Twisted Jacquet modules: a conjecture of D. Prasad

TL;DR

This paper proves a conjecture on twisted Jacquet modules for Speh representations of GL_2(D) over division algebras, and computes dimensions and structures of these modules for depth-zero principal series.

Contribution

It proves D. Prasad's conjecture for Speh representations of GL_2(D) over quaternionic division algebra and analyzes twisted Jacquet modules for general division algebras.

Findings

Proof of Prasad's conjecture for quaternionic case

Explicit dimension formulas for twisted Jacquet modules

Structural analysis of modules for depth-zero principal series

Abstract

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