If it doesn't then there is a homogeneous polynomial on V which is invariant under the semisimple part of G. In the setting of Sato, G is an algebraic group and V is a rational representation of G which has a (nonempty) open orbit in the Zariski topology.
In this case, a theorem of Élie Cartan shows that is a reductive group, with a centre that is at most one-dimensional.
The classification of irreducible reduced PVS (G, V) splits into two cases: those for which G is semisimple, and those for which it is reductive with one-dimensional centre.
If G is semisimple, it is (perhaps a covering of) a subgroup of SL(V), and hence G × GL(1) acts prehomogenously on V, with one-dimensional centre.
In other words, in the case that G has one-dimensional center, we assume that the semisimple part does not act prehomogeneously; it follows that there is a relative invariant, i.e., a function invariant under the semisimple part of G, which is homogeneous of a certain degree d. This makes it possible to restrict attention to semisimple G ≤ SL(V) and split the classification as follows: However, it turns out that the classification is much shorter, if one allows not just products with GL(1), but also with SL(n) and GL(n).
Sato and Kimura establish this classification by producing a list of possible irreducible prehomogeneous (G, V), using the fact that G is reductive and the dimensional restriction.
Indeed, in 1974, Richardson observed that if H is a semisimple Lie group with a parabolic subgroup P, then the action of P on the nilradical
This shows in particular (and was noted independently by Vinberg in 1975) that the Levi factor G of P acts prehomogeneously on V :=
Almost all of the examples in the classification can be obtained by applying this construction with P a maximal parabolic subgroup of a simple Lie group H: these are classified by connected Dynkin diagrams with one distinguished node.
One reason that PVS are interesting is that they classify generic objects that arise in G-invariant situations.
For example, if G = GL(7), then the above tables show that there are generic 3-forms under the action of G, and the stabilizer of such a 3-form is isomorphic to the exceptional Lie group G2.
Another example concerns the prehomogeneous vector spaces with a cubic relative invariant.