Study on prehomogeneous vector spaces

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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.