The cohomology of $BPU(p^m)$ and invariant polynomials

Abstract

Let $p$ be an odd prime. For a compact Lie group $G$ and an elementary abelian $p$-group $A$ of $G$, one may define the Weyl group $W_A$ of $A$ in a similar fashion as defining the Weyl group of a maximal torus, such that $W_A$ acts on $H^*(BA;R)$ for any coefficient ring $R$, and the image of the restriction $H^*(BG;R)\to H^*(BA;R)$ lies in $H^*(BA;R)^{W_A}$, the sub-algebra of $H^*(BA:R)$ of $W_A$-invariant elements. In this paper, we consider the projective unitary group $PU(p^m)$ and one of its maximal elementary abelian $p$-subgroup $A_m$, of which the Weyl group is isomorphic to $Sp_{2m}(\mathbb{F}_p)$. Then the theory of $Sp_{2m}(\mathbb{F}_p)$-invariant polynomials over $\mathbb{F}_p$ may be applied to study the cohomology of $BPU(p^m)$, the classifying space of $PU(p^m)$. Following a theorem by Quillen, we deduce several theorems on $H^*(BPU(p^m);\mathbb{F}_p)$ modulo the nilradical from results on invariant polynomials.