  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.
