Consider the space of pairwise commuting –tuples of elements in a compact Lie group . This forms a real algebraic variety, which is generally singular. In this paper, we construct a desingularization of the generic component of , which allows us to derive formulas for its ordinary and equivariant cohomology in terms of the Lie algebra of a maximal torus in and the action of the Weyl group. This is an application of a general theorem concerning –spaces for which every element is fixed by a maximal torus.
"Cohomology of the space of commuting $n$–tuples in a compact Lie group." Algebr. Geom. Topol. 7 (2) 737 - 754, 2007. https://doi.org/10.2140/agt.2007.7.737