{
  "id": "1710.06847",
  "version": "v1",
  "published": "2017-10-18T17:54:20.000Z",
  "updated": "2017-10-18T17:54:20.000Z",
  "title": "Dirac induction for rational Cherednik algebras",
  "authors": [
    "Dan Ciubotaru",
    "Marcelo De Martino"
  ],
  "comment": "32 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "We introduce the local and global indices of Dirac operators for the rational Cherednik algebra $\\mathsf{H}_{t,c}(G,\\mathfrak{h})$, where $G$ is a complex reflection group acting on a finite-dimensional vector space $\\mathfrak{h}$. We investigate precise relations between the (local) Dirac index of a simple module in the category $\\mathcal{O}$ of $\\mathsf{H}_{t,c}(G,\\mathfrak{h})$, the graded $G$-character of the module, the Euler-Poincar\\'e pairing, and the composition series polynomials for standard modules. In the global theory, we introduce integral-reflection modules for $\\mathsf{H}_{t,c}(G,\\mathfrak{h})$ constructed from finite-dimensional $G$-modules. We define and compute the index of a Dirac operator on the integral-reflection module and show that the index is, in a sense, independent of the parameter function $c$. The study of the kernel of these global Dirac operators leads naturally to a notion of dualised generalised Dunkl-Opdam operators.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-18T17:54:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G99",
      "20F55",
      "20C08"
    ],
    "keywords": [
      "rational cherednik algebra",
      "dirac induction",
      "integral-reflection module",
      "global dirac operators",
      "finite-dimensional vector space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}