Cohomology of the space of commuting $n$–tuples in a compact Lie group

Abstract

Consider the space $Hom\left({\mathbb{Z}}^n,G\right)$ of pairwise commuting $n$–tuples of elements in a compact Lie group $G$. This forms a real algebraic variety, which is generally singular. In this paper, we construct a desingularization of the generic component of $Hom\left({\mathbb{Z}}^n,G\right)$, which allows us to derive formulas for its ordinary and equivariant cohomology in terms of the Lie algebra of a maximal torus in $G$ and the action of the Weyl group. This is an application of a general theorem concerning $G$–spaces for which every element is fixed by a maximal torus.