Mathematical Sciences: Analysis on Semisimple Groups

Knapp教授将系统地利用半单李群的可约和不可约表示中的相互交织的算子，以便更好地理解这些表示。两个具体的研究领域是表示的代数形式和解析形式的关系，以及引入一种更直接的方法来对半单对称空间进行调和分析。该奖项支持的博士后助理将使用微局部分析来更好地理解半单李群的对偶。该奖项支持的研究是在群的表征理论领域。群通常作为保持某个系统的结构的变换而出现。例如，围绕中心的旋转组保留球体。在应用中，抽象群并不是所需的全部，而是该群的具体实现，称为表示。本课题将研究由一族称为李群的光滑群所产生的表示。

Professor Knapp will make systematic use of intertwining operators into and out of reducible and irreducible representations of semisimple Lie groups in order to understand the representations better. Two specific areas of investigation are the relationship of algebraic and analytic forms of representations and the introduction of a more direct approach to harmonic analysis on semisimple symmetric spaces. The postdoctoral associate supported by this award will use micro-local analysis to gain a better understanding of the dual of a semisimple Lie group. The research supported by this award is in the area of the representation theory of groups. Groups usually arise as transformations which preserve the structure of some system. For example, the group of rotations about the center preserves the sphere. In applications the abstract group is not all that is needed but in addition a concrete realization of this group, called a representation. This project will study the representations which arise from a family of smooth groups called Lie groups.