{
  "id": "2007.11267",
  "version": "v1",
  "published": "2020-07-22T08:40:02.000Z",
  "updated": "2020-07-22T08:40:02.000Z",
  "title": "Affine category O, Koszul duality and Zuckerman functors",
  "authors": [
    "Ruslan Maksimau"
  ],
  "comment": "71 pages. This represents a portion of arXiv:1512.04878 which was split into two parts, this is the second part. This paper is rewritten (compared to arXiv:1512.04878) in a way that we never use KLR algebras explicitly. This makes the paper more independent from the first part",
  "categories": [
    "math.RT"
  ],
  "abstract": "The parabolic category $\\mathcal{O}$ for affine ${\\mathfrak{gl}}_N$ at level $-N-e$ admits a structure of a categorical representation of $\\widetilde{\\mathfrak{sl}}_e$ with respect to some endofunctors $E$ and $F$. This category contains a smaller category $\\mathbf{A}$ that categorifies the higher level Fock space. We prove that the functors $E$ and $F$ in the category $\\mathbf{A}$ are Koszul dual to Zuckerman functors. The key point of the proof is to show that the functor $F$ for the category $\\mathbf{A}$ at level $-N-e$ can be decomposed in terms of the components of the functor $F$ for the category $\\mathbf{A}$ at level $-N-e-1$. To prove this, we use the following fact: a category with an action of $\\widetilde{\\mathfrak sl}_{e+1}$ contains a (canonically defined) subcategory with an action of $\\widetilde{\\mathfrak sl}_{e}$. We also prove a general statement that says that in some general situation a functor that satisfies a list of axioms is automatically Koszul dual to some sort of Zuckerman functor.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-22T08:40:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10"
    ],
    "keywords": [
      "zuckerman functor",
      "koszul duality",
      "affine category",
      "higher level fock space",
      "parabolic category"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 71,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}