Whittaker supports for representations of reductive groups

Gomez, Raul; Gourevitch, Dmitry; Sahi, Siddhartha

doi:10.5802/aif.3372

Whittaker supports for representations of reductive groups
[Supports de Whittaker pour les représentations des groupes réductifs]

Gomez, Raul ¹ ; Gourevitch, Dmitry ² ; Sahi, Siddhartha ³

¹ UANL FCFM Av. Universidad, San Nicolas de los Garza, N.L., Mexico
² Faculty of Mathematics and Computer Science, Weizmann Institute of Science, 234 Herzl Street, Rehovot 7610001 Israel
³ Department of Mathematics, Rutgers University, Hill Center – Busch Campus, 110 Frelinghuysen Road Piscataway, NJ 08854-8019, USA

Annales de l'Institut Fourier, Tome 71 (2021) no. 1, pp. 239-286.

Résumé
Abstract

Soit $F$ le corps $ℝ$ , ou une extension finie de $ℚ_{p}$ , et soit $G$ une extension centrale finie du groupe des $F$ -points d’un groupe réductif fini sur $F$ . Soit aussi $π$ une representation lisse de $G$ (Fréchet à croissance modérée dans le cas $F = ℝ$ ). Pour chaque orbite nilpotente $𝒪$ , on considère un certain quotient de Whittaker $π_{𝒪}$ de $π$ . Nous définissons le support de Whittaker $WS (π)$ comme étant l’ensemble des $𝒪$ maximales parmi celles pour lesquelles $π_{𝒪} \neq 0$ .

Dans cet article, nous prouvons que toutes les $𝒪 \in WS (π)$ sont des orbites nilpotentes quasi-admissibles, généralisant les résultats de Mœglin et de Jiang–Liu–Savin. Si $F$ est p-adique et $π$ est quasi-cuspidale, alors nous montrons que toutes les $𝒪 \in WS (π)$ sont F-distinguées, c’est-à-dire qu’elles ne rencontrent l’algèbre de Lie d’aucun sous-groupe de Levi de G défini sur $F$ .

Nous donnons aussi une adaptation de nos méthodes aux représentations automorphes, généralisant ainsi des résultats de Ginzburg–Rallis–Soudry, Shen et Cai, et confirmant certaines conjectures de Ginzburg.

Nos méthodes combinent celles des auteurs susmentionnés et de nos propres travaux antérieurs.

Let $F$ be either $ℝ$ or a finite extension of $ℚ_{p}$ , and let $G$ be a finite central extension of the group of $F$ -points of a reductive group defined over $F$ . Also let $π$ be a smooth representation of $G$ (Fréchet of moderate growth if $F = ℝ$ ). For each nilpotent orbit $𝒪$ we consider a certain Whittaker quotient $π_{𝒪}$ of $π$ . We define the Whittaker support $WS (π)$ to be the set of maximal $𝒪$ among those for which $π_{𝒪} \neq 0$ .

In this paper we prove that all $𝒪 \in WS (π)$ are quasi-admissible nilpotent orbits, generalizing results of Mœglin and Jiang–Liu–Savin. If $F$ is $p$ -adic and $π$ is quasi-cuspidal then we show that all $𝒪 \in WS (π)$ are $F$ -distinguished, i.e. do not intersect the Lie algebra of any proper Levi subgroup of $G$ defined over $F$ .

We also give an adaptation of our argument to automorphic representations, generalizing results of Ginzburg–Rallis–Soudry, Shen, and Cai, and confirming some conjectures of Ginzburg.

Our methods are a synergy of the methods of the above-mentioned authors, and of our own earlier work.

Reçu le : 2018-05-07
Révisé le : 2018-12-18
Accepté le : 2019-06-14
Publié le : 2021-06-07

DOI : 10.5802/aif.3372

Classification : 20G05, 20G20, 20G25, 20G30, 20G35, 22E27, 22E46, 22E50, 22E55, 17B08
Keywords: Fourier coefficient, wave-front set, oscillator representation, Heisenberg group, metaplectic group, admissible orbit, distinguished orbit, cuspidal representation, automorphic form
Mot clés : coefficient de Fourier, front d’onde, représentation oscillateur, groupe de Heisenberg, groupe métaplectique, orbite admissible, orbite distinguée, représentation cuspidale, forme automorphe

Affiliations des auteurs :

Gomez, Raul ¹ ; Gourevitch, Dmitry ² ; Sahi, Siddhartha ³

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Whittaker supports for representations of reductive groups},
     journal = {Annales de l'Institut Fourier},
     pages = {239--286},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Gomez, Raul; Gourevitch, Dmitry; Sahi, Siddhartha. Whittaker supports for representations of reductive groups. Annales de l'Institut Fourier, Tome 71 (2021) no. 1, pp. 239-286. doi : 10.5802/aif.3372. https://aif.centre-mersenne.org/articles/10.5802/aif.3372/

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