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# Multiplicity Free Jacquet Modules

Published:2011-06-29
Printed: Dec 2012
• Avraham Aizenbud,
Massachussetts Institute of Technology, Cambridge, MA 02139, USA
• Dmitry Gourevitch,
Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, Rehovot 76100, Israel
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## Abstract

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 $G:=\operatorname{GL}_{n+k}(F)$ and let $M:=\operatorname{GL}_n(F) \times \operatorname{GL}_k(F)\lt G$ be a maximal Levi subgroup. Let $U\lt G$ be the corresponding unipotent subgroup and let $P=MU$ be the corresponding parabolic subgroup. Let $J:=J_M^G: \mathcal{M}(G) \to \mathcal{M}(M)$ 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., $\dim \operatorname{Hom}_M(J(\pi),\rho)\leq 1$, 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.
 Keywords: multiplicity one, Gelfand pair, invariant distribution, finite group
 MSC Classifications: 20G05 - Representation theory 20C30 - Representations of finite symmetric groups 20C33 - Representations of finite groups of Lie type 46F10 - Operations with distributions 47A67 - Representation theory

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