Irreducible Representations of a Product of Real Reductive Groups
Journal of Lie Theory, Volume 23 (2013) no. 4, pp. 1005-1010
\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 $\Delta H \subset G \times H$.
@article{JOLT_2013_23_4_a5,
author = {D. Gourevitch and A. Kemarsky},
title = {Irreducible {Representations} of a {Product} of {Real} {Reductive} {Groups}},
journal = {Journal of Lie Theory},
pages = {1005--1010},
year = {2013},
volume = {23},
number = {4},
doi = {10.5802/jolt.761},
zbl = {1284.22007},
url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.761/}
}
TY - JOUR AU - D. Gourevitch AU - A. Kemarsky TI - Irreducible Representations of a Product of Real Reductive Groups JO - Journal of Lie Theory PY - 2013 SP - 1005 EP - 1010 VL - 23 IS - 4 UR - https://jolt.centre-mersenne.org/articles/10.5802/jolt.761/ DO - 10.5802/jolt.761 ID - JOLT_2013_23_4_a5 ER -
D. Gourevitch; A. Kemarsky. Irreducible Representations of a Product of Real Reductive Groups. Journal of Lie Theory, Volume 23 (2013) no. 4, pp. 1005-1010. doi: 10.5802/jolt.761
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