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
dimagur@weizmann.ac.il
Alexander Kemarsky
Mathematics Department, Technion, Israel Institute of Technology, Haifa 32000, Israel
alexkem@tx.technion.ac.il
[Abstract-pdf]
\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
[ Fulltext-pdf (241 KB)] for subscribers only.