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

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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.