Ideals of Heisenberg Type and Minimax Elements of Affine Weyl Groups

Let g be a simple Lie algebra over C. Fix a Borel subalgebra b and a Cartan subalgebra t ⊂ b. The corresponding set of positive (resp. simple) roots is ∆ (resp. Π). Write θ for the highest root in∆. An ideal of b is called ad-nilpotent, if it is contained in [b, b]. Consequently, such a subspace is fully determined by the corresponding set of positive roots, and this set is called an ideal (in ∆). Let Ad denote the set of all ideals of ∆. We regard ∆ as poset with respect to the standard root order ‘4’ (see Section 1). Given I ∈ Ad, the minimal roots in it are said to be the generators. The set of generators of an ideal form an antichain in (∆,4), and this correspondence sets up a bijection between the ideals and the antichains of ∆. An ideal or antichain is called strictly positive, if it contains no simple roots. Another interesting class consists of Abelian ideals, i.e., those with the property (I + I) ∩ ∆ = ∅. We write Ad0 (resp. Ab) for the set of strictly positive (resp. Abelian) ideals. Various results for Ad, Ad0, and Ab were recently obtained in [2], [3], [4], [5], [10], [11], [12], [15]. It was shown by Cellini and Papi [4] that there is a bijection between the ideals and certain elements of Ŵ , the affine Weyl group associated with g. Then Sommers [15] discovered a bijection between the strictly positive ideals (or antichains) and another class of elements of Ŵ . Following [15], the elements in these two classes are said to be minimal and maximal, respectively. Furthermore, the minimal elements of Ŵ are in a bijection with the points of the coroot lattice lying in a certain simplex. The same assertion also holds for the maximal elements (with another simplex!) A different approach to (ad-nilpotent) ideals relates them with the theory of hyperplane arrangements. By a result of Shi [13], there is a bijection between Ad and the dominant regions of the Shi (or Catalan) arrangement. It was then observed by Athanasiadis [2] and Panyushev [11] that the Shi bijection induces the bijection between Ad0 and the bounded dominant regions. We survey these results in Section 2. The goal of this paper is two-fold. First, we study the ideals lying inside H, the set of all positive roots that are not orthogonal to θ. Such ideals are said to be of Heisenberg type. Second, we study the minimax elements of Ŵ , i.e., those that are simultaneously minimal and maximal. The corresponding strictly positive ideals are said to be minimax, too. Both minimal and maximal elements of Ŵ are particular instances of dominant elements. To