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The Kirillov model in families

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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. By extending recent results of the second author regarding the Rankin–Selberg functional equations, 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 Vignéras from (Compos Math 72(1):33–66, 1989).

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Acknowledgements

The first author thanks the Abdus Salam School of Mathematical Sciences where parts of the paper were written for its hospitality, and the CNRS for giving him a “délégation.” The second author thanks the CNRS for its “poste-rouge” visitor funding, and the laboratoire de mathématiques de Versailles, where the paper started, for their hospitality, as well as the University of Utah. Both authors thank Olivier Fouquet and Rob Kurinczuk for their interest, and the referees for their useful comments and corrections.

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Authors and Affiliations

  1. Laboratoire de Mathématiques et Applications, Université de Poitiers, Téléport 2, Boulevard Marie et Pierre Curie, BP 30179, 86962, Futuroscope Chasseneuil Cedex, France

    Nadir Matringe

  2. Department of Mathematics, University of Utah, 155 South 1400 East JWB233, Salt Lake City, UT, 84103, USA

    Gilbert Moss

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  1. Nadir Matringe
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  2. Gilbert Moss
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Correspondence to Gilbert Moss.

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Communicated by Alberto Minguez.

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Matringe, N., Moss, G. The Kirillov model in families. Monatsh Math 198, 393–410 (2022). https://doi.org/10.1007/s00605-022-01675-4

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