Wave-front sets for $p$-adic Lie algebras

We study the wave-front set of an element in a $p$-adic reductive Lie algebra (for $p\gg \mathrm{rank}$), namely the set of maximal nilpotent orbits appearing in its Shalika germ expansion. By adapting an algorithm of Waldspurger that computes orbital integrals, we obtain an inductive algorithm to compute an invariant that determines the wave-front set. This gives an algorithm to compute the wave-front sets for regular supercuspidal representations and reveals examples whose wave-front sets are not contained in a single geometric orbit, for arbitrarily large $p$ within a fixed rank.