Vol. 2, No. 2, 2020

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Tame multiplicity and conductor for local Galois representations

Colin J. Bushnell and Guy Henniart

Vol. 2 (2020), No. 2, 337–357
Abstract

Let F be a non-Archimedean locally compact field of residual characteristic p. Let σ be an irreducible smooth representation of the absolute Weil group WF of F and sw(σ) the Swan exponent of σ. Assume sw(σ) 1. Let F be the inertia subgroup of WF and PF the wild inertia subgroup. There is an essentially unique, finite, cyclic group Σ, of order prime to p, such that σ(F) = Σσ(PF). In response to a query of Mark Reeder, we show that the multiplicity in σ of any character of Σ is bounded by sw(σ).

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Keywords
Local field, tame multiplicity, conductor bound, primitive representation
Mathematical Subject Classification 2010
Primary: 11S15, 11S37, 22E50
Milestones
Received: 16 September 2018
Revised: 8 May 2019
Accepted: 27 May 2019
Published: 2 August 2019
Authors
Colin J. Bushnell
Department of Mathematics
King’s College London
Strand
London
United Kingdom
Guy Henniart
Laboratoire de Mathématiques d’Orsay
Univ. Paris-Sud
CNRS, Université Paris-Saclay
Orsay
France