  We show there exist representations of each maximal compact subgroup $K$ of
the $p$-adic group $G=\mathrm{SL}(2,F)$, attached to each nilpotent coadjoint
orbit, such that every irreducible representation of $G$, upon restriction to a
suitable subgroup of $K$, is a sum of these five representations in the
Grothendieck group. This is a representation-theoretic analogue of the analytic
local character expansion due to Harish-Chandra and Howe. Moreover, we show for
general connected reductive groups that the wave front set of many irreducible
positive-depth representations of $G$ are completely determined by the
nilpotent support of their unrefined minimal $K$-types.
