Constructible sheaves on affine Grassmannians and geometry of the dual nilpotent cone

TL;DR

This paper explores the derived category of sheaves on the affine Grassmannian of a complex reductive group, linking it to coherent sheaves on the dual group's nilpotent cone, and studies the effects of restriction to Levi subgroups.

Contribution

It provides a new description of the derived category of sheaves on affine Grassmannians in terms of coherent sheaves on the dual group's nilpotent cone, extending previous ideas.

Findings

Category described via coherent sheaves on the nilpotent cone

Restriction to Levi subgroup corresponds to hyperbolic localization

Results connect geometric representation theory with dual group structures

Abstract

In this paper we study the derived category of sheaves on the affine Grassmannian of a complex reductive group G, contructible with respect to the stratification by G(C[[x]])-orbits. Following ideas of Ginzburg and Arkhipov-Bezrukavnikov-Ginzburg, we describe this category (and a mixed version) in terms of coherent sheaves on the nilpotent cone of its Langlands dual reductive group. We also show, in the mixed case, that restriction to the nilpotent cone of a Levi subgroup corresponds to hyperbolic localization on affine Grassmannians.