Mathematical Sciences: Geometric Aspects of Semisimple Lie Group Representations

在半单李群的表示理论中，一个重要而鼓舞人心的观点是试图将其表示为某些几何对象，通常是基础群的各种齐次空间上的某些向量丛的整体截面的空间。Zierau和Chang教授的项目将进一步研究几何运算和相应的表示理论含义之间的相互作用。Chang将使用D-模技术研究半单对称空间的离散级数，而Zierau将研究Kahler半单对称空间上的调和形式空间，以便将某些表示统一。这个项目与李群的表示理论有关，李群以挪威数学家索菲斯·李的名字命名。这些基本的数学对象以几种方式自然产生。李群的一个基本例子是球体的旋转群，其中群运算包括跟随一个运动接着另一个运动。关于这个群的详细信息对解决存在球对称的数学或物理问题非常有帮助。其他运动组捕捉到了其他类型的对称性。李群的例子的一个更代数的(相对于几何的)来源来自于矩阵的乘法。所有给定大小的可逆实(或复)矩阵组成的群是李群，就像它的任何可以用自然方式描述的子群一样。希望能够在几何和代数观点之间来回移动，例如考虑可以将球体的旋转群实现为一组可逆矩阵的多种方式。粗略地说，这就是表象理论。它很有用，因为有关给定群体的表述的事实往往会非常经济地存储大量信息。Zierau和Chang教授的研究将集中在如何通过让群作用于特定类型的几何结构来建立群的表示。

An important and inspiring viewpoint in the representation theory of semisimple Lie groups has been the attempt of realizing representations as certain geometric objects, typically as the space of global sections of certain vector bundles on various homogeneous spaces of the underlying group. The project of Professors Zierau and Chang will investigate further this interplay between geometric operations and the corresponding representation-theoretic implications. Chang will study discrete series for semisimple symmetric spaces using D-module techniques, while Zierau will study the space of harmonic forms on Kahler semisimple symmetric spaces in order to unitarize certain representations. This project has to do with the representation theory of Lie groups, which bear the name of the Norwegian mathematician Sophus Lie. These fundamental mathematical objects arise naturally in several ways. One basic example of a Lie group is the group of rotations of a sphere, where the group operation consists of following one motion by another. Detailed information concerning this group is very helpful in solving mathematical or physical problems in which spherical symmetry is present. Other groups of motions capture other kinds of symmetry. A more algebraic (as opposed to geometric) source of examples of Lie groups comes from the multiplication of matrices. The group of all invertible real (or complex) matrices of a given size is a Lie group, as is just about any subgroup thereof that can be described in a natural manner. It is desirable to be able to go back and forth between the geometric and algebraic points of view, for instance to consider the numerous ways in which the rotation group of the sphere can be realized as a group of invertible matrices. This, roughly, is representation theory. It is useful because facts about the representations of a given group tend to store a lot of information very economically. The research of Professors Zierau and Chang will concentrate on ways to build representations of a group by letting it act on certain kinds of geometric structures.