{
  "id": "1309.6993",
  "version": "v2",
  "published": "2013-09-26T18:01:41.000Z",
  "updated": "2014-12-15T20:33:36.000Z",
  "title": "The symmetric invariants of the centralizers and Slodowy grading",
  "authors": [
    "Jean-Yves Charbonnel",
    "Anne Moreau"
  ],
  "comment": "60 pages. We thank Alexander Premet for pointing out to us, in the previous version, a mistake in the proof of the main theorem. The proof has been so significantly modified",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let g be a finite-dimensional simple Lie algebra of rank r over an algebraically closed field of characteristic zero, and let e be a nilpotent element of g. Denote by g^e the centralizer of e in g and by S(g^e)^{g^e} the algebra of symmetric invariants of g^e. We say that e is good if the nullvariety of some r homogeneous elements of S(g^e)^{g^e} in the dual of g^{e} has codimension r. If e is good then S(g^e)^{g^e} is polynomial. The main result of this paper stipulates that if for some homogeneous generators of S(g^e)^{g^e}, the initial homogeneous component of their restrictions to e+g^f are algebraically independent, with (e,h,f) an sl2-triple of g, then e is good. As applications, we obtain new examples of nilpotent elements that verify the above polynomiality condition, in in simple Lie algebras of both classical and exceptional types. We also give a counter-example in type D_7.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-09-26T18:01:41.000Z",
      "title": "The symmetric invariants of the centralizers and finite W-algebras",
      "abstract": "Let g be a finite-dimensional simple Lie algebra of rank r over an algebraically closed field of characteristic zero, and let e be a nilpotent element of g. Denote by g^e the centralizer of e in g and by S(g^e)^{g^e} the algebra of symmetric invariants of g^e. We say that e is good if the nullvariety of some r homogeneous elements of S(g^e)^{g^e} in the dual of g^{e} has codimension r. If e is good then S(g^e)^{g^e} is polynomial. The main result of this paper stipulates that if for some homogeneous generators of S(g^e)^{g^e}, the initial homogeneous component of their restrictions to e+g^f are algebraically independent, with (e,h,f) an sl2-triple of g, then e is good. The proof is strongly based on the theory of finite W-algebras. As applications, we obtain new examples of nilpotent elements that verify the above polynomiality condition in simple Lie algebras of both classical and exceptional types. We also give a counter-example in type D7.",
      "comment": "45 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-12-15T20:33:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B35",
      "17B20",
      "13A50",
      "14L24"
    ],
    "keywords": [
      "finite w-algebras",
      "symmetric invariants",
      "centralizer",
      "finite-dimensional simple lie algebra",
      "nilpotent element"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 60,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1309.6993C"
    }
  }
}