Speaker: George McNinch (Tufts)
Title: Nilpotent orbits of a reductive group in positive characteristic
Abstract: Let K be a field and let G be a reductive group over K. Examples include the general linear group of a (finite dimensional) K-vector space, the group of units of a central simple K-algebra, orthogonal and symplectic groups, as well as some "exceptional groups". In the study of the structure and representations of G, one is often interested in conjugacy classes. The Jordan decomposition essentially reduces one to a study of semisimple classes and of unipotent classes. The "discrete part" of this study is the unipotent variety - there are only finitely many "geometric" unipotent classes. Under mild assumptions on the characteristic, the unipotent classes may be identified with the nilpotent variety in Lie(G), permitting one to "linearize" the problem somewhat. The talk will discuss the Bala-Carter theorem classifying the geometric nilpotent oribts in "good" characteristic; we will mention Premet's recent proof of that theorem.
We then explain how Galois cohomology of the centralizer CG(X) of a K-rational nilpotent element X permits one to ponder the orbits of the group of rational points G(K) on the nilpotent elements of Lie(G)(K). This motivates interest in the structure of such a centralizer; we will then describe various results about CG(X) (which are mainly interesting when the field K has positive characteristic).