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表示实现为某些几何对象，通常是底层群的各种齐次空间上某些向量束的整体截面空间。Zierau教授和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模技术研究半简单对称空间的离散级数，而Zierau将研究Kahler半简单对称空间上的调和形式空间，以统一某些表示。这个项目与李群的表示理论有关，它以挪威数学家Sophus Lie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，其中群运算包括一个运动跟随另一个运动。关于这一群体的详细信息对于解决球面对称存在的数学或物理问题非常有帮助。其他组的运动捕捉到其他种类的对称性。李群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。给定大小的所有可逆实（或复）矩阵的群是李群，就像它的任何子群一样，可以用自然的方式来描述。我们希望能够在几何和代数的观点之间来回转换，例如考虑球体的旋转群可以被实现为一组可逆矩阵的许多方法。粗略地说，这就是表征理论。这是有用的，因为关于给定群体表示的事实倾向于非常经济地存储大量信息。Zierau教授和Chang教授的研究将集中在如何通过让群体作用于某些几何结构来构建群体的表征。

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.