素特征域上李代数的表示与上同调理论是李理论及表示理论的核心课题之一。本项目将在我们已有研究基础上开展如下工作: (1) 借鉴Andersen-Kaneda等在代数群模表示及其李代数限制表示研究中的上同调技巧，在简约代数群李代数的非限制表示中，围绕Lusztig关于单模特征标的猜想, 给出与之相关的几个等价猜想, 从上同调角度理解Lusztig猜想。(2) 利用代数群的繁衍代数以及单广义限制模的提升技巧研究Cartan型李代数的支柱簇，给出支柱簇的刻画及实现，建立不同的表示范畴的支柱簇之间的关联。(3) 借助典型李代数上同调研究的技巧以及特征零的无穷维向量场李代数上同调的计算手法，研究限制Cartan型李代数的限制上同调环与幂零锥的坐标代数之间的关系，以及与之密切相关的幂零锥、幂零轨道、幂零交换簇、基本子代数簇的几何结构等。本项目对模李代数的上同调理论及表示理论的进一步研究有重要的理论意义。

Representation and cohomology theory of Lie algebras in prime characteristic is one of the key problems in Lie theory and representation theory. Based on our previous research, we will study the following problems in this project: (1) In the category of non-restricted representations of Lie algebras of reductive algebraic groups, we will do some research on Lusztig’s conjectures on character formulas for simple modules, and give several equivalent conjectures, so that we can understand Lusztig’s conjectures from the viewpoint of cohomology. We will use the cohomology techniques of Andersen-Kaneda in the study of modular representations of algebraic groups and restricted representations of their Lie algebras. (2) With aid of distribution algebras of algebraic groups and techniques in extension of simple generalized restricted modules, we will give detailed description of support varieties and their realization for Lie algebras of Cartan type. Moreover, the relationship among support varieties in different representation categories will be established. (3) With aid of the cohomology techniques for classical Lie algebras and the methods of cohomology computation for infinite dimensional Lie algebras of vector fields in characteristic zero, we will study the relation between the restricted cohomology rings of restricted Lie algebras of Cartan type and the coordinate algebras of their nullcones. The geometric structure of the related nullcones, nilpotent orbits, nilpotent commuting varieties and the varieties of elementary subalgebras will also be investigated. This project will play an important theoretic role for further study of cohomology and representations of modular Lie algebras.