Studies in Representation Theory

表示论研究

• 批准号: 1300185
• 负责人: Wilfried Schmid
• 金额: $ 22.5万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1300185&HistoricalAwards=false
• 关键词: Studies; Representation; Theory

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.

这个数学研究项目由威尔弗里德施密德由两个松散相关的部分。在与卡里Vilonen合作，施密德将完成证明他们最近的猜想约化李群的不可约酉表示。对这样一个群G的不可约酉表示进行分类被认为等价于一个代数问题：在具有不变但可能是不定内积的不可约哈里什-钱德拉模中，确定那些内积是正定的。Vogan和他在AIM Atlas项目中的合作者指出，这个内积与某个不定内积直接而明确地相关，这个不定内积在G的复化的紧致真实的形式U下是无穷小不变的。根据上述猜想，U-不变内积可以根据Morihiko Saito的Hodge滤子在所讨论的Harish-Chandra模的Beilinson-Bernstein D-模实现上计算。这个猜想不会明确地对不可约的酉表示进行分类，但会使霍奇理论的函子装置对酉性问题产生影响。Schmid和Vilnoen将证明这个猜想，并研究它的各种含义。该项目的其他组成部分是与史蒂夫米勒联合。Schmid和米勒提出了一种新的方法来证明函数方程和Langlands L-函数的全纯性。与现有的方法相比，它的优点是使Gamma因子直接可计算，至少在所有的情况下，我们已经检查到目前为止。这使他们能够排除所有其他方法无法排除的L函数的意外极点。原则上，它应该适用于所有可以通过积分表示方法访问的L函数。米勒和施密德计划完善他们的方法，并扩大其适用范围。这个数学研究项目由威尔弗里德施密德是在一般领域的表示理论，特别是对表示的所谓非紧群的对称性。 对称性是日常生活中常见的现象。对称性的概念是由19世纪世纪的数学家们形式化的，他们引入了“对称群”的概念;在这种情况下，三维空间的旋转群是一个基本但重要的例子，因为经典力学的定律在空间旋转下不会改变，这一事实有助于组织，简化，从而更好地理解解决许多实际问题的其他学科，如相对论。虽然在世纪的四分之三的时间里，紧致群的表示已经被很好地理解了，但对于非紧致群的表示却不是这样，这是本项目研究的主题之一。除了这项工作，施密德还参与了有关K-12教育的几项活动，如在咨询小组任职，并发表公开演讲。