Let G be a semisimple group over an arbitrary field K. Interesting examples include the special orthogonal group SO(V, q) for a quadratic form q on a Kvector space V and the group Aut(V) for an 8-dimensional Cayley algebra V . In these examples, the Lie algebra of G is a Lie subalgebra of gl(V), and the nilpotent variety the variety of nilpotent elements of Lie(G) are just those elements of Lie(G) which are nilpotent endomorphisms of V . The group G acts “by conjugation” on its Lie algebra, and when K is algebraically closed it is a basic fact that the number of orbits of G on the nilpotent variety is finite. When the field K is complete with respect to a discrete valuation and has finite or algebraically closed residue field, one is often interested in the group G(K) of K-rational points. In the talk, Ill sketch an argument that the number of orbits of G(K) on K-points of the nilpotent variety remains finite.