{
  "abstract": "Let $G$ be a split connected semi-simple group over a local non-archimedean field $F$. In this paper we study the action of Hecke operators on the Schwartz space of half-densities on the moduli stack $\\bpo$ of $G$-bundles on ${\\mathbb P}^1$ endowed with a trivialization at $0$ and $\\infty$. This space is a representation of $G(F)\\times G(F)$ which contains the regular representation of $G(F)$. We show that this inclusion is an isomorphism on the cuspidal part. In addition we show that every irreducible square-integrable representation $π$ of $G(F)$ defines a Hecke eigenvalue. As a byproduct we show that every irreducible square-integrable representation of $G(F)$ with an Iwahori fixed vector defines a ${\\mathbb C}$-point on the Calogero-Moser space associated to $G$. We also show that for $G=PGL(N)$ the eigenvalues of certain \"simplest non-trivial\" Hecke operators are essentially equal to the values of the character of $π$ on regular elliptic elements. These results are parallel to some of the results of our previous publication where some analogous statements are proved in the case when $F$ is a finite field.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17317",
  "author_total": 2,
  "authors": [
    {
      "name": "Alexander Braverman"
    },
    {
      "name": "David Kazhdan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2zw6tmcpqk1dr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17317",
  "fulltext_url": null,
  "id": "2zw6tmcpqk1dr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17317",
  "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.AG"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7221809557",
  "pdf_url": "https://arxiv.org/pdf/2610.17317",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2zw6tmcpqk1dr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2610.17317",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17317"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Square integrable representations and Hecke eigenvalues over a local non-archimedian field",
  "title_text": "Square integrable representations and Hecke eigenvalues over a local non-archimedian field",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/2zw6tmcpqk1dr",
  "venue": "arXiv",
  "versions": []
}