{
  "id": "0909.1735",
  "version": "v1",
  "published": "2009-09-09T16:17:59.000Z",
  "updated": "2009-09-09T16:17:59.000Z",
  "title": "Infinite Dimensional Multiplicity Free Spaces III: Matrix Coefficients and Regular Functions",
  "authors": [
    "Joseph A. Wolf"
  ],
  "categories": [
    "math.RT",
    "math.DG"
  ],
  "abstract": "In earlier papers we studied direct limits $(G,K) = \\varinjlim (G_n,K_n)$ of two types of Gelfand pairs. The first type was that in which the $G_n/K_n$ are compact Riemannian symmetric spaces. The second type was that in which $G_n = N_n\\rtimes K_n$ with $N_n$ nilpotent, in other words pairs $(G_n,K_n)$ for which $G_n/K_n$ is a commutative nilmanifold. In each we worked out a method inspired by the Frobenius--Schur Orthogonality Relations to define isometric injections $\\zeta_{m,n}: L^2(G_n/K_n) \\hookrightarrow L^2(G_m/K_m)$ for $m \\geqq n$ and prove that the left regular representation of $G$ on the Hilbert space direct limit $L^2(G/K) := \\varinjlim L^2(G_n/K_n)$ is multiplicity--free. This left open questions concerning the nature of the elements of $L^2(G/K)$. Here we define spaces $\\cA(G_n/K_n)$ of regular functions on $G_n/K_n$ and injections $\\nu_{m,n} : \\cA(G_n/K_n) \\to \\cA(G_m/K_m)$ for $m \\geqq n$ related to restriction by $\\nu_{m,n}(f)|_{G_n/K_n} = f$. Thus the direct limit $\\cA(G/K):= \\varinjlim \\{\\cA(G_n/K_n), \\nu_{m,n}\\}$ sits as a particular $G$--submodule of the much larger inverse limit $\\varprojlim \\{\\cA(G_n/K_n), \\text{restriction}\\}$. Further, we define a pre Hilbert space structure on $\\cA(G/K)$ derived from that of $L^2(G/K)$. This allows an interpretation of $L^2(G/K)$ as the Hilbert space completion of the concretely defined function space $\\cA(G/K)$, and also defines a $G$--invariant inner product on $\\cA(G/K)$ for which the left regular representation of $G$ is multiplicity--free.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-09-09T16:17:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E65",
      "17B65",
      "22E70",
      "43A85"
    ],
    "keywords": [
      "infinite dimensional multiplicity free spaces",
      "regular functions",
      "matrix coefficients",
      "hilbert space",
      "left regular representation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0909.1735W"
    }
  }
}