{
  "id": "2006.11659",
  "version": "v1",
  "published": "2020-06-20T21:29:23.000Z",
  "updated": "2020-06-20T21:29:23.000Z",
  "title": "Complexity of actions over perfect fields",
  "authors": [
    "Friedrich Knop",
    "Vladimir S. Zhgoon"
  ],
  "categories": [
    "math.AG",
    "math.RT",
    "math.SG"
  ],
  "abstract": "Let $G$ be a connected reductive group over a perfect field $k$ acting on an algebraic variety $X$ and let $P$ be a minimal parabolic subgroup of $G$. For $k$-spherical $G$-varieties we prove finiteness result for $P$-orbits that contain $k$-points. This is a consequence of an equality on $P$-complexities of $X$ and of any $P$-invariant $k$-dense subvariety in $X$, which generalizes a corresponding result of E.B.Vinberg in the case of algebraically closed field $k$. Also we introduce an action of the restricted Weyl group $W$ on the set of $k$-dense $P$-invariant closed subvarieties of $X$ of maximal $P$-complexity and $k$-rank in the case of ${\\rm char}\\ k =0$ and on the set of all $k$-dense $P$-orbits in the case of real spherical variety which generalizes the action on $B$-orbits introduced by F.Knop in the algebraically closed field case. We also introduce a little Weyl group related with this action and describe its generators in terms of the generators of $W$ which generalize the description of M.Brion in algebraically closed field case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-20T21:29:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "perfect field",
      "complexity",
      "algebraically closed field case",
      "minimal parabolic subgroup",
      "little weyl group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}