p-Character Orbits and Reduced Enveloping Algebras for Lie Algebras of Cartan Type

Let k be an algebraically closed field of characteristic p>0 and let g be a finite-dimensional centerless restricted Lie algebra over k. We give polynomial criteria for the Aut(g)-orbits of p-characters on g* and for isomorphism among the reduced enveloping algebras u_chi(g). A primitive element in the reduced coordinate algebra of a finite affine linear section yields a normal-form map whose fibers are precisely the coadjoint orbits. For restricted simple Lie algebras of Cartan type this gives the p-character orbits, a complete set of orbit representatives, and the isomorphism classes of the corresponding reduced enveloping algebras requested in Problem 3 of Benkart and Feldvoss.