Studies on Group Representations and Related Special Functions

从表示论的角度出发，研究了代数群、李代数、对称空间、准齐次空间、Hecke代数等相关的各种特殊函数。在研究的过程中，许多有趣的结果，其中一些是有关数论，或数学物理，获得。加藤研究Hecke代数。首先，他展示了如何“双”的陈述Hecke代数。其次，他利用Hecke代数构造了一种新的R-矩阵，明确定义了量子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-方程之间的某些关系。Saito从数论的角度研究了由对称矩阵构成的准齐次向量空间。他明确地确定了这些PV的zeta函数，并将这一结果应用于西格尔模形式的研究。Gyoja研究PV与表示和D-模。特别是，他研究之间的关系约化的广义Verma模块和b-功能的PV.松木调查轨道分解的对称空间和其他类似的空间。这一研究对于几何描述具有重要意义。西山研究了超级李代数的酉表示，特别是对偶对理论的超级版本。高崎研究数学物理中出现的非线性可积系统。他认为对称性隐藏在这些系统和调查之间的关系，这些对称性和无限维李代数等。

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.