Since these principal bundles correspond via our identification, and since the line bundles are associated to these principal bundles and the same representations of B , they are the same line bundle. Thus φ and ˜ φ agree on the R factor as well. As mentioned, the S -equivariant cohomology of any S -variety X is an algebra for Λ$_{S}$, the S -equivariant cohomology of a point. We have the following standard localization theorem for actions of tori, one reference for which is [Bri98]: Theorem 1.2.2. Let X be an S -variety, and let i : X S ↪ → X be the inclusion of the S -fixed locus of X . The pullback map of Λ$_{S}$ -modules i ∗ : H ∗ $_{S}$( X ) → H ∗ $_{S}$( X $^{S}$) is an isomorphism after a localization which inverts finitely many characters of S . In par- ticular, if H ∗ $_{S}$( X ) is free over Λ$_{S}$ , then i ∗ is injective. The last statement is what is relevant for us, since when X is the flag variety, H ∗ $_{S}$( X ) = R ′ ⊗$_{R}$ W R is free over R $^{′}$. Thus in the case of the flag variety, the localization theorem tells us that any equivariant class is entirely determined by its image under i $^{∗}$. As noted in the next section (cf. Proposition 1.3.1), the locus of S -fixed points is finite, and indexed by the Weyl group W , even in the event that S is a proper subtorus of the maximal torus T of G . Thus in our setup, H ∗ $_{S}$( X $^{S}$) ∼ = ⊕ w ∈ W Λ$_{S}$ , so that in fact a class in H ∗ $_{S}$( X ) is determined by its image under i ∗ w for each w ∈ W , where here i$_{w}$ denotes the inclusion of the S -fixed point wB . Given a class β ∈ H ∗ $_{S}$( X ) and an S -fixed point wB , we will typically denote the restriction i ∗ $_{w}$( β ) at wB by β |$_{wB}$ , or simply by β |$_{w}$ if no confusion seems likely to arise. Suppose that Y is a closed K -orbit. We denote by [ Y ] ∈ H ∗ $_{S}$( X ) its S -equivariant 18