Reductivity and finiteness of pseudo-reflections of algebraic groups and homogeneous fiber bundles

Abstract

Let G be an affine algebraic group defined over an algebraically closed field K of any characteristic. For an affine Krull scheme X over SpecK and a regular action G on X, a pseudo-reflection group R(X,G) of the action (X,G) is defined to be a generalization of the subgroup of G generated by all pseudo-reflections in case of a finite group action. We show the equivalence for G between its (geometric) reductivity and finiteness of pseudo-reflection groups on X’s of all actions (X,G0)’s on affine Krull schemes over SpecK. For a finite normal subgroup H of G0, we will give a pseudo-reflection model(X,G0) for {G0,H}, i.e., a faithful regular action of G0 on an affine normal variety X satisfying R(X,G0)=H. Further remarks on pseudo-reflection groups of actions are obtained.