Irreducible Local Systems on Nilpotent Orbits

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 $\mathcal O$ 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some cases by Barbasch and Vogan in their study of unipotent representations for complex groups and, in general, in an unpublished part of the author's doctoral thesis.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 $\mathcal O$. A key aspect of the latter application is some new cohomological statements that generalize those in earlier work of the author.