Abstract
In recent work, Graham has defined a variety which maps to the nilpotent cone, and which shares many properties with the Springer resolution. However, Graham's map is not an isomorphism over the principal orbit, and for type A in particular, its fibers have a nice relationship with the fundamental groups of the nilpotent orbits. The goal of this dissertation is to determine which simple perverse sheaves appear when the Decomposition Theorem for perverse sheaves is applied in Graham's setting for type A, and to begin to answer this question in the other types as well. In Chapter 1, we give some motivation and a brief description of this project. Then, Chapter 2 is a summary of several background topics. In Chapter 3, we review Graham's construction of his variety. In Chapter 4, we use results of Tymozcko to study the fibers of Graham's map in type A. Chapter 5 contains the conclusions in the perverse sheaf setting, and lastly, Chapter 6 contains results pertaining to Graham's fibers in the other types.
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