{
  "id": "2205.15632",
  "version": "v1",
  "published": "2022-05-31T09:21:26.000Z",
  "updated": "2022-05-31T09:21:26.000Z",
  "title": "Projective representations of real reductive Lie groups and the gradient map",
  "authors": [
    "Leonardo Biliotti"
  ],
  "comment": "29 pages. arXiv admin note: text overlap with arXiv:2012.14858",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $G$ be a connected semisimple noncompact real Lie group and let $\\rho: G \\longrightarrow \\mathrm{SL}(V)$ be a representation on a finite dimensional vector space $V$ over $\\mathbb R$, with $\\rho(G)$ closed in $\\mathrm{SL}(V)$. Identifying $G$ with $\\rho(G)$, we assume there exists a $K$-invariant scalar product $\\mathtt g$ such that $G=K\\exp(\\mathfrak p)$, where $K=\\mathrm{SO}(V,\\mathtt g)\\cap G$, $\\mathfrak p=\\mathrm{Sym}_o (V,\\mathtt g)\\cap \\mathfrak g$ and $\\mathfrak g$ denotes the Lie algebra of $G$. Here $\\mathrm{Sym}_o (V,\\mathtt g)$ denotes the set of symmetric endomorphisms with trace zero. Using the $G$-gradient map techniques we analyze the natural projective representation of $G$ on $\\mathbb P(V)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-31T09:21:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E45",
      "53D20",
      "14L24"
    ],
    "keywords": [
      "real reductive lie groups",
      "gradient map",
      "projective representation",
      "semisimple noncompact real lie group",
      "finite dimensional vector space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}