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

Abstract: Let $F$ be a non-archimedean local field or a finite field. Let $π$ be a principal series representation of $GL_{2n}(F)$ induced from any of its maximal standard parabolic subgroups. Let $N$ be the unipotent radical of the maximal parabolic subgroup $P$ of $GL_{2n}(F)$ corresponding to the partition $(n,n).$ In this article, we describe the structure of the twisted Jacquet module $π{N,ψ}$ of $π$ with respect to $N$ and a non-degenerate character $ψ$ of $N.$ We also provide a necessary and sufficient condition for $π{N,ψ}$ 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 $GL_{2n}(F)$ and establish the existence of their Shalika models. We conclude our article with a conjecture by Dipendra Prasad classifying the smooth irreducible representations of $GL_{2n}(F)$ with a non-zero twisted Jacquet module.