Research on a classification theory of prehomogeneous vector spaces

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The basic theory of prehomogeneous vector spaces was first obtained by Mikio Sato, and later many mathematicians have been developing this theory. This has been investigated by number theoretical method, algebra-analytic methos or representation theoretical method. There are applications to the theory of zeta functions etc. Hence it is an important problem to classify them to know what kind of examples exist.The first result was a classification of irreducible prehomogeneous vector spaces obtained by Mikio Sato and Tatsuo Kimura. However a classification problem is very difficult and some mathematicians say impossible. But Mikio Sato has some result even for non-irreducible case. In our work, we developed a lot his results. We have also classified some prehomogeneous vector spaces with the different view point from M. Sato’s research. We also find some tools such as PV-equivalences.