Multiplicity Free Jacquet Modules

Canadian mathematical bulletin

Dmitry Gourevitch²  Avraham Aizenbud¹ 

[1] Massachussetts Institute of Technology, Cambridge, MA 02139, USA;Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, Rehovot 76100, Israel

关键词: multiplicity one;    Gelfand pair;    invariant distribution;    finite group;   

DOI  :  10.4153/CMB-2011-127-8

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

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$M:=operatorname{GL}_n(F) imes operatorname{GL}_k(F)lt G$ be a maximal Levi subgroup. Let $Ult G$ be the corresponding unipotent subgroup and let $P=MU$ be the corresponding parabolic subgroup.Let $J:=J_M^G: mathcal{M}(G) o mathcal{M}(M)$ 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.,$dim operatorname{Hom}_M(J(pi),ho)leq 1$,for any irreducible representations $pi$ of $G$ and $ho$ 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.

【 授权许可】

Unknown