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An admissible representation is a smooth representation for which every fixed-vector space $V^K$ for a compact-open subgroup has finite dimension. Admissibility can hold for an infinite-dimensional representation; it bounds each local fixed level, not their union.

For the upper-triangular subgroup of ${\mathrm{GL}}_n(F)$, conjugation by $\mathrm{diag}({\varpi }^{-(n-1)},\dots ,1)$ expands the upper-unipotent coordinates. For a decomposed compact-open subgroup $K=T_0{U}_0$, its conjugates eventually contain $K$, while their fixed-vector spaces are isomorphic. Finite dimension therefore forces those spaces to be equal. Inverse conjugation contracts any fixed unipotent element into $U_0$, proving that the unipotent radical acts trivially in every admissible representation of this triangular group.

The fixed-vector space is $V^K=\{v:\pi (k)v=v\text{ for all }k\in K\}$. Conjugation gives an isomorphism $V^K\to V^{gK{g}^{-1}}$, $v\mapsto \pi (g)v$. If $K\subseteq L$, then $V^L\subseteq V^K$.

Write $d=\mathrm{diag}({\varpi }^{-(n-1)},\dots ,1)$ for the diagonal matrix called $\delta$ in this question. If $v\in V^K$ and $k_i=d^{i}k{d}^{-i}\in K_i$, thenThus $\rho (d^i)$ maps the fixed-vector space $V^K$ into $V^{K_i}$. Its inverse $\rho (d^{-i})$ gives the reverse map, so

For an upper-unipotent matrix $u$, inverse conjugation changes each entry above the diagonal byEvery such factor tends to zero as $i\to \infty$. Because $U_0$ is compact, its finitely many coordinate functions are uniformly bounded; because it is open, it contains a neighbourhood of the identity in $U$. Therefore this contraction is uniform on $U_0$ and gives $d^{-i}{U}_0{d}^i\subseteq U_0$ for every sufficiently large $i$. The diagonal group commutes with $d$, soConjugating this inclusion gives $K\subseteq K_i$ for all sufficiently large $i$. This is an eventual inclusion; arbitrary $U_0$ need not give a monotone sequence at every earlier step.

Consequently $V^{K_i}\subseteq V^K$ eventually. Admissibility makes these spaces finite-dimensional, while the isomorphism above makes their dimensions equal. HenceThis proves contraction and admissible fixed spaces by both inclusion and dimension, not by an assumption that conjugate fixed spaces are literally equal from the outset.

The right action on compactly supported locally constant functions is a smooth representation. For a compact-open subgroup $K$, its fixed-vector space has basis the characteristic functions of cosets in $G/K$, since compact support on that discrete quotient is finite support. Consequently this representation is admissible exactly when $G$ is compact.