Mathematical Sciences: Complex Geometry and Representation Theory of Lie Groups

李群理论是以挪威数学家索菲斯·李的名字命名的，一直是20世纪数学的主要主题之一。作为利用系统固有对称性的数学工具，李理论对数学本身和理论物理，特别是量子力学和基本粒子物理产生了深远的影响。李群表示的抽象理论提供了群的最小线性实现--或者在技术术语中，是不可约表示--的列表。这些构成了所有此类实现的构建块。通常，李群作为几何对象的运动而出现，例如，正交群是球体保持距离的运动的群。为了研究这种表示的精细结构或群本身的精细结构，有必要获得反映固有几何结构的表示的实现。几十年来，这一直是非对易调和分析的基础研究领域。罗西教授是表象理论和几何学，特别是复杂几何学的接口专家。对于一大类李群，最可达的不可约表示集是全纯离散级数，由Harish-Chandra在1950‘S中发现。然而，还有更多这种类型的表示--称为最高权重表示--直到最近才被具体实现。其中一些表示，据说是解析延拓，直接与基本粒子物理有关，也与微分方程式的研究有关。罗西教授一直站在发现和调查此类陈述的最前列。最近，他观察到，这些几何背景将允许相当大的普遍性。这一观点将表象理论与几个复变量的分析和几何联系起来。在他目前的研究中，Rossi教授打算探索这些结构，并精确地确定哪些表示出现在Grassmanian中不变区域上的形式向量丛上，并尽可能地开发分析工具来完整地研究这些表示的结构。例如，以这种方式产生的表示是全纯离散级数和伴随的张量积的子表示，并且它们应该是非全纯离散级数的解析延拓。

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.