{
  "id": "0812.1343",
  "version": "v1",
  "published": "2008-12-07T13:12:22.000Z",
  "updated": "2008-12-07T13:12:22.000Z",
  "title": "Actions of the derived group of a maximal unipotent subgroup on $G$-varieties",
  "authors": [
    "Dmitri I. Panyushev"
  ],
  "comment": "23 pages",
  "journal": "Intern. Math. Res. Notices, 2010, no. 4, 674--700",
  "categories": [
    "math.AG",
    "math.RT"
  ],
  "abstract": "Let $U$ be a maximal unipotent subgroup of a connected semisimple group $G$ and $U'$ the derived group of $U$. We study actions of $U'$ on affine $G$-varieties. First, we consider the algebra of $U'$ invariants on $G/U$. We prove that $k[G/U]^{U'}$ is a polynomial algebra of Krull dimension $2r$, where $r=rk(G)$. A related result is that, for any simple finite-dimensional $G$-module $V$, $V^{U'}$ is a cyclic $U/U'$-module. Second, we study \"symmetries\" of Poincare series for $U'$-invariants on affine conical $G$-varieties. Third, we obtain a classification of simple $G$-modules $V$ with polynomial algebras of $U'$-invariants (for $G$ simple).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-12-07T13:12:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14L30"
    ],
    "keywords": [
      "maximal unipotent subgroup",
      "derived group",
      "polynomial algebra",
      "invariants",
      "connected semisimple group"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0812.1343P"
    }
  }
}