Study on prehomogeneous vector spaces

首先，我们研究了紧李群关于两个对称子群的双陪集分解。我们得到了一个精确的分解定理，分类和根系理论。我们还计算了旗流形在几种典型球面子群作用下的轨道结构。其次，我们研究了约化预齐次向量空间上的zeta函数。在各向同性子群不含二维环面的条件下，证明了zeta函数的收敛性。在Kimura-Sato意义下，我们给出了阿基米德域上zeta函数关于15号预齐性向量空间的函数方程。第三，我们研究了p-adic球面齐性空间。给出了轨道分解，在一定条件下得到了球函数的唯一性和一个显式公式，研究了射影空间上的复杂动力系统，并给出了处理三维动态场景的计算模式投影方法。

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.