{
  "id": "1103.3545",
  "version": "v1",
  "published": "2011-03-18T02:36:29.000Z",
  "updated": "2011-03-18T02:36:29.000Z",
  "title": "Maximal eigenvalues of a Casimir operator and multiplicity-free modules",
  "authors": [
    "Gang Han"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $\\g$ be a finite-dimensional complex semisimple Lie algebra and $\\b$ a Borel subalgebra. Then $\\g$ acts on its exterior algebra $\\w\\g$ naturally. We prove that the maximal eigenvalue of the Casimir operator on $\\w\\g$ is one third of the dimension of $\\g$, that the maximal eigenvalue $m_i$ of the Casimir operator on $\\w^i\\g$ is increasing for $0\\le i\\le r$, where $r$ is the number of positive roots, and that the corresponding eigenspace $M_i$ is a multiplicity-free $\\g$-module whose highest weight vectors corresponding to certain ad-nilpotent ideals of $\\b$. We also obtain a result describing the set of weights of the irreducible representation of $\\g$ with highest weight a multiple of $\\rho$, where $\\rho$ is one half the sum of positive roots.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-03-18T02:36:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10"
    ],
    "keywords": [
      "maximal eigenvalue",
      "casimir operator",
      "multiplicity-free modules",
      "finite-dimensional complex semisimple lie algebra",
      "highest weight vectors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1103.3545H"
    }
  }
}