Equivariant Satake category and Kostant–Whittaker reduction

We explain (following V. Drinfeld) how the $G(\mathbb C[[t]])$ equivariant derived category of the affine Grassmannian can be described in terms of coherent sheaves on the Langlands dual Lie algebra equivariant with respect to the adjoint action, due to some old results of V. Ginzburg. The global cohomology functor corresponds under this identification to restriction to the Kostant slice. We extend this description to loop rotation equivariant derived category, linking it to Harish-Chandra bimodules for the Langlands dual Lie algebra, so that the global cohomology functor corresponds to the quantum Kostant–Whittaker reduction of a Harish-Chandra bimodule. We derive a conjecture by the authors and I. Mirković, which identifies the loop-rotation equivariant homology of the affine Grassmannian with quantized Toda lattice.