The Kirillov model in families

Nadir Matringe, Gilbert Moss

Published 2020-05-27Version 1

Let $F$ be a non-archimedean local field, let $k$ be an algebraically closed field of characteristic $\ell$ different from the residual characteristic of $F$, and let $A$ be a commutative Noetherian $W(k)$-algebra, where $W(k)$ denotes the Witt vectors. Using the Rankin-Selberg functional equations and extending recent results of the second author, we show that if $V$ is an $A[\text{GL}_n(F)]$-module of Whittaker type, then the mirabolic restriction map on its Whittaker space is injective. In the special case where $A=k=\overline{\mathbb{F}_{\ell}}$ and $V$ is irreducible generic, our result in particular answers a question of Vign\'eras from 1989.