Topics in Representation Theory of Real and p-adic Groups

摘要：本课题涉及表征理论中的两个主题。首先是通过研究仿射Hecke代数在标量扩展下的行为，对p进群的表示有更深的理解。这类代数似乎将完全控制p进群的表示理论。第二个项目是研究GLn表示的张量积中最高权向量的显式家族。这似乎是第一个这样明确的家庭。主要的问题是，它们是否真的张成了所有最大权向量的空间。表征理论是利用对称性存在的主要数学技术。例如，氢原子的结构，量子力学的基本计算之一，是由表征理论控制的。另一种思考表征理论的方法是将特征值理论和矩阵理论一般化很多人在大学基本线性代数课程中都看到过。这个项目解决了表征理论中的两个重要问题。其中一个对数论有潜在的重要应用。另一个是关于量子力学中角动量理论的推广。它将有助于描述一对相互作用的一般类型的量子力学系统如何分解成更简单的系统。

AbstractRoger HoweYale9970626This project concerns two topics in representation theory. The first is to attain a deeper understanding of representations of p-adic groups by studying the behavior of affine Hecke algebras under extension of scalars. It appears that such algebras will completely control the representation theory of p-adic groups. The second project is to investigate an explicit family of highest weight vectors in tensor products of representations of GLn. These appear to be the first such explicitly given family. The main question is, whether they infact span the space of all highest weight vectors.Representation theory is a major mathematical technique for exploiting the presence of symmetry. For example, the structure of the hydrogen atom, one of the fundamental computations in quantum mechanics, is controlled by representation theory. Another way of thinking about representation theory is as a generalization of the theory of eigenvalues and matrices that many people see in a college course on basic linear algebra. This project addresses two important questions in representation theory. One potentially has important applications to number theory. The other concerns a generalization of the theory of angular momentum in quantum mechanics. It will help describe how pair of interacting quantum mechanical systems of a rather general type breaks up into simpler systems.