{
  "id": "1003.0169",
  "version": "v2",
  "published": "2010-02-28T08:46:26.000Z",
  "updated": "2025-01-17T02:14:39.000Z",
  "title": "First extension groups of Verma modules and $R$-polynomials",
  "authors": [
    "Noriyuki Abe"
  ],
  "comment": "16 pages, one of the main theorems was false, erratum is added",
  "categories": [
    "math.RT"
  ],
  "abstract": "We study the first extension groups between Verma modules. There was a conjecture which claims that the dimensions of the higher extension groups between Verma modules are the coefficients of $R$-polynomials defined by Kazhdan-Lusztig. This conjecture was known as the Gabber-Joseph conjecture (although Gebber and Joseph did not state.) However, Boe gives a counterexample to this conjecture. In this paper, we study how far are the dimensions of extension groups from the coefficients of $R$-polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-02-28T08:46:26.000Z",
      "comment": "15 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2025-01-17T02:14:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10",
      "17B55"
    ],
    "keywords": [
      "first extension groups",
      "verma modules",
      "polynomials",
      "higher extension groups",
      "dimensions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1003.0169A"
    }
  }
}