Explicit asymptotic expansions in p-adic harmonic analysis II

TL;DR

This paper develops a method to extend local character expansions for supercuspidal representations of p-adic groups to arbitrary positive-depth cases, involving Fourier transforms of orbital integrals.

Contribution

It provides a systematic way to derive Kim–Murnaghan-type asymptotic expansions from depth-zero expansions, generalizing previous results in p-adic harmonic analysis.

Findings

Extended asymptotic expansions to positive-depth supercuspidal characters.

Derived new results for Fourier transforms of orbital integrals.

Connected local character expansions with orbital integral analysis.

Abstract

We unwind the induction implicit in the "one-step" asymptotic expansions of arXiv:1701.02417 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.