Dynamical Weyl groups and equivariant cohomology of transversal slices on affine Grassmannians

Let G be a reductive group and let Ǧ be its Langlands dual. We give an interpretation of the dynamical Weyl group of Ǧ defined in [5] in terms of the geometry of the affine Grassmannian Gr of G. In this interpretation the dynamical parameters of [5] correspond to equivariant parameters with respect to certain natural torus acting on Gr. We also present a conjectural generalization of our results to the case of affine Kac-Moody groups.