Cuspidal character sheaves on graded Lie algebras

TL;DR

This paper demonstrates that in graded Lie algebras, all cuspidal character sheaves can be constructed via a specific nearby-cycle and Fourier--Sato transform process, completing their classification for certain classical Lie algebras.

Contribution

It provides a complete classification of cuspidal character sheaves in the context of Vinberg's type I graded classical Lie algebras using a novel construction approach.

Findings

All cuspidal character sheaves arise from a specific nearby-cycle and Fourier--Sato transform process.

The classification of cuspidal character sheaves for Vinberg's type I graded classical Lie algebras is completed.

The results unify previous partial classifications with a new construction method.

Abstract

We show in this paper that in the context of graded Lie algebras, all cuspidal character sheaves arise from a nearby-cycle construction followed by a Fourier--Sato transform in a very specific manner. Combined with results of the last two named authors, this completes the classification of cuspidal character sheaves for Vinberg's type I graded classical Lie algebras.