{
  "id": "2410.20183",
  "version": "v1",
  "published": "2024-10-26T13:44:57.000Z",
  "updated": "2024-10-26T13:44:57.000Z",
  "title": "A note on characters and endoscopy for $p$-adic $\\mathrm{SL}_2$",
  "authors": [
    "Anne-Marie Aubert",
    "Roger Plymen"
  ],
  "comment": "21 pages",
  "categories": [
    "math.RT",
    "math.NT"
  ],
  "abstract": "We consider the depth-zero supercuspidal $L$-packets of $\\mathrm{SL}_2$. With the aid of the classical character formulas of Sally-Shalika, we prove the endoscopic character identities. For the depth-zero $L$-packet of cardinality $4$, we find that, in order to comply with the Sally-Shalika formulas, the three endoscopic groups must be paired carefully with semisimple elements in the dual group $\\mathrm{PGL}_2(\\mathbb{C})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-26T13:44:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E50"
    ],
    "keywords": [
      "endoscopic character identities",
      "dual group",
      "semisimple elements",
      "endoscopic groups",
      "classical character formulas"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}