Supercuspidal unipotent representations: L-packets and formal degrees
[Représentations unipotentes supercuspidales : L-paquets et degrés formels]
Journal de l’École polytechnique — Mathématiques, Tome 7 (2020) , pp. 1133-1193.

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 :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/jep.138
Classification : 22E50,  11S37,  20G25,  43A99
Mots clés : Groupes réductifs p-adiques, représentation cuspidale, degré formel, correspondance de Langlands locale
@article{JEP_2020__7__1133_0,
     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},
     pages = {1133--1193},
     publisher = {\'Ecole polytechnique},
     volume = {7},
     year = {2020},
     doi = {10.5802/jep.138},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.138/}
}
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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