{
  "id": "2412.01742",
  "version": "v1",
  "published": "2024-12-02T17:36:14.000Z",
  "updated": "2024-12-02T17:36:14.000Z",
  "title": "Character of Irreducible Representations Restricted to Finite Order Elements -- An Asymptotic Formula",
  "authors": [
    "Shrawan Kumar",
    "Dipendra Prasad"
  ],
  "comment": "10 pages",
  "categories": [
    "math.RT",
    "math.AG",
    "math.GR"
  ],
  "abstract": "Let $G$ be a connected reductive group over the complex numbers and let $T\\subset G$ be a maximal torus. For any $t\\in T$ of finite order and any irreducible representation $V(\\lambda)$ of $G$ of highest weight $\\lambda$, we determine the character $ch(t, V(\\lambda))$ by using the Lefschetz Trace Formula due to Atiyah-Singer and explicitly determining the connected components and their normal bundles of the fixed point subvariety $(G/P)^t\\subset G/P$ (for any parabolic subgroup $P$). This together with Wirtinger's theorem gives an asymptotic formula for $ch(t, V(n\\lambda))$ when $n$ goes to infinity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-02T17:36:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E46",
      "22F30"
    ],
    "keywords": [
      "finite order elements",
      "asymptotic formula",
      "irreducible representation",
      "lefschetz trace formula",
      "highest weight"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}