Twisted Jacquet modules: A conjecture of D. Prasad

Nadimpalli, Santosh; Sheth, Mihir

doi:10.1090/proc/17086

Twisted Jacquet modules: A conjecture of D. Prasad
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by Santosh Nadimpalli and Mihir Sheth;

Proc. Amer. Math. Soc. 153 (2025), 1349-1361

DOI: https://doi.org/10.1090/proc/17086

Published electronically: January 21, 2025

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Abstract:

In this note, we study the twisted Jacquet modules of subquotients 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 of D. Prasad on twisted Jacquet modules of Speh representations of $\mathrm { GL}_2(D)$ when $D$ is the quaternionic division algebra. For arbitrary division algebras $D$ over $F$, 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 $D^{\times }$-representations in the depth-zero situation.

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