| Co-Investigator(Kenkyū-buntansha) |
HARAOKA Yoshishige Kumamoto University, Faculty of Sciences, Professor, 理学部, 教授 (30208665)
KOHNO Mitsuhiko Kumamoto University, Faculty of Sciences, Professor, 理学部, 教授 (30027370)
YAMAKI Hiroyoshi Kumamoto University, Faculty of Sciences, Professor, 自然科学研究科, 教授 (60028199)
TAKANO Kyoichi Kobe University, Faculty of Sciences, Professor, 理学部, 教授 (10011678)
IWASAKI Katsunori Kyushu University, Graduate school of Mathematics, Professor, 数理学研究院, 教授 (00176538)
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| Research Abstract |
The objective of this research is 1) the study of the general hypergeometric functions and Okubo systems, 2) the study of nonlinear integrable systems including Painleve equations. The conjugacy classes of the centralizers of regular elements of GL(N,C) are determined by partitions of N. The general hypergeometric functions are functions on the Grassmannian manifold Gr(n,N) obtained by the Radon transformation of characters of universal covering groups of centralizers. We explicitly determined the algebraic de Rham cohomology groups associated with the integral representation of the general hypergeometric functions. This problem has been isolved in the case n=2 and in the case n>2 with the partitions (1,…,1), (N). For the case of partitions (q, 1,…,1), we proved the purity of the cohomology group, determined the dimension of the top cohomology and gave an explicit basis for it. This result will be important in constructing the Gauss-Manin system characterizing the general hypergeometri
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c functions. In the case where the partition is (N), we constructed the intersection theory of de Rham cohomology and expressed the intersection numbers in terms of skew Schur polynomials. In this computation, we recognized that an analogue of flat basis plays an important roles which appears in the theory of singularity. For the differential equation of Schlesinger type on P^1 without accessory parameters, we showed that the solutions have integral representations using the corresponding result for Okubo system. This integral representation is a particular case of that of GKZ hypergeometric functions. Thus it may be an interesting problem to understand the accessory parameter free equations in the framework of GKZ hypergeometric functions and to generalize this problem to the equations with irregular singularities.
For the Painleve equations, we showed that there is a symplectic structure for the space of initial conditions for each Painleve equation and also showed that the geometry of the space of initial conditions determines the Painleve equation. We found an interesting phenomenon that a generating function for a series of rational solutions of Painleve II coincides with the asymptotic expansion at infinity of the function obtained from Airy function. Less
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