Graded Hecke algebras and equivariant constructible sheaves on the nilpotent cone

TL;DR

This paper establishes a geometric construction of graded Hecke algebras using equivariant sheaves on the nilpotent cone, and provides a categorification linking these algebras to derived categories of sheaves.

Contribution

It proves the isomorphism between geometric graded Hecke algebras and endomorphism algebras of equivariant sheaves, and describes their derived categories algebraically.

Findings

Geometric graded Hecke algebras are isomorphic to endomorphism algebras of equivariant sheaves.

The derived category of constructible sheaves is equivalent to modules over graded Hecke algebras.

Provides a categorification of graded Hecke algebras.

Abstract

Graded Hecke algebras can be constructed geometrically, with constructible sheaves and equivariant cohomology. The input consists of a complex reductive group G (possibly disconnected) and a cuspidal local system on a nilpotent orbit for a Levi subgroup of G. We prove that every such "geometric" graded Hecke algebra is naturally isomorphic to the endomorphism algebra of a certain G x C*-equivariant semisimple complex of sheaves on the nilpotent cone $g_N$. From there we provide an algebraic description of the G x C*-equivariant bounded derived category of constructible sheaves on $g_N$. Namely, it is equivalent with the bounded derived category of finitely generated differential graded modules of a suitable direct sum of graded Hecke algebras. This can be regarded as a categorification of graded Hecke algebras.