Long Abelian Ideals

Let b be a Borel subalgebra of a simple Lie algebra g. LetAb 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 ∈ ∆+. A nice result of D. Peterson says that the cardinality of Ab is 2rkg. His approach uses a one-to-one correspondence between the Abelian ideals and the so-called ‘minuscule’ elements of the affine Weyl group Ŵ (see Section 1 for precise definitions). An exposition of Peterson’s results is found in [3]. Peterson’s work appeared to be the point of departure for active recent investigations of Abelian ideals, and related problems of representation theory and combinatorics [1],[2],[3],[7],[8]. Our definition of minuscule elements follows Kostant’s paper [3], so that w ∈ Ŵ is minuscule in our sense if and only if w−1 is minuscule in the sense of Cellini–Papi [1], [2]. An elegant proof of Peterson’s theorem is given in [1]. Let A be the fundamental alcove of Ŵ . Cellini and Papi show that w ∈ Ŵ is minuscule if and only if w−1·A ⊂ 2A . Since 2A consists of 2rkg alcoves and Ŵ acts simply transitively on the set of alcoves, Peterson’s theorem follows. In this paper, we first show that methods of [1] can be adapted to solving the following problems: Suppose g has two root lengths.