Orbital Varieties and Unipotent Representations Attached to Spherical Orbits
We examine a construction of W. Graham and D. Vogan in the case of spherical nilpotent orbits.This paper consists of two parts. The first parameterizes orbital varieties contained in a given nilpotent orbit using N. Spaltenstein's work on the unipotent variety. The key to the proof is understanding certain closed cycles as defined by D. Garfinkle. The second part addresses the representations constructed by Graham and Vogan from orbital varieties of spherical nilpotent orbits. We show that when it is possible to carry out the Graham-Vogan construction for an orbital variety of an orbit $\mathcal{O}$, the corresponding infinitesimal character lies in the set of characters attached to $\mathcal{O}$ by W. M. McGovern. Furthermore, we show that it is possible to carry out the Graham-Vogan construction for a sufficient number of orbital varieties to account for all the infinitesimal characters attached to $\calO$ in this way.