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".
Read more about this topic: Homogeneous Space
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