Nilpotent orbits of certain simple Lie algebras over truncated polynomial rings

Let F be an algebraically closed field of positive characteristic p > 3 , and A the divided power algebra in one indeterminate, which, as a vector space, coincides with the truncated polynomial ring of F [ T ] by T p n . Let g be the special derivation algebra over A which is a simple Lie algebra, and additionally non-restricted as long as n > 1 . Let N be the nilpotent cone of g , and G = Aut ( g ) , the automorphism group of g . In contrast with only finitely many nilpotent orbits in a classical simple Lie algebra, there are infinitely many nilpotent orbits in g . In this paper, we parameterize all nilpotent orbits, and obtain their dimensions. Furthermore, the nilpotent cone N is proven to be reducible and not normal. There are two irreducible components in N . The dimension of N is determined.