{
  "id": "2501.17832",
  "version": "v1",
  "published": "2025-01-29T18:25:12.000Z",
  "updated": "2025-01-29T18:25:12.000Z",
  "title": "Orthogonality relations for Poincaré series",
  "authors": [
    "Sonja Žunar"
  ],
  "comment": "11 pages",
  "categories": [
    "math.NT",
    "math.RT"
  ],
  "abstract": "Let $ G $ be a connected semisimple Lie group with finite center. We prove a formula for the inner product of two cuspidal automorphic forms on $ G $ that are given by Poincar\\'e series of $ K $-finite matrix coefficients of an integrable discrete series representation of $ G $. As an application, we give a new proof of a well-known result on the Petersson inner product of certain vector-valued Siegel cusp forms. In this way, we extend results previously obtained by G. Mui\\'c for cusp forms on the upper half-plane, i.e., in the case when $ G=\\mathrm{SL}_2(\\mathbb R) $.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-01-29T18:25:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F70",
      "11F46"
    ],
    "keywords": [
      "orthogonality relations",
      "vector-valued siegel cusp forms",
      "petersson inner product",
      "integrable discrete series representation",
      "finite matrix coefficients"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}