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Multiplicity Free Jacquet Modules

Published online by Cambridge University Press:  20 November 2018

Avraham Aizenbud  and
Dmitry Gourevitch
Avraham Aizenbud
Affiliation:
Massachussetts Institute of Technology, Cambridge, MA 02139, USAe-mail: aizner@gmail.com
Dmitry Gourevitch
Affiliation:
Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, Rehovot 76100, Israele-mail: dimagur@weizmann.ac.il
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Abstract

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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.

Keywords

20G0520C3020C3346F1047A67multiplicity oneGelfand pairinvariant distributionfinite group
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 55 , Issue 4 , 01 December 2012 , pp. 673 - 688
Copyright
Copyright © Canadian Mathematical Society 2012

Footnotes

The authors were partially supported by a BSF grant, a GIF grant, and an ISF Center of excellency grant. The first author was also supported by ISF grant No. 583/09 and the second author by NSF grant DMS-0635607.

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