Cartan型李代数的模表示与Springer对应

几何表示是表示理论的核心主题之一，但Cartan型李代数的几何表示尚未建立起来。本项目拟研究Cartan型李代数的模表示，建立并发展Cartan型李代数的Springer对应与几何表示理论。.模表示方面，本项目拟研究诱导模、单模和投射模之间的关系(W型)；Chevalley限制定理及其应用（W、S和H型）以及一类模伪反射群(Weyl群的推广)的不变量。研究方法上主要借鉴Premet，Jantzen，Nakano，Lin-Nakano的工作。.Springer对应方面，我们已经刻画了W(n)的旗簇和Springer簇。进一步，本项目拟研究W(n)和半简约李代数的Springer对应。研究方法上借鉴Lusztig关于分片光滑簇上导范畴的研究、代数几何中关于奇点解消的一般性理论以及半简约李代数的结构。.本项目将填补Cartan型李代数的几何表示的空白，发展模表示理论。

Geometric representation theory is one of the core themes in representation theory. However, the geometric representation of Cartan type Lie algebra is not yet established. In this program, we will consider the modular representations of Cartan type Lie algebras, and establish as well as develop the Springer correspondence and geometric representation theory of Cartan type Lie algebra...For the modular representations, we will focus on the relations among induced modules, simple modules and projective modules (for type W); the Chevalley’s Restriction Theorem and its applications (for type W, S and H) and the invariants for a family of modular finite pseudo-reflection groups which are generalizations of Weyl groups. Our methods are based on the research of modular representations from Premet, Jantzen, Nakano and Lin-Nakano...For the Springer correspondence, we have described the flag variety and Springer variety of W(n). Furthermore, we will study the Springer correpondence for W(n) as well as semi-reductive Lie algebras. The methods are based on the derived category of piecewise smooth variety studied by Lusztig, general theory about resolution of singularities in algebraic geometry, and the structure of semi-reductive Lie algebras...This project will fill the gaps of geometric representation in Cartan type Lie algebra and develop the modular representation theory.