Normalizers of ad-nilpotent ideals

Let g be a complex simple Lie algebra. Fix a Borel subalgebra b and a Cartan subalgebra t ⊂ b. The nilpotent radical of b is denoted by u. The corresponding set of positive (resp. simple) roots is ∆ (resp. Π). An ideal of b is called ad-nilpotent, if it is contained in [b, b]. The theory of ad-nilpotent ideals has attracted much recent attention in the work of Kostant, Cellini-Papi, Sommers, and others. The goal of this paper is to study the normalizer of an ad-nilpotent ideal. We obtain several general descriptions of the normalizer, and present a number of more explicit results for g = sln or sp2n. Let c be an ad-nilpotent ideal. Being a t-stable subspace of u, it is a sum of root spaces. The sum of the corresponding roots is an integral weight, denoted |c|. We show that |c| is a dominant weight and that the normalizer of c, ng(c), is completely determined by |c|. Since ng(c) contains b, it suffices to realize which root spaces g−α (α ∈ Π) are contained in ng(c). We prove that g−α ⊂ ng(c) if and only if (|c|, α) = 0. Another type of descriptions is based on a relationship between the ad-nilpotent ideals and certain elements of the affine Weyl group Ŵ . Let Ad = Ad(g) denote the set of all ad-nilpotent ideals of b. By [3], to each c ∈ Ad one associates an element of Ŵ , which we denote by wmin,c. An ad-nilpotent ideal is called strictly positive, if it is contained in [u, u]. The set of strictly positive ideals is denoted by Ad0. By [15], to each c ∈ Ad0 one associates an element of Ŵ , which we denote by wmax,c. The group Ŵ acts linearly on a vector space V̂ , containing affine root system, and we prove that