{
  "id": "1811.12867",
  "version": "v1",
  "published": "2018-11-30T16:15:52.000Z",
  "updated": "2018-11-30T16:15:52.000Z",
  "title": "Normalizers of maximal tori and real forms of Lie groups",
  "authors": [
    "Anton A. Gerasimov",
    "Dmitry R. Lebedev",
    "Sergey V. Oblezin"
  ],
  "comment": "17 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "For a complex reductive Lie group $G$ Tits defined an extension $W_G^T$ of the corresponding Weyl group $W_G$. The extended group is supplied with an embedding into the normalizer $N_G(H)$ of the maximal torus $H\\subset G$ such that $W_G^T$ together with $H$ generate $N_G(H)$. We give an interpretation of the Tits classical construction in terms of the maximal split real form $G(\\mathbb{R})\\subset G(\\mathbb{C})$, leading to a simple topological description of $W^T_G$. We also propose a different extension $W_G^U$ of the Weyl group $W_G$ associated with the compact real form $U\\subset G(\\mathbb{C})$. This results into a presentation of the normalizer of maximal torus of the group extension $U\\ltimes {\\rm Gal}(\\mathbb{C}/\\mathbb{R})$ by the Galois group ${\\rm Gal}(\\mathbb{C}/\\mathbb{R})$. We also describe explicitly the adjoint action of $W_G^T$ and $W^U_G$ on the Lie algebra of $G$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-30T16:15:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "maximal torus",
      "normalizer",
      "maximal split real form",
      "complex reductive lie group",
      "compact real form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}