Irreducible Representations Of Graded Cartan Type Special Algebra S(m;n)

In this paper, we consider the (?)-module category theory in graded Cartan type special algebra S(m; n) which was firstly introduced by Skryabin to study representations of the generalized witt algebra. We prove that as a generalized restricted Lie algebra, the induced modules of S(m;n) are objects of the (?)-module category. Irreducible modules with generalized p-character no more than min are determined. In the nonexceptional case, all irreducible modules are induced modules. In the exceptional case, irrducible modules are the unique quotient modules of induced modules. For the latter case, irreducible modules are concretely constructed through the Koszul complex of induced modules. Furthermore, all isomorphism classes of irreducible modules are determined when the height of the character is 0, and the dimensions of all exceptional irreducible modules are given.