Invariant Theory and Transformation Groups

研究了单复李代数上不变微分算子代数的模理论，特别研究了有限维模。他目前正在考虑二阶李代数的情况，在A型李代数的情况下取得了成功。研究人员和他的同事研究了约化代数群上的交换对合，以及由对合的固定群生成的相应的双陪集空间。利用适当的Weyl群确定了二重陪集空间的结构，以及它们与特殊类型的分裂环面的商空间的关系。对称群，如球面所有旋转的集合，在数学和物理中起着重要的作用。了解这些群体和他们的行为方式是我们了解物理和数学宇宙的基础。在过去的75年里，关于对称群的研究一直非常活跃和富有成效。研究人员和同事研究了这些对称群、相关的数学对象，以及它们可能起作用和相互作用的不同方式。

The investigator studies the module theory of the algebra of invariant differential operators on a simple complex Lie algebra, in particular, he studies finite dimensional modules. He is currently considering the case of rank two Lie algebras, with success in the case of the Lie algebra of type A. The investigator and a colleague study commuting involutions on reductive algebraic groups, and the corresponding double coset spaces by the fixed groups of the involutions. We determine the structure of the double coset spaces, and their relations to quotient spaces of special kinds of split tori by appropriate Weyl groups.Symmetry groups, such as the collection of all rotations of a sphere, play a fundamental role in mathematics and physics. Understanding these groups and the ways they can act is basic to our knowledge of the physical and mathematical universes. Research on symmetry groups has been very active and productive for the last 75 years. The investigator and colleagues study these symmetry groups, related mathematical objects, and the different ways they can act and interact.