Study on a theory of Fourirer transformation by using the thory of prehomogeneous vector paces

研究前期与筑波大学的Tatsuo Kimura和Kazunari Sugiyama组成了两个简单预齐次向量空间的相对不变性对应的所有有理性指标。此外，与金泽大学的Satoshi Wakatsuki共同研究了E8型的Dynkin-Kostant型预齐次向量空间，并在此基础上构造了该空间的基本相对不变性类型、b函数和Igusa局部zeta函数。此外，美国普林斯顿大学Bhargava最近的一项研究的解释能够从东京大学的Takashi Taniguchi那里得到关于这项研究的详细研究资料，并获得非常重要的信息。此外，在立教大学的佐藤文博的指导和合作下，在研究的后半段能够在一定程度上达到最初的目的。虽然它不是预齐次向量空间的相对不变量，但它成功地得到了许多用来填充函数方程的多项式，并找到了新的预齐次向量空间的分类问题。这些研究正在进行中。本文的研究成果如下：(1)明确地计算了2-简单预齐次向量空间的相对不变量及其有变性质。(2)显式计算E8、Dynkin-Kostant型预齐次向量空间的相对不变量b函数和Igusa局部zeta函数。(3)利用Clifford代数的表示对Clerc的结果进行重构。(4)从Clifford环Cl（p,0）和Cl(q,0) n （n=p+q）的表示角度，给出了满足泛函方程的多项式的大量实例。(5)我们的系统具有一些预齐次向量空间的无穷实数形式。(6)本研究的组合观察。

The first half of the research period was able to compose all rational indices corresponding to a relative invariability type of two simple prehomogeneous vector spaces with Tatsuo Kimura and Kazunari Sugiyama of Tsukuba University. Moreover, respectively the Dynkin-Kostant type prehomogeneous vector space of the E8 type was studied by a joint research with Satoshi Wakatsuki of Kanazawa University in parallel with it, and a basic, relative invariability type of the space, b-function, and Igusa local zeta function were able to be composed specifying it.Moreover, the explanation of a recent research of Bhargava of the Princeton university in the United States was able to be received from Takashi Taniguchi of The University of Tokyo in detail research for informations about this research, and acquire very significant information.Moreover, an original purpose was able to be achieved in some degree in the latter half of the research period by guidance and the cooperation of Fumihiro Sato of Rikkyo University. It succeeded in obtaining a lot of polynomials by which the function equation was filled though it was not a relative invariant of prehomogeneous vector space and new classification problem of prehomogeneous vector spaces was able to be found. These researches are progressing now. We list up the results in this research period as follows :(1)Explicitly calculation of relative invariants and thir rarional character of 2-sipmle prehomogeneous vector spaces.(2)Explicitly calculation relative invariants b-function, and Igusa local zeta function of E8, Dynkin-Kostant type prehomogeneous vector spaces.(3)The re-construction of Clerc's resut by using representations of Clifford algebra.(4)A lot of examples of polynomials which satisfy the functional equation from the view point of representatio of Clifford ring Cl(p,0) and Cl(q,0) n (n=p+q).(5)Our system have the infinity class of real form of some prehomogeneous vector spaces.(6)Combinatric observation of our research.