TITLE: non-artificial examples of non-smooth and non-admissible representations of GL_2
QUESTION [10 upvotes]: Let $F$ be a finite degree extension over $\mathbf{Q}_p$ and consider the locally profinite group $G:=GL_2(\mathbf{Q}_p)$. 
P1: Give an interesting example (non-artificial one, i.e., one that arises in real life for a representation theorist) of a non-smooth representation $\rho$ of $G$ on a topological $\mathbf{C}$-vector space $V$. 
So here I would like the representation $\rho:G\rightarrow Aut(V)$ to be at least continuous in the following sense: for every $v\in V$ I want the orbit map $\pi^v:G\rightarrow V$, $g\mapsto \pi(g)(v)$ to be continuous.
P2: Give an interesting example of a smooth representation of $G$ on a topological $\mathbf{C}$-vector space which is not admissible. 
added: Is it possible to construct such representations by inducing over an appropriate closed subgroup of $G$?

REPLY [6 votes]: Another example, this time of a natural representation which is smooth, but not admissible. The group $G$ acts naturally on its Bruhat-Tits tree $X$. Let $C(X)$ be the space of complex-valued functions on (the set of vertices of) $X$ with finite support. Then $G$ has a natural
representation on $C(X)$, which is smooth (the stabilizer of a point is a maximal compact subgroup is a compact maximal, the stabilizer of a function is an intersection of finitely many such maximal compact subgroup hence is open) but not admissible (for example, the space of invariants in $C(X)$ by a compact maximal is the underlying space of the unramified (or spherical) Hecke algebra of $G$, which has infinite dimension over $\mathbb C$.