Spherical Orbits and Abelian Ideals

Abstract

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 algebra of B and Bu by b and bu , respectively. The group B acts on any ideal of b by means of the adjoint representation. After a preliminary section we study a relationship between spherical nilpotent orbits and abelian ideals a of b, using the structure theory for these orbits from [10]. The principal result of this section is that, for an abelian ideal a of b, any nilpotent orbit meeting a is a spherical G-variety, see Theorem 2.3. As a consequence of this we obtain a short conceptual proof of a finiteness theorem from [14]. Namely, for a parabolic subgroup P of G and an abelian ideal a of p=Lie P in the nilpotent radical pu=Lie Pu , the group P operates on a with finitely many orbits. The proof of this fact in [14] involved long and tedious case by case considerations. We also prove a partial converse to the result just mentioned. Following [4], we say that an ideal of b is ad nilpotent whenever it consists of nilpotent elements. In case G is simply laced, we show that an ad nilpotent ideal c of b is abelian provided any nilpotent orbit meeting c is spherical, see Proposition 2.7. In Section 3 we consider some properties of ad nilpotent ideals of b. In Theorem 3.2 we give a description of the normaliser of such ideals. This applies in particular to abelian ideals of b. A remarkable theorem of doi:10.1006 aima.2000.1959, available online at http: www.idealibrary.com on