On the canonical embeddings of certain homogeneous spaces

Abstract

We study equivariant affine embeddings of homoge- neous spaces and their equivariant automorphisms. An example of a quasiaffine, but not affine, homogeneous space with finitely ma ny equivariant automorphisms is presented. We prove the solvability of any connected group of equivariant automorphisms for an affine embedding with a fixed point and finitely many orbits. This is ap- plied to studying the orbital decomposition for algebraic monoids and canonical embeddings of quasiaffine homogeneous spaces,i.e., those affine embeddings associated with the coordinate algebras of homogeneous spaces, provided the latter algebras are finitely gen- erated. We pay special attention to the canonical embeddings of quotient spaces of reductive groups modulo the unipotent radicals of parabolic subgroups. For these varieties, we describe the orbital decomposition, compute the modality of the group action, and find out which of them are smooth. We also describe minimal ambient modules for these canonical embeddings provided that the acting group is simply connected.