Let G G be a connected semisimple group over C \mathbb C , whose simple components have type A \mathsf A or D \mathsf D . We prove that wonderful G G -varieties are classified by means of combinatorial objects called spherical systems. This is a generalization of a known result of Luna for groups of type A \mathsf A ; thanks to another result of Luna, this implies also the classification of all spherical G G -varieties for the groups G G we are considering. For these G G we also prove the smoothness of the embedding of Demazure.

We complete the classification of wonderful varieties initiated by D. Luna. We review the results that reduce the problem to the family of primitive varieties, and report the references where some of them have already been studied. Finally, we analyze the rest case-by-case

Abstract. The classification of spherical varieties is already known for semisimple groups of types A and D. Adding type E, we complete the classification for all semisimple groups with a simply laced Dynkin diagram.

Abstract. Let G be a semisimple complex algebraic group, and H ⊆ G a wonderful subgroup. We prove several results relating the subgroup H to the properties of a combinatorial invariant S of G/H, called its spherical system. It is also possible to consider a spherical system S as a datum defined by purely combinatorial axioms, and under certain circumstances our results prove the existence of a wonderful subgroup H associated to S . As a byproduct, we reduce for any group G the proof of the classification of wonderful G-varieties, known as the Luna conjecture, to its verification on a small family of cases, called primitive.

Let G be a connected complex reductive group. A well known theorem of I. Losev's says that a smooth affine spherical G-variety X is uniquely determined by its weight monoid, which is the set of irreducible representations of G that occur in the coordinate ring of X. In this paper, we use the combinatorial theory of spherical varieties and a smoothness criterion of R. Camus to characterize the weight monoids of smooth affine spherical varieties.