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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)研究了约化李代数标量型广义Verma模的湮灭子。澄清了湮灭子给出广义Verma模与一般Verma模之间存在差距的条件，并给出了其最高权占主导时的充要条件。并在一般情况下得到了一个较好的充分条件。(2)通过还原李代数的忠实有限维表示及其无限维表示的对偶映射，定义了矩阵的特征多项式和极小多项式，构造了标量型广义Verma模的湮灭子的生成系统。将理想给出间隙的一个充分条件应用于积分几何中的一些问题。(3)一般研究了非紧型黎曼对称空间上不变微分算子一般特征值的Fatou定理和hardy型空间。只有当空间的秩等于1时，这些定理才能局域化。(4)在由经典型根格定义的欧几里得空间的紧化上，给出了允许交换四阶微分算子的Shrodinger算子的分类，其势在无限点处是亚纯的。(5) R、C或H上n维空间的k维线性子空间的总和是格拉斯曼流形。这些格拉斯曼流形上的Radon变换自然地扩展了Gelfand-Aomoto的广义超几何函数理论。研究了全测地线子流形上的扭曲Radon变换，并给出了用微分方程表征其图像的有趣例子。