{
  "id": "2310.14956",
  "version": "v1",
  "published": "2023-10-23T13:57:25.000Z",
  "updated": "2023-10-23T13:57:25.000Z",
  "title": "Action of $w_0$ on $V^L$: the special case of $\\mathfrak{so}(1,n)$",
  "authors": [
    "Ilia Smilga"
  ],
  "comment": "Main text: 9 pages, 1 figure. Appendix: 7 pages, of which 4.5 pages are occupied by 2 tables. The ancillary files contain an implementation of the algorithm in the LiE software package (with an explanation)",
  "categories": [
    "math.RT",
    "math.GR"
  ],
  "abstract": "In this note, we present an algorithm that allows to answer any individual instance of the following question. Let $G_{\\mathbb{R}}$ be a semisimple real Lie group, and $V$ an irreducible representation of $G_{\\mathbb{R}}$. How does the longest element $w_0$ of the restricted Weyl group $W$ act on the subspace $V^L$ of $V$ formed by vectors that are invariant by $L$, the centralizer of a maximal split torus of $G_{\\mathbb{R}}$? This algorithm comprises two parts. First we describe a complete answer to this question in the particular case where $G_{\\mathbb{R}} = \\operatorname{SO}(1,n)$ for any $n \\geq 2$. Then, for an arbitrary $G_{\\mathbb{R}}$, we show that it suffices to do the computation in a well-chosen subgroup $S_{\\mathbb{R}} \\subset G_{\\mathbb{R}}$ which is (up to isogeny) the product of several groups that are either compact, abelian or isomorphic to $\\operatorname{SO}(1,n)$ for some $n \\geq 2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-23T13:57:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10",
      "17B20",
      "22E46",
      "22E47"
    ],
    "keywords": [
      "special case",
      "semisimple real lie group",
      "maximal split torus",
      "algorithm comprises",
      "well-chosen subgroup"
    ],
    "tags": [
      "research tool"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}