Studies on Group Representations and Related Special Functions

从表征理论的角度，研究了代数群、李代数、对称空间、预齐次空间、Hecke代数等相关的各种特殊函数。在研究过程中，获得了许多有趣的结果，其中一些与数论或数学物理有关。加藤研究赫克代数。首先，他展示了如何给出赫克代数的“对偶”表示。接着，他利用Hecke代数构造了一种新的r矩阵，明确定义了一个量子Knizhnik-Zamolodchikov方程，一个q-差分方程组，并证明了Macdonald差分算子的本征函数与我们的kz方程之间存在一定的关系。斋藤从数论的角度研究了对称矩阵组成的预齐次向量空间。他明确地确定了这些PV的ζ函数，并将这一结果应用于西格尔模形式的研究。Gyoja研究了PV与表示和d模块的关系。特别地，他研究了广义Verma模的可约性与PV的b函数之间的关系。Matsuki研究了对称空间和其他类似空间的轨道分解。这一研究对于用几何方法描述表征具有重要意义。Nishiyama研究了超李代数的酉表示，特别是对偶对理论的超版本。高崎研究数学物理中出现的非线性可积系统。他考虑了这些系统中隐藏的对称性，并研究了这些对称性与无限维李代数等的关系。

We studied various kind of special functions associated with aigebraic groups, Lie algebras, symmetric spaces, prehomogeneous spaces, Hecke algebras and so on, form the view point of representation theory. In the course of the research, many interesting results, some of which are related to number theory, or mathematical physics, are obtained.Kato studied Hecke algebras. First he showed how the "dual" of representations of Hecke algebras are given. Next, he constructed a new kind of R-matrix by using Hecke algebras, defined explicitly a quantum Knizhnik-Zamolodchikov equations, a system of q-difference equations, and showed certain relation between eigenfunctions of Macdonald's difference operators and our KZ-equations. Saito investigated the prehomogeneous vector spaces (PV) consisting of symmetric matrices from number theoretical view point. He determined the zeta function of these PV explicitly, and applied this result to the study of Siegel modular forms. Gyoja studied PV in connection with representations and D-modules. Particulary, he studies the relation between reducibility of generalized Verma modules and b-functions of PV.Matsuki investigated orbital decomposition of symmetric spaces and other similar spaces. This research is important in descriving representations geometrically. Nishiyama studied unitary representations of super Lie algebras, especially super version of the theory of dual pairs. Takasaki studies nonlinear integrable systems which appear in mathematical physics. He considered the symmetry hidden in these systems and investigated the relation between these symmetries and infinite dimensional Lie algebras, etc.