Equivariant Satake category and Kostant-Whittaker reduction. (English) Zbl 1205.19005

Bezrukavnikov, Roman; Finkelberg, Michael

Mosc. Math. J. 8, No. 1, 39-72 (2008).

Summary: We explain (following V. Drinfeld) how the \(G(C[[t]])\) 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.

MSC:

19E08	\(K\)-theory of schemes
22E65	Infinite-dimensional Lie groups and their Lie algebras: general properties
37K10	Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)

Keywords:

affine Grassmannian; Langlands dual group; Toda lattice