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

本文的研究目的在于确立根系超几何函数理论与积分理论的统一观点。本工作的具体主题如下：De Rham理论（与Selberg型积分相关的同调和上同调的研究，它表现为A型的球函数），2。2 .几种代数（Hecke代数等）的表示与积分的关系。3 .在painlevel方程（特殊多项式如Okamoto多项式）中的应用。数学物理的应用（Calogero系统，共形场论中的相关函数或可解晶格模型）。首席调查员的结果主要在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。松井的帮助在四垒的研究中很有价值，落合在二垒和四垒，和歌山在二垒，加藤在一垒。无论如何，通过目前的研究项目期间，我们已经取得了很多成果。作为证据，许多论文已经出现在优秀水平的杂志上。

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.