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Local Langlands correspondence and ramification for Carayol representations

Part of: Algebraic number theory: local and $p$-adic fields Lie groups

Published online by Cambridge University Press:  06 September 2019

Colin J. Bushnell  and
Guy Henniart
Colin J. Bushnell
Affiliation:
King’s College London, Department of Mathematics, Strand, London WC2R 2LS, UK email [email protected]
Guy Henniart
Affiliation:
Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France email [email protected]
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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.

Keywords

local Langlands correspondencecuspidal representationsimple characterendo-classramification groupHerbrand functionCarayol representation

MSC classification

Primary: 22E50: Representations of Lie and linear algebraic groups over local fields 11S37: Langlands-Weil conjectures, nonabelian class field theory 11S15: Ramification and extension theory
Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 10 , October 2019 , pp. 1959 - 2038
Copyright
© The Authors 2019 

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