{
  "id": "1308.4464",
  "version": "v3",
  "published": "2013-08-21T01:30:49.000Z",
  "updated": "2013-12-09T11:51:09.000Z",
  "title": "Remarks on the abelian ideals of a Borel subalgebra",
  "authors": [
    "Chao-Ping Dong"
  ],
  "comment": "The results were known already",
  "categories": [
    "math.RT"
  ],
  "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.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-12-09T11:51:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "abelian ideals",
      "finite-dimensional complex simple lie algebra",
      "shi bijection associates",
      "affine weyl group",
      "ad-nilpotent ideal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1308.4464D"
    }
  }
}