Distributions and Representations: The Kirillov Conjecture, Special Values Of L-functions and Fourier-Jacobi Models

分布和表示：基里洛夫猜想、L 函数的特殊值和 Fourier-Jacobi 模型

• 批准号: 0070762
• 负责人: Ehud Baruch
• 金额: $ 7.48万
• 依托单位: University of California-Santa Cruz
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-15 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070762&HistoricalAwards=false
• 关键词: Distributions; Representations; Kirillov; Conjecture; Special

The investigator and his collaborators study several problems in therepresentation theory of reductive groups and its applications to numbertheory. The first problem is to prove the Kirillov conjecture which statesthat every irreducible unitary representation of Gl(n,R) or GL(n,C)remains irreducible when restricted to a certain subgroup. The secondproblem is to find a formula relating fourier coefficients of halfintegral weight modular forms with special values of L-functions ofintegral weight modular forms. This formula generalizes a classicalformula of Kohnen and Zagier and is different from a formula givenby Waldspurger. Other problems which are considered are the study ofBessel distributions and Bessel functions for quasi-split groups and thestudy of Fourier Jacobi models for representations of symplectic andunitary groups. The line of attack on these problems is to study certaininvariant distributions and the spherical functions associated to them viaregularity theorems. Unitary representations of semisimple Lie groups were studied by the famous physicists Wigner and Dirac among others in an attempt to understand anddevelop the theory of quantum mechanics. Later, mathematicians such as Gelfandand Harish Chandra developed and rigorized this beautiful theory. Manyapplications were found to physics, geometry, number theory and otherfields. The purpose of the current proposal is to advance the understanding of representation theory and to continue to explore the connectionsbetween representation theory and number theory.

研究者及其合作者研究了还原群体的说明理论的几个问题及其在数字理论中的应用。第一个问题是证明Kirillov的猜想，即当限于某个亚组时，gl（n，r）或gl（n，c）的每个不可还原的统一表示。第二个问题是找到一个公式，该公式与半积体重量模块化形式的傅立叶系数具有特殊的构成重量重量模块化形式的值。该公式概括了Kohnen和Zagier的经典形式，与给定的Waldspurger公式不同。被认为的其他问题是贝塞尔分布的研究和贝塞尔函数的准分组组和傅立叶雅各比模型的特征，用于表示符号和统一组的表示。对这些问题的攻击线是研究某些变化的分布以及与它们相关的球形函数通过定理。著名的物理学家Wigner和Dirac等人研究了半神经谎言群体的单一表示，以了解和开发量子力学理论。后来，诸如Gelfandand Harish Chandra之类的数学家发展并严格了这一美丽的理论。在物理学，几何学，数理论和其他场地上发现了许多申请。当前建议的目的是提高对表示理论的理解，并继续探索表示理论与数字理论之间的联系。