Let $(\rho,V)$ be an irreducible admissible infinite-dimensional representation of $GL_2(\mathbb{Q}_p)$, with central character $\omega$.
Fact: there exists an $i$ such that there is a non-zero $v \in V$ such that
$$ \rho\begin{pmatrix} a & b \\c & d \end{pmatrix} v = \omega(a)v \text{ for all }\begin{pmatrix} a & b \\c & d \end{pmatrix} \in \Gamma_0(p^i). $$
I know how to check this using some non-trivial results in the representation theory of $GL_2(\mathbb{Q}_p)$, e.g. the classification of admissible irreducibles and the theory of the Kirillov model.
But is there a purely elementary/formal way to see that such a $v$ must exist? (Casselman claims that it is immediate from the definition of admissibility: see the first paragraph of the proof of Theorem 1 in https://link.springer.com/article/10.1007%2FBF01428197.)
Here's an attempt to get started: we can certainly find a finite-dimensional space on which the action of $GL_2(\mathbb{Z}_p)$ factors through $GL_2(\mathbb{Z}/p^r)$ for some $r$. But is it necessarily the case that the nilpotent radical must have a fixed vector in such a representation? Even if so, is it necessarily the case that the resulting representation of the torus acts through the central character on some vector?