Harmonic Analysis on Symmetric Spaces

1. 在这个项目下，大岛于1988年1月在东京大学组织了一次专题讨论会，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，我们讨论了项目的当前阶段及其未来发展。Oshima发表了几个重要的结果，这些结果将用于获得半简单对称空间谐波分析的主要目标，即Plancherel公式。它们是：Oshima实现了对称空间在紧流形中的光滑嵌入，并利用它构造了不变微分算子的特征函数的边值映射，发现了特征函数在无穷远处的渐近具有一个简单的几何结构。用同样的方法，Oshima证明了在齐次空间上实现的可一元Harish-Chandra模的有界性。Kobayashi计算了一个齐次空间上的拉普拉斯谱，这个空间是Riemann上的一个纤维束，更对称的空间。这是对Sunada猜想的一个反例。Kobayashi还证明了几个半简单对称空间序列中一致格的存在性。对于具有S^1作用的几乎复杂流形上的线束，当流形的维数很小时，Hattori证明了Kodaira型的消失定理。Masuda证明了一些由Gierer-Meinhard提出的作为生物形成数学模型的反应扩散系统解的存在性和有界性。Ihara研究了有理数域上的绝对伽罗瓦群及其对某代数流形基本群补全的自然作用。Ihara清楚地表明，这些动作给出了伽罗瓦群的充分一般的非阿贝尔表示，并在数论中得到了几个应用。Kataoka和Tose推广了Sjostrand的微局部双曲边值问题规律的微局部传播理论。其结果还包含了解的存在性定理。少

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