Quotients of group completions by spherical subgroups

Let G be a semi-simple algebraic group and let H be a spherical subgroup. The ground field k is algebraically closed and of characteristic zero. This article is concerned with projective embeddings Y of spherical homogeneous spaces G/H. Our approach in the study of such a variety Y is to realize them as quotients under the action of H of projective embeddings of G. First, we give a more precise sense to this project by defining the quotient of a G-variety by a spherical subgroup H. Then, we give a condition, in terms of G-invariant valuations, under which Y can be obtained by quotient of an embedding of G. Finally, if the index of H in its normalizer is finite, we show that an important class of embeddings of G/H (toroidal and liftable) geometric quotients of embeddings of G.