{
  "id": "1205.2362",
  "version": "v1",
  "published": "2012-05-10T19:41:31.000Z",
  "updated": "2012-05-10T19:41:31.000Z",
  "title": "Coadjoint structure of Borel subgroups and their nilradicals",
  "authors": [
    "Bertram Kostant"
  ],
  "comment": "8 pages in plain tex; Representation theory, symplectic manifolds, coadjoint orbits",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $G$ be a complex simply-connected semisimple Lie group and let $\\frak{g}= Lie G$. Let $\\frak{g} = \\frak{n}_- +\\frak{h} + \\frak{n}$ be a triangular decomposition of $\\frak{g}$. One readily has that $Cent\\,U({\\frak n})$ is isomorphic to the ring $S({\\frak n})^{{\\frak\\n}}$ of symmetric invariants. Using the cascade ${\\cal B}$ of strongly orthogonal roots, some time ago we proved that $S({\\frak n})^{{\\frak n}$ is a polynomial ring $\\Bbb C [\\xi_1,...,\\xi_m]$ where $m$ is the cardinality of ${\\cal B}$. Using this result we establish that the maximal coadjoint of $N = exp \\frak {n}$ has codimension $m$. Let $\\frak {b}= \\frak {h} + \\frak {n}$ so that the corresponding subgroup $B$ is a Borel subgroup of $G$. Let $\\ell = rank \\frak{g}$. Then in this paper we prove the theorem that the maximal coadjoint orbit of $B$ has codimension $\\ell - m$ so that the following statements (1) and (2) are equivalent: (1) -1 is in the Weyl group of $G$ (i.e., $\\ell = m$), and (2), B has a nonempty open coadjoint orbit. We remark that a nilpotent or a semisimple group cannot have a nonempty open coadjoint orbit. Celebrated examples where a solvable Lie group has a nonempty open coadjoint orbit are due to Piatetski--Shapiro in his counterexample construction of a bounded complex homogeneous domain which is not of Cartan type.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-05-10T19:41:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B08",
      "17B22",
      "17B30",
      "22Exx",
      "53D05"
    ],
    "keywords": [
      "nonempty open coadjoint orbit",
      "borel subgroup",
      "coadjoint structure",
      "nilradicals",
      "complex simply-connected semisimple lie group"
    ],
    "note": {
      "typesetting": "Plain TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1205.2362K"
    }
  }
}