Twisted Satake Category

TL;DR

This paper generalizes the geometric Satake equivalence to a twisted setting, describing equivariant derived categories via coherent sheaves on twisted dual Lie algebras and extending related computations.

Contribution

It introduces a twisted version of the Satake category, extending existing descriptions to incorporate twisting, and develops a new functor analogous to global equivariant cohomology.

Findings

Description of the twisted Satake category in terms of coherent sheaves

Extension of loop rotation equivariant derived category computations

Introduction of a Kostant-Whittaker reduction for the twisted setting

Abstract

We extend Bezrukavnikov and Finkelberg's description of the G(\C[[t]])-equivariant derived category on the affine Grassmannian to the twisted setting of Finkelberg and Lysenko. Our description is in terms of coherent sheaves on the twisted dual Lie algebra. We also extend their computation of the corresponding loop rotation equivariant derived category, which is described in terms of Harish-Chandra bimodules for the twisted dual Lie algebra. To carry this out, we have to find a substitute for the functor of global equivariant cohomology. We describe such a functor, and show as in Bezrukavnikov-Finkelberg that it is computed in terms of Kostant-Whittaker reduction on the dual side.