Modern development of special functions - approach from the representation theory and the integrals

本研究的目的是解决超几何函数理论与积分理论的统一问题。这项工作的具体主题如下：1.de Rham理论(与Selberg型积分有关的同调和上同调的研究)，2.几类代数(Hecke代数等)的表示与积分的关系，3.在Painleve方程(特殊的多项式，如Okamoto多项式)中的应用，4.在数学物理中的应用(Calogero系统，共形场理论中的关联函数或可解的格子模型)。主要研究人员的研究结果主要在1和2左右，花村的结果在2左右，野美和山田的结果在3左右。松井的研究对4的研究有很大的帮助，Ochiai的2和4，和歌山的2，Kato的1。总之，在本研究期间，我们取得了许多成果。作为证据，许多论文都发表在优秀水平的期刊上。

The purpose of the present research was to settle the viewpoint to unify the theory of hypergeometic function associated with the root system and the theory of integrals. Concrete theme of this work was the following : 1. De Rham theory (Study of homology and cohomology associated with Selberg type integrals, which appear as the spherical functions of A type), 2. Relationship between the representations of several kinds of algebras (Hecke algebras and so on) and the integrals, 3. Application to Painleve equations (special polynomials such as Okamoto polynomials), 4. Application to mathematical physics (Calogero system, correlation functions in conformal field theory or solvable lattice models). The results of the head investigator were mainly about 1 and 2, those of Hanamura were about 2, those of Noumi and Yamada were about 3. Matsui's help was valuable in the study of 4, Ochiai's in 2 and 4, Wakayama's in 2, Kato's in 1.Anyway, we have obtained a lot of results through the period of the present research project. As an evidence, many of papers had appeared in the journal of excellent level.