The local character expansion as branching rules: nilpotent cones and the case of $\mathrm{S}\mathrm{L}(2)$

We show there exist representations of each maximal compact subgroup $K$ of the $p$-adic group $G=\mathrm{S}\mathrm{L}(2,F)$, 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.