Theory of prehomogeneous vector spaces (algebraic part)—the English translation of Sato’s lecture from Shintani’s note

Mikio Sato; Takuro Shintani

doi:10.1017/S0027763000003214

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The purpose of this paper is to introduce a-functions and b-functions of prehomogeneous vector spaces in the original way of M. Sato and give a proof of the structure theorem of them. All the results were obtained by M. Sato when he constructed the theory of prehomogeneous vector spaces in 60’s. However he did not write a paper on his outcomes at that time. His theory was distributed through his lectures and informal seminars. Only small number of people could know it. The only publication left for us is a mimeographed note of his lecture [Sa-Sh1] written by T. Shintani in Japanese.

References

[A1] Aomoto, K., Finiteness of a cohomology associated with certain Jackson integrals, preprint 1989.Google Scholar

[A2] Aomoto, K., Connection coefficients for Jackson integrals of extended Selberg type, preprint 1989.Google Scholar

[Sa-Ki] Sato, M. and Kimura, T., A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J., 65 (1977), 1–155.Google Scholar

[Sa-Sh1] Sato, M. and Shintani, T., Gaikinshitsu bekutoru kuukan no riron (Theory of prehomogeneous vector spaces) (in Japanese), Sugaku no Ayumi, 15-1 (1970), 85–157.Google Scholar

[Sa-Sh2] Sato, M. and Shintani, T., On zeta functions associated with prehomogeneous vector spaces, Ann. of Math., 100 (1974), 131–170.Google Scholar

[Sa-Ka-ki-Os] Sato, M., Kashiwara, M., Kimura, T. and Oshima, T., Micro-local analysis of prehomogeneous vector spaces, Invent. Math., 62 (1980), 117–179.Google Scholar

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