Twisted Jacquet modules: a conjecture of D. Prasad

Abstract

In this note, we study the twisted Jacquet modules of sub-quotients of principal series representations of ${\mathrm{GL}}_2(D)$ where $D$ is a division algebra over a non-archimedean local field $F$. We begin with a proof of a conjecture due to D. Prasad on twisted Jacquet modules of Speh representations of ${\mathrm{GL}}_2(D)$ when $D$ is the quaternionic division algebra. Later, when $D$ is an arbitrary division algebra over $F$, we focus on depth-zero principal series and compute the dimensions of twisted Jacquet modules of generalised Speh representations and investigate their structure explicitly.

Small corrections and further explanations in the proof of Theorem 4.8