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Explicit asymptotic expansions for tame supercuspidal characters

Part of: Lie groups

Published online by Cambridge University Press:  12 October 2018

Loren Spice
Loren Spice*
Affiliation:
2840 W. Bowie St, Texas Christian University, Fort Worth, TX 76129, USA email l.spice@tcu.edu
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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.

Keywords

p-adic groupsupercuspidal representationcharacter computation

MSC classification

Primary: 22E50: Representations of Lie and linear algebraic groups over local fields 22E35: Analysis on $p$-adic Lie groups
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 11 , November 2018 , pp. 2305 - 2378
Copyright
© The Author 2018 

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Footnotes

The author was partially supported by Simons Foundation Collaboration Grant 246066.

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