Let G be a simple algebraic group and P a parabolic subgroup of G with abelian unipotent radical P u , and let B be a Borel subgroup of G contained in P . Let p u be the Lie algebra of P u and L a Levi factor of P , then L is a Hermitian symmetric subgroup of G and B acts with finitely many orbits both on p u and on G/L. In this paper we study the Bruhat order of the B-orbits in p u and in G/L, proving respectively a conjecture of Panyushev and a conjecture of Richardson and Ryan.

Abstract. We prove that the multiplication of sections of globally generated line bundles on a model wonderful variety M of simply connected type is always surjective. This follows by a general argument which works for every wonderful variety and reduces the study of the surjectivity for every couple of globally generated line bundles to a finite number of cases. As a consequence, the cone defined by a complete linear system over M or over a closed G-stable subvariety of M is normal. We apply these results to the study of the normality of the compactifications of model varieties in simple projective spaces and of the closures of the spherical nilpotent orbits. Then we focus on a particular case proving two specific conjectures of Adams, Huang and Vogan on an analogue of the model orbit of the group of type E 8 .

We review in these notes the theory of equivariant embeddings of spherical homogeneous spaces. Given a spherical homogeneous space G/H, the normal equivariant embeddings of G/H are classified by combinatorial objects called colored fans, which generalize the fans appearing in the classification of toric varieties and which encode several geometric properties of the corresponding variety. Contents 1. Introduction 1 2. Characterizations of sphericality 3 3. Examples 9 4. Invariant valuations on a spherical homogeneous space 12 5. The B-stable affine open subset associated to a G-orbit 15 6. Simple spherical embeddings and colored cones 17 7. The classification of spherical embeddings 23 8. Morphisms between spherical embeddings 24 9. Co-connected inclusions and colored subspaces 28 10. The valuation cone of a spherical homogeneous space 30 11. Equivariant automorphisms of a spherical homogeneous space 32 12. Wonderful embeddings and spherical roots 36 References 38

Let G G be a quasi-simple algebraic group over an algebraically closed field k \mathsf {k} whose characteristic is not very bad for G G , and let B B be a Borel subgroup of G G with Lie algebra b \mathfrak {b} . Given a B B -stable abelian subalgebra a \mathfrak {a} of the nilradical of b \mathfrak {b} , we parametrize the B B -orbits in a \mathfrak {a} and we describe their closure relations.

If G is a complex simply connected semisimple algebraic group and if λ is a dominant weight, we consider the compactification X λ ⊂ P End(V (λ)) obtained as the closure of the G × Gorbit of the identity and we give necessary and sufficient conditions on the support of λ so that X λ is normal; as well, we give necessary and sufficient conditions on the support of λ so that X λ is smooth.