Let $(G,V)$ be a regular prehomogeneous vector space (abbreviated to $PV$), where $G$ is a reductive algebraic group over $ℂ$. If $V={\oplus }_{i=1}^n{V}_i$ 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 $Q$-isotypic components are intrinsically defined, as in ordinary representation theory. We also show that, in an appropriate...