Tame multiplicity and conductor for local Galois representations

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