  Using Bruhat-Tits theory, we analyse 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.
