{
  "id": "1201.4494",
  "version": "v1",
  "published": "2012-01-21T17:42:02.000Z",
  "updated": "2012-01-21T17:42:02.000Z",
  "title": "Center of U(n), Cascade of Orthogonal Roots, and a Construction of Lipsman-Wolf",
  "authors": [
    "Bertram Kostant"
  ],
  "comment": "10 pages, plain.tex; key words: cascade of orthogonal roots, Borel subgroups, nilpotent coadjoint action, dedicated to Joe Wolf; msc keywords: Representation theory, invariant theory",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $G$ be a complex simply-connected semisimple Lie group and let $\\g=\\hbox{\\rm Lie}\\,G$. Let $\\g = \\n_- +\\hh + \\n$ be a triangular decomposition of $\\g$. One readily has that $\\hbox{\\rm Cent}\\,U(\\n)$ is isomorphic to the ring $S(\\n)^{\\n}$ of symmetric invariants. Using the cascade ${\\cal B}$ of strongly orthogonal roots, some time ago we proved (see [K]) that $S(\\n)^{\\n}$ is a polynomial ring $\\Bbb C[\\xi_1,...,\\xi_m]$ where $m$ is the cardinality of ${\\cal B}$. The authors in [LW] introduce a very nice representation-theoretic method for the construction of certain elements in $S(\\n)^{\\n}$. A key lemma in [LW] is incorrect but the idea is in fact valid. In our paper here we modify the construction so as to yield these elements in $S(\\n)^{\\n}$ and use the [LW] result to prove a theorem of Tony Joseph.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-01-21T17:42:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "construction",
      "complex simply-connected semisimple lie group",
      "lipsman-wolf",
      "nice representation-theoretic method",
      "strongly orthogonal roots"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1201.4494K"
    }
  }
}