Nilpotent cones and their representation theory

We describe two approaches to classifying the possible monodromy cones C arising from nilpotent orbits in Hodge theory. The first is based upon the observation that C is contained in the open orbit of any interior point N in C under an associated Levi subgroup determined by the limit mixed Hodge structure. The possible relations between the interior of C its faces are described in terms of signed Young diagrams. The second approach is to understand the Tannakian category of nilpotent orbits via a category D introduced by Deligne in a letter to Cattani and Kaplan. In analogy with Hodge theory, there is a functor from D to a subcategory of SL(2)-orbits. We prove that these fibers are, roughly speaking, algebraic. We also give a correction to a result of K. Kato.