Periodicity in the cohomology of finite general linear groups via $q$-divided powers

Abstract:

We show that $\bigoplus _{n \ge 0} \mathrm {H}^t(\mathbf {GL}_{n}(\mathbf {F}_q), \mathbf {F}_{\ell })$ canonically admits the structure of a module over the $q$-divided power algebra (assuming $q$ is invertible in $\mathbf {F}_{\ell }$), and that, as such, it is free and (for $q \neq 2$) generated in degrees $\le t$. As a corollary, we show that the cohomology of a finitely generated $\mathbf {VI}$-module in non-describing characteristic is eventually periodic in $n$. We apply this to obtain a new result on the cohomology of unipotent Specht modules.