Remarks on the abelian ideals of a Borel subalgebra

Let $\frb$ be a fixed Borel subalgebra of a finite-dimensional complex simple Lie algebra $\frg$. The Shi bijection associates to every ad-nilpotent ideal $\fri$ of $\frb$ a region $V_{\fri}$. In this paper, we show that $\fri$ is abelian if and only if $V_{\fri}\cap 2A$ is empty, if and only if the volume of $V_{\fri}\cap 2A$ equals to that of $A$, where $A$ is the fundamental alcove of the affine Weyl group. For certain flag of abelian ideals, we record an ascending property of their associated regions. We also determine the maximal eigenvalue $m_{r-1}$ of the Casimir operator on $\wedge^{r-1} \frg$ and the corresponding eigenspace $M_{r-1}$, where $r$ is the number of positive roots.