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

考文教授和古德曼教授将在李群和相关对象的表征理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进群的超尖表示。Goodman将继续研究当前代数和环群的表示。利用经典群的张量不变量理论研究了高阶Sugawara算子的代数性质。圆环群的正能量表示的几何模型的解析方面将被检查。古德曼还将寻找新的惠特克模型来代表真实的还原群体。这个项目与李群的表示理论（以及它们的近亲，李代数）有关，它以挪威数学家Sophus Lie的名字命名。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。关于特定群体表征的事实倾向于非常经济地存储大量信息。根据所讨论的群体来自哪里，这些信息几乎可以影响从数论到数学物理的任何学科。考文教授和古德曼教授的项目涵盖了当代表征理论的相当一部分应用和重点

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