#### 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 $\sigma$ be an irreducible smooth representation of the absolute Weil group ${\mathsc{W}}_{F}$ of $F$ and $sw\left(\sigma \right)$ the Swan exponent of $\sigma$. Assume $sw\left(\sigma \right)\ge 1$. Let ${\mathsc{ℐ}}_{F}$ be the inertia subgroup of ${\mathsc{W}}_{F}$ and ${\mathsc{P}}_{F}$ the wild inertia subgroup. There is an essentially unique, finite, cyclic group $\Sigma$, of order prime to $p$, such that $\sigma \left({\mathsc{ℐ}}_{F}\right)=\Sigma \sigma \left({\mathsc{P}}_{F}\right)$. In response to a query of Mark Reeder, we show that the multiplicity in $\sigma$ of any character of $\Sigma$ is bounded by $sw\left(\sigma \right)$.

##### Keywords
Local field, tame multiplicity, conductor bound, primitive representation
##### Mathematical Subject Classification 2010
Primary: 11S15, 11S37, 22E50