{
  "id": "2208.05139",
  "version": "v1",
  "published": "2022-08-10T04:33:35.000Z",
  "updated": "2022-08-10T04:33:35.000Z",
  "title": "Gelfand-Kirillov dimension of Representations of $\\mathrm{GL}_n$ over a non-archimedean local field",
  "authors": [
    "Kenta Suzuki"
  ],
  "categories": [
    "math.RT",
    "math.NT"
  ],
  "abstract": "We calculate the asymptotic behavior of the dimension of the fixed vectors of $\\pi$ with respect to compact open subgroups $1+ M_n(\\mathfrak{p}^N)\\subset\\mathrm{GL}_n(F)$ for $\\pi$ an admissible representation of $\\mathrm{GL}_n(F)$, and $F$ a nonarchimedean local field. Such dimensions can be calculated by germs of the character of $\\pi$. We also make some observations on how those dimensions behave under instances of Langlands functoriality, such as the Jacquet-Langlands correspondence and cyclic base change, where relations between characters are known.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-10T04:33:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "non-archimedean local field",
      "gelfand-kirillov dimension",
      "representation",
      "cyclic base change",
      "compact open subgroups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}