The Index And Representations Of Restricted Jacobson-Witt Algbra W(n;1)-Premet-Skryabin Theorem's Application

In this paper, we consider irreducible representations of graded Cartan type Lie algebrasof W series. The structure of Cartan type Lie algebras is not as symmetric as the case of classical Lie algebras arising from algebraic groups. There is no satisfactory results on the representation theory of Cartan type Lie algebras. For the special case of restricted Cartan type Lie algebras, irreducible representations have been determined when the height of the character is not bigger than 1. In the paper, we apply Premet-Skryabin'Theorem to the case of the Jacobson-Witt algebra,obtain the following result: Assume L=W(nï¼›(?)),(?)₁:={χ∈L^*|(?)L(χ) is a torus},(?)₃={χ∈L^*|dim(?)L(χ)is minimal},then:1)Whenn≥3,(?)₁ is empty.(2)Whenn = 1,(?)₁ is nonempty,(?)₁ (?) (?)₃and dim(?)₁ = p, dim(?)₃ = p.(3)Whenn = 2,(?)₁ is nonempty,(?)₁ (?) (?)₃and dim(?)₁ = 2p², dim(?)₃ = 2p².