Explicit asymptotic expansions for tame supercuspidal characters

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.

74 pp

References in corpus (1)

• Reductive groups, epsilon factors and Weil indices

Cited by in corpus (5)

• Supercuspidal L-packets
• The local character expansion as branching rules: nilpotent cones and the case of $\mathrm{SL}(2)$
• The Bernstein projector determined by a weak associate class of good cosets
• Progrès récents sur les représentations supercuspidales
• A Gelfand-Graev Formula and Stable Transfer Factors in the Unramidied Case for ${\text{SL}}_{\ell }(F)$ and ${\text{GL}}_{\ell }(F)$, $\ell$ an odd Prime