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, Kbiequivariant functions f : G(F ) EndV . These Hecke algebras were first considered by Barthel–Livné 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.