{
  "id": "1406.1453",
  "version": "v1",
  "published": "2014-06-05T17:29:16.000Z",
  "updated": "2014-06-05T17:29:16.000Z",
  "title": "On Lusztig's $q$-analogues of all weight multiplicities of a representation",
  "authors": [
    "Dmitri I. Panyushev"
  ],
  "comment": "15 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $\\mathfrak g$ be a complex semisimple Lie algebra. We obtain new properties of the $q$-analogue of weight multiplicities in finite-dimensional representations of $\\mathfrak g$. In particular, it is proved that certain weighted sum of $q$-analogues of all weights of a representation $V$ equals the $q$-analogue of the zero weight multiplicity in the reducible representation $V\\otimes V^*$. This also provides another formula for the $\\mathbb Z[q]$-valued symmetric bilinear form on the character ring of $\\mathfrak g$ that was introduced by R.Gupta (Brylinski) in 1987.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-05T17:29:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10",
      "17B20",
      "20G10"
    ],
    "keywords": [
      "complex semisimple lie algebra",
      "zero weight multiplicity",
      "valued symmetric bilinear form",
      "finite-dimensional representations",
      "properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.1453P"
    }
  }
}