Mathematical Sciences: Topics in Analysis on Real and p-adicLie Groups

数学科学：实数和 p 进李群分析主题

• 批准号: 8902993
• 负责人: Lawrence Corwin
• 金额: $ 12.73万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-06-15 至 1992-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8902993&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Topics; Analysis; Real

Professors Corwin and Goodman will pursue a wide-ranging program of research in the representation theory of Lie groups and related objects. Part of Corwin's work will deal with invariant differential operators on homogeneous spaces of nilpotent groups. He will also study supercuspidal representations of p-adic groups. Goodman will continue investigating the representations of current algebras and loop groups. The algebraic properties of higher-order Sugawara operators will be studied using the theory of tensor invariants for the classical groups. Analytic aspects of the geometric models for positive-energy representations of loop groups will be examined. Goodman will also seek new types of Whittaker models for representations of real reductive groups. This project has to do with the representation theory of Lie groups (and of their cousins, Lie algebras), which bear the name of the Norwegian mathematician Sophus Lie. 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. Facts about the representations of a given group tend to store a lot of information very economically. Depending on where the group in question comes from, this information can impinge on almost any subject from number theory to mathematical physics. The project of Professors Corwin and Goodman spans a fair portion of the applications and emphases of contemporary representation theory

科温教授和古德曼教授将在李群及相关物体的表示论方面进行广泛的研究。 Corwin 的部分工作将处理幂零群齐次空间上的不变微分算子。他还将研究 p-adic 群的上尖峰表示。古德曼将继续研究当前代数和循环群的表示。将使用经典群的张量不变量理论来研究高阶菅原算子的代数性质。将检查环群正能量表示的几何模型的分析方面。古德曼还将寻求新型惠特克模型来表示真实的还原群。 该项目与李群（及其近亲李代数）的表示论有关，该表示论以挪威数学家 Sophus Lie 的名字命名。李群的一个基本示例是球体的旋转群，其中群运算由一个运动跟随另一个运动组成。有关该群的详细信息对于解决存在球对称性的数学或物理问题非常有帮助。其他组的运动捕捉其他类型的对称性。 李群例子的一个更代数（而不是几何）的来源来自矩阵的乘法。给定大小的所有可逆实数（或复数）矩阵的群是李群，几乎可以以自然方式描述的任何子群也是如此。我们希望能够在几何和代数的观点之间来回切换，例如考虑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，这就是表示论。 有关给定群体的表征的事实往往会非常经济地存储大量信息。根据相关群体的来源，这些信息可能会影响从数论到数学物理的几乎所有学科。 科温和古德曼教授的项目涵盖了当代表征理论的大部分应用和重点