Homogeneous Space - Prehomogeneous Vector Spaces

Prehomogeneous Vector Spaces

The idea of a prehomogeneous vector space was introduced by Mikio Sato.

It is a finite-dimensional vector space V with a group action of an algebraic group G, such that there is an orbit of G that is open for the Zariski topology (and so, dense). An example is GL₁ acting on a one-dimensional space.

The definition is more restrictive than it initially appears: such spaces have remarkable properties, and there is a classification of irreducible prehomogeneous vector spaces, up to a transformation known as "castling".