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We study the Lusztig-Vogan bijection for the case of a local system. Wecompute the bijection explicitly in type A for a local system and then showthat the dominant weights obtained for different local systems on the sameorbit are related in a manner made precise in the paper. We also give aconjecture (putatively valid for all groups) detailing how the weighted Dynkindiagram for a nilpotent orbit in the dual Lie algebra should arise under thebijection.

Local systems on nilpotent orbits and weighted Dynkin diagrams