Compact-Unit Types and Fixed Vectors for Symplectic Similitudes

For every prime \(p\) and every rank \(r\geq1\), let \(H_r=\{\operatorname{diag}(I_r,aI_r):a\in\mathbb Z_p^\times\}\) inside \(\operatorname{GSp}_{2r}(\mathbb Q_p)\). 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 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 \(p=2\).