Equivariant Satake category and Kostant-Whittaker reduction

TL;DR

This paper connects the equivariant derived category of the affine Grassmannian with coherent sheaves on the Langlands dual Lie algebra, extending to loop rotation equivariance and linking to quantum Kostant-Whittaker reduction, with implications for Toda lattice quantization.

Contribution

It extends the description of the equivariant derived category of the affine Grassmannian to include loop rotation equivariance and relates it to Harish-Chandra bimodules and quantum Kostant-Whittaker reduction.

Findings

Identification of the equivariant derived category with coherent sheaves on the dual Lie algebra.

Extension to loop rotation equivariant derived category and its link to Harish-Chandra bimodules.

Conjecture relating loop-rotation equivariant homology to quantized Toda lattice.

Abstract

We explain (following V. Drinfeld) how the 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 restricti on 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 from math.AG/0306413 which identifies the loop-rotation equivariant homology of the affine Grassmannian with quantized completed Toda lattice.