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Representation Theory

Published by the American Mathematical Society since 1997, this electronic-only journal is devoted to research in representation theory and seeks to maintain a high standard for exposition as well as for mathematical content. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.71.

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Generalized Whittaker functions and Jacquet modules
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by Nadir Matringe PDF
Represent. Theory 27 (2023), 62-79 Request permission

Abstract:

Let $F$ be a non-Archimedean local field and $G$ be (the $F$-points of) a connected reductive group defined over $F$. Fix $U_0$ to be the unipotent radical of a minimal parabolic subgroup $P_0$ of $G$, and $\psi :U_0\rightarrow \mathbb {C}^\times$ be a non-degenerate character of $U_0$. Let $P=MU\supseteq P_0$ be a standard parabolic subgroup of $G$ so that the restriction $\psi _M$ of $\psi$ to $M\cap U_0$ is non-degenerate. We denote by $\mathcal {W}(G,\psi )$ the space of smooth $\psi$-Whittaker functions on $G$ and by $\mathcal {W}_c(G,\psi )$ its $G$-stable subspace consisting of functions with compact support modulo $U_0$. In this situation Bushnell and Henniart identified $\mathcal {W}_c(M,\psi _M^{-1})$ to the Jacquet module of $\mathcal {W}_c(G,\psi ^{-1})$ with respect to $P^-$ (Bushnell and Henniart [Amer. J. Math. 125 (2003), pp. 513–547]). On the other hand Delorme defined a constant term map from $\mathcal {W}(G,\psi )$ to $\mathcal {W}(M,\psi _M)$ which descends to the Jacquet module of $\mathcal {W}(G,\psi )$ with respect to $P$ (Delorme [Trans. Amer. Math. Soc. 362 (2010), pp. 933–955]). We show (as we surprisingly could not find a proof of this statement in the literature) that the descent of Delorme’s constant term map is the dual map of the isomorphism of Bushnell and Henniart, in particular the constant term map is surjective. We also show that the constant term map coincides on admissible submodules of $\mathcal {W}(G,\psi )$ with the inflation of the “germ map” defined by Lapid and Mao [Represent. Theory 13 (2009), pp. 63–81] following earlier works of Casselman and Shalika [Compositio Math. 41 (1980), pp. 207–231]. From these results we derive a simple proof of a slight generalization of a theorem of Delorme and Sakellaridis–Venkatesh ([Ast’erisque 396 (2017), pp. viii+360] for quasi-split $G$) on irreducible discrete series with a generalized Whittaker model to the setting of admissible representations with a central character under the split component of $G$, and similar statements in the cuspidal case (also generalizing a result of Delorme) and in the tempered case. We also show that the germ map of Lapid and Mao is injective, answering one of their questions. Finally using a result of Vignéras [Contributions to automorphic forms, geometry, and number theory, Johns Hopkins Univ. Press, Baltimore, MD, 2004, pp. 773–801] and recent results of Dat, Helm, Kurinczuk, and Moss [Finiteness for hecke algebras of p-adic groups, arXiv:2203.04929, 2022], we show in the context $\ell$-adic representations that the asymptotic expansion of Lapid and Mao can be chosen to be integral for functions in integral $G$-submodules of $\mathcal {W}(\pi ,\psi )$ of finite length.
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Additional Information
  • Nadir Matringe
  • Affiliation: Université Paris Cité, IMJ-PRG, 8 Pl. Aurélie Nemours, 75013 Paris, France
  • MR Author ID: 852792
  • ORCID: 0000-0003-3064-2525
  • Email: nadir.matringe@imj-prg.fr
  • Received by editor(s): September 4, 2020
  • Received by editor(s) in revised form: March 11, 2021, March 16, 2021, April 18, 2022, and April 27, 2022
  • Published electronically: March 28, 2023
  • © Copyright 2023 American Mathematical Society
  • Journal: Represent. Theory 27 (2023), 62-79
  • MSC (2020): Primary 22E50, 22E35
  • DOI: https://doi.org/10.1090/ert/636