Project/Area Number  08454030 
Research Category 
GrantinAid for Scientific Research (B)

Section  一般 
Research Field 
解析学

Research Institution  Kobe University 
Principal Investigator 
NOUMI Masatoshi Kobe Univ., Dept.Math., Professor, 理学部, 教授 (80164672)

CoInvestigator(Kenkyūbuntansha) 
TAKAYAMA Nobuki Kobe Univ., Dept.Math., Professor, 理学部, 教授 (30188099)
樋口 保成 神戸大学, 理学部, 教授 (60112075)
YAMAZAKI Tadashi Kobe Univ., Dept.Math., Professor, 理学部, 教授 (30011696)
SASAKI Takeshi Kobe Univ., Dept.Math., Professor, 理学部, 教授 (00022682)
TAKANO Kyoichi Kobe Univ., Dept.Math., Professor, 理学部, 教授 (10011678)
SEKIGUCHI Hideko Kobe Univ., Dept.Math., Instructor, 理学部, 助手 (50281134)

Project Fiscal Year 
1996 – 1996

Project Status 
Completed(Fiscal Year 1996)

Budget Amount *help 
¥5,600,000 (Direct Cost : ¥5,600,000)
Fiscal Year 1996 : ¥5,600,000 (Direct Cost : ¥5,600,000)

Keywords  hypergeometric function / Macdonald polynomial / spherical function / Grassmannian / confluent hypergeometric function / configuration space / 超幾何函数 / マクドナルド多項式 / 球函数 / グラスマン多様体 / 合流型超幾何函数 / 配置空間 / Macdonald多項式 / Grassmann多様体 
Research Abstract 
The main subject of the this research project has been to construct a new prototype of the theory of special functions, by originating a systematic study of hypergeometric special functions in many variables. In this repect, the following results have been obtained from the viewpoints of (1) quantum group symmetry of difference systems, (2) confluent hypergeometric functions and Hamiltonian systems, (3) geometry of configurations spaces, and (4) representation theory and integral transformations, respectively. (1) M.Noumi developed a theory of sperical functions on quantum symmetric spaces in relation to quantum group symmetry. In terms of quantum groups, he gave a representationtheoretic realization of commuting families of qdifference operators and of the qhypergeometric orthogonal polynomials of Macdonald type. (2) K.Takano studied in detail the procedure of confluence for hypergeometric functions over the Grassmannians, namely the degeneration of a general regular singularity to a confluent singularity. He also clarified the structure of the spaces of initial values and the mechanism of degeneration in Hamiltonian systems of Painleve' type. (3) T.Sasaki investigated the spaces of configurations of one nondegenerate quadratic hypersurface and n hyperplanes in the projective space. He determined the differential system for the associated hypergeometric integrals and described the symmetry of them. For the configurations in the projective plane, in particular, he consturcted explicit power series solutions and independent cycles, and clarified the relationship with Appell's hypergeometric functions. (4) From the viewpoint of Penrose transformations in symmetric domains, H.Sekiguchi studied the generalization of hypergeometric integrals and hypergeometric differential equations to higher ranks. She also established the finite dimensionality of their solution spaces by means of the method of unitary representations.
