Principal-unit filtration of a local field ($U_m=1+{\mathfrak{p}}^m(m\ge 1)$)

The groups $U_m$ form a compact-open neighbourhood basis at one inside the units of a non-Archimedean local field. A smooth character trivial on $U_{i+1}$ but nontrivial on $U_i$ cannot have a nonzero fixed vector under $U_m$ for $m\le i$. This elementary conductor comparison controls admissibility and smooth duals of direct sums.