Abelian Ideals of a Borel Subalgebra and Long Positive Roots

Let b be a Borel subalgebra of a simple Lie algebra g. Let Ab denote the set of all Abelian ideals of b. It is easily seen that any a ∈ Ab is actually contained in the nilpotent radical of b. Therefore a is determined by the the corresponding set of roots. More precisely, let t be a Cartan subalgebra of g lying in b and let ∆ be the root system of the pair (g, t). Choose ∆, the system of positive roots, so that the roots of b are positive. Then a = ⊕γ∈Igγ, where I is a suitable subset of ∆ and gγ is the root space for γ ∈ ∆ . It follows that there are finitely many Abelian ideals and that any question concerning Abelian ideals can be stated in terms of combinatorics of the root system. An amazing result of D.Peterson says that the cardinality of Ab is 2. His approach uses a one-to-one correspondence between the Abelian ideals and the so-called ‘minuscule’ elements of the affine Weyl group Ŵ . An exposition of Peterson’s results is found in [5]. Peterson’s work appeared to be the point of departure for active recent investigations of Abelian ideals, ad -nilpotent ideals, and related problems of representation theory and combinatorics [1],[2],[3],[4],[5],[6],[7],[8]. We consider Ab as poset with respect to inclusion, the zero ideal being the unique minimal element of Ab. Our goal is to study this poset structure. It is easily seen that Ab is a ranked poset; the rank function attaches to an ideal its dimension. It was shown in [8] that there is a one-to-one correspondence between the maximal Abelian ideals and the long simple roots of g. (For each simple Lie algebra, the maximal Abelian ideals were determined in [10].) This correspondence possesses a number of nice properties, but the very existence of it was demonstrated in a case-by-case fashion. Here we give a conceptual explanation of that empirical observation. More generally, we prove that