Rationality Properties of Nilpotent Orbits in Good Characteristic

Let X be an F -rational nilpotent element in the Lie algebra of a connected and reductive group G defined over the ground field F . One may associate toX certain cocharacters of Gwith favorable properties; this is an essential feature of the classification of geometric nilpotent orbits due to Bala-Carter, Pommerening, and, more recently, Premet. Suppose that the Lie algebra has a non-degenerate invariant bilinear form. We prove that there is a suitable cocharacter associated with X which is defined over F . We use an F -defined cocharacter to show that the unipotent radical of the centralizer of X is F -split. This property has several consequences. When F is complete with respect to a non-trivial discrete valuation with either finite or algebraically closed residue field, we deduce a uniform proof that G(F ) has finitely many nilpotent orbits in g(F ). When the residue field is finite, we obtain a proof that nilpotent orbital integrals converge; this was proved by Deligne and Ranga Rao when F has characteristic 0.