Geodesic
Irreducible Representations of a Product of Real Reductive Groups
Journal of Lie theory, Tome 23 (2013) no. 4, pp. 1005-101.

Voir la notice de l'article provenant de la source Heldermann Verlag

\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 $\Delta H \subset G \times H$.
Classification : 20G05, 22D12, 22E47
Mots-clés : Gelfand pair 
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     author = {D. Gourevitch and A. Kemarsky },
     title = {Irreducible {Representations} of a {Product} of {Real} {Reductive} {Groups}},
     journal = {Journal of Lie theory},
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D. Gourevitch; A. Kemarsky . Irreducible Representations of a Product of Real Reductive Groups. Journal of Lie theory, Tome 23 (2013) no. 4, pp. 1005-101. http://geodesic.mathdoc.fr/item/JLT_2013_23_4_JLT_2013_23_4_a5/