## Generalized Whittaker functions and Jacquet modules

HTML articles powered by AMS MathViewer

- 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.

## References

- Colin J. Bushnell and Guy Henniart,
*Generalized Whittaker models and the Bernstein center*, Amer. J. Math.**125**(2003), no. 3, 513–547. MR**1981032** - W. Casselman,
*Introduction to the theory of admissible representations of p-adic reductve groups*, https://www.math.ubc.ca/~cass/research/publications.html. - W. Casselman and J. Shalika,
*The unramified principal series of $p$-adic groups. II. The Whittaker function*, Compositio Math.**41**(1980), no. 2, 207–231. MR**581582** - J.-F. Dat, D. Helm, R. Kurinczuk, and G. Moss,
*Finiteness for hecke algebras of p-adic groups*, arXiv:2203.04929, 2022. - J.-F. Dat,
*$v$-tempered representations of $p$-adic groups. I. $l$-adic case*, Duke Math. J.**126**(2005), no. 3, 397–469. MR**2120114**, DOI 10.1215/S0012-7094-04-12631-4 - Patrick Delorme,
*Constant term of smooth $H_\psi$-spherical functions on a reductive $p$-adic group*, Trans. Amer. Math. Soc.**362**(2010), no. 2, 933–955. MR**2551511**, DOI 10.1090/S0002-9947-09-04925-3 - Patrick Delorme,
*Théorème de Paley-Wiener pour les fonctions de Whittaker sur un groupe réductif $p$-adique*, J. Inst. Math. Jussieu**11**(2012), no. 3, 501–568 (French, with English and French summaries). MR**2931317**, DOI 10.1017/S1474748011000193 - L. Fargues and S. Scholze,
*Geometrization of the local langlands correspondence*, arXiv:2102.13459, 2021. - Erez Lapid and Zhengyu Mao,
*On the asymptotics of Whittaker functions*, Represent. Theory**13**(2009), 63–81. MR**2495561**, DOI 10.1090/S1088-4165-09-00343-4 - David Renard,
*Représentations des groupes réductifs $p$-adiques*, Cours Spécialisés [Specialized Courses], vol. 17, Société Mathématique de France, Paris, 2010 (French). MR**2567785** - Yiannis Sakellaridis and Akshay Venkatesh,
*Periods and harmonic analysis on spherical varieties*, Astérisque**396**(2017), viii+360 (English, with English and French summaries). MR**3764130** - Marie-France Vignéras,
*Représentations $l$-modulaires d’un groupe réductif $p$-adique avec $l\ne p$*, Progress in Mathematics, vol. 137, Birkhäuser Boston, Inc., Boston, MA, 1996 (French, with English summary). MR**1395151** - Marie-France Vignéras,
*On highest Whittaker models and integral structures*, Contributions to automorphic forms, geometry, and number theory, pp. 773–801, Johns Hopkins Univ. Press, Baltimore, MD, 2004.

## 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