Study on prehomogeneous vector spaces

预齐次向量空间的研究

• 批准号: 09640029
• 负责人: MATSUKI Toshihiko
• 金额: $ 2.11万
• 依托单位: KYOTO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640029/
• 关键词: prehomogeneous vector space; zeta function; Lie group; algebraic group; homogeneous space; リー群; ヘッケ環; 球関数; 対称空間; 軌道分解; 進化ゲノム理論; 動的パターン投影法

First, we studied double coset decompositions of compact Lie groups with respect to two symmetric subgroups. We got a precise decomposition theorem, classification and a theory of root systems. We also computed orbit structure on flag manifolds under the action of some typical spherical subgroups.Secondly, we studied zeta functions on reduced prehomogeneous vector spaces. We showed the convergence of the zeta function under the condition that the isotropy subgroup contains no 2-dimensional torus. We also gave the functional equation for the zeta function on archimedian fields with respect to the prehomogeneous vector space no. 15 in the sense of Kimura-Sato.Thirdly, we studied p-adic spherical homogeneous spaces. We gave the orbit decomposition and got the uniqueness and an explicit formula of the spherical functions under some comditions.We also studied complex dynamical system over projective spaces and a computational pattern projection method to handle dynamical three-dimensional scenes.

首先，我们研究了两个对称亚组的紧凑型谎言组的双固定分解。我们获得了精确的分解定理，分类和根系理论。我们还根据一些典型的球形子组的作用计算了标志歧管上的轨道结构。首先，我们研究了Zeta在降低的均匀载体空间上的功能。我们在各向同性亚组中含有不二维圆环的条件下显示了Zeta函数的收敛性。我们还提供了相对于耶和华前矢量空间编号的Archimedian字段上ZETA函数的功能方程。 15从木村 - 撒托（Kimura-Sato）的意义上。三分之二，我们研究了p-adic球形均匀空间。我们给出了轨道分解，并在某些合并下获得了球形函数的独特性和明确公式。我们还研究了射影空间上的复杂动力学系统和计算模式投影方法，以处理动态的三维场景。

项目成果

N.Kono: "How big are the increments of a two-parameter Gaussian process?" J.Theoretical Probability. Vol.12. No.1 (to appear). (1999)

N.Kono：“二参数高斯过程的增量有多大？”

H.Saito: "Explicit form of zeta functions of prehomogeneous vector spaces" Math. Ann. (to appear). (1999)

H.Saito：“预齐次向量空间 zeta 函数的显式形式”

T.Ueda: "Critically finite maps on projective spaces" the Journal of Geometric Analysis. (to appear).

T.Ueda：“射影空间上的临界有限映射”，《几何分析杂志》。

松木敏彦: "Classification of two involutions on compact semisimple Lie groups and root systems"

Toshihiko Matsuki：“紧半单李群和根系统上的二次对合分类”

齋藤, 裕: "On zeta functions associated to symmetric matricesII: Functional equations and special values." 未定. (未定). (1999)

Saito, Hiroshi：“关于与对称矩阵相关的 zeta 函数：函数方程和特殊值。”（待定）。