Just like the title. I want a simple proof of the statment in the title.
$\mathbb{Q}\_p$ is the p-adic field.
I wonder which module (or vector space) will be chosen as the space for the representation. Is this statement true for arbitarily module/vector space?
Thanks!



Any finite dimensional admissible(smooth) irreducible representation of GL(2,Q_p) is 1-dim
What does "admissible" mean for you? Does it imply smoothness (stabilisers are open)? If not then I think the statement might be false (choose some hopelessly discontinuous injection from $\mathbf{Q}\_p$ into $\mathbf{C}$ and then consider the induced "natural" representation of $GL(2,\mathbf{Q}\_p)$ on $\mathbf{C}^2$; that's definitely irreducible, and the space of vectors fixed by some compact open will definitely be finite-dimensional).


But assuming smoothness too, it is true that any finite-dimensional smooth irreducible representation of $GL(2,\mathbf{Q}\_p)$ is 1-dimensional. The proof is: if $V$ is such
a thing, then choose a basis for $V$ and for each basis vector choose a compact
open subgroup stabilising it. The intersection of these guys is still compact and
open, and fixes everything. So the kernel of the representation contains a compact
open subgroup. But this is a bit worrying because the kernel is normal. Now use
the fact that the normal subgroup generated by matrices $(1,e;0,1)$ and $(1,0;e,1)$
for $e$ small is still the whole of $SL(2,\mathbf{Q}\_p)$ to deduce that $SL(2,\mathbf{Q}\_p)$
is in the kernel, and now the action has to factor through $GL(2,\mathbf{Q}\_p)/SL(2,\mathbf{Q}\_p)=\mathbf{Q}\_p^\times$. But now Schur's Lemma, which works for smooth irreducible
representations, says $V$ is 1-dimensional. 


