{
  "id": "1309.5313",
  "version": "v1",
  "published": "2013-09-20T16:33:04.000Z",
  "updated": "2013-09-20T16:33:04.000Z",
  "title": "Hitchin's conjecture for simply-laced Lie algebras implies that for any simple Lie algebra",
  "authors": [
    "Nathaniel Bushek",
    "Shrawan Kumar"
  ],
  "comment": "14 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $\\g$ be any simple Lie algebra over $\\mathbb{C}$. Recall that there exists an embedding of $\\mathfrak{sl}_2$ into $\\g$, called a principal TDS, passing through a principal nilpotent element of $\\g$ and uniquely determined up to conjugation. Moreover, $\\wedge (\\g^*)^\\g$ is freely generated (in the super-graded sense) by primitive elements $\\omega_1, \\dots, \\omega_\\ell$, where $\\ell$ is the rank of $\\g$. N. Hitchin conjectured that for any primitive element $\\omega \\in \\wedge^d (\\g^*)^\\g$, there exists an irreducible $\\mathfrak{sl}_2$-submodule $V_\\omega \\subset \\g$ of dimension $d$ such that $\\omega$ is non-zero on the line $\\wedge^d (V_\\omega)$. We prove that the validity of this conjecture for simple simply-laced Lie algebras implies its validity for any simple Lie algebra. Let G be a connected, simply-connected, simple, simply-laced algebraic group and let $\\sigma$ be a diagram automorphism of G with fixed subgroup K. Then, we show that the restriction map R(G) \\to R(K) is surjective, where R denotes the representation ring over $\\mathbb{Z}$. As a corollary, we show that the restriction map in the singular cohomology H^*(G)\\to H^*(K) is surjective. Our proof of the reduction of Hitchin's conjecture to the simply-laced case relies on this cohomological surjectivity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-09-20T16:33:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E46",
      "14D20"
    ],
    "keywords": [
      "simple lie algebra",
      "hitchins conjecture",
      "simple simply-laced lie algebras implies",
      "restriction map",
      "primitive element"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1309.5313B"
    }
  }
}