{
  "id": "2111.02077",
  "version": "v2",
  "published": "2021-11-03T08:59:30.000Z",
  "updated": "2022-02-09T08:56:08.000Z",
  "title": "Periodicity for subquotients of the modular category $\\mathcal{O}$",
  "authors": [
    "Peter Fiebig"
  ],
  "comment": "14 pages; second version without essential changes",
  "categories": [
    "math.RT"
  ],
  "abstract": "In this paper we study the category $\\mathcal{O}$ over the hyperalgebra of a reductive algebraic group in positive characteristics. For any locally closed subset $\\mathcal{K}$ of weights we define a subquotient $\\mathcal{O}_{[\\mathcal{K}]}$ of $\\mathcal{O}$. It has the property that its simple objects are parametrized by elements in $\\mathcal{K}$. We then show that $\\mathcal{O}_{[\\mathcal{K}]}$ is equivalent to $\\mathcal{O}_{[\\mathcal{K}+p^l\\gamma]}$ for any dominant weight $\\gamma$ if $l>0$ is an integer such that $\\mathcal{K}\\cap (\\mathcal{K}+p^l\\eta)=\\emptyset$ for all dominant weights $\\eta$. This allows one, for example, to restrict attention to subquotients inside the dominant (or the antidominant) chamber.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-02-09T08:56:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "modular category",
      "periodicity",
      "dominant weight",
      "simple objects",
      "reductive algebraic group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}