$'', Selecta Mathematica N. S. 15 (2009) 271--294], 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 $\Delta H \subset G \times H$.

Keywords: Gelfand pair.

MSC: 20G05, 22D12, 22E47

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