LetGbe a connected linear algebraic group, andpa rational representation ofGon a finite-dimensional vector spaceV, all defined over the complex number fieldC.We call such a triplet (G, p, V) aprehomogeneous vector spaceifVhas a Zariski-denseG-orbit. The main purpose of this paper is to classify all prehomogeneous vector spaces whenpis irreducible, and to investigate their relative invariants and the regularity.

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IntroductionThe purpose of this paper is to give an explicit method to calculate the bfunctions of the relative invariants of regular prehomogeneous vector spaces by using the theory of simple hotonomic systems of micro-differential equations.It is proved in [7] (P(s,x,O)Therefore, in principle, we can calculate b(s) if we know the system of differential equations to which f(x) ~ is a solution. When f(x) is a relative invariant of a regular prehomogeneous vector space, f(x) ~ satisfies the system of the first-order differential equations derived from the relative invariance off(x). This is the case that we treat in this paper. Now we shall explain how the micro-local analysis is applied to obtain b(s).

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