Universal transitivity of simple and 2-simple prehomogeneous vector spaces
We denote by k a field of characteristic zero satisfying \(H^ 1(k,Aut(SL_ 2))\neq 0\). Let G be a connected k-split linear algebraic group acting on \(X=Aff^ n\) rationally by \(\rho\) with a Zariski-dense G- orbit Y. A prehomogeneous vector space (G,\(\rho\),X) is called ''universally transitive'' if the set of k-rational points Y(k) is a single \(\rho\) (G)(k)-orbit for all such k. Such prehomogeneous vector spaces are classified by J. Igusa when \(\rho\) is irreducible. We classify them when G is reductive and its commutator subgroup [G,G] is either a simple algebraic group or a product of two simple algebraic groups.
scientific article; zbMATH DE number 3979256