Mathematical Sciences: Lie Groups

本课题是李群表示理论的数学研究。后者的一个合适的例子是一个球体的旋转群。像这样的群很重要，因为它们作为对称群出现在数学的许多领域（例如几何、微分方程、代数数论、数学物理）。表征理论允许人们利用对称性来解决问题。更具体地说，沃尔夫教授将完成将哈里什-钱德拉理论扩展到一般实半单李群的工作。他将完成最近发展的哈里什-钱德拉模块的两个结构之间的二元性的某些重要结果的发展。他将完成并应用一个项目的结果，该项目以一种相当明确的方式显示了所有通常的可容许表示的解析和几何结构对于约化李群是等效的。他的研究计划的另一个方面涉及表征理论和复杂分析之间的相互作用。

This project is mathematical research in the representation theory of Lie groups. A suitable example of the latter is the group of rotations of a sphere. Groups like this are important because they occur in many areas of mathematics (e.g. geometry, differential equations, algebraic number theory, mathematical physics) as groups of symmetries. Representation theory allows one to take advantage of symmetries in solving problems. More specifically, Professor Wolf will complete work on an extension of the theory of Harish - Chandra to general real semisimple Lie groups. He will complete the development of certain important consequences of the recently developed duality between two constructions of Harish - Chandra modules. He will complete and apply the results of a project that shows in a rather explicit way how all the usual analytic and geometric constructions of admissible representations are equivalent for reductive Lie groups. A further aspect of his research plan concerns the interplay between representation theory and complex analysis.