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29 March 2024
 
  » arxiv » math.RT/0312183

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Acyclicity of Schneider and Stuhler's coefficient systems:another approach in the level 0 case
Paul Broussous ;
Date 9 Dec 2003
Subject Representation Theory; K-Theory and Homology MSC-class: 22E50; 18G10 | math.RT math.KT
AbstractLet F be a non archimedean local field and G be the locally profinite group GL(N,F), N>0. We denote by X the Bruhat-Tits building of G. For all smooth complex representation V of G and for all level n>0, 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 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.
Source arXiv, math.RT/0312183
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