Supercuspidal unipotent representations: L-packets and formal degrees

Feng, Yongqi; Opdam, Eric; Solleveld, Maarten

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

Let $K$ be a non-archimedean local field and let $\mathbf{G}$ be a connected reductive $K$-group which splits over an unramified extension of $K$. We investigate supercuspidal unipotent representations of the group $\mathbf{G}\left(K\right)$. We establish a bijection between the set of irreducible $\mathbf{G}\left(K\right)$-representations of this kind and the set of cuspidal enhanced L-parameters for $\mathbf{G}\left(K\right)$, 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 $\gamma$-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 $\mathbf{G}$. Finally, we prove that our bijection satisfies the conjecture of Hiraga, Ichino and Ikeda about the formal degrees of the representations.

Soit $K$ un corps local non archimédien et soit $\mathbf{G}$ un $K$-groupe connexe, réductif et deployé sur une extension non ramifiée de $K$. Nous étudions des représentations unipotentes supercuspidales du groupe $\mathbf{G}\left(K\right)$. Nous établissons une bijection entre l’ensemble de telles $\mathbf{G}\left(K\right)$-représentations irréductibles et l’ensemble des L-paramètres étendus pour $\mathbf{G}\left(K\right)$, 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 $\gamma$ 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 $\mathbf{G}$. Finalement, nous prouvons que notre bijection satisfait à la conjecture de Hiraga, Ichino et Ikeda sur les degrés formels des représentations.

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

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