On branching rules of depth-zero representations

Using Bruhat-Tits theory, we analyze the restriction of depth-zero representations of a semisimple simply connected p-adic group G to a maximal compact subgroup K. We prove the coincidence of branching rules within classes of Deligne-Lusztig supercuspidal representations. Furthermore, we show that under obvious compatibility conditions, the restriction to K of a Deligne-Lusztig supercuspidal representation of G intertwines with the restriction of a depth-zero principal series representation in infinitely many distinct components of arbitrarily large depth. Several qualitative and quantitative results are obtained, and their use is illustrated in an example.