{
  "id": "2201.09742",
  "version": "v1",
  "published": "2022-01-24T15:19:58.000Z",
  "updated": "2022-01-24T15:19:58.000Z",
  "title": "Action of $w_0$ on $V^L$ for orthogonal and exceptional groups",
  "authors": [
    "Ilia Smilga"
  ],
  "comment": "8 pages. arXiv admin note: substantial text overlap with arXiv:2002.09378",
  "categories": [
    "math.RT",
    "math.GR"
  ],
  "abstract": "In this note, we present some results that partially answer the following question. Let $G$ be a simple real Lie group; what is the set of representations $V$ of $G$ in which the longest element $w_0$ of the restricted Weyl group $W$ acts nontrivially on the subspace $V^L$ of $V$ formed by vectors that are invariant by $L$, the centralizer of a maximal split torus of $G$? We give a conjectural answer to that question, as well as the experimental results that back this conjecture, when $G$ is either an orthogonal group (real form of $\\operatorname{SO}_n(\\mathbb{C})$ for some $n$) or an exceptional group.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-01-24T15:19:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10",
      "17B20",
      "22E46",
      "22E47"
    ],
    "keywords": [
      "exceptional group",
      "simple real lie group",
      "maximal split torus",
      "orthogonal group",
      "real form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}