{
  "id": "1508.06763",
  "version": "v1",
  "published": "2015-08-27T09:19:24.000Z",
  "updated": "2015-08-27T09:19:24.000Z",
  "title": "Quantization commutes with singular reduction: cotangent bundles of compact Lie groups",
  "authors": [
    "Jord Boeijink",
    "Klaas Landsman",
    "Walter van Suijlekom"
  ],
  "comment": "37 pages",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "We analyse the `quantization commutes with reduction' problem (first studied in physics by Dirac, and known in the mathematical literature also as the Guillemin-Sternberg Conjecture) for the conjugate action of a compact connected Lie group G on its own cotangent bundle T*G. This example is interesting because the momentum map is not proper and the ensuing symplectic (or Marsden-Weinstein quotient) T*G/Ad G is typically singular. In the spirit of (modern) geometric quantization, our quantization of T*G (with its standard Kaehler structure) is defined as the kernel of the Dolbeault-Dirac operator (or, alternatively, the spin Dirac operator) twisted by the pre-quantum line bundle. We show that this quantization of T*G reproduces the Hilbert space found earlier by Hall (2002) using geometric quantization based on a holomorphic polarisation. We then define the quantization of the singular quotient T*G/Ad G$ as the kernel of the (twisted) Dolbeault-Dirac operator on the principal stratum, and show that quantization commutes with reduction in the sense that either way one obtains the same Hilbert space L^2(T)^{W(G,T)}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-08-27T09:19:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53D50",
      "81S10"
    ],
    "keywords": [
      "compact lie groups",
      "quantization commutes",
      "cotangent bundle",
      "singular reduction",
      "hilbert space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150806763B"
    }
  }
}