A counterexample in the theory of prehomogeneous vector spaces (Q757600)

Let (G,\(\rho\),V) be a complex prehomogeneous vector space, i.e., there exists an open dense G-orbit \(\Omega\) in V. If G is reductive and if (G,\(\rho\),V) is regular, then \(\Omega\) is an affine variety. It seemed that the regularity condition is essential but the reductivity condition is not. The author proves that the two conditions of this statement are essential by giving an example. Namely, he constructs an example of a regular prehomogeneous vector space (G,\(\rho\),V) such that G is not reductive and \(\Omega\) is not affine.