A Satake isomorphism in characteristic p

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 )→End V . These Hecke algebras were first considered by Barthel and Livné 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.