A Satake isomorphism for representations modulo p of reductive groups over local fields

Guy Henniart; Marie-France Vignéras

doi:10.1515/crelle-2013-0021

Article

A Satake isomorphism for representations modulo p of reductive groups over local fields

Published/Copyright: May 22, 2013

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2015 Issue 701

Abstract

Let F be a local field with finite residue field of characteristic p. Let G be a connected reductive group over F and B a minimal parabolic subgroup of G with Levi decomposition B=ZU. Let K be a special parahoric subgroup of G, in good position relative to (Z,U). Fix an absolutely irreducible smooth representation of K on a vector space V over some field C of characteristic p. Writing ℋ(G,K,V) for the intertwining Hecke algebra of V in G, we define a natural algebra homomorphism from ℋ(G,K,V) to ℋ(Z,Z∩K,VU∩K), we show it is injective and identify its image. We thus generalize work of F. Herzig, who assumed F of characteristic 0, G unramified and K hyperspecial, and took for C an algebraic closure of the prime field 𝔽_p. We show that in the general case ℋ(G,K,V) need not be commutative; that is in contrast with the cases Herzig considers and with the more classical situation where V is trivial and the field of coefficients is the field of complex numbers.

It is a pleasure to thank F. Herzig for communicating his results to us in a preprint form.

Received: 2011-7-11

Revised: 2013-1-16

Published Online: 2013-5-22

Published in Print: 2015-4-1

You are currently not able to access this content.

Articles in the same Issue

https://doi.org/10.1515/crelle-2013-0021