Project/Area Number  09440064 
Research Category 
GrantinAid for Scientific Research (B)

Section  一般 
Research Field 
解析学

Research Institution  KYUSHU UNIVERSITY 
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)

CoInvestigator(Kenkyūbuntansha) 
前園 宜彦 九州大学, 経済学部, 助教授 (30173701)
木塚 崇 九州大学, 大学院・数理学研究科, 講師 (70186279)
宮脇 伊佐夫 九州大学, 大学院・数理学研究科, 教授 (40028254)
KAZAMA Hideaki Kyushu University, Graduate School of Mathematics, Professor, 大学院・数理学研究科, 教授 (10037252)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥5,500,000 (Direct Cost : ¥5,500,000)
Fiscal Year 1998 : ¥2,400,000 (Direct Cost : ¥2,400,000)
Fiscal Year 1997 : ¥3,100,000 (Direct Cost : ¥3,100,000)

Keywords  Selberg type integral / deformed Solberg type / twisted (co) homology group / GaussMarin system / Jack polynomial / Ahypergeometric function / Ahypergeometric ideal / weakly 1complete manifold / Selberg型積分 / 変形Selberg型積分 / ツイスト(コ)ホモロジー群 / GaussManin系 / Jack多項式 / A超幾何関数 / A超幾何イデアル / weakly 1complete多様体 / 変型Selberg型積分 / Joak多項式 / CokenMacaulay環 / weakly 1corplete 多様体 / twisted cohomology群 / 超平面配置 / Coxeter配置 / BakerForrester予想 
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 betanbc bases due to Falk and Terao. We also calculated the GaussManin system explicitly in graphtheoretical terms (Duke Math. J.). Our deformed Selberg type integral is also an example of Atype hypergeometric functions due to GelfandKapranovZelevinsky. We have been investigating the socalled Ahypergeometric ideal describing the holonomic system of the integral, especially on its CohenMacaulay 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 longstanding conjecture of S.Nakano concerning **problem on weakly 1complete manifolds (Nagoya Math. J., to appear). They also investigated related problems on complex Lie groups (Nagoya Math. 3., to appear).
