Mathematical Sciences: Complex Geometry and Representation Theory of Lie Groups

数学科学：复几何与李群表示论

• 批准号: 8701194
• 负责人: Hugo Rossi
• 金额: $ 13.51万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-07-01 至 1991-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8701194&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Complex; Geometry; Representation

The theory of Lie groups, named in honor of the Norwegian mathematician Sophus Lie, has been one of the major themes in twentieth century mathematics. As the mathematical vehicle for exploiting the symmetries inherent in a system, Lie theory has had a profound impact upon mathematics itself and theoretical physics, especially quantum mechanics and elementary particle physics. The abstract theory of representations of Lie groups provides a list of the minimal linear realizations -- or in technical jargon, the irreducible representations -- of the group. These form the building blocks for all such realizations. Often, Lie groups arise as motions of geometric objects, as for example, the orthogonal group is the group of distance preserving motions of the sphere. In order to study the fine structure of such representations, or of the group itself, it is necessary to obtain realizations of the representation that reflect the inherent geometric structure. This has been an area of fundamental research in noncommutative harmonic analysis for several decades. Professor Rossi is an expert in this interface of representation theory and geometry, especially complex geometry. For a large class of Lie groups, the most accessible set of irreducible representations is the holomorphic discrete series, discovered in the 1950's by Harish-Chandra. However, there are many more representations of this type -- called highest weight representations -- that only recently have been realized concretely. Some of these representations, said to lie in the analytic continuation, relate directly to elementary particle physics and also to the study of differential equations. Professor Rossi has been in the forefront of the discovery and investigation of such representations. Recently, he observed that these geometric contexts would allow considerable generalization. This point of view connects representation theory with analysis and geometry of several complex variables. In his present research, Professor Rossi intends to explore these constructions and make precise which representations arise on vector bundles of forms on invariant domains in Grassmanians, and as far as possible develop the analytic tools to make a complete study of the structure of these representations. For example, the representations which arise in this way are subrepresentations of the tensor product of holomorphic discrete series and adjoints, and they should be in the analytic continuation of non-holomorphic discrete series.

李群理论，以挪威人的荣誉命名 数学家Sophus Lie，一直是 世纪数学。 作为数学工具， 利用系统中固有的对称性，李理论 对数学本身和理论产生了深远的影响 物理学，特别是量子力学和基本粒子 物理学 李群表示的抽象理论 提供了一个最小线性实现的列表--或者在 技术术语，不可约表示， 组 这些构成了所有这些实现的基石。 通常，李群作为几何对象的运动而出现，例如， 例如，正交群是距离保持群， 球体的运动。 为了研究其精细结构， 这种代表，或该集团本身，有必要 获得反映 固有几何结构。 这是一个领域， 非对易调和分析基础研究 几十年 罗西教授是这方面的专家 表示论和几何学的结合，特别是复杂的 几何 对于一大类李群，最易接近的 不可约表示是全纯离散级数， 1950年由Harish-Chandra发现。 但有 更多的这种类型的表示--称为最高权重 直到最近才意识到 具体地说 其中一些表示，据说在于 解析延拓，与基本粒子直接相关 物理学和微分方程的研究。 罗西教授一直站在这一发现的最前沿， 调查这些陈述。 最近，他观察到 这些几何学背景可以让我们 一般化 这种观点将代表性 理论与分析和几何几个复杂的变量。 在他目前的研究中，罗西教授打算探索这些 结构和精确的表示出现在 Grassmanians中不变域上形式的向量丛，以及 尽可能开发分析工具以进行完整的 研究这些表征的结构。 比如说， 以这种方式出现的表示是 全纯离散张量积的子表示 级数和伴随，它们应该在解析的 非全纯离散级数的延拓