{
  "id": "2408.07581",
  "version": "v1",
  "published": "2024-08-14T14:25:44.000Z",
  "updated": "2024-08-14T14:25:44.000Z",
  "title": "The ($Γ$-asymptotic) wavefront sets: $GL_n$",
  "authors": [
    "Dan Ciubotaru",
    "Ju-Lee Kim"
  ],
  "comment": "14 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $G$ be a connected reductive $p$-adic group. As verified for unipotent representations, it is expected that there is a close relation between the (Harish-Chandra-Howe) wavefronts sets of irreducible smooth representations and their Langlands parameters in the local Langlands correspondence via the Lusztig-Spaltenstein duality and the Aubert-Zelevinsky duality. In this paper, we define the $\\Gamma$-asymptotic wavefront sets generalizing the notion of wavefront sets via the $\\Gamma$-asymptotic expansions (in the sense of Kim-Murnaghan), and then study the their relation with the Langlands parameters. When $G=GL_n$, it turns out that this reduces to the corresponding relation of unipotent representations of the appropriate twisted Levi subgroups via Hecke algebra isomorphisms. For unipotent representations of $GL_n$, we also describe the Harish-Chandra-Howe (HCH) local character expansions of irreducible smooth representations using Kazhdan-Lusztig theory, and give another computation of the coefficients in the HCH expansion and the wavefront sets.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-14T14:25:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E50"
    ],
    "keywords": [
      "unipotent representations",
      "irreducible smooth representations",
      "langlands parameters",
      "local character expansions",
      "hecke algebra isomorphisms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}