1998 Fiscal Year Final Research Report Summary
Many-Sided study of Selberg type integrals
| Project/Area Number |
09440064
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
解析学
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| Research Institution | KYUSHU UNIVERSITY |
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Principal Investigator |
KANEKO Jyoichi Kyushu University, Graduate School of Mathematics, Associate Professor, 大学院・数理学研究科, 助教授 (10194911)
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| Co-Investigator(Kenkyū-buntansha) |
KAZAMA Hideaki Kyushu University, Graduate School of Mathematics, Professor, 大学院・数理学研究科, 教授 (10037252)
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| Project Period (FY) |
1997 – 1998
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| Keywords | Selberg type integral / deformed Solberg type / twisted (co-) homology group / Gauss-Marin system / Jack polynomial / A-hypergeometric function / A-hypergeometric ideal / weakly 1-complete manifold |
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| Research Abstract |
In our research, we mainly studied the twisted (co-)homology groups and the holonomic systems attached to the (deformed) Selberg type integrals. We regard the integral as the dual pairing of the twisted homology group and the twisted cohomology group, and so the fundamental problem is to construct the bases of these (co-)homology groups. We explicitly constructed the bases of the (co-)homology groups, and noticed that these bases are nothing but the ones obtained from the beta-nbc bases due to Falk and Terao. We also calculated the Gauss-Manin system explicitly in graph-theoretical terms (Duke Math. J.). Our de-formed Selberg type integral is also an example of A-type hypergeometric functions due to Gelfand-Kapranov-Zelevinsky. We have been investigating the so-called A-hypergeometric ideal describing the holonomic system of the integral, especially on its Cohen-Macaulay property and construction of the basis.
The conjecture of Forrester predicts that the value of certain generalization of the original Selberg integral is also given by an explicit GAMMA product. We verified this in some cases by using the integration formula of Jack polynomials (Contemporary Math., to appear). We have been currently, working on this conjecture by employing Dunkl operators.
Kazama (with S.Takayama) solved in the negative the long-standing conjecture of S.Nakano concerning **-problem on weakly 1-complete manifolds (Nagoya Math. J., to appear). They also investigated related problems on complex Lie groups (Nagoya Math. 3., to appear).
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