{
  "id": "2301.09812",
  "version": "v1",
  "published": "2023-01-24T04:41:41.000Z",
  "updated": "2023-01-24T04:41:41.000Z",
  "title": "Characterization of supercuspidal representations and very regular elements",
  "authors": [
    "Charlotte Chan",
    "Masao Oi"
  ],
  "comment": "75 pages",
  "categories": [
    "math.RT",
    "math.NT"
  ],
  "abstract": "We prove that regular supercuspidal representations of $p$-adic groups are uniquely determined by their character values on very regular elements -- a special class of regular semisimple elements on which character formulae are very simple -- provided that this locus is sufficiently large. As a consequence, we resolve a question of Kaletha by giving a description of Kaletha's $L$-packets of regular supercuspidal representations which mirrors Langlands' construction for real groups following Harish-Chandra's characterization theorem for discrete series representations. Our techniques additionally characterize supercuspidal representations in general, giving $p$-adic analogues of results of Lusztig on reductive groups over finite fields. In particular, we establish an easy, non-cohomological characterization of unipotent supercuspidal representations when the residue field of the base field is sufficiently large.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-24T04:41:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E50",
      "11S37",
      "11F70"
    ],
    "keywords": [
      "regular elements",
      "regular supercuspidal representations",
      "sufficiently large",
      "harish-chandras characterization theorem",
      "techniques additionally characterize supercuspidal representations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 75,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}