Let G be a connected semi-simple algebraic group over the algebraically closed characteristic zero field k, and let H be an algebraic subgroup of G (H is usually required to be self-normalizing). This paper considers embeddings of the algebraic homogeneous space G/H. The Demazure embedding X is obtained as the closure of the G-orbit of Lie(H) in \(Grass_ r(Lie(G))\), where \(r=\dim (H)\). When H is spherical (that is, when each Borel subgroup of G has an open orbit in G/H), then there is the mangificent embedding (in French: “plongement magnifique”), which is complete, normal, has a unique closed orbit, and dominates all other such. In a spherical algebraic homogeneous space one can find tori C such that the closure of the C-orbit of Lie(H) in X contains an open affine Z such that each G-orbit of X meets Z in a unique C-orbit (so that the G- orbits of X are classified by the C-orbits on Z).
In the first section of this paper, the author establishes that, for a spherical algebraic homogeneous space G/H the magnificent embedding is the normalization of the Demazure embedding. - In the second one, with suitable choice of torus C and affine Z with the above properties, he explicitly determines the monoid -\({\mathcal M}\) of weights of C on k[Z] (and hence the orbit structure of the C-action on Z). - In section three, he determines the geometric structure of the cone generated by \({\mathcal M}:\) among other things, this shows that the only singularities in the magnificent embedding of G/H are quotients by finite abelian groups. Finally, a root system and Weyl group are introduced and it is shown that the cone of valuations of G/H is the negative chamber of this Weyl group. The paper concludes with three conjectures and a question.