Local Langlands correspondence and ramification for Carayol representations

Let $F$ be a non-Archimedean locally compact field of residual characteristic $p$ with Weil group $\Cal W_F$. Let $\sigma$ be an irreducible smooth complex representation of $\Cal W_F$, realized as the Langlands parameter of an irreducible cuspidal representation $\pi$ of a general linear group over $F$. In an earlier paper, we showed that the ramification structure of $\sigma$ is determined by the fine structure of the endo-class $\varTheta$ of the simple character contained in $\pi$, in the sense of Bushnell-Kutzko. The connection is made via the {\it Herbrand function} $\Psi_\varTheta$ of $\varTheta$. In this paper, we concentrate on the fundamental Carayol case in which $\sigma$ is totally wildly ramified with Swan exponent not divisible by $p$. We show that, for such $\sigma$, the associated Herbrand function satisfies a certain symmetry condition or functional equation, a property that essentially characterizes this class of representations. We calculate $\Psi_\varTheta$ explicitly, in terms of a classical Herbrand function coming from the Bushnell-Kutzko classification of simple characters. We describe exactly the class of functions arising as Herbrand functions $\Psi_\varXi$, as $\varXi$ varies over totally wild endo-classes of Carayol type. In a separate argument, we get a complete description of $\sigma$ restricted to any ramification subgroup. This provides a different, more Galois-centred, view on $\Psi_\varTheta$.