2009/10/31 by Florian Herzig
Mathematics · #Advanced Algebra and Geometry #Algebra over a field #Algebraic Geometry and Number Theory #Algebraic closure #Algebraic group #Algebraic number #Algebraic structures and combinatorial models #Conjecture #Field (mathematics) #Galois module #Group theory #Hecke algebra #Isomorphism (crystallography) #Mathematics #Pure mathematics #Reductive group #Residue field #math.NT #math.RT #msc:20C08 #msc:22E50
paper · pdf · doi:10.1112/s0010437x10004951
published as Compositio Math. 147 (2011) 263-283 · 24 pages, revised
openalex publication_date 2010/08/23 · arxiv created 2011/02/02 · openalex created_date 2016/06/24 · arxiv updated 2019/02/20 · openalex updated_date 2026/08/05
Abstract 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 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.