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Steinberg module and top-dimensional cohomology for arithmetic Chevalley groups
Abstract
We prove that for a Chevalley--Demazure group scheme 𝒢 of type G2, F4 or E8 and a Euclidean ring 𝒪 with field of fractions K, the Steinberg module of 𝒢(K) is generated by integral apartment classes. As a consequence, the top-dimensional rational cohomology of 𝒢(𝒪) vanishes if 𝒪 is a ring of integers, extending results of Tóth and of Brück--Santos Rego--Sroka.
Related
- A resolution of the Steinberg module of a parabolic subgroup of GL_n
- Chevalley-Eilenberg cohomology of linearly reductive Lie algebras in the Verlinde category
- Fixed point properties and cohomology of Banach representations of arithmetic groups
- Computations of Cohomology of Arithmetic Groups, Part 1
- Maximal proper extremal subset in the crystal basis of a finite-dimensional irreducible module
BibTeX
@misc{gentili2026steinberg,
title = {{Steinberg module and top-dimensional cohomology for arithmetic Chevalley groups}},
author = {Miguel Gentili},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08381},
eprint = {2611.08381},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08381},
}
RIS
TY - UNPB
TI - Steinberg module and top-dimensional cohomology for arithmetic Chevalley groups
AU - Miguel Gentili
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08381
UR - https://arxiv.org/abs/2611.08381
AB - We prove that for a Chevalley--Demazure group scheme 𝒢 of type G_2, F_4 or E_8 and a Euclidean ring 𝒪 with field of fractions K, the Steinberg module of 𝒢(K) is generated by integral apartment classes. As a consequence, the top-dimensional rational cohomology of 𝒢(𝒪) vanishes if 𝒪 is a ring of integers, extending results of Tóth and of Brück--Santos Rego--Sroka.
ER -