Lie Groups

摘要对于美国国家科学基金会的提案DMS 99 88643“李群”，加州大学伯克利分校的约瑟夫·A·沃尔夫，研究人员提出了六个相互交织的研究项目，所有这些项目都是关于群论和几何之间的界面。在很大程度上，这些建议是对现代调和分析中特定问题的几何激励方法，与半单李群的表示理论有关。第一步是构造和分析实约化李群的奇异么正表示，分两步进行：(I)几何构造这些表示，作为旗域上齐次向量丛的上同调的Frechet空间表示；(Ii)通过从旗域到其线性循环空间的双重纤维变换(复彭罗斯变换是特例)，将表示空间变换到由微分方程组定义的Stein流形上的泛函空间。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，有限维李群的直接极限，特别是有限维实李群和复约化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对Schwartz空间的结构分析，完成对一般半单李群的Harish-Chandra Schwartz空间的研究工作。这位研究人员的第四个项目是完成对有限维实可约李群的可容许表示的特征和增长性质(渐近性)的直接读数的发展，这些基础数据规定了它们在齐次向量丛上的旗标域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实现这一点。第五个项目是继续一些早期在控制理论和数值分析方面的工作，特别是观测点布置问题和数值求积方案，基于结构结果和来自实约化李群表示理论的先验估计。第六个项目不是直接解析的，是对研究者最近发现的Grassmann流形中等宿球之间的奇怪关系、二次型的合成、反对称双线性形式空间的标准形以及平齐伪黎曼流形的构造的后续研究。这六个研究项目都依赖于使用对称性来澄清解析(在一个情况下是几何)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，它们被用来允许使用几何和分析的洞察力、工具和结果。对称性体现在群论中，它实际上是对称性概念的代数抽象。但现代李群理论融合了经典分析(微积分、微分方程式……)并与几何(黎曼几何、辛几何、凯勒几何等)密切相关。群在其上充当对称性的空间。几何和分析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，并由数学家详细地开发出来，反过来在数学和物理的各种环境中也非常有用。在六个项目中的五个项目中就是这种情况，其中几何量子化与现代微分几何相结合，以解决各种分析问题。这一点在第一个项目中尤其明显，其目标可以被视为一种去量子化。

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.