{
  "id": "math/0610613",
  "version": "v5",
  "published": "2006-10-20T05:45:42.000Z",
  "updated": "2008-10-02T17:17:10.000Z",
  "title": "Kirillov's character formula, the holomorphic Peter-Weyl theorem, and the Blattner-Kostant-Sternberg pairing",
  "authors": [
    "Johannes Huebschmann"
  ],
  "comment": "29 pages; prompted by some reactions, a number of comments have been incorporated",
  "journal": "Journal of Geometry and Physics 58 (2008) 833-848",
  "doi": "10.1016/j.geomphys.2008.02.004",
  "categories": [
    "math.DG",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Let K be a compact Lie group, endowed with a bi-invariant Riemannian metric. The complexification G of K inherits a Kaehler structure having twice the kinetic energy of the metric as its potential, and left and right translation turn the Hilbert space of square-integrable holomorphic functions on G relative to a suitable measure into a unitary (KxK)-representation. We establish the statement of the Peter-Weyl theorem for this Hilbert space to the effect that this Hilbert space contains the vector space of representative functions on G as a dense subspace and that the assignment to a holomorphic function of its Fourier coefficients yields an isomorphism of Hilbert algebras from the convolution algebra on G onto the algebra which arises from the endomorphism algebras of the irreducible representations of K by the appropriate operation of Hilbert space sum. Consequences are (i) a holomorphic Plancherel theorem and the existence of a uniquely determined unitary isomorphism between the space of square-integrable functions on K and the Hilbert space of holomorphic functions on G, and (ii) a proof that this isomorphism coincides with the corresponding Blattner-Kostant-Sternberg pairing map, multiplied by a suitable constant. We then show that the spectral decomposition of the energy operator on the Hilbert space of holomorphic functions on G associated with the metric on K refines to the Peter-Weyl decomposition of this Hilbert space in the usual manner and thus yields the decomposition of this Hilbert space into irreducible isotypical (KxK)-representations. Among our crucial tools is Kirillov's character formula. Our methods are geometric and independent of heat kernels, which are used by B. C. Hall to obtain many of these results [Journal of Functional Analysis 122 (1994), 103--151], [Comm. in Math. Physics 226 (2002), 233--268].",
  "revisions": [
    {
      "version": "v5",
      "updated": "2008-10-02T17:17:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B63",
      "17B81",
      "22E30",
      "22E46",
      "22E70",
      "32W30",
      "53D50",
      "81S10",
      "14L35",
      "17B65",
      "17B66",
      "32Q15",
      "53D17",
      "53D20"
    ],
    "keywords": [
      "hilbert space",
      "kirillovs character formula",
      "holomorphic peter-weyl theorem",
      "holomorphic function",
      "blattner-kostant-sternberg pairing"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}