{
  "id": "1908.10704",
  "version": "v1",
  "published": "2019-08-28T13:10:20.000Z",
  "updated": "2019-08-28T13:10:20.000Z",
  "title": "Character varieties for real forms of classical complex groups",
  "authors": [
    "Miguel Acosta"
  ],
  "comment": "25 pages, preliminary version, comments are welcome!",
  "categories": [
    "math.GT",
    "math.GR"
  ],
  "abstract": "Let $\\Gamma$ be a finitely generated group, $G_\\mathbb{C}$ be a classical complex group and $G_\\mathbb{R}$ a real form of $G_\\mathbb{C}$. We propose a definition of the $G_\\mathbb{R}$-character variety of $\\Gamma$ as a subset $\\mathcal{X}_{G_\\mathbb{R}}(\\Gamma)$ of the $G_\\mathbb{C}$-character variety $\\mathcal{X}_{G_\\mathbb{C}}(\\Gamma)$. We prove that these subsets cover the set of irreducible $G_\\mathbb{C}$-characters fixed by an anti-holomorphic involution $\\Phi$ of $\\mathcal{X}_{G_\\mathbb{C}}(\\Gamma)$. Whenever $G_\\mathbb{R}$ is compact, we prove that $\\mathcal{X}_{G_\\mathbb{R}}(\\Gamma)$ is homeomorphic to the topological quotient $\\mathrm{Hom}(\\Gamma,G_\\mathbb{R})/G_\\mathbb{R}$. Finally, we identify the reducible points of $\\mathcal{X}_{\\mathrm{GL}(n,\\mathbb{C})}(\\Gamma)$ fixed by an anti-holomorphic involution $\\Phi$ as coming from direct sums of representations with values in real groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-28T13:10:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20C15",
      "14L24",
      "14D20"
    ],
    "keywords": [
      "classical complex group",
      "character variety",
      "real form",
      "anti-holomorphic involution",
      "real groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}