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A Satake isomorphism in characteristic p

Part of: Lie groups Representation theory of groups

Published online by Cambridge University Press:  23 August 2010

Florian Herzig
Florian Herzig*
Affiliation:
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston IL 60208-2730, USA (email: herzig@math.northwestern.edu)
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Abstract

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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 and Livné for GL 2. They play a role in the recent mod p andp-adic Langlands correspondences for GL 2 (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.

Keywords

Hecke algebrasSatake isomorphismcharacteristic p

MSC classification

Secondary: 20C08: Hecke algebras and their representations 22E50: Representations of Lie and linear algebraic groups over local fields
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 1 , January 2011 , pp. 263 - 283
Copyright
Copyright © Foundation Compositio Mathematica 2010

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