![](https://d3ilqtpdwi981i.cloudfront.net/2E_m-eTyayu-Alo7WKdsk_p2peE=/425x550/smart/https://bepress-attached-resources.s3.amazonaws.com/uploads/2b/16/b0/2b16b059-a4e9-415e-baba-b50eb02a91e6/thumbnail_e3590ed2-2807-4378-9c78-d232b7e892d1.jpg)
For an almost simple complex algebraic group G with affine Grassmannian $\text{Gr}_G=G(\mathbb{C}(({\rm t})))/G(\mathbb{C}[[{\rm t}]])$, we consider the equivariant homology $H^{G(\mathbb{C}[[{\rm t}]])}(\text{Gr}_G)$ and K-theory $K^{G(\mathbb{C}[[{\rm t}]])}(\text{Gr}_G)$. 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 $\check G$, and we relate the spectrum of K-homology ring to the universal group-group centralizer of G and of $\check G$. 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 $G(\mathbb{C}[[{\rm t}]])$-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 GrG 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 $G(\mathbb{C}[[{\rm t}]])$-modules.
Available at: http://works.bepress.com/ivan_mirkovic/17/
This is the pre-published version harvested from arXiv. The published version is located at http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=297084