Character Sheaves for Graded Lie Algebras: Stable Gradings

TL;DR

This paper constructs full support character sheaves for stably graded Lie algebras, linking them to cuspidal sheaves and analyzing Fourier transforms of nearby cycle sheaves, with implications for representation theory.

Contribution

It introduces a new construction of character sheaves for graded Lie algebras and explores their relation to cuspidal sheaves and complex reflection groups.

Findings

Construction of full support character sheaves for stably graded Lie algebras

Identification of these sheaves with conjectural cuspidal character sheaves

Analysis of Fourier transforms of nearby cycle sheaves

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].