TWO RESULTS ON THE ABELIAN IDEALS OF A BOREL SUBALGEBRA

Abstract

Let \(\mathcal{b}\) a fixed Borel subalgebra of a finite-dimensional complex simple Lie algebra \(\mathcal{g}\).  The Shi bijection associates to every ad-nilpotent ideal \(\mathcal{i}\) of \(\mathcal{b}\) a region V_i.  In this paper, we show that \(\mathcal{i}\) is abelian if and only if V_i ∩ 2A is nonempty, if and only if the volume of V_i ∩ 2A equals to that of A, where A is the fundamental alcove of the affine Weyl group.  We also determine the maximal eigenvalue m_r−1 of the Casimir operator on Λ^r−1\(\mathcal{g}\) and the corresponding eigenspace M_r−1, where r is the number of positive roots.