Speculations on the mod $p$ representation theory of $p$-adic groups

Harris, Michael

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 25 (2016) no. 2-3, pp. 403-418.

Starting with the hypothesis, inspired by recent work on the geometric Langlands correspondence, that the analogue for mod p representations of p-adic groups of the local Langlands correspondence might be an equivalence of (higher) categories rather than a bijection of sets, this paper presents a series of speculations and questions about the properties of such a hypothetical equivalence. The Galois side of the correspondence would be a category of sheaves on the ind-algebraic stack constructed by Emerton and Gee, or perhaps a derived variant thereof ; the automorphic side of the correspondence would be a derived category of dg-modules over the derived Hecke algebra studied by Schneider. Both sides are quite mysterious, but some of the questions proposed in this paper may be accessible.

Partant de l’hypothèse, inspirée par des travaux récents sur la correspondance de Langlands géométrique, que l’analogue de la correspondance de Langlands locale, pour les représentations modulo p de groupes p-adiques, pourrait prendre la forme d’une équivalence de catégories (supérieures) plutôt qu’une bijection d’ensembles, cet article présente une séries de spéculations et de questions sur les propriétés d’une telle équivalence hypothétique. Du côté galoisien de la correspondance, on trouverait une catégorie de faisceaux sur le champ ind-algébrique construit par Emerton et Gee, ou éventuellement une version dérivée de ce champ ; du côté automorphe de la correspondance, on trouverait une catégorie dérivée de dg-modules sur l’algèbre de Hecke dérivée étudiée par Schneider. Les deux côtés sont assez mystérieux, mais certaines des questions proposées dans cet article pourraient être accessibles.

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