Computations of Cohomology of Arithmetic Groups, Part 1

Abstract

The key part of the current paper is the computation of boundary and Eisenstein cohomology of $GL_4({\mathbb Z})$ with coefficient in any highest weight representations. The method we develop let us compute in an alternative way the cohomology of $SL_3({\mathbb Z})$ and of $GL_3({\mathbb Z})$ with coefficients in any highest weight representation. This is done in a simpler, faster and in a more structured way compared to \cite{BHHM}. We state a duality for the boundary cohomology of $GL_m({\mathbb Z})$ of the type of Serre's duality, where the dualizing sheaf is a power of the determinant representation. We refine this duality to a duality on the level of the spectral sequence for the boundary cohomology $E_\infty^{p,q}$. We state it as a conjecture. However, all the computations, 30 different families of representations, satisfy this conjecture. We compute the Eisenstein cohomology of $GL_4({\mathbb Z})$ with coefficients in the symmetric powers and their twist by the determinant representation. For several other representations, we compute the Eisenstein cohomology, based a few conjectures. Based on those conjectured, one can compute the Eisenstein cohomology in most of the cases. They will be included in the next version of the paper.