素特征域上李(超)代数与代数(超)群的表示及相关课题是重要数学研究领域。各种新的深刻数学思想正在孕育与发展。这给我们研究带来了极大机会。项目组拟在其原有成果基础上继续相关研究，期望在主流问题研究上获得新突破，取得更大成绩，同时期望在新的理论与方法上也有实质性的发展。 所开展的研究主要包括如下: (1)围绕模表示中简约李代数与基本典型李超代数不可约模的完全分类,特征标确定以及给定表示范畴的块结构等关键问题开展研究； 这些课题与有限W代数、Springer纤维、旗簇上的等变K理论及不变量理论研究密切关联。(2)利用代数D-模理论, 研究素特征域上Cartan李(超) 代数的不可约表示及相关范畴上的深刻关系。(3)对以上课题中表示的变形与伴随商映射纤维的几何,特定表示范畴引起的代数簇、相关代数群伴随、余伴随作用的几何系列问题开展研究。这是一个涉及代数群、代数簇与李代数表示理论之间深刻关联的课题。

Representations of Lie (super) algebras and algebraic (super) groups in prime characteristic and related topics are important research fields. Many kinds of new and deep mathematical thinkingness are in pregnancy and in fast development, which is a chance for all researchers. This project is aimed at great achivement in some mainstreams of these research topics, as a sequence of the former one undertook by the applicants. The main research contents are as follows:(1) modular representatons of reductive Lie algebras and of basic classical Lie superalgebras with focus on the classification of irreducible modules, character formulas, and block structure of related representation categories. These topics are in some close connecton to finite W-algebras, Springer fiber，equivarient K-theory and invariant theory.(2)With use of algebraic D-modules, to study irreducible representations of Cartan type Lie algebras and the connection among some related categories.(3)To study some geometric aspects arising from the deformation of representations and from the fibers of adjoint quotient map, and from the action of related algebraic groups in the topics mentioned above. In summary, this is a project involving the deep connection between modular representation theory of Lie algebras with related algebraic varieties and with geometric aspects arsing from the action of algebraic groups.