Resolutions of the Steinberg module for GL(n)

JOURNAL OF ALGEBRA

卷:349

Article

Ash, Avner²  Gunnells, Paul E.¹  McConnell, Mark³ 

[1] Univ Massachusetts, Amherst, MA 01003 USA

[2] Boston Coll, Chestnut Hill, MA 02445 USA

[3] Ctr Commun Res, Princeton, NJ 08540 USA

关键词: Cohomology of arithmetic groups;    Voronoi complex;    Steinberg module;    Modular symbols;   

DOI  :  10.1016/j.jalgebra.2011.09.018

来源: Elsevier

【 摘 要 】

We give several resolutions of the Steinberg representation St(n) for the general linear group over a principal ideal domain, in particular over Z. We compare them, and use these results to prove that the computations in Avner Ash et al. (2011) [AGM11] are definitive. In particular, in Avner Ash et al. (2011) [AGM11] we use two complexes to compute certain cohomology groups of congruence subgroups of SL(4, Z). One complex is based on Voronoi's polyhedral decomposition of the symmetric space for SL(n, R), whereas the other is a larger complex that has an action of the Hecke operators. We prove that both complexes allow us to compute the relevant cohomology groups, and that the use of the Voronoi complex does not introduce any spurious Hecke eigenclasses. (C) 2011 Published by Elsevier Inc.

【 授权许可】

Free