Let X be a topological space upon which a compact connected Lie group G acts. It is well-known that the equivariant cohomology H_G^*(X,Q) is isomorphic to the subalgebra of Weyl group invariants of the equivariant cohomology H_T^*(X,Q), where T is a maximal torus of G. We establish a similar relationship for coefficient rings more general than Q.
Our results rely on work of Grothendieck and Demazure concerning the intersection theory of flag varieties and have applications to the cohomology of homogeneous spaces and, potentially, symplectic quotients.