Summary: For an almost simple complex algebraic group \(G\) with affine Grassmannian \(\text{Gr}_G=G(\mathbb{C}((t)))/G(\mathbb{C}[[t]])\), we consider the equivariant homology \(H^{G(\mathbb{C}[[t]])}(\text{Gr}_G)\) and the \(K\)-theory \(K^{G(\mathbb{C} [[t]])}(\text{Gr}_G)\). They both have a commutative ring structure with respect to convolution. We identify the spectrum of the homology ring with the universal group-algebra centralizer of the Langlands dual group \(\check G\), and we relate the spectrum of the \(K\)-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 \(G(\mathbb{C} [[t]])\)-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 \(\text{Gr}_G\) 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 \(G(\mathbb{C}[[t]])\)-modules.