Whittaker Vectors and the Helgason Conjecture on Riemannian Manifolds

惠特克向量和黎曼流形上的赫尔加森猜想

• 批准号: 9970762
• 负责人: Richard Penney
• 金额: $ 7.63万
• 依托单位: Purdue Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-08-01 至 2003-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970762&HistoricalAwards=false
• 关键词: Whittaker; Vectors; Helgason; Conjecture; Riemannian

AbstractPenneyOne of the most fundamental theorems in harmonic analysis on Riemannian symmetric spaces is the Helgason theorem which states that any harmonic function may be represented as a Poisson integral over the Furstenberg boundary. Here, "harmonic" means that the function is annihilated by all of the invariant differential operators which have no constant term. For a Hermitian symmetric space, there are generalizations of this theorem which replace the Furstenberg boundary with the Shilov boundary and the invariant differential operators with larger systems such as the Hua-Johnson-Koranyi operators or the Berline-Vergne operators or the Lassalle operators. In this proposal we propose generalizing the Helgason theory, as well as the Hua-Johnson-Koranyi theory, to the context of homogeneous Kaehler manifolds.Harmonic functions occur repeatedly in science and engineering, in contexts such as heat flow, electricity, wave propagation and particle theory, just to name a few. Many attempts to understand the fundamental properties of nature are based on ever more complicated spaces. The solutions to many important physical and engineering problems on such spaces involve harmonic functions. While the current techniques work reasonably well for Riemanniansymmetric spaces, it it is clear that fundamental advances in the current technology must be made if the spaces become even slightly more complicated. Furthermore, the required new technology will certainly yield new information and new techniques in harmonic theory on Riemannian symmetric spaces as well. We propose continuing our previous work which is aimed at the development of this technology.

在Riemannian对称空间的谐波分析中，最基本定理的摘要是Helgason定理，该定理指出，任何谐波函数都可以表示为在Furstenberg边界上的Poisson积分。 在这里，“谐波”意味着该函数被没有恒定术语的所有不变差分运算符歼灭。 对于Hermitian对称空间，该定理有一些概括，这些定理用Shilov边界代替Furstenberg边界，而不变的差速器运算符则使用较大的系统（例如Hua-Johnson-Koranyi操作员）或Berline-vergne操作员或Lassalle操作员。 在这项建议中，我们提出将赫尔加森理论以及Hua-Johnson-Koranyi理论推广到同质Kaehler歧管的背景下。谐波功能在科学和工程中反复出现在热流，电，电，波传播和粒子理论等上下文中。 许多试图了解自然的基本特性是基于越来越复杂的空间。 在此类空间上，许多重要的物理和工程问题的解决方案涉及谐波功能。 尽管目前的技术在Riemanniansymmemmetric的空间方面运作良好，但很明显，如果空间变得更加复杂，则必须在当前技术的基本进步。 此外，所需的新技术肯定会在Riemannian对称空间的谐波理论中产生新的信息和新技术。我们建议继续我们以前的工作，该工作旨在开发这项技术。