Harmonic Analysis on Symmetric Spaces

对称空间的调和分析

• 批准号: 62460004
• 负责人: OSHIMA Toshio
• 金额: $ 3.46万
• 依托单位: University of Tokyo, Faculty of Science, Department of Mathematics
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for General Scientific Research (B)
• 财政年份: 1987
• 资助国家: 日本
• 起止时间: 1987 至 1988
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-62460004/
• 关键词: Symmetric space; Homogeneous space; Harmonic analysis; Unitary representation; Lie group; Boundary value; 代数解析; 半単純リー群; 境界値問題; 主系列表現; 離散系列表現; 半単純対称空間

1. Under this project, Oshima organized a simposium at University of Tokyo on January 1988 and also summer seminars at Institute of Vocational training on August 1988 and August 1989, and we discussed the present stage of the project and its future development.2. Oshima published several important results which will be used to obtain the main aim in harmonic analysis on semisimple symmetric spaces, the Plancherel formula. They are as follows: Oshima realized a smooth imbedding of the symmetric space in a compact manifold and by using it, Oshima constructed boundary value maps for eigenfunctions of the invariant differential operators and discovered that the asymptotic of the eigenfunctions at infinity are characterized by a simple geometric structure. By the same method Oshima proved a certain boundedness of unitarizable Harish-Chandra modules realized on a homogeneous space.3. Kobayashi calculated the spectra of the Laplacian on a homogeneous space which is a fibre bundle over Riemann … More ian symmetric space. This gives a counter example of a conjecture given by Sunada. Kobayashi also proved the existence of uniform lattices in several series of semisimple symmetric spaces.4. Hattori proved a certain vanishing theorem of Kodaira type for a line bundle over a almost complex manifold with S^1-action when the dimension of the manifold is small.5. Masuda shows the existence and boundedness of solutions for some reaction-diffusion systems posed by Gierer-Meinhard as mathematical models of biological formation.6. Ihara studied the absolute Galois group over the rational number field and its natural actions on the completion of the fundamental group of a certain algebraic manifold. Ihara is making clear that the actions give sufficiently general non-abelian representations of the Galois group and obtained several applications to number theory.7. Kataoka and Tose extended the theory of microlocal propagation of regularities for microlocal hyperbolic boundary value problems originated by Sjostrand. Their result also contains and existence theorem of the solutions. Less

1．在该项目下，大岛于1988年1月在东京大学组织了一次研讨会，并于1988年8月和1989年8月在职业培训学院组织了夏季研讨会，讨论了该项目的现阶段及其未来的发展。2．大岛发表了几项重要成果，这些成果将用于获得半简单对称空间调和分析的主要目标，即 Plancherel 公式。它们如下：Oshima实现了对称空间在紧流形中的平滑嵌入，并利用它构造了不变微分算子的本征函数的边值图，并发现无穷远本征函数的渐近性可以用简单的几何结构来表征。用同样的方法Oshima证明了在齐次空间上实现的可单化Harish-Chandra模的一定有界性。 3．小林计算了均匀空间上的拉普拉斯光谱，该均匀空间是黎曼对称空间上的纤维丛。这给出了砂田提出的猜想的反例。小林还证明了数列半单对称空间中均匀格子的存在性。 4． Hattori证明了当流形维数小时，几乎复流形上具有S^1作用的线丛的小平型消失定理。 5． Masuda证明了Gierer-Meinhard作为生物形成数学模型提出的一些反应扩散系统解的存在性和有界性。6．伊原研究了有理数域上的绝对伽罗瓦群及其对某个代数流形的基本群的完备性的自然作用。 Ihara 明确表示，这些行为给出了伽罗瓦群的足够普遍的非阿贝尔表示，并获得了数论的一些应用。7。 Kataoka 和 Tose 将 Sjostrand 提出的微局部双曲边值问题的微局部规律传播理论进行了扩展。他们的结果还包含解的存在定理。较少的

项目成果

Toshio Oshima: Advanced Studies in Pure Math.14. 561-601 (1988)

大岛敏夫：纯数学高级研究.14。

Toshiyuki,Kobayashi;T.Sunada;K.Ono: Forum Mathematicum. 1. 69-79 (1989)

Toshiyuki,Kobayashi;T.Sunada;K.Ono：数学论坛。

Toshio,Oshima: Advanced Studies in Pure Math.14. 561-601 (1988)

Toshio，Oshima：纯数学高级研究.14。

Yasatake,Ihara: Ann.of Math. 128. 271-293 (1988)

Yasatake，Ihara：数学安。

Toshio Oshima: "A realization of semisimple symmetric spaces and construction of boundary value maps" Advanced Studies in Pure Math.14. 603-650 (1988)

Toshio Oshima：“半简单对称空间的实现和边值图的构造”纯数学高级研究.14。