{
  "id": "2207.05830",
  "version": "v1",
  "published": "2022-07-12T20:46:12.000Z",
  "updated": "2022-07-12T20:46:12.000Z",
  "title": "Spectral equivalence of smooth group schemes over principal ideal local rings",
  "authors": [
    "Itamar Hadas"
  ],
  "comment": "26 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $\\mathcal{G}$ be a smooth linear group scheme of finite type. For any positive integer $k$ and a finite field $\\mathbb{F}$, let $W_k(\\mathbb{F})$ be the ring of Witt vectors of length $k$ over $\\mathbb{F}$. We show that the group algebras of $\\mathcal{G}(\\mathbb{F}[t]/(t^k))$ and $\\mathcal{G}(W_k(\\mathbb{F}))$ are isomorphic (i.e. the multi-sets of the dimensions of the irreducible representations are equal) for any positive integer $k$ and finite field $\\mathbb{F}$ with large enough characteristic. We also prove that if $\\mathrm{char}\\mathbb{F}$ is large enough, then the cardinality of the set $\\{\\dim\\rho\\big|\\rho\\in \\mathrm{irr}(\\mathcal{G}(\\mathbb{F}))\\}$ is bounded uniformly in $\\mathbb{F}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-12T20:46:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20C15",
      "20G25",
      "11U07",
      "20J06",
      "11E72"
    ],
    "keywords": [
      "principal ideal local rings",
      "smooth group schemes",
      "spectral equivalence",
      "smooth linear group scheme",
      "finite field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}