Cohomology of Commuting Varieties of Connected Compact Reductive Lie Groups

Abstract

We calculate the rational cohomology of the commuting variety $X_{G,n}$ consisting of $n$-tuples of commuting elements of a compact reductive group $G$. This is done by studying a map from a related variety $Y_{G,n}$, which has easily calculated cohomology. The proof studies the fibers of the map and uses the Vietoris-Begle theorem to prove that the induced map on rational cohomology is an isomorphism.