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Local Langlands correspondence and ramification for Carayol representations

  • Colin J. Bushnell (a1) and Guy Henniart (a2)

Abstract

Let $F$ be a non-Archimedean locally compact field of residual characteristic $p$ with Weil group ${\mathcal{W}}_{F}$ . Let $\unicode[STIX]{x1D70E}$ be an irreducible smooth complex representation of ${\mathcal{W}}_{F}$ , realized as the Langlands parameter of an irreducible cuspidal representation $\unicode[STIX]{x1D70B}$ of a general linear group over $F$ . In an earlier paper we showed that the ramification structure of $\unicode[STIX]{x1D70E}$ is determined by the fine structure of the endo-class $\unicode[STIX]{x1D6E9}$ of the simple character contained in $\unicode[STIX]{x1D70B}$ , in the sense of Bushnell and Kutzko. The connection is made via the Herbrand function $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}$ of $\unicode[STIX]{x1D6E9}$ . In this paper we concentrate on the fundamental Carayol case in which $\unicode[STIX]{x1D70E}$ is totally wildly ramified with Swan exponent not divisible by $p$ . We show that, for such $\unicode[STIX]{x1D70E}$ , the associated Herbrand function satisfies a certain functional equation, and that this property essentially characterizes this class of representations. We calculate $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}$ explicitly, in terms of a classical Herbrand function arising naturally from the classification of simple characters. We describe exactly the class of functions arising as Herbrand functions $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6EF}}$ , as $\unicode[STIX]{x1D6EF}$ varies over the set of totally wild endo-classes of Carayol type. In a separate argument, we derive a complete description of the restriction of $\unicode[STIX]{x1D70E}$ to any ramification subgroup and hence a detailed interpretation of the Herbrand function. This gives concrete information concerning the Langlands correspondence.

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Local Langlands correspondence and ramification for Carayol representations

  • Colin J. Bushnell (a1) and Guy Henniart (a2)

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