Supercuspidal unipotent representations: L-packets and formal degrees

Yongqi Feng; Eric Opdam; Maarten Solleveld

doi:10.5802/jep.138

Supercuspidal unipotent representations: L-packets and formal degrees
[Représentations unipotentes supercuspidales : L-paquets et degrés formels]

Yongqi Feng ¹ ; Eric Opdam ² ; Maarten Solleveld ³

¹ Department of Mathematics, Shantou University Daxue Road 243, 515063 Shantou, China
² Korteweg-de Vries Institute for Mathematics, Universiteit van Amsterdam Science Park 105-107, 1098 XG Amsterdam, The Netherlands
³ Institute for Mathematics, Astrophysics and Particle Physics, Radboud Universiteit Nijmegen Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands

Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 1133-1193.

Résumé
Abstract

Soit $K$ un corps local non archimédien et soit $G$ un $K$ -groupe connexe, réductif et deployé sur une extension non ramifiée de $K$ . Nous étudions des représentations unipotentes supercuspidales du groupe $G (K)$ . Nous établissons une bijection entre l’ensemble de telles $G (K)$ -représentations irréductibles et l’ensemble des L-paramètres étendus pour $G (K)$ , qui sont triviaux sur le sous-groupe d’inertie du groupe de Weil de $K$ . Le bijection est caractérisée par quelques propriétés simples et une comparaison des degrés formels des représentations avec des facteurs $γ$ adjoints des L-paramètres.

On peut considérer cela comme une correspondance de Langlands locale pour toutes les représentations unipotentes supercuspidales. Nous comptons les L-paquets résultants en termes de données déduites du diagramme de Dynkin affine de $G$ . Finalement, nous prouvons que notre bijection satisfait à la conjecture de Hiraga, Ichino et Ikeda sur les degrés formels des représentations.

Let $K$ be a non-archimedean local field and let $G$ be a connected reductive $K$ -group which splits over an unramified extension of $K$ . We investigate supercuspidal unipotent representations of the group $G (K)$ . We establish a bijection between the set of irreducible $G (K)$ -representations of this kind and the set of cuspidal enhanced L-parameters for $G (K)$ , which are trivial on the inertia subgroup of the Weil group of $K$ . The bijection is characterized by a few simple equivariance properties and a comparison of formal degrees of representations with adjoint $γ$ -factors of L-parameters.

This can be regarded as a local Langlands correspondence for all supercuspidal unipotent representations. We count the ensuing L-packets, in terms of data from the affine Dynkin diagram of $G$ . Finally, we prove that our bijection satisfies the conjecture of Hiraga, Ichino and Ikeda about the formal degrees of the representations.

Reçu le : 2019-11-20
Accepté le : 2020-10-03
Publié le : 2020-10-27

DOI : 10.5802/jep.138

Classification : 22E50, 11S37, 20G25, 43A99
Keywords: Reductive $p$-adic group, cuspidal representation, formal degree, local Langlands correspondence
Mots-clés : Groupes réductifs $p$-adiques, représentation cuspidale, degré formel, correspondance de Langlands locale

Affiliations des auteurs :

Yongqi Feng ¹ ; Eric Opdam ² ; Maarten Solleveld ³

¹ Department of Mathematics, Shantou University Daxue Road 243, 515063 Shantou, China
² Korteweg-de Vries Institute for Mathematics, Universiteit van Amsterdam Science Park 105-107, 1098 XG Amsterdam, The Netherlands
³ Institute for Mathematics, Astrophysics and Particle Physics, Radboud Universiteit Nijmegen Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Yongqi Feng and Eric Opdam and Maarten Solleveld},
     title = {Supercuspidal unipotent representations: {L-packets} and formal degrees},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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     doi = {10.5802/jep.138},
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Yongqi Feng; Eric Opdam; Maarten Solleveld. Supercuspidal unipotent representations: L-packets and formal degrees. Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 1133-1193. doi : 10.5802/jep.138. https://jep.centre-mersenne.org/articles/10.5802/jep.138/

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