Embeddings of homogeneous spaces into irreducible modules

Let G be a connected reductive algebraic group. We find a necessary and sufficient condition for a quasi-affine homogeneous space G/H to have an embedding into an irreducible G-module. For reductive H we also find a necessary and sufficient condition for a closed embedding of G/H into an irreducible module to exist. These conditions are stated in terms of the group of central automorphisms of G/H.