Representation Theory of Reductive Groups

Herb教授计划继续她在表示理论和约化实李群和p进群的调和分析领域的工作。在早期关于实数半单李群的离散级数表示的特征公式的工作中，她引入了双结构的概念。她现在提议扩展这一双结构理论，通过将群从局部同构提升到群的直积，得到任意半单实李群上的离散级数特征公式，这些群是SL（2，C）或Sp（2，C）的实形式。她也有兴趣进一步研究这种提升对应的性质，并继续研究非连接线性约化p进群的诱导表示。傅立叶级数理论是从18世纪开始发展起来的，用于研究实变量的周期函数。我们的想法是把任意一个周期函数写成三角函数的和，也就是用“谐波”展开一个函数这一理论在科学、工程和数学中有许多应用。傅里叶分析中涉及的许多基本思想可以扩展到分析任何具有足够对称性的空间上的函数。对于给定的空间，问题是找到一组特别好的初等函数，然后研究如何将任意表现良好的函数展开为这些初等函数的和或积分（连续和）。傅里叶级数的这些推广被称为表示理论或谐波分析。在物理和数学中有一类特别有趣的空间是半单实李群。赫伯教授计划继续研究半简单实李群的双结构理论。这一理论可以简化一般半单李群谐波分析的一些问题，使之更容易理解这两个最简单的例子。

Abstract Herb Professor Herb plans to continue her work in the field of representation theory and harmonic analysis on reductive real Lie groups and p-adic groups. In earlier work on character formulas for discrete series representations of real semisimple Lie groups, she introduced the notion of two-structures. She now proposes to extend this theory of two-structures to obtain discrete series character formulas on arbitrary semisimple real Lie groups via lifting from groups locally isomorphic to direct products of groups which are real forms of SL(2,C) or Sp(2,C). She is also interested in investigating further properties of this lifting correspondence, and continuing to study induced representations for non-connected linear reductive p-adic groups. The theory of Fourier series was developed, starting in the 18th century, to study periodic functions of a real variable. The idea is to write an arbitrary periodic function as a sum of the well-understood trigonometric functions, that is expand a function in terms of its "harmonics." This theory has many applications in the sciences, in engineering, and in mathematics. Many of the basic ideas involved in Fourier analysis can be extended to analyze functions on any space with sufficient symmetry. For a given space, the problem is to find a collection of especially nice elementary functions, and then study how to expand arbitrary well-behaved functions as a sum or integral (continuous sum) of these elementary functions. These generalizations of Fourier series are known as representation theory or harmonic analysis. One class of spaces of special interest in physics and mathematics is the class of semisimple real Lie groups. Professor Herb plans to continue her work on the theory of two-structures for semisimple real Lie groups. This theory can be used to reduce some problems of harmonic analysis on general semisimple Lie groups to understanding the two easiest examples.