  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 $\Cal W_F$ of $F$ and $\sw(\sigma)$ the Swan exponent of $\sigma$. Assume
$\sw(\sigma) \ge1$. Let $\Cal I_F$ be the inertia subgroup of $\Cal W_F$ and
$\Cal P_F$ the wild inertia subgroup. There is an essentially unique, finite,
cyclic group $\varSigma$, of order prime to $p$, so that $\sigma(\Cal I_F) =
\sigma(\Cal P_F)\varSigma$. In response to a query of Mark Reeder, we show that
the multiplicity in $\sigma$ of any character of $\varSigma$ is bounded by
$\sw(\sigma)$.
