# Explicit asymptotic expansions for tame supercuspidal characters

## TL;DR

This paper develops explicit asymptotic expansions for tame supercuspidal characters around semisimple elements, combining local character and Kim--Murnaghan expansions, with an inductive method for coefficient computation.

## Contribution

It introduces a new explicit, inductive approach to compute asymptotic expansion coefficients for tame supercuspidal representations near semisimple elements.

## Key findings

- Validates Kim--Murnaghan-type expansions near arbitrary semisimple elements.
- Provides an explicit recipe for coefficient calculation in asymptotic expansions.
- Highlights potential applications in stability and endoscopic transfer proofs.

## Abstract

We combine the ideas of a Harish-Chandra--Howe local character expansion, which can be centred at an arbitrary semisimple element, and a Kim--Murnaghan asymptotic expansion, which so far has been considered only around the identity. We show that, for most smooth, irreducible representations (those containing a good, minimal K-type), Kim--Murnaghan-type asymptotic expansions are valid on explicitly defined neighbourhoods of nearly arbitrary semisimple elements. We then give an explicit, inductive recipe for computing the coefficients in an asymptotic expansion for a tame supercuspidal representation. The only additional information needed in the inductive step is a fourth root of unity, which we expect to be useful in proving stability and endoscopic-transfer identities.