Studies in Representation Theory

Wilfried Schmid的这个数学研究项目由两个松散相关的部分组成。在与Kari Vilonen的合作中，Schmid将完成他们最近关于可约李群的不可约酉表示的猜想的证明。对这样一个群G的不可约酉表示进行分类等价于一个代数问题：在具有不变但可能不定的内积的不可约Harish-Chandra模中，确定那些内积为正定的模。Vogan和他的合作者在Atlas项目目标指出,这个内积是直接和明确相关的某些不定内积,是极其不变量在一个紧凑的形式U g .根据上述猜想的复杂性,U-invariant内部产品是可计算的“一村齐藤的霍奇过滤Beilinson-Bernstein d模块实现Harish-Chandra模块的问题。这个猜想不会明确地对不可约的酉表示进行分类，但会将霍奇理论的泛函装置引入到酉性问题中。Schmid和Vilnoen将证明这一猜想，并研究其各种含义。该项目的另一个组成部分是与史蒂夫·米勒共同完成的。Schmid和Miller等人提出了一种证明朗兰兹l函数泛函方程和全纯性的新方法。与现有的方法相比，它的优点是可以直接计算伽马因子，至少在我们目前所研究的所有情况下是这样。这使它们能够排除其他方法无法排除的l函数的所有意想不到的极点。原则上，它应该适用于所有可以用积分表示方法访问的l函数。米勒和施密德计划改进他们的方法，扩大其适用范围。威尔弗里德·施密德的这个数学研究项目是在表示理论的一般领域，特别是关于所谓的对称的非紧群的表示。对称是日常生活中常见的现象。对称性的概念是由19世纪的数学家形式化的，他们引入了“对称群”的概念；在这种情况下，三维空间的旋转组是一个基本但重要的例子，因为经典力学定律在空间旋转下不会改变，这一事实有助于组织、简化并因此更好地理解其他学科（如相对论）中许多实际问题的解决方案。虽然紧凑群的表示已经被很好地理解了四分之三个世纪，但对于非紧凑群的表示，这是本项目研究的主题之一，情况并非如此。除了这项工作，施密德还参与了一些与K-12教育有关的活动，比如在咨询小组任职，并发表公开演讲。

This mathematics research project by Wilfried Schmid consists of two loosely related parts. In collaboration with Kari Vilonen, Schmid will complete the proof of their recent conjecture about irreducible unitary representations of reductive Lie Groups. Classifying the irreducible unitary representations of such a group G is known to be equivalent to an algebraic problem: among the irreducible Harish-Chandra modules with an invariant, but possibly indefinite inner product, determine those for which the inner product is positive definite. Vogan and his collaborators on the AIM Atlas project have pointed out that this inner product is directly and explicitly related to a certain indefinite inner product, one that is infinitesimally invariant under a compact real form U of the complexification of G. According to the above-mentioned conjecture, the U-invariant inner product is computable in terms of Morihiko Saito's Hodge filtration on the Beilinson-Bernstein D-module realization of the Harish-Chandra module in question. The conjecture would not explicitly classify the irreducible unitary representations, but would bring the functorial apparatus of Hodge theory to bear on the unitarity problem. Schmid and Vilnoen will prove the conjecture, and also investigate its various implications. The other component of the project is joint with Steve Miller. Schmid and Miller and have developed a new method for proving the functional equations and holomorphy for Langlands L-functions. Compared to the existing methods, it has the advantage of making the Gamma factors directly computable, at least in all the cases we have examined so far. This enables them to exclude all unexpected poles of L-functions that the other methods cannot rule out. In principle, it should apply to all L-functions accessible by the method of integral representations. Miller and Schmid plan to refine their method and extend its range of applicability.This mathematics research project by Wilfried Schmid is in the general area of representation theory, specifically on the representation of so-called non-compact groups of symmetries. Symmetry is a familiar phenomenon that occurs in everyday life. The concept of symmetry was formalized by 19th century mathematicians, who introduced the notion of "group of symmetries"; in this context the group of rotations of three dimensional space is a basic but important example because the laws of classical mechanics do not change under space rotations, and this fact helps to organize, simplify and thus better understand the solution of many practical problems in other disciplines such as relativity theory. While representations of compact groups have been well understood for three quarters of a century, the same is not true for the representations of non-compact groups, which are among the topics studied by this project. In addition to this work, Schmid is involved in several activities pertaining K-12 education, such as serving in advisory panels, and giving public lectures.