The realizations of primitive P-envelopes and the support varieties for graded Cartan type Lie algebras

Let L be a graded Lie algebra of Cartan type over an algebraically closed field of characteristic p ≥ 3, which has been proved to be generalized restricted in the sense of [Shu1, Shu2]. For a generalized restricted L-module M, the homological support variety ||L||_M is defined to be that of the primitive p-envelope P(L). A realization £ of P(L) is given in Der(Θ(m : n)). Furthermore, a class of generalized restricted highest weight L-modules lift to Dist(T_X)V(£)-module structures and their support varieties can be computed by using algebraic group techniques developed in [LN].