Let X be a topological space upon which a compact connected Lie group G acts. It is well-known that the G-equivariant cohomology with rational coefficients H_G^*(X,Q) is isomorphic to the subalgebra of Weyl group invariants of the T-equivariant cohomology H_T^*(X,Q), where T is a maximal torus of G. Easy examples show that this relationship may fail for coefficient rings k other than Q. Instead, using work of Demazure we show that (under a mild condition on k) the algebra H_G^*(X,k) is isomorphic to the subalgebra of H_T^*(X,k) annihilated by Newton's divided difference operators.
We also have a version of this result for the equivariant intersection theory of algebraic varieties acted upon by linear algebraic groups.
If there is enough time I will discuss some applications to homogeneous spaces and Hamiltonian G-manifolds.