{
  "id": "1606.02262",
  "version": "v1",
  "published": "2016-06-07T18:53:41.000Z",
  "updated": "2016-06-07T18:53:41.000Z",
  "title": "On commuting varieties of parabolic subalgebras",
  "authors": [
    "Russell Goddard",
    "Simon M. Goodwin"
  ],
  "comment": "17 pages",
  "categories": [
    "math.RT",
    "math.AG"
  ],
  "abstract": "Let $G$ be a connected reductive algebraic group over an algebraically closed field $k$, and assume that the characteristic of $k$ is zero or a pretty good prime for $G$. Let $P$ be a parabolic subgroup of $G$ and let $\\mathfrak p$ be the Lie algebra of $P$. We consider the commuting variety $\\mathcal C(\\mathfrak p) = \\{(X,Y) \\in \\mathfrak p \\times \\mathfrak p \\mid [X,Y] = 0\\}$. Our main theorem gives a necessary and sufficient condition for irreducibility of $\\mathcal C(\\mathfrak p)$ in terms of the modality of the adjoint action of $P$ on the nilpotent variety of $\\mathfrak p$. As a consequence we give a classification of when $\\mathcal C(\\mathfrak b)$ is irreducible for the case $P = B$ a Borel subgroup of $G$; this builds on a partial classification given by Keeton. Further, in cases where $\\mathcal C(\\mathfrak p)$ is irreducible, we consider whether $\\mathcal C(\\mathfrak p)$ is a normal variety.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-06-07T18:53:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "commuting variety",
      "parabolic subalgebras",
      "borel subgroup",
      "nilpotent variety",
      "adjoint action"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}