Cohomology of GL(4,Z) with Non-trivial Coefficients

Abstract

In this paper we compute the cohomology groups of GL(4,Z) with coefficients in symmetric powers of the standard representation twisted by the determinant. This problem arises in Goncharov's approach to the study of motivic multiple zeta values of depth 4. The techniques that we use include Kostant's formula for cohomology groups of nilpotent Lie subalgebras of a reductive Lie algebra, Borel-Serre compactification, a result of Harder on Eisenstein cohomology. Finally, we need to show that the ghost class, which is present in the cohomology of the boundary of the Borel-Serre compactification, disappears in the Eisenstein cohomology of GL(4,Z). For this we use a computationally effective version for the homological Euler characteristic of GL(4,Z) with non-trivial coefficients.