Question

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?

Answer 1

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.