Lie Groups

美国国家科学基金会提案DMS 99 88643“李群”约瑟夫·a·沃尔夫，加州大学伯克利分校研究员提出了六个相互交织的研究项目，都是关于群论和几何之间的接口。在大多数情况下，这些建议是对与半单李群表示理论有关的现代谐波分析中的具体问题的几何动机方法。第一个是构造和分析实约李群的奇异酉表示的方法，分为两个步骤：(i)这些表示的几何构造，作为标志域上齐次向量束上同调的Frechet空间表示，以及（ii）通过从标志域到其线性循环空间的双重纤维变换（复彭罗斯变换是特殊情况），将表示空间转换为由微分方程组定义的Stein流形上的空间函数。第二个项目是构造和分析一类无限维李群的酉表示和其他表示的方法，有限维李群的直接极限，特别是有限维实数和复约李群在解析范畴和代数范畴中的严格直接极限。第三个项目是通过综合相关Schwartz空间的结构分析，完成研究者对一般半单李群的Harish-Chandra Schwartz空间的研究工作。研究者的第四个项目是完成他对有限维实约李群的可容许表示的特征和增长性质（渐近性）的直接阅读的发展，这些基本数据指定了它们在齐次向量束在旗域上的上同调空间上的构造。这里的一个目标是以一种直接适用于第一个和第二个项目的方式来做这件事。第五项计划继续在控制理论和数值分析方面的一些早期工作，特别是观测点放置问题和数值正交方案，基于结构结果和来自实约化李群表示理论的先验估计。第六个项目，本质上不是直接解析的，是跟进研究者最近发现的格拉斯曼流形中的等斜球之间的奇怪关系，二次形式的组成，反对称双线性形式空间的正规形式，以及平坦齐次伪黎曼流形的构造。这六个研究项目都依赖于使用对称来澄清解析问题（在一个案例中是几何问题）。传统上，对称性考虑被用来通过减少变量的数量来简化问题，但在这里，它们被用来使洞察力、工具和几何学和分析结果的使用成为可能。对称性体现在群论中，群论实际上是对称概念的代数抽象。但现代李群论结合了经典分析（微积分、微分方程……），并与群作为对称作用的空间的几何（黎曼、辛、凯勒等）密切相关。这种几何和分析的综合的一个重要方面是量化的几何形式，它特别适合于在几个项目中考虑的有限维群。这种几何量子化，最初受到物理学的启发，并由数学家在一些细节上加以发展，反过来在数学和物理学的各种设置中非常有用。这在六个项目中的五个项目中非常常见，在这些项目中，几何量化与现代微分几何相结合，以理解各种解析问题。这在第一个项目中尤其明显，其目标可以被视为一种去量化。

Abstract for NSF Proposal DMS 99 88643 "Lie Groups" Joseph A. Wolf, U. C. Berkeley The investigator proposes six 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 semisimple 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 strictdirect limits of finite dimensional real and complex reductive Lie groups,both in the analytic category and in the algebraic category. The thirdproject 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 complete his development of a 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 bundlesover flag domains. One goal here is to do this in such a way that isdirectly applicable to the first and second projects. The fifth projectis to continue some earlier work on control theory and numerical analysis, especially observation point placement problems and numerical quadrature schemes, based on structural results and a priori estimates that come out ofthe representation theory of real reductive Lie groups. The sixth project, not directly analytic in nature, is to follow up on the investigator'srecent discovery of a strange relation between isoclinic spheres inGrassmann manifolds, composition of quadratic forms, normal forms for spaces of antisymmetric bilinear forms, and the construction of flathomogeneous pseudo-riemannian manifolds.These six 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 in fact is the algebraic abstraction of the notionof symmetry. But modern Lie group theory incorporates classical analysis(calculus, differential equations...) and is closely tied to the geometry(riemannian, symplectic, kaehler, ...) of the spaces on which the groupsact 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 five of the six projects, where the geometric quantization is combined with modern differential geometry tounderstand various analytic problems. This is especially evident in the first project, whose objective can be viewed as a sort of dequantization.