  Let (G,V) be a regular prehomogeneous vector space (abbreviated to PV), where
G is a connected reductive algebraic group over C. If $V=
\oplus_{i=0}^{n}V_{i}$ is a decomposition of V into irreducible
representations, then, in general, the PV's $(G,V_{i})$ are no longer regular.
In this paper we introduce the notion of quasi-irreducible PV (abbreviated to
Q-irreducible), and show first that for completely Q-reducible PV's, the
Q-isotopic components are intrinsically defined, as in ordinary representation
theory. We also show that, in an appropriate sense, any regular PV is a direct
sum of quasi-irreducible PV's. Finally we classify the quasi-irreducible PV's
of parabolic type.
