Nilpotent orbits of certain simple Lie algebras over truncated polynomial rings

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