Prehomogeneous vector space

In mathematics, a prehomogeneous vector space (PVS) is a finite-dimensional vector space V together with a subgroup G of the general linear group GL(V) such that G has an open dense orbit in V. Prehomogeneous vector spaces were introduced by Mikio Sato in 1970 and have many applications in geometry, number theory and analysis, as well as representation theory. The irreducible PVS were classified by Sato and Tatsuo Kimura in 1977, up to a transformation known as "castling". They are subdivided into two types, according to whether the semisimple part of G acts prehomogeneously or not. If it doesn't then there is a homogeneous polynomial on V which is invariant under the semisimple part of G.