Local Langlands correspondence and ramification for Carayol representations

TL;DR

This paper investigates the ramification properties of certain wildly ramified Weil group representations associated with Carayol types, establishing symmetry conditions of Herbrand functions and explicit calculations within the local Langlands correspondence framework.

Contribution

It characterizes Herbrand functions for Carayol-type representations, proving symmetry properties and providing explicit formulas, thus deepening understanding of ramification in the local Langlands correspondence.

Findings

Herbrand functions satisfy a specific symmetry condition.

Explicit formulas for Herbrand functions in the Carayol case.

Complete description of the restriction of representations to ramification subgroups.

Abstract

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 $\Psi_\varTheta$.