{
  "id": "2312.12238",
  "version": "v1",
  "published": "2023-12-19T15:23:53.000Z",
  "updated": "2023-12-19T15:23:53.000Z",
  "title": "Parabolic induction in the homotopy category of pro-$p$ Iwahori-Hecke modules",
  "authors": [
    "Nicolas Dupré"
  ],
  "comment": "54 pages, comments welcome",
  "categories": [
    "math.RT",
    "math.CT",
    "math.NT"
  ],
  "abstract": "Let $G$ be the group of rational points of a split connected reductive group over a non-archimedean local field of residue characteristic $p$, and let $\\mathcal{H}$ denote the pro-$p$ Iwahori-Hecke algebra of $G$ over a field of characteristic $p$. We study the parabolic induction functor for $\\mathcal{H}$-modules in terms of the Gorenstein projective model structures introduced by Hovey. Let $\\text{Ho}(\\mathcal{H})$ denote the associated homotopy category of this model structure. We show that $\\text{Ho}(\\mathcal{H})$ and its thick subcategory generated by the essential images of finitely many parabolic induction functors are related via a recollement of triangulated categories. We then investigate the isomorphism classes of simple $\\mathcal{H}$-modules in $\\text{Ho}(\\mathcal{H})$ and give a complete classification when $G=\\mathrm{GL}_n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-19T15:23:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20C08",
      "18N40",
      "18N55"
    ],
    "keywords": [
      "homotopy category",
      "iwahori-hecke modules",
      "parabolic induction functor",
      "non-archimedean local field",
      "gorenstein projective model structures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 54,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}