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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 multiplicity one, Gelfand pair, invariant distribution, finite group
MSC Classifications: 20G05, 20C30, 20C33, 46F10, 47A67 show english descriptions Representation theory
Representations of finite symmetric groups
Representations of finite groups of Lie type
Operations with distributions
Representation theory
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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