| Project/Area Number |
16340034
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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 |
Basic analysis
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| Research Institution | The University of Tokyo |
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Principal Investigator |
OSHIMA Toshio The University of Tokyo, Graduate School of Mathematical Sciences, Professor (50011721)
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| Co-Investigator(Kenkyū-buntansha) |
ODA Takayuki The University of Tokyo, Graduate School of Mathematical Sciences, Professor (10109415)
KOBAYASHI Toshiyuki The University of Tokyo, Graduate School of Mathematical Sciences, Professor (80201490)
MATUMOTO Hisayosi The University of Tokyo, Graduate School of Mathematical Sciences, Associate Professor (50272597)
SEKIGUCHI Hideko The University of Tokyo, Graduate School of Mathematical Sciences, Associate Professor (50281134)
TERADA Itaru The University of Tokyo, Graduate School of Mathematical Sciences, Associate Professor (70180081)
関口 次郎 東京農工大学, 大学院・共生科学技術研究部, 教授 (30117717)
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| Project Period (FY) |
2004 – 2007
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| Project Status |
Completed (Fiscal Year 2007)
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| Budget Amount *help |
¥16,800,000 (Direct Cost: ¥15,600,000、Indirect Cost: ¥1,200,000)
Fiscal Year 2007: ¥5,200,000 (Direct Cost: ¥4,000,000、Indirect Cost: ¥1,200,000)
Fiscal Year 2006: ¥3,700,000 (Direct Cost: ¥3,700,000)
Fiscal Year 2005: ¥3,800,000 (Direct Cost: ¥3,800,000)
Fiscal Year 2004: ¥4,100,000 (Direct Cost: ¥4,100,000)
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| Keywords | Differential equation / Representation theory / Completely integrable system / Hypergeometric function / Integral geometry / Regular singularities / Whittaker function / Lie group / 解析学 / 対称空間 / 不確定特異点 / 境界値問題 / ルート系 / 超幾何微分方程式 / 完全積分可能量子系 / Verma加群 / Whittaker模型 / 退化系列表現 / Whittakerモデル / 普遍包絡環 / Calogeo-Moser系 / 戸田系 / ベキ零軌道 |
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| Research Abstract |
1. A conjecture for the classification of completely integrable quantum systems related to classical root systems is given and it is proved under a suitable condition. In particular the classification is complete if the systems have a regular singularity at an infinite point, which are most important cases. Higher order operators corresponding to the integrable Schrodinger operators are explicitly given and the complete integrability is proved. The relation between the systems are cleared.
2. The generators of the annihilator of a generalized Verma module of a scalar type for reductive Lie algebra are constructed in two ways by quatization of elementary divisors and by that of minimal polynomials in linear algebra. These correspond to generalization of Capelli identity and Hua operators. These also give the differential equations for degenerate series representations on generalized flag manifolds and some applications to integral geometry including Radon and Poisson transformations.
3. The condition for the existence of Whittaker model for degenerate series is obtained and the multiplicity of the realization is calculated under algebraic sense and also under the moderate growth condition. The differential equations satisfied by K-finite vectors in the realization is also obtained and the condition that the vectors are expressed by classical Whittaker functions is obtained.
4. A general theory of systems of partial differential equations of a little wider class than those with regular singularities is studied and their multi-valued holomorphic solutions are constructed.
5. The subsystems of a root system are classified and the homomorphisms between subsystems are classified.
6. Confluent limits, restrictions to singular sets and different real forms of Heckman-Opdam hypergeometric systems are studied. It is proved that the Whittaker vector with the moderate growth is obtained by this limit of Heckman-Opdam hypergeometric function.
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