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some cases by Barbasch and Vogan in their study of unipotent representationsfor complex groups and, in general, in an unpublished part of the author'sdoctoral thesis.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

Irreducible local systems on nilpotent orbits