= Contraction and admissible fixed spaces
For the upper-triangular subgroup of $\operatorname{GL}_n(F)$, conjugation by $\operatorname{diag}(\varpi^{-(n-1)},\ldots,1)$ expands the upper-unipotent coordinates. For a decomposed <compact-open subgroup> $K=T_0U_0$, 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.
Back to article page
