The local character expansion as branching rules : nilpotent cones and the case of SL(2)

We show there exist representations of each maximal compact subgroup K of the padic group G = 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.where G x,r+ is the Moy-Prasad filtration subgroup of G x of depth r+, and the sum is over all nilpotent orbits in g * .