{
  "id": "2106.05336",
  "version": "v1",
  "published": "2021-06-09T18:46:49.000Z",
  "updated": "2021-06-09T18:46:49.000Z",
  "title": "Spectra of non-regular elements in irreducible representations of simple algebraic groups",
  "authors": [
    "Donna M Testerman",
    "Alexandre Zalesski"
  ],
  "comment": "22 pages",
  "categories": [
    "math.RT",
    "math.GR"
  ],
  "abstract": "We study the spectra of non-regular semisimple elements in irreducible representations of simple algebraic groups. More precisely, we prove that if G is a simply connected simple linear algebraic group and f is a non-trivial irreducible representation of G in some GL(V) for which there exists a non-regular non-central semisimple element s in G such that f(s) has almost simple spectrum, then, with few exceptions, G is of classical type and dim V is minimal possible. Here the spectrum of a diagonalizable matrix is called simple if all eigenvalues are of multiplicity 1, and almost simple if at most one eigenvalue is of multiplicity greater than 1. This yields a kind of characterization of the natural representation (up to their Frobenius twists) of classical algebraic groups in terms of the behavior of semisimple elements.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-09T18:46:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20G05"
    ],
    "keywords": [
      "simple algebraic groups",
      "irreducible representation",
      "non-regular elements",
      "connected simple linear algebraic group",
      "non-regular non-central semisimple element"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}