TWO RESULTS ON THE ABELIAN IDEALS OF A BOREL SUBALGEBRA - Zhan, Qian (Italian Journal of Pure and Applied Mathematics, 2015)
Let \(b\) a fixed Borel subalgebra of a finite-dimensional complex simple Lie algebra \(g\). The Shi bijection associates to every ad-nilpotent ideal \(i\) of \(b\) a region Vi. In this paper, we show that \(i\) is abelian if and only if Vi ∩ 2A is nonempty, if and only if the volume of Vi ∩ 2A equals to that of A, where A is the fundamental alcove of the aﬃne Weyl group. We also determine the maximal eigenvalue mr−1 of the Casimir operator on Λr−1\(g\) and the corresponding eigenspace Mr−1, where r is the number of positive roots.