{
  "abstract": "Recently, Girsch-Kurinczuk have obtained new formulas for local newforms in depth-zero and minimax integral-depth cuspidal representations of $p$-adic $\\mathrm{GL}(n)$. One of their main results is an averaging formula which provides an integral representation of the Whittaker newform in terms of the Paškūnas-Stevens Whittaker function. In the present paper, we first prove a reverse averaging formula by establishing an integral representation of the Paškūnas-Stevens Whittaker function in terms of the Whittaker newform. Together, these formulas give an explicit two-way passage between these two distinguished vectors. Finally, we use our averaging formula and earlier work of Kurinczuk-Matringe to give new explicit test vectors in terms of Whittaker newforms for such cuspidal $\\mathrm{GL}(n)\\times\\mathrm{GL}(n)$ pairs, whenever their Rankin-Selberg $L$-factor is not identically equal to $1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.08633",
  "author_total": 1,
  "authors": [
    {
      "name": "Alexandros Groutides"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9mwghpxbvtsna.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.08633",
  "fulltext_url": null,
  "id": "9mwghpxbvtsna",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.08633",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7221027473",
  "pdf_url": "https://arxiv.org/pdf/2610.08633",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9mwghpxbvtsna.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.08633",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.08633"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Local newforms, Paškūnas-Stevens Whittaker functions, and explicit test vectors for $\\mathrm{GL}(n)\\times \\mathrm{GL}(n)$",
  "title_text": "Local newforms, Paškūnas-Stevens Whittaker functions, and explicit test vectors for GL(n)× GL(n)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/9mwghpxbvtsna",
  "venue": "arXiv",
  "versions": []
}