Let $F$ be a number field and let $o$ be its ring of integers. Let $o\_p$ resp. $F\_p$ be the completion at a prime ideal $p$ in $o$. Let $B$ be the group of upper triangular matrices in $GL\_2$. Let $\pi $ be a character of $B(F\_p)$.


How can we describe the irreducible which occur in the restriction of the induced representations $$Res\_{GL\_2(o\_p)} Ind\_{B(F\_p)}^{GL\_2(F\_p)} \pi = Ind\_{B(o\_p)}^{GL\_2(o\_p)} \pi = lim\_r Ind\_{B(o/p^r)}^{GL\_2(o/p^r)} \pi.$$


Is there a nice description of the irreducibles?



Spectral decomposition of parabolic induced for GL2(Zp)
This is done in :


Casselman, William The restriction of a representation of ${\rm GL}\_2 (k)$ 
to ${\rm GL}\_{2}({\mathfrak o})$. Math. Ann. 206 (1973), 311–318.


Also see :


Silberger, A.: $PGL\_2$ over the p-adics. Lecture Notes in Mathematics 166, Berlin-
Heidelberg-New York: Springer 1970


Very roughly speaking, the idea is the following. When you restrict an irreducible smooth representation of ${\rm GL}(2)$ to ${\rm GL}(2,{\mathfrak o})$ you get two types of constituents. The first ones (infinitely many) are uninteresting : they appear in the restriction of many other representations. These representations are described by Casselman. The second ones are more interesting : if an ireeducible representation contains such constituents then it must belong to a single component of the Bernstein decomposition of the category. They are called typical. Actually the situation is a bit more complicate. I've made it simpler. For more detail, you may read the paper by Henniart :


Breuil, Christophe; Mézard, Ariane Multiplicités modulaires et représentations de ${\rm GL}\_2({\bf Z}\_p)$ et de ${\rm Gal}(\overline{\bf Q}\_p/{\bf Q}\_p)$ en $l=p$. (French) [Modular multiplicities and representations of ${\rm GL}\_2({\bf Z}\_p)$ and ${\rm Gal}(\overline{\bf Q}\_p/{\bf Q}\_p)$ at $l=p$] With an appendix by Guy Henniart. Duke Math. J. 115 (2002), no. 2, 205–310. 


