Equivariant homology and K-theory of affine Grassmannians and Toda lattices

For an almost simple complex algebraic group G with affine Grassmannian Gr_G = G(ℂ((t)))/G(ℂ[[t]]), we consider the equivariant homology H^G(ℂ[[t]])(Gr_G) and K-theory K ^G(ℂ[[t]])(GrG). They both have a commutative ring structure with respect to convolution. We identify the spectrum of homology ring with the universal group-algebra centralizer of the Langlands dual group G, and we relate the spectrum of K-homology ring to the universal group-group centralizer of G and of Ǧ. If we add the loop-rotation equivariance, we obtain a noncommutative deformation of the (K-)homology ring, and thus a Poisson structure on its spectrum. We relate this structure to the standard one on the universal centralizer. The commutative subring of G(ℂ[[t]])-equivariant homology of the point gives rise to a polarization which is related to Kostant's Toda lattice integrable system. We also compute the equivariant K-ring of the affine Grassmannian Steinberg variety. The equivariant K-homology of Gr _G is equipped with a canonical basis formed by the classes of simple equivariant perverse coherent sheaves. Their convolution is again perverse and is related to the Feigin-Loktev fusion product of G(ℂ[[t]])-modules.