Systems of differential equations attached to representations of Lie groups

（1）研究了约化李代数的标量型广义Verma模的零化子。明确了零化子给出广义Verma模与通常Verma模之间的差距的条件，并给出了零化子的最大权占优的充要条件.并且得到了一个较好的充分条件。(2)By定义了约化李代数的忠实有限维表示与无限维表示的对偶映射，矩阵的特征多项式和极小多项式，并构造了标量型广义Verma模的零化子的生成系，将理想给出差距的一个充分条件应用于积分几何中的一些问题。（3）一般地研究了非紧型黎曼对称空间上不变微分算子广义特征值的Fatou定理和Hardy型空间。这些定理只有当空间的秩等于1时才能局部化。(4)On Shrodinger算子是由经典型根格定义的欧氏空间的紧化，它允许交换四阶微分算子，条件是它们的势在无穷远点是亚纯的。（5）R、C或H上n维空间的k维线性子空间的全体是格拉斯曼流形。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。研究了全测地子流形上的扭Radon变换，给出了一些有趣的例子，它们的象可用微分方程刻画。

(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.