Graded Hecke algebras and equivariant constructible sheaves on the nilpotent cone
Maarten Solleveld
TL;DR
This work provides a geometric construction and categorification of twisted graded Hecke algebras via equivariant constructible sheaves on the nilpotent cone of a complex reductive group. It identifies the graded Hecke algebra $\mathbb H(G,M,q\mathcal E)$ with the endomorphism algebra of a $G\times\mathbb C^\times$-equivariant complex $K$, and shows that the $G\times GL_1$-equivariant derived category on the nilpotent cone is equivalent to the derived category of finitely generated differential graded modules over $\mathbb H$, constituting a geometric categorification. The nilpotent-cone category decomposes orthogonally into blocks indexed by cuspidal quasi-supports, giving a direct sum of DG-module categories for distinct Hecke algebras. The paper further extends these constructions to positive characteristic and establishes a triangulated equivalence between the geometric category and DG-modules over $\mathbb H$, providing a robust bridge between geometric representation theory and graded Hecke algebras with potential Langlands-parameter applications.
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.