{
  "id": "2401.02515",
  "version": "v1",
  "published": "2024-01-04T19:50:03.000Z",
  "updated": "2024-01-04T19:50:03.000Z",
  "title": "Limits of Bessel functions for root systems as the rank tends to infinity",
  "authors": [
    "Dominik Brennecken",
    "Margit Rösler"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "We study the asymptotic behaviour of Bessel functions associated of root systems of type $A_{n-1}$ and type $B_n$ with positive multiplicities as the rank $n$ tends to infinity. In both cases, we characterize the possible limit functions and the Vershik-Kerov type sequences of spectral parameters for which such limits exist. In the type $A$ case, this generalizes known results about the approximation of the (positive-definite) Olshanski spherical functions of the space of infinite-dimensional Hermitian matrices over $\\mathbb F = \\mathbb R, \\mathbb C, \\mathbb H$ (with the action of the associated infinite unitary group) by spherical functions of finite-dimensional spaces of Hermitian matrices. In the type B case, our results include asymptotic results for the spherical functions associated with the Cartan motion groups of non-compact Grassmannians as the rank goes to infinity, and a classification of the Olshanski spherical functions of the associated inductive limits.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-04T19:50:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C67",
      "33C52",
      "43A90",
      "22E66"
    ],
    "keywords": [
      "root systems",
      "bessel functions",
      "rank tends",
      "olshanski spherical functions",
      "infinite-dimensional hermitian matrices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}