Lie Groups

李群

• 批准号: 0652840
• 负责人: Joseph Wolf
• 金额: --
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0652840&HistoricalAwards=false
• 关键词: Lie; Groups

AbstractWolfThe investigator proposes seven intertwined research projects, all on the interface between group theory and geometry. For the most part these proposals are geometrically motivated approaches to specific problems in modern harmonic analysis connected to the representation theory of semi-simple Lie groups. The first is an approach to construction and analysis of singular unitary representations of real reductive Lie groups, in two steps: (i) geometric construction of those representations, as Frechet space representations on cohomology of homogeneous vector bundles over flag domains, and (ii) transforming the representation space to a space offunctions on a Stein manifold defined by a system of differential equations,by means of a double fibration transform (the complex Penrose transform is a particular case) from the flag domain to its linear cycle space. The second project is an approach to construction and analysis of unitary and other representations for a class of infinite dimensional Lie groups, the direct limits of finite dimensional Lie groups, especially strict direct limits of finite dimensional real and complex reductive Lie groups, both in the analytic category and in the algebraic category. The third project is to complete the investigator's work on the Harish-Chandra Schwartz space of a general semisimple Lie group, by synthesizing structural analyses of the relative Schwartz spaces. The investigator's fourth project is to extend the notion of Dirac cohomology to a notion of partial Diraccohomology so that it applies to all the representations of a semisimpleLie group that appear in the Plancherel formula. The fifth project is toextend the investigator's isospectral group method from the setting of spherical space forms to the setting of locally symmetric Riemannian spaces.The sixth project is to investigate certain restrictions of discrete seriesrepresentations to a class of subgroups of great geometric interest. And,seventh, the investigator will continue his development of a method for direct reading of the character and growth properties (asymptotics) of admissible representations of finite dimensional real reductive Lie groups from the basic data that specify their construction on cohomology spaces of homogeneous vector bundles over flag domains. One goal here is to do this in such a way that is directly applicable to the first and second projects. These seven research projects all depend on the use of symmetry to clarifyanalytic (and in one case geometric) problems. Traditionally symmetryconsiderations are used to simplify matters by decreasing the number ofvariables, but here they are used to enable the use of insight, tools andresults from geometry and analysis. The symmetries are embodied in group theory, which is the algebraic abstraction of the notion of symmetry. But modern Lie group theory incorporates classical analysis (calculus, differential equations...) and is closely tied to the geometry (Riemannian, symplectic, Kaehler, ...) of the configurations on which the groups act as symmetries. An important aspect of this synthesis of geometry and analysis is a geometric form of quantization that is particularly well suited to the sorts of finite dimensional groups considered in several of the projects. This geometric quantization, originally inspired by physics and developed in some detail by mathematicians, has in turn been very useful in a variety of settings in mathematics and physics. This is very much the case in six of the seven projects, where the geometric quantization is combined with modern differential geometry to understand various analytic problems. This is especially evident in the first and fourth projects, whose objectives can be viewed as a sort of dequantization.

研究者提出了七个相互交织的研究项目，都是关于群论和几何学之间的接口。 在大多数情况下，这些建议是几何动机的具体问题，在现代调和分析连接到代表性理论的半单李群。 第一个是构造和分析真实的约化李群的奇异酉表示的方法，分两步：（i）这些表示的几何构造，作为旗域上齐次向量丛上同调的Frechet空间表示，以及（ii）将表示空间变换为由微分方程组定义的Stein流形上的函数空间，通过从标志域到其线性循环空间的双纤维化变换（复彭罗斯变换是一种特殊情况）。 第二个项目是一种方法的建设和分析酉和其他表示的一类无限维李群，直接限制有限维李群，特别是严格的直接限制有限维真实的和复杂的还原李群，无论是在分析范畴和代数范畴。 第三个项目是完成研究者对一般半单李群的Harish-Chandra Schwartz空间的工作，通过综合相关Schwartz空间的结构分析。 调查员的第四个项目是扩大概念的狄拉克上同调的概念部分Diracohomology，使它适用于所有的陈述一个semisimpleLie群出现在Plancherel公式。 第五个项目是将研究者的等谱群方法从球面空间形式的情形推广到局部对称黎曼空间的情形，第六个项目是研究离散级数表示对一类具有极大几何意义的子群的某些限制。 第七，研究人员将继续发展一种方法，用于直接阅读有限维真实的还原李群的可接受表示的特征和增长特性（渐近性），这些基本数据指定它们在旗域上的齐次向量丛的上同调空间上的构造。 这里的一个目标是以直接适用于第一个和第二个项目的方式做到这一点。 这七个研究项目都依赖于使用对称性来澄清分析（在一个情况下是几何）问题。 传统上，通过减少变量的数量来简化问题，但在这里，它们被用来使来自几何和分析的洞察力、工具和结果的使用成为可能。 对称性体现在群论中，群论是对称性概念的代数抽象。 但现代李群理论结合了经典分析（微积分，微分方程.）并密切联系到几何（黎曼，辛，Kaehler，.） 这些群作为对称的构型。 这种几何和分析的综合的一个重要方面是量子化的几何形式，它特别适合于在几个项目中考虑的有限维群的种类。 这种几何量子化最初受到物理学的启发，并由数学家进行了一些详细的发展，反过来在数学和物理学的各种设置中非常有用。 这在七个项目中的六个项目中非常常见，其中几何量子化与现代微分几何相结合，以理解各种分析问题。 这在第一个和第四个项目中尤其明显，其目标可以被视为一种去量化。