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Research on prehomogeneous vector spaces and micro-local analysis

Research Project

Project/Area Number 13640163
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionGifu University

Principal Investigator

MURO Masakazu  Gifu University, Faculty of Engineerings, Professor, 工学部, 教授 (70127934)

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)
尼野 一夫  岐阜大学, 工学部, 教授 (40021761)
Project Period (FY) 2001 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥4,000,000 (Direct Cost: ¥4,000,000)
Fiscal Year 2002: ¥1,900,000 (Direct Cost: ¥1,900,000)
Fiscal Year 2001: ¥2,100,000 (Direct Cost: ¥2,100,000)
KeywordsPrehomogeneous vector space / Micro-local analysis / Representation of Lie groups and algebraic groups / Invariant theory / Invariant hyperfunctions / Grobner basis / Differential equations / Algebraic analysis / 線形微分方程式 / 群論 / 非線形解析
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.

Report

(3 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report
  • Research Products

    (27 results)

All Other

All Publications (27 results)

  • [Publications] M.Muro: "Singular invariant hyperfunctions on the square matrix space and the alternationg matrix space"Nagoya Math. J.. 169. 19-75 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Muro: "Invariant hyperfunction solutions to invariant differential equations on the space of real symmetric matrices"J. of Funct. Analy.. 193. 346-384 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Muro: "Hyperfunction solutions to invariant differential on the space of real symmetric matrices"RIMS Kokyuroku. 1238. 83-142 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Muto: "Construction of hyperfunction solutions to invariant linear differential equations"RIMS Kokyuroku. 1211. 143-154 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Akihiko Gyoja: "Certain unipotent representations of finite Chevalley groups and Picard-Lefschetz monodromy"Ann. Sci. Ecole Norm. Sup.. 35. 437-444 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] E.Bannnai, M.Koike, A.Munemasa, J.Sekiguchi: "Some results on modular forms-Subgroups of the modular group whose ring of modular forms is a polynomial ring"Advanced Studies in Pure Mathematics. 32. 245-254 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] J.Gray著, 関口次郎, 室政和訳: "リーマンからポアンカレにいたる線形微分方程式と群論"シュプリンガーフェアラーク東京. 450 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Muro, M.: "Singular invariant hyperfunctions on the square matrix space and the alternating matrix space"Nagoya Math. J.. Vol.169. 19-75 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Muro, M.: "Invariant hyperfunction solutions to invariant differential equations on the space of real symmetric matrices"J. Funct. Anal.. Vol.193. 346-384 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Muro, M.: "Hyperfunction solutions to invariant differential equations on the space of real symmetric matrices"RIMS Kokyuroku. Vol.1238. 83-142 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Muro, M.: "Construction of hyperfunction solutions to invariant linear differential equations"RIMS Kokyuroku. Vol.1211. 143-154 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Akihiko Gyoja: "Certain unipotent representations of finite Chevalley groups and Picard-Lefschetz monodromy"Ann. Sci. Ecole Norm.. Sup. Vol. 35. 437-444 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] E. Bannnai, M. Koike, A. Munemasa and J. Sekiguchi: "Some results on modular forms - Subgroups of the modular group whose ring of modular forms is a polynomial ring"Advanced Studies in Pure Mathematics. Vol. 32. 245-254 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] J. J. Gray (translation from English to Japanese by J. Sekiguchi and M. Muro): "Linear differential equations and group theory from Riemann to Poincare"Springer-Verlag. 450 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Muro: "Singular invariant hyperfunctions on the square matrix space and the alternating matrix space"Nagoya Math. J.. 169. (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Muro: "Invariant hyperfunction solutions to invariant differential equations on the space of real symmetric matrices"J. of Funct. Analy.. 193. 346-384 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] S.Kanemitsu, T.Kuzumaki: "On a generalization of the Maillet determinant II"Acta Arith.. XCIX. 343-361 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] H.Asakawa: "On nonresonant singular two-point boundary value problems"Nonlinear Analysis T.M.A.. 47-7. 4849-4860 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] Akihiko Gyoja: "Certain unipotent representations of finite Chevalley groups and Picard-Lefschetz monodromy"Ann. Sci. Ecole Norm. Sup.. 35. 437-444 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] E.Bannnai, M.Koike, A.Mune-masa, J.Sekiguchi: "Some results on modular forms -Subgroups of the modular group whose ring of modular forms is a polynomial ring"Advanced Studies in Pure Mathematics. 32. 245-254 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] J.Gray 著, 関口次郎, 室 政和 訳: "リーマンからポアンカレにいたる線形微分方程式と群論"シュプリンガーフェアラーク 東京. 450 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] M. Muro: "Invariant hyperfunction solutions to invariant differential equations on the space of real symmetric matrices"to appear in J. of Functional Analysis. (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] M. Muro: "Singular invariant hyperfunctions on the square matrix space and the alternating matrix space"to appear in Nagoya Math. J.. (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] H. Asakawa: "Nonresonant singular two-point boundary value problems"Nonlinear Analysis T.M.A.. 44-6. 791-809 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] H. Asakawa: "On nonresonant singular two-point boundary value problems"Nonlinear Analysis T.M.A.. 47-7. 4849-4860 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] E. Bannnai, M. Koike, A. Munemasa, J. Sekiguchi: "Some results on modular forms-Subgroups of the modular group whose ring of modular forms is a polynomial ring"Advanced Studies in Pure Mathematics. 32. 245-254 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] D. Z. Djokovic, N. Lemire, J. Sekiguchi: "The closure ordering of adjoint nilpotent orbits in so(p,q)"Tohoku Math. J.. 53. 395-442 (2001)

    • Related Report
      2001 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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