Equivariant ($K$-)homology of affine Grassmannian and Toda lattice

Abstract

For an almost simple complex algebraic group $G$ with affine Grassmannian $G{r}_G=G(C((t)))/G(C[[t]])$ we consider the equivariant homology $H^{G(C[[t]])}(G{r}_G)$, and $K$-theory $K^{G(C[[t]])}(G{r}_G)$. 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 identify this structure with 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 $G{r}_G$ is equipped with a canonical base 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.

28 pages. v4: Basic definitions in sections 2.1--2.5 are corrected. The proof of Proposition 2.8 in section 4 is corrected. v5: corrected description of the set up in 2.5, expanded proof of Proposition 4.2