Restriction of smooth representaions of SL(2,Q_p) to the maximal compact 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?
 A: 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, Can. J. Math. 57, No. 3, 648-672 (2005). ZBL1071.22008.
Nevins, Monica, Branching rules for supercuspidal representations of $\mathrm{SL}_2(k)$, for $k$ a $p$-adic field, J. Algebra 377, 204-231 (2013). ZBL1282.22011.
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.
