{
  "id": "2502.04368",
  "version": "v1",
  "published": "2025-02-05T09:11:44.000Z",
  "updated": "2025-02-05T09:11:44.000Z",
  "title": "Cartan motion groups: regularity of K-finite matrix coefficients",
  "authors": [
    "Guillaume Dumas"
  ],
  "comment": "21 pages. arXiv admin note: substantial text overlap with arXiv:2409.07944",
  "categories": [
    "math.GR",
    "math.FA",
    "math.RT"
  ],
  "abstract": "If $G$ is a connected semisimple Lie group with finite center and $K$ is a maximal compact subgroup of G, then the Lie algebra of $G$ admits a Cartan decomposition $\\mathfrak{g}=\\mathfrak{k}\\oplus\\mathfrak{p}$. This allows us to define the Cartan motion group $H=\\mathfrak{p}\\rtimes K$. In this paper, we study the regularity of $K$-finite matrix coefficients of unitary representations of $H$. We prove that the optimal exponent $\\kappa(G)$ for which all such coefficients are $\\kappa(G)$-H\\\"older continuous coincides with the optimal regularity of all $K$-finite coefficients of the group $G$ itself. Our approach relies on stationary phase techniques that were previously employed by the author to study regularity in the setting of $(G,K)$. Furthermore, we provide a general framework to reduce the question of regularity from $K$-finite coefficients to $K$-bi-invariant coefficients.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-02-05T09:11:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E46",
      "43A85",
      "43A90"
    ],
    "keywords": [
      "cartan motion group",
      "k-finite matrix coefficients",
      "regularity",
      "finite coefficients",
      "stationary phase techniques"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}