{
  "id": "1707.02565",
  "version": "v1",
  "published": "2017-07-09T11:49:15.000Z",
  "updated": "2017-07-09T11:49:15.000Z",
  "title": "Gelfand-Kirillov Dimensions of Highest Weight Harish-Chandra Modules for $SU(p,q)$",
  "authors": [
    "Zhanqiang Bai",
    "Xun Xie"
  ],
  "comment": "19 pages. Comments are welcome",
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "abstract": "Let $ (G,K) $ be an irreducible Hermitian symmetric pair of non-compact type with $G=SU(p,q)$, and let $ \\lambda $ be an integral weight such that the simple highest weight module $ L(\\lambda) $ is a Harish-Chandra $ (\\mathfrak{g},K) $-module. We give a combinatoric algorithm for the Gelfand-Kirillov dimension of $ L(\\lambda) $. This enables us to prove that the Gelfand-Kirillov dimension of $ L(\\lambda) $ decreases as the integer $ \\langle\\lambda+\\rho,\\beta^\\vee\\rangle $ increases, where $\\rho$ is the half sum of positive roots and $\\beta$ is the maximal noncompact root. As a byproduct, we obtain a description on the associated variety of $ L(\\lambda) $.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-09T11:49:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E47",
      "17B10"
    ],
    "keywords": [
      "highest weight harish-chandra modules",
      "gelfand-kirillov dimension",
      "simple highest weight module",
      "maximal noncompact root",
      "irreducible hermitian symmetric pair"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}