Structure of Twisted Jacquet Modules of principal series representations of $S{p}_4(F)$

Abstract

Let $F$ be a non-archimedean local field. For the symplectic group $S{p}_4(F),$ let $P$ and $Q$ denote respectively its Siegel and Klingen parabolic subgroups with respective Levi decompositions $P=MN$ and $Q=LU.$ For a non-trivial character $\psi$ of the unipotent radical $N$ of $P,$ let $M_{\psi }$ denote the stabilizer of the character $\psi$ in $M$ under the conjugation action of $M$ on characters of $N.$ For an irreducible representation of the Levi subgroups $M$ or $L,$ let $\pi$ denote the respective representation of $S{p}_4(F)$ parabolically induced either from $P$ or from $Q.$ Let $\psi$ be a character of the group $N$ given by a rank one quadratic form. In this article, we determine the structure of the twisted Jacquet module $r_{N,\psi }(\pi )$ as an $M_{\psi }$-module. We also deduce the analogous results in the case where $F$ is a finite field of order $q.$