Contraction and admissible fixed spaces

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.