# Verification and scope report

Title: Compact-Unit Types and Fixed Vectors for Symplectic Similitudes

Author: Alper Ferudun; affiliation: Mercury Software GmbH.

Source target: AIM-REPRESENTATION_THEORY-0007, AIM automorphic-forms
problem list, section 3, Problem 3.2 (Saha), indexed in
ulamai/UnsolvedMath v1.6.0.

## Claim and scope

Every smooth compact-unit character occurs in every irreducible
infinite-dimensional smooth complex representation of GSp_{2r}(Q_p),
for every r>=1 and prime p. A nonzero vector is therefore fixed by
R_r(m)=H_r K_r(m) at some level m. Admissibility is unnecessary.

The source uses the same n for rank and level. This result fixes the rank
and allows the level to vary, in agreement with the subgroup definition in
Saha, arXiv:1303.5246v2, section 3.1. It does not claim m=r or give a
minimal-level/dimension/newform-uniqueness formula. Rank two is established
prior art, Roberts--Schmidt, Documenta Math. 21 (2016), section 7.2.

## Proof basis

The compact affine-group lemma starts with a nontrivial character on a
compact additive subgroup. Enlarging the subgroup raises its character
order, shrinking the unit stabilizer until it fixes the input component
and lies in the kernel of the desired character. The latter is essential:
the projected average equals the positive Haar mass of that stabilizer
times a nonzero vector. No global additive eigenvector, genericity,
unitarity, Bessel model or twisted-Jacquet nonvanishing is assumed.

A long-root subgroup of GSp has exactly the required faithful unit action.
If it acted trivially, all symplectic transvections would act trivially,
forcing an irreducible representation to factor through the abelian
similitude quotient and be one-dimensional. An elementary induction proving
transvection generation is included.

## Checks

The standard-library exact checker passed with both system and bundled
Python: 4,203 finite character cases (1,408 nonzero averages and 2,795
necessary stabilizer exclusions), 175 conductor restrictions, 150 matrix
conjugations, 40 rational symplectic reductions in ranks 1--5, 22 weight-two
obstruction checks, and three one-dimensional exceptions. No floating-point
numerics occur. The finite tests are diagnostic and not formal verification
of the p-adic theorem.

The detailed originating-researcher self-audit checks the order of choices,
projection signs, dyadic edge case, transvection induction, quotient
argument and final congruence-level step. It identifies no remaining
mathematical gap in the stated theorem. This is not independent review.
Compilation and full-page visual checks are recorded separately in the
build receipt. A bounded search did not find a matching all-rank statement;
folklore or unlocated prior work is not excluded.

This unrefereed preprint used AI assistance in derivation, literature search,
checking and exposition. Publication and DOI assignment do not certify
correctness, novelty or peer review. No third-party source PDF is included
in the public artifacts.