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

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