2002 Fiscal Year Final Research Report Summary
Research on prehomogeneous vector spaces and micro-local analysis
| Project/Area Number |
13640163
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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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 | Gifu University |
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Principal Investigator |
MURO Masakazu Gifu University, Faculty of Engineerings, Professor, 工学部, 教授 (70127934)
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| Co-Investigator(Kenkyū-buntansha) |
ASAKAWA Hidekazu Gifu University, Faculty of Engineerings, Research Assistant, 工学部, 助手 (00211003)
KOBAYASHI Takako Gifu University, Faculty of Engineerings, Associate Professor, 工学部, 助教授 (40252126)
SHIGA Kiyoshi Gifu University, Faculty of Engineerings, Professor, 工学部, 教授 (10022683)
GYOJA Akihiko Nagoya University, Institute of Mathematics, Professor, 大学院・多元数理科学研究科, 教授 (50116026)
SEKIGUCHI Jiro Tokyo University of Agriculture and Technology, Faculty of Engineerings, Professor, 工学部, 教授 (30117717)
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| Project Period (FY) |
2001 – 2002
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| Keywords | Prehomogeneous vector space / Micro-local analysis / Representation of Lie groups and algebraic groups / Invariant theory / Invariant hyperfunctions / Grobner basis / Differential equations / Algebraic analysis |
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| Research Abstract |
(a) (From the abstract of the paper "Singular invariant hyperfunctions on the square matrix space and the alternating matrix space".) Fundamental calculations on singular invariant hyperfunctions on the n × n square matrix space and on the 2n × 2n alternating matrix space are considered in this paper. By expanding the complex powers of the determinant function or the Pfaffian function into the Laurent series with respect to the complex parameter, we can construct singular invariant hyperfunctions as their Laurent expansion coefficients. The author presents here the exact orders of the poles of the complex powers and determines the exact supports of the Laurent expansion coefficients. By applying these results, we prove that every quasi-relatively invariant hyperfunction can be expressed as a linear combination of the Laurent expansion coefficients of the complex powers and that every singular quasi-relatively invariant hyperfunction is in fact relatively invariant on the generic points of its support. In the last section, we give the formula of the Fourier transforms of singular invariant tempered distributions.
(b) (From the abstract of the paper "Invariant Hyperfunction Solutions to Invariant Differential Equations on the Space of Real Symmetric Matrices".) The real special linear group of degree n naturally acts on the vector space of n × n real symmetric matrices. How to determine invariant hyperfunction solutions of invariant linear differential equations with polynomial coefficients on the vector space of n × n real symmtric matrices is discussed in this paper. We prove that every invariant hyperfunction solution is expressed as a linear combination of Laurent expansion coefficients of the complex power of the determinant function with respect to the parameter of the power. Then the problem is reduced to the determination of Laurent expansion coefficients.
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