Algebro-analytic and/or representation-theoretic study of hypergeometric differential systems

通过计算机上的大量例子的支持，以及与世界上几个领域的专家的交流，我们得到了以下结果：斋藤睦美研究了A-超几何系统。他与Bernd Sturmfels和Nobuki Takayama合作，发现并研究了A-超几何系统与整数规划之间的一种意想不到的关系，证明了正则完整系统的秩在Grobner变形下的不变性，得到了A-超几何系统的秩等于A所张凸船体体积的三个充分条件.他根据A-超几何系统的D-同构类对参数进行了分类，Hiro-Fumi Yamada研究了Q-函数与仿射李代数之间的关系。他证明了表示为幂和对称函数的多项式的Q-函数是在多项式环上实现的某个仿射李代数的基本表示的权向量，并通过Young图说明了相应的权。他还发现了一个意想不到的关系舒尔的S-职能和Q-职能。山下弘研究哈里什-钱德拉模块。他指定嵌入Borel-德Siebenthal离散系列的主要系列表示。他还利用其核实现对偶Harish-Chandra模的梯度型不变微分算子的主符号描述了一些重要表示（如离散级数和酉最高权表示）的相关循环。有关其拉克斯介绍，他与川澄成也合作，获得了所有亚纯解的奇-Calogero微分方程。

With support of many examples by a computer, and by communication with world-wide experts in several fields, we obtained the following results.Mutsumi Saito has studied A-hypergeometric systems. He, in collaboration with Bernd Sturmfels and Nobuki Takayama, found and studied an unexpected relationship between A-hypergeometric systems and integer programmings, and showed the invariance of the rank of a regular holonomic system under Grobner deformations, and obtained three sufficient conditions for the rank of an A-hypergeometric system to equal the volume of the convex hull spanned by A. He classified parameters according to D-isomorphism classes of their corresponding A-hypergeometric systems.Hiro-Fumi Yamada has studied the relationship between Q-functions and affine Lie algebras. He showed a Q-function expressed as a polynomial of power sum symmetric functions is a weight vector for the basic representation of a certain affine Lie algebra realized on the polynomial ring, and illustrated the corresponding weight by Young diagrams. He also found an unexpected relation of Schur's S-functions and Q-functions.Hiroshi Yamashita has studied Harish-Chandra modules. He specified the embedding of Borel-de Siebenthal discrete series into the principal series representations. He also described the associated cycles of some important representations, such as discrete series and unitary highest weight representations, by using the principal symbols of invariant differential operators of gradient type whose kernels realize their dual Harish-Chandra modules.Youichi Shibukawa has worked on Ruijsenaars-Schneider dynamical integrable system. Related to its Lax presentation, he, in collaboration with Nariya Kawazumi, obtained all meromorphic solutions to the Bruschi-Calogero differential equation.