On spherical ideals of Borel subalgebras

The goal of this paper is to extend some previous results on abelian ideals of Borel subalgebras to so-called spherical ideals of b. These are ideals c of b such that their G-saturation G c is a spherical G-variety. We classify all maximal spherical ideals of b for all simple G. Let G be a connected reductive complex algebraic group with Lie algebra Lie G = g. Let B be a Borel subgroup of G with unipotent radical Bu. We denote the Lie algebras of B and Bu by b and bu, respectively. The group B acts on any ideal of b by means of the adjoint representation. There has been quite a lot of activity recently in the study of various aspects of ad-nilpotent ideals and, in particular, abelian ideals of b , for instance, see (5), (6), (9), (12), (14), and (16), and the additional references therein. After a preliminary section, we recall our main finiteness results on abelian ideals from (14). The goal of this paper is to extend these results to so-called spherical ideals of b in the next section. These are ideals c of b such that their G-saturation G c is a spherical G-variety. Our aim is to classify all maximal spherical ideals of b for all simple G.