The cohomology of $BPU(p^m)$ and polynomial invariants

Abstract: For an odd prime $p$, we explore some connections between the cohomology algebra $H^*(BPU{p^m};\mathbb{F}_p)$ and the subrings of the polynomial ring $\mathbb{F}p[\xi_1,\eta_1,\cdots,\xi_l,\eta_l]$ that are invariant under some $Sp{2m}(\mathbb{F}p)$-action and $GL{2m}(\mathbb{F}_p)$-action.