{
  "id": "1101.3463",
  "version": "v1",
  "published": "2011-01-18T14:33:57.000Z",
  "updated": "2011-01-18T14:33:57.000Z",
  "title": "The Segal-Bargmann Transform on Compact Symmetric Spaces and their Direct Limits",
  "authors": [
    "Gestur Olafsson",
    "Keng Wiboonton"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "We study the Segal-Bargmann transform, or the heat transform, $H_t$ for a compact symmetric space $M=U/K$. We prove that $H_t$ is a unitary isomorphism $H_t : L^2(M) \\to \\cH_t (M_\\C)$ using representation theory and the restriction principle. We then show that the Segal-Bargmann transform behaves nicely under propagation of symmetric spaces. If $\\{M_n=U_n/K_n,\\iota_{n,m}\\}_n$ is a direct family of compact symmetric spaces such that $M_m$ propagates $M_n$, $m\\ge n$, then this gives rise to direct families of Hilbert spaces $\\{L^2(M_n),\\gamma_{n,m}\\}$ and $\\{\\cH_t(M_{n\\C}),\\delta_{n,m}\\}$ such that $H_{t,m}\\circ \\gamma_{n,m}=\\delta_{n,m}\\circ H_{t,n}$. We also consider similar commutative diagrams for the $K_n$-invariant case. These lead to isometric isomorphisms between the Hilbert spaces $\\varinjlim L^2(M_n)\\simeq \\varinjlim \\mathcal{H} (M_{n\\mathbb{C}})$ as well as $\\varinjlim L^2(M_n)^{K_n}\\simeq \\varinjlim \\mathcal{H} (M_{n\\mathbb{C}})^{K_n}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-01-18T14:33:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "compact symmetric space",
      "direct limits",
      "hilbert spaces",
      "similar commutative diagrams",
      "heat transform"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1101.3463O"
    }
  }
}