Mathematical Sciences: Unitary Representations and Zuckerman Modules

数学科学：酉表示和祖克曼模块

• 批准号: 9706922
• 负责人: Susana Salamanca-Riba
• 金额: $ 7.63万
• 依托单位: New Mexico State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-15 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706922&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Unitary; Representations; Zuckerman

Abstract Salamanca-Riba During the duration of this grant Dr. Salamanca-Riba intends to investigate some extensions to her proof of the following conjecture of Vogan and Zuckerman on the representation theory of Lie groups: 'An irreducible unitary Harish-Chandra module of a group G whose infinitesimal character satisfies some regularity assumption is isomorphic to a Zuckerman functor module derived from a one dimensional unitary representation of a subgroup of G'. Relaxing this regularity assumption to include all genuine unitary representations of the metaplectic group, the PI plans to use the Adams-Barbasch dual pair correspondence of these representations to realize them as Zuckerman modules that are induced from some other representations easy to describe. The PI also has a program already in progress in collaboration with David Vogan. The problem they want to solve is the following conjecture: There is a bijection between the set of unitary representations of G whose lowest K type is associated to a fixed parameter and the set of unitary representations of a certain subgroup associated to the same parameter and whose K types are in turn associated to that same parameter. A fourth problem she wants to study is those unitary representations of a Lie group G which are predicted by the orbit method. These representations, when restricted to the maximal compact subgroup K are multiplicity free and are related to multiplicity free representations on the ring of regular functions of a nilpotent orbit. Lie groups have connections in many areas of pure and applied Mathematics like Differential Equations, Harmonic Analysis, Topology, Geometry, Ergodic Theory; and in other areas of Science and Engineering as well, like Materials Science, Quantum Field Theory, Particle Physics, Control Theory and Robotics. For example, the PI has become interested in the applications of Lie Theory to the field of Geometric Control Theory. She is now exploring some mechanical systems that are interesting for applications in several very different areas, such as robotic assisted surgery, waste management and spacecraft control. These systems seem to have some common grounds that allow engineers to be able to control the systems and could possibly be explained with Lie Theory. She is also interested in exploring these questions a bit further to see if they lead to interesting mathematical questions and if the answers to those questions lead to practical applications in industry.

在此期间，Salamanca-Riba博士打算研究她对Vogan和Zuckerman关于李群表示理论的下列猜想的证明的一些推广：群G的不可约酉Harish-Chandra模，其无穷小特征标满足某种正则性假设，同构于由G的子群的一维么正表示导出的Zuckerman函子模。松弛这一正则性假设以包括亚普勒群的所有真么正表示，PI计划使用这些表示的Adams-Barbasch对应来实现它们作为Zuckerman模，这些Zuckerman模是由其他容易描述的表示导出的。PI还与David Vogan合作，已经在进行一个项目。他们想要解决的问题是以下猜想：G的酉表示集与其最低K型与固定参数相关的集合与某个子群与同一参数相关且其K类型又与该参数相关的某个子群的酉表征集之间存在双射。她想研究的第四个问题是用轨道方法预测的李群G的酉表示。当限制到极大紧子群K时，这些表示是无重数的，并且与幂零轨道的正则函数环上的无重数表示有关。李群在纯数学和应用数学的许多领域都有联系，如微分方程、调和分析、拓扑学、几何学、遍历理论；以及科学和工程的其他领域，如材料科学、量子场论、粒子物理、控制论和机器人学。例如，PI已经开始对Lie理论在几何控制理论领域的应用感兴趣。她现在正在探索一些机械系统，这些系统在几个非常不同的领域都有应用价值，例如机器人辅助手术、废物管理和航天器控制。这些系统似乎有一些共同点，使工程师能够控制系统，并可能用谎言理论来解释。她还有兴趣进一步探索这些问题，看看它们是否会导致有趣的数学问题，以及这些问题的答案是否会导致在工业上的实际应用。