Representation Theory of Reductive Groups

抽象草药 赫伯教授计划继续她的工作领域的代表性理论和谐波分析约化真实的李群和p-adic群。在早期的工作特点公式离散系列表示的真实的半单李群，她介绍了概念的两个结构。她现在提出扩展这一理论的两个结构，以获得离散系列字符公式的任意半单真实的李群通过解除从群体局部同构的直接产品的群体是真实的形式的SL（2，C）或Sp（2，C）。 她还感兴趣的是调查进一步的性质，这提升对应，并继续研究诱导表示非连接线性还原p-adic组。 傅立叶级数理论的发展，开始于世纪，研究周期函数的一个真实的变量。 这个想法是写一个任意的周期函数作为一个众所周知的三角函数的总和，也就是说，展开一个函数的“谐波”。“这个理论在科学、工程和数学中有许多应用。 傅立叶分析中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，问题是找到一个特别好的初等函数的集合，然后研究如何将任意行为良好的函数展开为这些初等函数的和或积分（连续和）。 傅立叶级数的这些推广被称为表示论或调和分析。 在物理学和数学中特别感兴趣的一类空间是半单真实的李群。 赫伯教授计划继续她的工作理论的两个结构的半简单真实的李群。 这个理论可以用来减少一般半单李群上的调和分析的一些问题，以理解两个最简单的例子。

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.