
Journal of Lie Theory 23 (2013), No. 4, 10051010 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 dimagur@weizmann.ac.il Alexander Kemarsky Mathematics Department, Technion, Israel Institute of Technology, Haifa 32000, Israel alexkem@tx.technion.ac.il [Abstractpdf] \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 HarishChandra 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) 271294], 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 [ Fulltextpdf (241 KB)] for subscribers only. 