Compositio Mathematica

Research Article

A Satake isomorphism in characteristic p

Florian Herziga1

a1 Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston IL 60208-2730, USA (email:


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 andp-adic Langlands correspondences for GL 2 (xs211Ap) , 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.

(Received June 19 2009)

(Accepted March 31 2010)

(Online publication August 23 2010)

2000 Mathematics Subject Classification

  • 20C08;
  • 22E50


  • Hecke algebras;
  • Satake isomorphism;
  • characteristic p


The author was partially supported by NSF grant DMS-0902044.