Graded sheaves in geometric representation theory

Abstract

The recently constructed graded ℓ-adic sheaf theory by Ho-Li provides a uniform construction of a "mixed version" of the category of constructible sheaves in the sense of Beilinson-Ginzburg-Soergel which works for any Artin stacks of finite type over 𝔽 ¯ q . This paper explores the following two applications of graded sheaves in geometric representation theory. The theory of Springer correspondence relates representations of the Weyl group to perverse sheaves on the nilpotent cone. We extend this to an equivalence between the category of constructible graded sheaves generated by the graded Springer sheaf and the DG-category of modules over a ring related to the Weyl group in Vect^gr, which is a more streamlined proof of the result in [Rid13]. The classical geometric Satake equivalence is an equivalence between the category of equivariant perverse sheaves on the affine Grassmannian and the category of representations of the Langlands dual group. A derived version of this was proved by Bezrukavnikov-Finkelberg, we discuss the use of graded sheaves in providing a graded derived geometric Satake equivalence using Soergel’s method. This part is still a work in progress.

Date of Award	2024
Original language	English
Awarding Institution	The Hong Kong University of Science and Technology
Supervisor	Quoc HO (Supervisor)