Elsevier

Journal of Algebra

Volume 279, Issue 2, 15 September 2004, Pages 737-748
Journal of Algebra

Acyclicity of Schneider and Stuhler's coefficient systems: another approach in the level 0 case

https://doi.org/10.1016/j.jalgebra.2004.02.008Get rights and content
Under an Elsevier user license
open archive

Abstract

Let F be a non-archimedean local field and G be the locally profinite group GL(N,F), N⩾1. We denote by X the Bruhat–Tits building of G. For any smooth complex representation V of G and for any level n⩾1, Schneider and Stuhler have constructed a coefficient system C=C(V,n) on the simplicial complex X. They proved that if V is generated by its fixed vectors under the principal congruence subgroup of level n, then the augmented complex Cor(X,C)→V of oriented chains of X with coefficients in C is a resolution of V in the category of smooth complex representations of G. In this paper, we give another proof of this result, in the level-0 case, and assuming moreover that V is generated by its fixed vectors under an Iwahori subgroup I of G. Here “level-0” refers to Bushnell and Kutzko's terminology, that is to the case n=1+0. Our approach is different. We strongly use the fact that the trivial character of I is a type in the sense of Bushnell and Kutzko.

Keywords

Simplicial complex
Affine building
p-adic group
Smooth representation
Coefficient system
Resolution
Homological algebra

Cited by (0)