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

Patterns in Branching Rules for Irreducible Representations of SL_2(k), for k a p-adic field