Remarks on the abelian ideals of a Borel subalgebra

TL;DR

This paper characterizes abelian ideals of a Borel subalgebra in terms of geometric regions associated with the affine Weyl group and explores properties of these regions and related algebraic invariants.

Contribution

It provides a geometric criterion for abelian ideals using the Shi bijection and investigates properties of associated regions and Casimir eigenvalues.

Findings

Abelian ideals correspond to regions with empty intersection with 2A.

Volume of the intersection equals that of the fundamental alcove for abelian ideals.

Determination of the maximal eigenvalue of the Casimir operator and its eigenspace.

Abstract

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.