Mathematical Sciences: Geometric Aspects of Semisimple Lie Group Representations

一个重要的和鼓舞人心的观点，在代表性 半单李群理论一直是实现 表示为某些几何对象，通常为 上某些向量丛的整体截面空间 基本群的齐性空间。项目 Zierau教授和Chang教授将对此进行进一步研究 几何运算和相应的 代表理论的影响。Chang将学习离散 半单对称空间的D-模方法， 而Zierau将研究Kahler的调和形式空间 半单对称空间，以单位化某些 表示。 这个项目与谎言的表征理论有关 以挪威数学家索菲斯（Sophus）的名字命名的 撒谎这些基本的数学对象自然产生于 几种方式。李群的一个基本例子是 球体的旋转，其中组运算包括 一个动作接着另一个动作的详细资料 这个小组对解决数学或物理问题非常有帮助。 存在球面对称的问题。其他组 运动捕捉到了其他类型的对称性。一个更代数（如 李群例子的来源来自于 矩阵的乘法。所有可逆真实的群 (or一个给定大小的（复数）矩阵是一个李群，因为 关于其任何亚组，可以在自然界中描述， 方式希望能够在 几何和代数的观点，例如， 考虑到许多方式，其中旋转组的 球可以被实现为一组可逆矩阵。这个， 粗略地说，就是表征理论。这是有用的，因为事实 关于一个给定群体的表征往往会存储大量的 信息经济。Zierau教授的研究 而张将专注于如何构建一个 通过让它作用于特定的几何结构。

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.