Dirac上同调与广义Harish-Chandra模

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.

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。

Harish-Chandra模是一类极为重要的表示，而广义Harish-Chandra模作为它的自然推广，其研究也有着重要的意义。Dirac上同调是最近几十来年发展起来的研究李群和李代数表示论的新工具。对于酉表示分类等前沿问题有着重要的推动作用。在已有理论的基础上，我们首先研究广义Harish-Chandra模的结构问题。主要是关于权模，特别是最高权模的结构问题。我们成功的引入Jantzen系数，提出了广义Verma模可约性的实用判别法，我们还完全解决了范畴Op分块问题。另一方面，我们在权模这一类重要广义Harish-Chandra模的研究中引入了Dirac上同调，成功证明了Euler-Poincaré对合与Dirac上同调对合之间的等价性，从而证明了权模情形的Kazhdan正交化猜想。并且我们给出了全部权模的Dirac上同调。