On systems of Hecke eigenvalues in cohomology of certain subgroups of GL_n(F)

TL;DR

This paper investigates how systems of Hecke eigenvalues in the cohomology of certain subgroups of GL_n(F) can be transferred between different coefficient modules, revealing a natural action of Hecke operators.

Contribution

It demonstrates that eigenvalue systems in cohomology with complex modules also appear in cohomology with 1-dimensional modules after changing the subgroup, highlighting a transfer property.

Findings

Eigenvalue systems transfer between modules via subgroup changes.

Hecke operators act naturally on cohomology groups.

The transfer preserves eigenvalue systems across different coefficients.

Abstract

We show how there is a natural action on the cohomology groups attached to certain subgroups of GL_n(F) of the Hecke operators defined as elements in an adelic double coset algebra. Our main result is, that if a system of eigenvalues for Hecke operators occur in the cohomology groups with coefficients in certain modules, then by changing the groups the system also occur in the cohomology groups with coefficients in a 1-dimensional module.