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

JOURNAL OF ALGEBRA

卷:279

Article

Broussous, P

关键词: simplicial complex;    affine building;    p-adic group;    smooth representation;    coefficient system;    resolution;    homological algebra;   

DOI  :  10.1016/j.jalgebra.2004.02.008

来源: Elsevier

【 摘 要 】

Let F be a non-archimedean local field and G be the locally profinite group GL(N, F), N greater than or equal to 1. We denote by X the Bruhat-Tits building of G. For any smooth complex representation V of G and for any level n greater than or equal to 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 = I + 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. (C) 2004 Elsevier Inc. All rights reserved.

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