Affine open orbits, reductive isotropy groups, and dominant gradient morphisms; a theorem of Mikio Sato

Abstract

An algebraic proof is given for a theorem of M. Sato. The theorem gives criteria for the open orbit in a prehomogeneous vector space under a reductive group to be an affine variety. The following conditions are equivalent: 1. 0(G) the open orbit is an affine variety. 2. Gz the isotropy subgroup of X in O(G) is reductive. 3. There exists a semi-invariant form P of degree r ^ 2 such that gradP: V->V* is a dominant morphism of affine varieties.