{
  "id": "2409.07944",
  "version": "v1",
  "published": "2024-09-12T11:12:01.000Z",
  "updated": "2024-09-12T11:12:01.000Z",
  "title": "Regularity of K-finite matrix coefficients of semisimple Lie groups",
  "authors": [
    "Guillaume Dumas"
  ],
  "categories": [
    "math.GR",
    "math.FA"
  ],
  "abstract": "We consider $G$ a semisimple Lie group with finite center and $K$ a maximal compact subgroup of $G$. We study the regularity of $K$-finite matrix coefficients of unitary representations of $G$. More precisely, we find the optimal value $\\kappa(G)$ such that all such coefficients are $\\kappa(G)$-H\\\"older continuous. The proof relies on analysis of spherical functions of the symmetric Gelfand pair $(G,K)$, using stationary phase estimates from Duistermaat, Kolk and Varadarajan. If $U$ is a compact form of $G$, then $(U,K)$ is a compact symmetric pair. Using the same tools, we study the regularity of $K$-finite coefficients of unitary representations of $U$, improving on previous results obtained by the author.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-12T11:12:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E46",
      "43A85",
      "43A90"
    ],
    "keywords": [
      "semisimple lie group",
      "k-finite matrix coefficients",
      "regularity",
      "unitary representations",
      "stationary phase estimates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}