Resolutions of the Steinberg module for GL(n)

TL;DR

This paper provides multiple resolutions of the Steinberg module for GL(n) over principal ideal domains, confirming the validity of previous cohomology computations and clarifying the role of different complexes in these calculations.

Contribution

It introduces and compares several resolutions of the Steinberg module for GL(n), establishing the correctness of prior cohomology computations and analyzing the complexes used.

Findings

Both complexes accurately compute the cohomology groups.

Voronoi complex does not produce spurious Hecke eigenclasses.

Previous computations in AGM4 are confirmed as definitive.

Abstract

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 [AGM4] are definitive. In particular, in [AGM4] 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.