{
  "schema_version": 1,
  "problem_number": "AIM-REPRESENTATION_THEORY-0007",
  "title": "Compact-Unit Types and Fixed Vectors for Symplectic Similitudes",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For every prime p and every rank r >= 1, let H_r be the compact subgroup of GSp_{2r}(Q_p) consisting of diag(I_r,aI_r) for p-adic units a. We give an elementary proof that every smooth character of H_r occurs in every irreducible infinite-dimensional smooth complex representation of this group. In particular, such a representation has a nonzero vector fixed by the subgroup congruent to H_r modulo p^m for some m. This answers the varying-level fixed-vector question recorded as Problem 3.2 of the AIM automorphic-forms problem list. The proof isolates a nontrivial character on a compact additive subgroup and enlarges that subgroup until its character stabilizer is small enough to make compact-unit averaging nonzero. One long-root subgroup and elementary symplectic transvections then suffice in every rank. The rank-two result and averaging mechanism are due to Roberts and Schmidt; the argument here requires no genericity, Bessel model or twisted-Jacquet nonvanishing theorem, and includes p=2.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.RT",
    "math.NT"
  ],
  "keywords": [
    "p-adic representations",
    "symplectic similitudes",
    "fixed vectors",
    "compact-unit types",
    "smooth representations",
    "character projectors",
    "AIM-REPRESENTATION_THEORY-0007",
    "math.RT",
    "math.NT"
  ],
  "manuscript_version_date": "2026-09-28",
  "publication_date": "2026-09-28",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-28",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-representation-theory-0007/",
  "pdf_url": "https://eulersolve.org/papers/aim-representation-theory-0007/paper.pdf?v=8fd0f2882f4d",
  "doi": "10.5281/zenodo.23023079",
  "zenodo_record_url": "https://zenodo.org/records/23023079",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Fixed rank and varying congruence level are distinguished explicitly, following the subgroup definition in the cited paper by Saha. No assertion that the level equals the rank, no minimal-level formula, and no multiplicity or newform-uniqueness statement are made. The rank-two result and compact averaging mechanism are credited to Roberts and Schmidt. Self-audited and unrefereed; no independent certification, formal verification or absolute-priority claim. Unlocated prior art or folklore is not excluded.",
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    "source.zip": {
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    "verification_report.md": {
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}