{
  "id": "0801.3869",
  "version": "v1",
  "published": "2008-01-25T01:55:35.000Z",
  "updated": "2008-01-25T01:55:35.000Z",
  "title": "Infinite Dimensional Multiplicity Free Spaces I: Limits of Compact Commutative Spaces",
  "authors": [
    "Joseph A. Wolf"
  ],
  "comment": "23 pages",
  "categories": [
    "math.RT",
    "math.DG"
  ],
  "abstract": "We study direct limits $(G,K) = \\varinjlim (G_n,K_n)$ of compact Gelfand pairs. First, we develop a criterion for a direct limit representation to be a multiplicity--free discrete direct sum of irreducible representations. Then we look at direct limits $G/K = \\varinjlim G_n/K_n$ of compact riemannian symmetric spaces, where we combine our criterion with the Cartan--Helgason Theorem to show in general that the regular representation of $G = \\varinjlim G_n$ on a certain function space $\\varinjlim L^2(G_n/K_n)$ is multiplicity free. That method is not applicable for direct limits of nonsymmetric Gelfand pairs, so we introduce two other methods. The first, based on ``parabolic direct limits'' and ``defining representations'', extends the method used in the symmetric space case. The second uses some (new) branching rules from finite dimensional representation theory. In both cases we define function spaces $\\cA(G/K)$, $\\cC(G/K)$ and $L^2(G/K)$ to which our multiplicity--free criterion applies.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-01-25T01:55:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E45",
      "22E65",
      "22E25",
      "22G05",
      "43A85",
      "43A90",
      "53C35"
    ],
    "keywords": [
      "infinite dimensional multiplicity free spaces",
      "compact commutative spaces",
      "direct limit",
      "representation",
      "compact riemannian symmetric spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0801.3869W"
    }
  }
}