{
  "id": "2105.09486",
  "version": "v1",
  "published": "2021-05-20T03:13:06.000Z",
  "updated": "2021-05-20T03:13:06.000Z",
  "title": "Generic stabilizers for simple algebraic groups",
  "authors": [
    "Skip Garibaldi",
    "Robert M. Guralnick"
  ],
  "categories": [
    "math.RT",
    "math.GR"
  ],
  "abstract": "We prove a myriad of results related to the stabilizer in an algebraic group $G$ of a generic vector in a representation $V$ of $G$ over an algebraically closed field $k$. Our results are on the level of group schemes, which carries more information than considering both the Lie algebra of $G$ and the group $G(k)$ of $k$-points. For $G$ simple and $V$ faithful and irreducible, we prove the existence of a stabilizer in general position, sometimes called a principal orbit type. We determine those $G$ and $V$ for which the stabilizer in general position is smooth, or $\\dim V/G < \\dim G$, or there is a $v \\in V$ whose stabilizer in $G$ is trivial.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-20T03:13:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20G05",
      "14L24"
    ],
    "keywords": [
      "simple algebraic groups",
      "generic stabilizers",
      "general position",
      "principal orbit type",
      "lie algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}