Twisted Jacquet modules: a conjecture of D. Prasad

In this note, we study the twisted Jacquet modules of sub-quotients of principal series representations of ${\mathrm{G}\mathrm{L}}_2(D)$ 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 ${\mathrm{G}\mathrm{L}}_2(D)$ 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.