{
  "id": "2206.10854",
  "version": "v1",
  "published": "2022-06-22T05:39:41.000Z",
  "updated": "2022-06-22T05:39:41.000Z",
  "title": "Annihilator of $({\\mathfrak g},K)$-modules of ${\\mathrm O}(p,q)$",
  "authors": [
    "Takashi Hashimoto"
  ],
  "comment": "13 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let ${\\mathfrak g}$ denote the complexified Lie algebra of $G={\\mathrm O}(p,q)$ and $K$ a maximal compact subgroup of $G$. In the previous paper, we constructed $({\\mathfrak g},K)$-modules associated to the finite-dimensional representation of ${\\mathfrak sl}_2$ of dimension $m+1$, which we denote by $M^{+}(m)$ and $M^{-}(m)$. The aim of this paper is to show that the annihilator of $M^{\\pm}(m)$ is the Joseph ideal if and only if $m=0$. We shall see that an element of the symmetric of square $S^{2}({\\mathfrak g})$ that is given in terms of the Casimir elements of ${\\mathfrak g}$ and the complexified Lie algebra of $K$ plays a critical role in the proof of the main result.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-22T05:39:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E46",
      "17B20",
      "17B10"
    ],
    "keywords": [
      "annihilator",
      "complexified lie algebra",
      "maximal compact subgroup",
      "finite-dimensional representation",
      "joseph ideal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}