Compact-Unit Types and Fixed Vectors for Symplectic Similitudes

For every prime p and every r >= 1, let H_r = {diag(I_r, a I_r): a in Z_p^times} inside GSp_{2r}(Q_p). 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 the 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 fixed-vector result and the 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. Scope and status: Unrefereed, self-audited preprint with AI assistance; not independently reviewed or proof-assistant verified. Fixed rank and varying level are explicitly separated. No minimal-level formula or absolute priority is claimed. The elementary method may have unlocated or folklore antecedents. Source problem: AIM-REPRESENTATION_THEORY-0007 in ulamai/UnsolvedMath; original statement: http://aimpl.org/aagaautomorphic/3/ (Problem 3.2). The source archive contains the standalone LaTeX manuscript, bibliography, complete proof, source and novelty audits, and an exact-arithmetic diagnostic checker. Third-party source PDFs are not redistributed.