THE EXPLICIT EXPRESSION OF C FUNCTION ON SEMISIMPLE LIE GROUPS AND ITS APPLICATION

半单李群上C函数的显式表达及其应用

• 批准号: 09640190
• 负责人: EGUCHI Masaaki
• 金额: $ 1.86万
• 依托单位: HIROSHIMA UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640190/
• 关键词: C Function; Intertwining Operator; Representation Theory; Fourier Transform; Lie Group; Homogeneou Space; Hardy Theorem; Heisenberg-Weyl; 表理論; ハリシュ・チャンドラC関数; SU(n,1); 不確定性原理; ハ-ディーの定理; 自由場表現; 量子アフィン代数; ヴィラソロ代数; C Function; Intertwining Operator; Representation Theory; Fourier Transform; Lie Group; Homogeneou Space; Hardy Theorem; Heisenberg-Weyl; 表理論; ハリシュ・チャンドラC関数; SU(n,1); 不確定性原理; ハ-ディーの定理; 自由場表現; 量子アフィン代数; ヴィラソロ代数

1. The Harish-Chandra C-function plays an essential role in harmonic analysis on semisimple Lie groups, because it closely relates to the Plancherel measure, the reducibility of the principal series representations and gives many information for the analysis. After a time, many peoples studied the Harish-Chandra C-function. However, even now, the explicit expressions of the Harish-Chandra C-functions are not known except for a few semisimple Lie groups and special cases.In this research we succeeded to give the explicit formulae of the Harish-Chandra C-functions for Spin(m, 1) and SU(n, 1). By the product formula for the Harish-Chandra C-function, the problem of computing the Harish-Chandra C-functions of semisimple Lie groups of general rank is reduced to the real rank one case. For this reason, it is crucial to compute the Harish-Chandra C-function for Spin(n, 1) and SU(n, 1). The reason for restricting our attention to the cases Spin(n, 1) and SU(n, 1) is that no multiple irreducible unitary representations of M occur in any irreducible unitary representation of K.2. In Euclidean space, various forms of the uncertainty principle between a function and its Fourier transform are known. The Hardy theorem asserts that if a measurable function f on R satisfies |f(x)| <less than or equal> C exp{-ax^2} and |f(y)| <less than or equal> C exp{-by^2} then f = O(a.e.) whenever ab> 1/4. This result is generalized to some semisimple Lie groups by A.Sitaram and M.Sundari, and M.Sunclari, M.0. Cowling and J.F.Price We get an analogue of the Hardy theorem for the Cartan motion group and also an L^p version of the Hardy theorem for the motion group.

1。harish-chandra c函数在半圣像谎言组的谐波分析中起着至关重要的作用，因为它与plancherel量度密切相关，是主序列表示的降低性，并为分析提供了许多信息。一段时间后，许多人研究了Harish-Chandra C功能。然而，即使到现在，除了几个半密布的谎言组和特殊案例外，尚不清楚哈里什 - chandra c函数的明确表达。在这项研究中，我们成功地为旋转（m，1）和su（n，1）提供了harish-chandra c命令的明确配方。通过Harish-Chandra C功能的产品公式，将一般等级的半密布谎言群体的Harish-Chandra C函数降低到了实际等级的一个情况。因此，对于自旋（n，1）和su（n，1）计算Harish-Chandra c功能至关重要。限制我们注意案例的原因（n，1）和su（n，1）的原因是，在K.2的任何不可还原的统一表示中，M的多个不可还原统一表示。在欧几里得空间中，已知函数与其傅立叶变换之间的各种形式的不确定性原理。 Hardy定理断言，如果R上的可测量函数f满足| f（x）| <小于或等于> c exp {-ax^2}和| f（y）| <小于或等于> c exp {-by^2}，然后每当ab> 1/4时f = o（a.e。）。 A.Sitaram和M.Sundari和M.Sunclari，M.0将此结果推广到某些半神经谎言组。 COWLING和J.F. PRICE我们获得了Cartan运动组Hardy定理的类似物，也获得了运动组Hardy Therorem的L^P版本。