Let $(\pi, V)$ a (smooth) admissible representation of a locally profinite topological group $G$ and fix $K \le G$ an open compact subgroup. These notes claim that if $V^K$ is a nonzero irreducible module for the Hecke algebra $\mathcal{H}_K := C^\infty_{c}(K\backslash G/K)$, then $V$ is irreducible.
Surely this is not true as written? Since we have an equivalence of categories between smooth $G$-reps and nondegenerate $\mathcal{H} = \varinjlim_K \mathcal{H}_K$-modules, it is certainly true that if $V^K$ is $\mathcal{H}_K$-irreducible for all $K$, then $V$ is irreducible, but the necessity of considering all $K$ seems certain to me.