Multiplicity Free Jacquet Modules

Aizenbud, Avraham ; Gourevitch, Dmitry

Canadian mathematical bulletin, Tome 55 (2012) no. 4, pp. 673-688

Résumé

Let $F$ be a non-Archimedean local field or a finite field. Let $n$ be a natural number and $k$ be 1 or 2. Consider $G\,:=\,\text{G}{{\text{L}}_{n+k}}\left( F \right)$ and let $M\,:=\,\text{G}{{\text{L}}_{n}}\left( F \right)\,\times \,\text{G}{{\text{L}}_{k}}\left( F \right)\,<\,G$ be a maximal Levi subgroup. Let $U\,<\,G$ be the corresponding unipotent subgroup and let $P\,=\,MU$ be the corresponding parabolic subgroup. Let $J\,:=\,J_{M}^{G}\,:\,\mathcal{M}\left( G \right)\,\to \,\mathcal{M}\left( M \right)$ be the Jacquet functor, i.e., the functor of coinvariants with respect to $U$ . In this paper we prove that $J$ is a multiplicity free functor, i.e., $\dim\,\text{Ho}{{\text{m}}_{M}}\left( J\left( \pi\right),\,\rho\right)\,\le \,1$ , for any irreducible representations $\pi $ of $G$ and $\rho $ of $M$ . We adapt the classical method of Gelfand and Kazhdan, which proves the “multiplicity free” property of certain representations to prove the “multiplicity free” property of certain functors. At the end we discuss whether other Jacquet functors are multiplicity free.

Détail

DOI : 10.4153/CMB-2011-127-8

Mots-clés : 20G05, 20C30, 20C33, 46F10, 47A67, multiplicity one, Gelfand pair, invariant distribution, finite group