Dirac上同调是最近十来年发展起来的研究李群和李代数表示论的新工具。在已有理论的基础上，本项目主要研究广义Harish-Chandra模的Dirac上同调和相关问题。. 本项目主要围绕两个方面展开。一是通过Dirac上同调来研究约化李代数的权模结构。权模是较为简单的一类广义Harish-Chandra模，我们将考虑权模的Dirac上同调的计算，它与权模现有不变量的关系以及权模的Dirac上同调和它的结构之间的关系。另一方面，因为Penkov和Zuckerman近年来在广义Harish-Chandra模的构造和分类所做的工作，我们希望能够计算出这些模的Dirac上同调，同时能够建立它们的Dirac上同调与另一重要不变量Euler-Poincaré函数之间的联系。

Dirac cohomology is a new tool developed in recent decades for the representation theory of Lie groups and Lie algebras. Based on the previous work, the aim of the present research project is to study Dirac cohomology on generalized Harish-Chandra modules...There are two main topics in our project. Firstly, Dirac cohomology will be used to study the structure of weight modules. As generalized Harish-Chandra modules, weight modules are relatively easy to study. We will calculate the Dirac cohomology for weight modules, study its relation with other existing invariants and the structure of weight modules. Secondly, much has been done by Penkov and Zuckerman on the classification and construction of generalized Harish-Chandra modules. It is aimed to determine the Dirac cohomology of these modules and build a connection between their Dirac cohomology and Euler-Poincaré function.