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The theory of prehomogeneous vector spaces. Algebraic part. (English) Zbl 0703.22011
Nagoya Math. J. (to appear).
A b-function associated with a regular prehomogeneous vector space is defined as a polynomial in several complex variables $$s:=(s_ 1,s_ 2,...,s_ n)\in {\mathbb{C}}^ n$$. M. Sato proved the following fact: the polynomial b(s) is decomposed into the product of some polynomials of degree one. This is the fundamental theorem of Sato on b-functions associated with prehomogeneous vector spaces. Sato’s proof was based on a cocycle condition which is satisfied by b(s). The theorem is proved in a very general situation and seems to have many applications. First, it appeared in a special issue of the private journal published by the students of University of Tokyo in the 60’s. It was written by T. Shintani in Japanese as a mimeographical note of Sato’s lecture. This article is the English translation of a part of Shintani’s note. The last author reconstructs the structure of Sato’s original proof and translated in into English. However, the original ideas are all due to Sato and the second author.
Reviewer: M.Muro

##### MSC:
 22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods 14M17 Homogeneous spaces and generalizations 58J15 Relations of PDEs on manifolds with hyperfunctions 43A85 Harmonic analysis on homogeneous spaces 32A45 Hyperfunctions 32M10 Homogeneous complex manifolds
##### Keywords:
b-functions; prehomogeneous vector spaces