  Building on prior work, we analyze the decomposition of the restriction of an
irreducible representation of SL_2(k), for k a p-adic field of odd residual
characteristic, to a maximal compact subgroup K. The pattern of the
decomposition varies between principal series and different supercuspidal
representations, whereas the K-representations which occur in the "tail end" of
these decompositions are precisely those occurring in the decomposition of
depth-zero supercuspidal representations. Various applications are considered.
