Character Sheaves for Graded Lie Algebras: Stable Gradings
Kari Vilonen, Ting Xue
TL;DR
The paper develops a framework for character sheaves on Z/mZ-graded Lie algebras, focusing on stable gradings and the conjectural identification of full-support cuspidal character sheaves via nearby cycles and Fourier transforms. It constructs local systems M_chi on the regular semisimple locus, analyzes their monodromy through a reduction to semisimple rank one using Kostant slices, and encodes the monodromy into Hecke algebras of complex reflection groups. The authors provide explicit polynomials R_{chi,s} and detailed calculations across Coxeter and twisted Coxeter rank-one cases, including several classical types and a full set of twisted types, with an endoscopic interpretation guiding the construction. They also outline how to pass from general reductive G to almost simple components and present explicit descriptions of full-support character sheaves in classical stable gradings via Lg and Lg Hecke algebras, while noting conjectural aspects about cuspidality and a connection to DAHA representations. Overall, the work offers a concrete, endoscopy–guided, rank-one–driven program to classify cuspidal character sheaves in the stable grading setting and to connect geometric objects to representations of complex reflection Hecke algebras.
Abstract
In this paper we construct full support character sheaves for stably graded Lie algebras. Conjecturally these are precisely the cuspidal character sheaves. Irreducible representations of Hecke algebras associated to complex reflection groups at roots of unity enter the description. We do so by analysing the Fourier transform of the nearby cycle sheaves constructed in [GVX2].