Consider a local non archimedean field $k$ and its ring of integer $o$. To what extent, do we know the complex irreducible representations of $GL(2,o)$? Is there a specific list giving them all in terms of induction from certain "simple" subgroups?


I have had read, that they have been classified according to their characters, but I am not quite happy with the various presentations so far.


What I'd like to see is pretty simple to explain:


1. Classify all irreducibles of the group (Borel subrgoup) of upper triangular matrices $B$ (easy).
2. Given an irreducible representation $\pi$ of $B$, decompose $Ind\_{B}^{GL(2,o)} \pi$ in its irreducibles (hard).


Is there such a treatment available? If not, what is the reason why not to proceed along these lines?



To what extent do we know the representations of GL(2,Zp)
All irreducible representations of ${\rm GL}(2, {\mathcal O})$ have been constructed by Alexender Statinski :


Stasinski, Alexander (2009) The smooth representations of ${\rm GL}(2, {\mathcal O})$.


Reference in Math. Arxiv : <http://arxiv.org/abs/0807.4684>


But he does not follow the procedure that you propose at all. He uses much more generalizable tools : Clifford theory and an adapted version of Kirillov's orbit method. These ideas are now classical in type theory for $p$-adic reductive groups (see e.g. Howe and Kutzko's works). 


Some people are now working on the more general question of constructing the representations of ${\rm GL}(n,{\mathcal O})$ (A. Stasinski, A.--M. Aubert, F. Courtès, and others). This problem is known to be "wild"


