A Satake isomorphism in characteristic p

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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-Livne for GL₂. They play a role in the recent mod p and p-adic Langlands correspondences for GL₂(Qₚ), 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.