I am reformulating a question I asked earlier with no answer: Consider $SL(2, Q\_p)$ and $K$ a maximal compact subgroup. Let $\pi$ be an irreducible spherical representation of $SL(2, Q\_p)$ (in the principal or complementary series). What is it known about $\pi$ restricted to $K$? Is there any difference if we start from the principal or complementary series? I see a lot of theory around by Dijk, Casselman, Bushnell and Kutzko, but what is the concrete answer in this very particular case?



Restriction of smooth representaions of SL(2,Q_p) to the maximal compact
This question was treated by Monica Nevins in the following pair of papers. 


*Nevins, Monica*, [**Branching rules for principal series representations of SL(2) over a $p$-adic field**](http://dx.doi.org/10.4153/CJM-2005-026-1), Can. J. Math. 57, No. 3, 648-672 (2005). [ZBL1071.22008](https://zbmath.org/?q=an:1071.22008).


*Nevins, Monica*, [**Branching rules for supercuspidal representations of $\mathrm{SL}\_2(k)$, for $k$ a $p$-adic field**](http://dx.doi.org/10.1016/j.jalgebra.2012.12.003), J. Algebra 377, 204-231 (2013). [ZBL1282.22011](https://zbmath.org/?q=an:1282.22011).


There were earlier results for $PGL\_2$ and $GL\_2$, due to Silberger and Casselman, but I don't know those well.


If one takes $K = SL\_2({\mathbb Z}\_p)$ for the maximal compact, then smooth irreps of $G = SL\_2({\mathbb Q}\_p)$ will decompose as direct sums of finite-dimensional irreps of $K$. But what are these irreps? Fortunately, Shalika (in his thesis, I think) described and organized the representations of $SL\_2({\mathbb Z}\_p)$. So we have a nicely arranged list of irreps of $SL\_2({\mathbb Z}\_p)$ to work with. Other maximal compact subgroups are $GL\_2({\mathbb Q}\_p)$-conjugate to $K$, so there's not much difference there.


Nevins completely describes the branching. It's multiplicity-free (always, I think), and it's best to just look it up in her papers.


Incidentally, she works over any $p$-adic field with odd residue characteristic. I'm not sure if anyone's worked out the $p=2$ case.


