Computations of Cohomology of Arithmetic Groups, Part 1

@misc{horozov2026computations,
  title = {{Computations of Cohomology of Arithmetic Groups, Part 1}},
  author = {Ivan Horozov},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.28869},
  eprint = {2609.28869},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.28869},
}

TY  - UNPB
TI  - Computations of Cohomology of Arithmetic Groups, Part 1
AU  - Ivan Horozov
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.28869
UR  - https://arxiv.org/abs/2609.28869
AB  - The key part of the current paper is the computation of boundary and Eisenstein cohomology of GL_4(ℤ) with coefficient in any highest weight representations. The method we develop let us compute in an alternative way the cohomology of SL_3(ℤ) and of GL_3(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(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_∞^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(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.
ER  -