  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 $K^{G(C[[t]])}(Gr_G)$. 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
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 $Gr_G$ 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.
