Studies on prehomogeneous vector space and micro-local analysis

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Various results on zeta functions associated with prehomogeneous vector spaces are obtained by viewing from the point of micro-local analysis. Especially experimental calculation on zeta functions of prehomogeneous vector space of commutative parabolic type are carried out. Some other studies on micro-local analysis are done by collaborative researchers.It has been designed in this study, analysis and determination of the invariant hyperfunctions on the prehomogeneous vector space (1), application to the study of residues and the functional equation of the zeta function (2), and invariant differential equations on the prehomogeneous vector space (3). These were some basic problems in prehomogeneous vector spaces. The following describes the step-by-research results and their significance.We know that constructing fundamental solutions is one of the method to obtain all the solutions to the given equation. One method to obtain the fundamental solution is to use the complex power of the … More polynomial of degree two. By operating the Laplacian to the complex power of the polynomial we get the b-function and the complex power is multiplied by it. At the zero-point of the b-function, the complex power has a pole with respect to the parameter. Expanding the complex power at the point to the Laurent expansion, some singular hyperfunctions appears in the rincipal part as coefficients of the Laurent expansion. The support of the singular hyperfunctions appearing here are contained in the set of the zero points of the polynomial. One of the most important singular hyperfunctions is the delta function, whose support concentrates on the origin. The fundamental solution is the hyperfunction which becomes the delta function after operating the differential operator. By constructing the fundamental solution, we can calculate any solution using the integral of the fundamental solution.We have some various method to compute the fundamental solutions, but we can compute the fundamental solution directly from the complex power of the polynomial. For this purpose we have to find polynomials that changes the product of b-functions and the complex power after operating differential operators. Sato Mikio began to seek for finding such polynomials in a more wide class of polynomials. But his trials did not work out well for about half of the year though he tried in the various polynomials. After some trials he found that the success in the case of the Laplacian is attributed to the invariance of homogeneous polynomials of degree two under the rotation group. Then we have to be careful to the group-invariance but what kind of group-action is appropriate for the construction of the fundamental solution. The answer is "the group-action is prehomogeneous".The problem is what kind of group-action is prehomogeneous. The first problem is the classification of prehomogeneous action on the vector space. In order to make the problem too complicated, we suppose that the group is a semi-simple Lie groups (complex algebraic group) and that the group action is a linear action and we have to search for the prehomogeneous action. Sato Mikio first works on this problem and rounded off the classification of prehomogeneous vector spaces under the conditions "semisimplicity" and "irreducibiliy". Later Kimura Tatsuo addressed the problem for the exceptional Lie group and make the complete table of prehomogeneous vector spaces.Now we have seen that what kind of prehomogeneous vector spaces exists. Then what we have to do next for the computation of the fundamental solution? Sato Mikio seemed to turn sour the fundamental solutions. He first worked on the problem of the classification of prehomogeneous vector spaces, and the next problem he was attracted was the problem of zeta functions. He gave considerable attention to the fact that the complex power is used to compute the functional equation of Riemann's zeta function. For the functional equation of Riemann's zeta function we used the homogeneous polynomial of one variable and of degree one, and the relative invariants of prehomogeneous vector spaces correspond to the polynomial. Supposing the regularity as the sufficient condition for the existence of relative invariant, we considered the partial differential operators corresponding to the relative invariants and we can construct the fundamental solutions to the differential operators. On the other hand, the problem of the computation of the functional equation is equivalent to that of the computation of the Fourier transform of the complex power of the relative invariant.The computation of the Fourier transform of relative invariant is more important than that of fundamental solution if you think that zeta functions are more important. The corresponding object of the Riemann's zeta functions was not found at that time, but if you think that the complex power of the relative invariant is the "local" zeta function, this has good grounds for the candidate of the zeta function on the real field. The "global" zeta function to be expected may have the same functional equation. Then we have to study on the explicit computation of the Fourier transform of the relative invariant. Sato Mikio studied the functional equation with Shintani Takuro and proved that the functional equation can be written by using Gamma function and the polynomial of exponential functions of complex parameters. Also, he succeeded to define the global zeta function in the collaborative study with Shintani Takuro. Consequently we can see global zeta function under what conditions. Then the concept of zeta functions associated with prehomogeneous vector spaces was clearly formulated on the basis of the Fourier transforms of relative invariants. The meaning of zeta functions from the point of view of number theory was also made clear. For the application, the role of the zeta function is still unclear, but we see the connection between the representation theoretic object (prehomogeneous vector spaces) and the numaber theoretic one (zeta functions).In the practical research, we have been studying around the theory of phomogeneous vector spaces and the application of micro-local analysis. In particular, we carried out laborious calculation on the basic prehomogeneous vector spaces and the prehomogeneous vector spaces with neither reductive group nor irreducible representation. In actual calculation, many problems remain unresolved but some progress has been made on irreducible prehomogeneous vector space . However, it is proceeding slowly but steadily. In addition, we have studied the graphics of differential equations using computer algebra. The collaborative researchers are studying b-functions and related topics in their own area. We also wrote a book on the invitation to differential equation for the students of the University of Air. This is of course an elementary textbook of differential equations. Here, the analysis was biased towards the mainly ordinary differential equations in the introduction of traditional differential equations, with the aim to expand their perspectives of partial differential equations. 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