Study On Some Problems In Modular Representations Of Lie Algebras

In this dissertation, we study related problems in modular representation theory of Lie algebras. We mainly consider the determination of irreducible modules of graded Lie algebras of Cartan type over algebraically closed fields of prime characteristic, the determination of the support varieties of Verma mod-ules and the precise construction and geometry of nilpotent orbits in the basic Cartan type Lie algebra of rank one, from which we also give the basic property and characterization of nilpotent orbits for algebras of type W in the Cartan type series. Furthermore, we give some new results on representations of general restricted Lie algebras from the view of primitive ideals of enveloping algebras. More precisely:1. Let R= (?)(m;n) be the divided power algebra and L= X(m;n), X∈{W, S, H} be a generalized Jacobson-Witt algebra, special algebra or Hamil-tonian algebra in the graded Cartan type series over an algebraically closed field F of characteristic p> 0. In the generalized restricted Lie algebra setting, any simple module of L corresponds to a unique (generalized) p-characterχ.When the height ht(χ) ofχis no more than min{pni-pni-1| i=1,…, m}-2+δxw, simple modules of L with p-characterχare determined. This is done by intro-ducing a "modified" induced module structure and thereby endowing induced module with the so-called (?)-module structure. The so-called category (?) for the generalized Jacobson-Witt algebras by Skryabin will be constructed on induced modules of a more natural class of generalized restricted Lie algebras. Moreover, this construction is applicable to all four series of Cartan type Lie algebras.(1) We prove that all irreducible representations of L with characterχsatisfying the above condition are induced from irreducible submodules of the maximal subal-gebra L0, modulo some exceptional cases. The exceptional cases happen to theχof height lower than 1. Simple modules in the exceptional cases were deter-mined mainly by Guang-Yu Shen, Nai-Hong Hu and so on. For the case that ht(χ)= -1, simple exceptional modules were determined by Shen [66] for types W,S,H and by Hu [25] for type K (see also [21,19,20]). When ht(χ)= 0 and X= W, S, we precisely construct simple exceptional modules in this dissertation via a complex of "modified" induced modules, and their dimensions are also ob-tained. For the case that ht(χ)= 0 and X= H, simple exceptional modules were determined by Pu and Jiang [59]. For type K, we can also introduce the category (?) and "modified" induced representations. But unlike the other three series of Cartan type Lie algebras, we can not strictly prove that those "modified" induced modules belong to the category (?) due to the fact that the graded structure of the Contact algebra does not inherit from the gradation of the generalized Jacobson-Witt algebra. However, by some concrete computation, we could conjecture that this holds. Then parallel to the other series of Cartan type Lie algebras, we can also conjecture that all simple modules of the Contact algebra with p-characterχsuch that ht(χ)< min{pni-pni-1|i= 1,…, m} - 2 are "modified" induced modules except the exceptional cases. This conjecture is true for the restricted Contact algebra by Zhang's work [100] (I would like to give some explanation as follows:One needs to handle each case of the four classes of Cartan type Lie algebras respectively. Until now, there is no unified method to deal with them in an axiomatic way).2. The nilpotent orbits of the Witt algebra W1, which is the basic Cartan type Lie algebra of rank 1, are determined under the automorphism group over an algebraically closed field F of characteristic p> 3. In contrast with a finite number of nilpotent orbits in a classical simple Lie algebra (cf. [31]), there is an infinite number of nilpotent orbits in W1. A set of representatives of nilpotent orbits, as well as their dimensions, are clearly presented. We also obtain that there are infinitely many nilpotent orbits in the Jacobson-Witt algebras. For the other Cartan type Lie algebras, we conjecture the same results.3. Support varieties for Lie algebras of Cartan type are studied. We give some description of the support varieties for the so-called baby Verma modules and a class of modules with semisimple characters. 4. For an arbitrary restricted Lie algebra g, we give an estimate of the codi-mension of the ideal of U(g) generated by the so-called central character ideal associated with an irreducible g-module. Moreover, we describe the primitive ideals corresponding to simple modules of maximal dimension. For the case of Lie algebras of reductive algebraic groups, we further give some description on the so-called G-invariant ideals.