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?
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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.
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$\begingroup$ This is great stuff.For a better intuition, I wish I could see how exactly this goes in a more simple case, like spherical series. Could that be that in this case (if necessary assume unitarity) only reprsentations from finite quotients of PSL(2,Z) by modular subgroups? Is there a way to obtain Shalika's thesis? $\endgroup$ Commented Jan 26, 2019 at 12:38
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1$\begingroup$ No need to assume unitarity. Smooth reps of $SL_2(Z_p)$ are finite-dimensional and factor through $SL_2(Z / p^k Z)$ for some $k$. Branching from $G$ to $K$, these $k$'s are unbounded. For Shalika's thesis... I'd consult Nevins' papers for the reference. $\endgroup$– MartyCommented Jan 26, 2019 at 16:59