Irreducible local systems on nilpotent orbits

TL;DR

This paper proves that irreducible representations of the fundamental group of nilpotent orbits lift to parabolic subgroups in complex algebraic groups, with applications to Lusztig's question and local system cohomology.

Contribution

It establishes a general lifting result for fundamental group representations of nilpotent orbits, extending previous partial results and unpublished work.

Findings

Irreducible fundamental group representations lift to Jacobson-Morozov parabolics.

Applications to special pieces in exceptional groups.

New cohomological results for local system sections.

Abstract

Let G be a simple, simply-connected algebraic group over the complex numbers with Lie algebra $\mathfrak g$. The main result of this article is a proof that each irreducible representation of the fundamental group of the orbit O through a nilpotent element $e \in \mathfrak g$ lifts to a representation of a Jacobson-Morozov parabolic subgroup of G associated to e. This result was shown in some cases by Barbasch and Vogan in their study of unipotent representations for complex groups and, in general, in an unpublished part of the author's doctoral thesis. In the last section of the article, we state two applications of this result, whose details will appear elsewhere: to answering a question of Lusztig regarding special pieces in the exceptional groups (joint work with Fu, Juteau, and Levy); and to computing the G-module structure of the sections of an irreducible local system on O. A key aspect of the latter application is some new cohomological statements that generalize those in earlier work of the author.