David Sprehn: Cohomology of finite general linear groups

I will introduce the problem of computing the mod-p cohomology of \(GL_n(k)\) for k the finite field of order \(p^r\), then describe how to construct a new class in (lowest possible) degree r(2p-3), and show it’s nonzero (when n <= p) by restricting to a subgroup of commuting regular unipotent matrices. I’ll first explain how the number r(2p-3) comes out of the invariant theory of finite fields. Lastly, I’ll describe what’s necessary to generalize the result to finite groups of Lie type.