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

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 C • or (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.