Let be either or a finite extension of , and let be a finite central extension of the group of -points of a reductive group defined over . Also let be a smooth representation of (Fréchet of moderate growth if ). For each nilpotent orbit we consider a certain Whittaker quotient of . We define the Whittaker support to be the set of maximal among those for which .
In this paper we prove that all are quasi-admissible nilpotent orbits, generalizing results of Mœglin and Jiang–Liu–Savin. If is -adic and is quasi-cuspidal then we show that all are -distinguished, i.e. do not intersect the Lie algebra of any proper Levi subgroup of defined over .
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.
Soit le corps , ou une extension finie de , et soit une extension centrale finie du groupe des -points d’un groupe réductif fini sur . Soit aussi une representation lisse de (Fréchet à croissance modérée dans le cas ). Pour chaque orbite nilpotente , on considère un certain quotient de Whittaker de . Nous définissons le support de Whittaker comme étant l’ensemble des maximales parmi celles pour lesquelles .
Dans cet article, nous prouvons que toutes les sont des orbites nilpotentes quasi-admissibles, généralisant les résultats de Mœglin et de Jiang–Liu–Savin. Si est p-adique et est quasi-cuspidale, alors nous montrons que toutes les 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 .
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.
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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
Gomez, Raul 1; Gourevitch, Dmitry 2; Sahi, Siddhartha 3
@article{AIF_2021__71_1_239_0, author = {Gomez, Raul and Gourevitch, Dmitry and Sahi, Siddhartha}, 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}, volume = {71}, number = {1}, year = {2021}, doi = {10.5802/aif.3372}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3372/} }
TY - JOUR AU - Gomez, Raul AU - Gourevitch, Dmitry AU - Sahi, Siddhartha TI - Whittaker supports for representations of reductive groups JO - Annales de l'Institut Fourier PY - 2021 SP - 239 EP - 286 VL - 71 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3372/ DO - 10.5802/aif.3372 LA - en ID - AIF_2021__71_1_239_0 ER -
%0 Journal Article %A Gomez, Raul %A Gourevitch, Dmitry %A Sahi, Siddhartha %T Whittaker supports for representations of reductive groups %J Annales de l'Institut Fourier %D 2021 %P 239-286 %V 71 %N 1 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3372/ %R 10.5802/aif.3372 %G en %F AIF_2021__71_1_239_0
Gomez, Raul; Gourevitch, Dmitry; Sahi, Siddhartha. Whittaker supports for representations of reductive groups. Annales de l'Institut Fourier, Volume 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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