A Satake isomorphism in characteristic p

TL;DR

This paper establishes a Satake isomorphism analogue in characteristic p for Hecke algebras associated with reductive groups over p-adic fields, aiding progress in mod p Langlands and Galois representation theories.

Contribution

It introduces a Satake isomorphism in characteristic p for Hecke algebras of reductive groups over p-adic fields, extending classical results to mod p settings.

Findings

Defines the Satake isomorphism in characteristic p

Connects Hecke algebras to mod p Langlands correspondence

Provides tools for classifying mod p representations

Abstract

Suppose that G is a connected reductive group over a p-adic field F, that K is a hyperspecial maximal compact subgroup of G(F), and that V is an irreducible representation of K over the algebraic closure of the residue field of F. We establish an analogue of the Satake isomorphism for the Hecke algebra of compactly supported, K-biequivariant functions f: G(F) \to End V. These Hecke algebras were first considered by Barthel-Livne for GL_2. They play a role in the recent mod p and p-adic Langlands correspondences for GL_2(Q_p), in generalisations of Serre's conjecture on the modularity of mod p Galois representations, and in the classification of irreducible mod p representations of unramified p-adic reductive groups.