{
  "abstract": "Let $F$ be a non-archimedean local field and let $K_μ$ be a standard parahoric subgroup of $GL_n(F)$. We express the diagonal values of normalized parahoric Whittaker eigenfunctions with trivial central character using blockwise Casselman--Shalika formulas with the appropriate modulus factor. Every irreducible generic Iwahori-spherical representation has a one-dimensional fixed space for a suitable parahoric, with no invariants under a strictly larger parahoric. The full Hecke character of this line determines the representation. We compute its adjacent-block Hecke eigenvalues and, under additional irreducibility hypotheses, obtain an explicit finite formula on every Bruhat cell using an opposite induced model, Whittaker-preserving intertwiners, and Demazure--Whittaker operators. We apply these formulas to local Rankin--Selberg integrals for $GL_n\\times GL_1$ and for compatible parahorics in $GL_n\\times GL_m$. In the latter case, the lowest Laurent coefficient and the equal-rank functional equation determine the integral. Explicit translations and rescalings then give test vectors for the local $L$-factor.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40021",
  "author_total": 1,
  "authors": [
    {
      "name": "Markos Karameris"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8jvm5x2mj427q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40021",
  "fulltext_url": null,
  "id": "8jvm5x2mj427q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40021",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.40021",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8jvm5x2mj427q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.40021",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40021"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Parahoric Spherical Whittaker Functions",
  "title_text": "Parahoric Spherical Whittaker Functions",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/8jvm5x2mj427q",
  "venue": "arXiv",
  "versions": []
}