This is a report on joint work in progress with Gregory D. Landweber and Reyer Sjamaar. Let X be a G-space, where G is a compact connected Lie group acting on a topological space X. Atiyah proved that the G-equivariant K-theory of X is a direct summand of the T-equivariant K-group of X, where T is a maximal torus of G; however, he did not tell whether this direct summand is determined by the Weyl group action on K_T(X). We show that the action of the Weyl group W extends to an action of a ring D generated by divided difference operators. These operators were first introduced in the context of Schubert calculus by Demazure; ours is a generalization to general G-spaces. The ring D contains a left ideal I(D}, which we call the ``augmentation left ideal'' in analogy with the augmentation ideal of the group ring of W. As an application of these algebraic constructions, we show that K_G(X) is isomorphic to the subring of K_T(X) annihilated by I(D).