Lie Groups

NSF提案DMS 99 88643“李群”的摘要。沃尔夫，美国C.伯克利调查员提出了六个相互交织的研究项目，所有的群论和几何之间的接口。 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，特别是有限维真实的和复约化李群的严格直接极限，包括在分析范畴和代数范畴中。 第三个项目是通过综合有关Schwartz空间的结构分析，完成了研究者关于一般半单李群的Harish-Chandra Schwartz空间的工作。 调查员的第四个项目是完成他的发展的直接阅读的字符和增长特性（渐近）的容许表示有限维真实的还原李群的基本数据，指定其建设上同调空间的齐次向量fallesover旗域。 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。 第五个项目是继续一些早期的工作控制理论和数值分析，特别是观察点的位置问题和数值求积计划，基于结构的结果和先验估计出来的表示理论的真实的约化李群。 第六个项目，不直接分析性质，是跟进调查员最近发现的一个奇怪的关系等倾球在格拉斯曼流形，二次形式的合成，正规形式的空间反对称双线性形式，和建设flathomogeneous伪黎曼流形。这六个研究项目都依赖于使用对称性来澄清分析（在一种情况下几何）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，群论实际上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.）并且与几何学（黎曼、辛、凯勒等）密切相关。 of the spaces空间on which哪一个the groups群act行为as symmetries对称. 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。 这是非常的情况下，在五个六个项目，其中几何量子化是结合现代微分几何tunderstand各种分析问题。 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.