Functions on Nilpotent Orbit Covers and Birational Geometry

We use an analogue of the Springer resolution to describe the $G$-module structure on the ring of regular functions on the universal cover $\widetilde{\co}$ of any nilpotent orbit for $G = SL_n$. Building on previous work on the extended Springer resolution, we construct a variety $\widetilde{\mcM}$ that is finite over the cotangent bundle of a partial flag variety $G/P$, and proper and birational over the affinization $\mcM$ of $\widetilde{\co}$. We use techniques in birational geometry to show that $\widetilde{\mcM}$ has rational singularities, which provides the cohomology vanishing needed to describe the ring of functions on $\widetilde{\co}$ as an induced representation from a Levi subgroup of $G$. Our results also yield a description of the structure of $R(\widetilde{\co})$ as a graded $G$-module. We describe the minimal embedding of $\mcM$, study the lifting of characters of the component group of $\widetilde{\co}$ to parabolics and Levi subgroups, and make a more general vanishing conjecture.