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arXiv · 2005.13484

The Kirillov model in families

Nadir Matringe·Gilbert Moss
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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.

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Nadir Matringe, Gilbert Moss. 2020-05-27. The Kirillov model in families. https://arxiv.org/abs/2005.13484

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