A note on Jacquet modules of general linear groups

Let F be a non-Archimedean local field. Consider G_n:= GL_n(F) and let M:= G_l * Gₙ₋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.