Systems of differential equations attached to representations of Lie groups

附加到李群表示的微分方程组

基本信息

批准号：
12440034
负责人：
OSHIMA Toshio
金额：
$ 5.25万
依托单位：
The University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-12440034/
关键词：
symmetric space Poisson transform Verma module Hardy space completely integrable system Radon transform unversal enveloping algebra primitive ideal c-函数最小多項式 Verma 加群単因子量子化完約リー環

项目摘要

(1)The annihilator of the generalized Verma module of the scalar type for a reductive Lie algebra is invesitigated. The condition that the annihilator gives the gap between the generalized Verma module and the usual Verma module is clearified and its necessary and sufficient condition is shown when its highest weight is dominant. Moreover a good sufficient condition is obtained in general.(2)By the dual map of the faithful finite dimensional representation of the reductive Lie algebra and its infinite dimensional representation, the characteristic polynomials and minimal polynomials of matrices are defined and the generator system of the annihilators of generalized Verma modules of scalar type is constracted., A sufficient condition that the ideal gives the gap is applied to some problems in the integral geometry.(3)Fatou's theorems and Hardy-type spaces on a Riemannian symmetric space of the noncompact type for the general eigenvalue of the invariant differential operators are studied in general. These theorems can be localized only when the rank of the space is equals to one.(4)On a compactification of a Euclidean space defined by a root lattice of the classical type, Shrodinger operators are classified which allow commuting differential operators of the forth order under the condition that their potentials are meromorphic at an infinite point.(5)The totality of k-dimensional linear subspaces of n-dimensional spaces over R, C or H is the Grassmannian manifold. The Radon transforms on these Grassmannian manifolds naturally extend the Gelfand-Aomoto's theory of generalized hypergeometric functions. Twisted Radon transforms over totally geodesic submanifolds are studied and interesting examples are given whose images are characterized by differential equations.

（1）为还原性谎言代数的标量类型的广义VERMA模块的歼灭器被发明。歼灭器给出了广义的Verma模块和通常的Verma模块之间的差距，并在其最高权重时显示其必要和足够的条件。此外，通常可以通过忠实的有限尺寸表示谎言代数及其无限尺寸表示，矩阵的特征多项式和最小的多项式的矩阵的特征性多项式和最小的矩阵中定义了verma iS的生成器，构成了cons cons cons cons cons cons cons cons cons cons cons cons cons cons cons cons cons cons s cans s cans s cans，cons s cons cons cons cons cons cons cons cons cons cons cons cons cons cons cons s cans s cans s cans，则一般而言。（3）通常研究了不稳定类型的Riemannian对称空间上Fatou的定理和Hardy型空间，用于不变差分运算符的一般特征值。这些定理只有在空间等级等于一个时才才能定位。（4）在经典类型的根部晶格定义的欧几里得空间的压实时，将shrodinger运算符分类为分类，使其在潜在的情况下是在不合时宜的点上，允许其潜在的差异范围的差异。在R，C或H上是Grassmannian歧管。 ra在这些司法歧管上的转化自然扩展了gelfand-aomoto的广义超几何函数理论。研究了完全测量的亚曼福尔德的扭曲ra换变换，并给出了有趣的示例，其图像以微分方程为特征。