Research on prehomogeneous vector spaces and micro-local analysis

预齐次向量空间与微局部分析研究

• 批准号: 13640163
• 负责人: MURO Masakazu
• 金额: $ 2.56万
• 依托单位: Gifu University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640163/
• 关键词: Prehomogeneous 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

(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.

(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.

项目成果

Akihiko Gyoja: "Certain unipotent representations of finite Chevalley groups and Picard-Lefschetz monodromy"Ann. Sci. Ecole Norm.. Sup. Vol. 35. 437-444 (2002)

Akihiko Gyoja：“有限 Chevalley 群和 Picard-Lefschetz monodromy 的某些单能表示”Ann。

M.Muro: "Singular invariant hyperfunctions on the square matrix space and the alternating matrix space"Nagoya Math. J.. 169. (2003)

M.Muro：“方阵空间和交替矩阵空间上的奇异不变超函数”名古屋数学。

H. Asakawa: "Nonresonant singular two-point boundary value problems"Nonlinear Analysis T.M.A.. 44-6. 791-809 (2001)

H. Asakawa：“非共振奇异两点边值问题”非线性分析 T.M.A. 44-6。

Muro, M.: "Construction of hyperfunction solutions to invariant linear differential equations"RIMS Kokyuroku. Vol.1211. 143-154 (2001)

Muro, M.：“不变线性微分方程超函数解的构造”RIMS Kokyuroku。

M. Muro: "Invariant hyperfunction solutions to invariant differential equations on the space of real symmetric matrices"to appear in J. of Functional Analysis. (2002)

M. Muro：“实对称矩阵空间上不变微分方程的不变超函数解”出现在《泛函分析杂志》中。