{
  "id": "2312.14282",
  "version": "v1",
  "published": "2023-12-21T20:37:46.000Z",
  "updated": "2023-12-21T20:37:46.000Z",
  "title": "Duality in Derived Category $\\mathcal O^\\infty$",
  "authors": [
    "Cemile Kurkoglu"
  ],
  "categories": [
    "math.RT",
    "math.GR",
    "math.NT",
    "math.RA"
  ],
  "abstract": "Let $\\bf{G}$ be a split connected reductive group over a finite extension $F$ of $\\mathbb Q_p$, and let $\\bf{T} \\subset \\bf{B} \\subset \\bf{G}$ be a maximal split torus and a Borel subgroup, respectively. Denote by $G = {\\bf{G}}(F)$ and $B= {\\bf{B}}(F)$ their groups of $F$-valued points and by $\\mathfrak g = \\rm Lie(G)$ and $\\mathfrak b = \\rm Lie(B)$ their Lie algebras. Let $\\mathcal O^\\infty$ be the thick category $\\mathcal O$ for $(\\mathfrak g,\\mathfrak b)$, and denote by $\\mathcal{O}^\\infty_{\\rm alg} \\subset \\mathcal{O}^\\infty$ the full subcategory consisting of objects whose weights are in $X^*(\\bf{T})$. Both are Serre subcategories of the category of all $U$-modules, where $U = U(\\mathfrak g)$. We show first that the functor $\\mathbb D^\\mathfrak g = \\rm RHom_U(-,U)$ preserves $D^b(U)_{\\mathcal{O}^\\infty_{\\rm alg}}$, and we deduce from a result of Coulembier-Mazorchuk that the latter category is equivalent to $D^b(\\mathcal O^\\infty_{\\rm alg})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-21T20:37:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "derived category",
      "maximal split torus",
      "finite extension",
      "serre subcategories",
      "borel subgroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}