Models of quasiprojective homogeneous spaces for Hopf algebras

Given an affine group scheme G of finite type over a field k, a homogeneous space for G is a scheme X over k containing a rational point x such that G operates "transitively" onX. Assuming that G operates on the right, we may identify X with the quotient KnG of G by the left action of the stabilizer K of x in G. The representation-theoretic significance of KnG is that the induction functor indG K from K-modules to G-modules factors as a category equivalence © Walter de Gruyter Berlin . New York 2010