Tame multiplicity and conductor for local Galois representations

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 $W_{F}$ of $F$ and $sw (σ)$ the Swan exponent of $σ$ . Assume $sw (σ) \geq 1$ . Let $ℐ_{F}$ be the inertia subgroup of $W_{F}$ and $P_{F}$ the wild inertia subgroup. There is an essentially unique, finite, cyclic group $Σ$ , of order prime to $p$ , such that $σ (ℐ_{F}) = Σ σ (P_{F})$ . 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

Vol. 2, No. 2, 2020

Colin J. Bushnell and Guy Henniart

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification 2010

Milestones

Authors