Mathematical Sciences: Spectral Analysis of Differential Operators on Hermitian Symmetric Spaces

数学科学：厄米对称空间上微分算子的谱分析

• 批准号: 9005111
• 负责人: Siddhartha Sahi
• 金额: $ 4.27万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1990
• 资助国家: 美国
• 起止时间: 1990-07-01 至 1992-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9005111&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Spectral; Analysis; Differential

Professor Sahi will apply algebraic and analytic techniques to study the spectrum of certain invariant differential operators on Hermitian symmetric spaces. Among the intended applications are (i) invariant theory: to establish identities generalizing the classical Capelli identity, (ii) representation theory: to study certain reducibility and unitarizability questions, (iii) analysis: to describe the homogeneous distributions on a formally real Jordan algebra and to extend Maxwell's theory of poles of spherical harmonics to this setting, (iv) number theory: to extend results of Shimura on the arithmeticity of certain non- holomorphic vector-valued Eisenstein series. The work described above involves the theory of representations of groups. Group theory is basically the theory of symmetry. To take a simple example, when the system in question is invariant under a change in the position of the origin of space, the group of translations naturally arises. While groups are abstract objects, particular situations demand concrete realizations or "representations" of the symmetry group.

Sahi教授将运用代数和解析技术研究厄米对称空间中某些不变微分算子的谱。预期的应用包括：(i)不变理论：建立一般化经典Capelli恒等式的恒等式；（ii）表示理论：研究某些可约性和一元性问题；（iii）分析：(4)数论：推广Shimura关于某些非全纯向量值爱森斯坦级数的算术性的结果。上述工作涉及到群体表征理论。群论基本上就是对称理论。举一个简单的例子，当所讨论的系统在空间原点位置的变化下是不变的，平移组自然产生。虽然群是抽象的对象，但特殊情况需要对称群的具体实现或“表示”。