Explicit asymptotic expansions in $p$-adic harmonic analysis II

Abstract We unwind the induction implicit in the “one-step” asymptotic expansions of (the author in Compositio Mathematica 154(11):2305–2378, 2018) to describe how to turn asymptotic expansions, such as (but not limited to) the Harish-Chandra–Howe local character expansion, for depth-0 supercuspidal characters into Kim–Murnaghan-type asymptotic expansions for arbitrary positive-depth, tame, supercuspidal characters. Doing so requires analogous results for Fourier transforms of orbital integrals on the Lie algebra of a reductive group, which are of independent interest.