Structure of twisted Jacquet modules of principal series representations of $G{L}_{2n}(F)$

Abstract

Let $F$ be a non-archimedean local field or a finite field. Let $\pi$ be a principal series representation of $G{L}_{2n}(F)$ induced from any of its maximal standard parabolic subgroups. Let $N$ be the unipotent radical of the maximal parabolic subgroup $P$ of $G{L}_{2n}(F)$ corresponding to the partition $(n,n).$ In this article, we describe the structure of the twisted Jacquet module ${\pi }_{N,\psi }$ of $\pi$ with respect to $N$ and a non-degenerate character $\psi$ of $N.$ We also provide a necessary and sufficient condition for ${\pi }_{N,\psi }$ to be non-zero and show that the twisted Jacquet module is non-zero under certain assumptions on the inducing data. As an application of our results, we obtain the structure of twisted Jacquet modules of certain non-generic irreducible representations of $G{L}_{2n}(F)$ and establish the existence of their Shalika model in the non-archimedean case. We conclude our article with a conjecture by Dipendra Prasad classifying the smooth irreducible representations of $G{L}_{2n}(F)$ with a non-zero twisted Jacquet module.

Revised version with Section 6.3 presenting additional results. 40 pages