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 $G$?