Twisted Jacquet modules: A conjecture of D. Prasad

In this note, we study the twisted Jacquet modules of subquotients of principal series representations of G L 2 ( D ) GL_2(D) where D D is a division algebra over a non-archimedean local field F F . We begin with a proof of a conjecture of D. Prasad on twisted Jacquet modules of Speh representations of G L 2 ( D ) GL_2(D) when D D is the quaternionic division algebra. For arbitrary division algebras D D over F F , we focus on depth-zero principal series. We compute the dimensions of twisted Jacquet modules of generalized Speh representations and explicitly investigate their structure as D × D× -representations in the depth-zero situation.