A note on Jacquet modules of general linear groups

TL;DR

This paper computes the semisimplified Jacquet modules of certain representations of general linear groups over non-Archimedean fields, demonstrating multiplicity-free properties within a specific category using Zelevinsky classification.

Contribution

It provides explicit calculations of Jacquet modules for a class of GL_n representations and proves their multiplicity-free nature in a particular subcategory.

Findings

Jacquet modules are explicitly computed for the considered representations.

Jacquet modules are shown to be multiplicity-free in a specific subcategory.

Results are based on Zelevinsky classification of representations.

Abstract

Let F be a non-Archimedean local field. Consider G_n:= GL_n(F) and let M:= G_l * G_{n-l} be a maximal Levi subgroup of G_n. In this article, we compute the semisimplified Jacquet module of representations of G_n with respect to the maximal Levi subgroup M, belonging to a particular category of representations. Utilizing our results, we prove that the Jacquet module is multiplicity-free for a specific subcategory of representations. Our findings are based on the Zelevinsky classification.