Acyclicity of Schneider and Stuhlerʹs coefficient systems: another approach in the level 0 case

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 of G and for any level n 1, Schneider and Stuhler have constructed a coefficient system on the simplicial complex X. They proved that if is generated by its fixed vectors under the principal congruence subgroup of level n, then the augmented complex of oriented chains of X with coefficients in is a resolution of 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 is generated by its fixed vectors under an Iwahori subgroup I of G. Here “level-0” refers to Bushnell and Kutzkoʹs 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.