The Kirillov model in families

TL;DR

This paper proves the existence of Kirillov models for a broad class of representations of GL(n) over non-archimedean fields, extending to modular and family settings, using Rankin-Selberg methods.

Contribution

It provides a new, quick proof of Kirillov model existence for Whittaker type representations, including modular and family cases, generalizing previous results.

Findings

Injectivity of the mirabolic restriction map for Whittaker type modules.

Extension of Kirillov model existence to $ ext{mod} ext{ } ext{ell}$-representations and families.

Answer to Vignéras's 1989 question in the irreducible generic case.

Abstract

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. This gives a new quick proof of the existence of Kirillov models for representations of Whittaker type, including complex representations, which generalizes to the $\ell$-modular and families setting, in contrast with the previous proofs. 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.