Local Langlands correspondence and ramification for Carayol representations

Abstract

Let F be a non-Archimedean locally compact field of residual characteristic p with Weil group W F . Let σ be an irreducible smooth complex representation of W F , realized as the Langlands parameter of an irreducible cuspidal representation π of a general linear group over F . In an earlier paper, we showed that the ramification structure of σ is determined by the fine structure of the endo-class Θ of the simple character contained in π, in the sense of Bushnell-Kutzko. The connection is made via the Herbrand function Ψ Θ of Θ. In this paper, we concentrate on the fundamental Carayol case in which σ is totally wildly ramified with Swan exponent not divisible by p. We show that, for such σ, the associated Herbrand function satisfies a certain functional equation, and that this property essentially characterizes this class of representations. We calculate Ψ Θ 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 Ψ Ξ , as Ξ varies over the set of totally wild endoclasses of Carayol type. In a separate argument, we derive a complete description of the restriction of σ to any ramification subgroup and hence a detailed interpretation of the Herbrand function. This gives concrete information concerning the Langlands correspondence.2000 Mathematics Subject Classification. 22E50, 11S37, 11S15.

Comparison of the depths on both sides of the local Langlands correspondence for Weil-restricted groups (with an appendix by Jessica Fintzen)

“…Proof. If r > 0 then this is the second statement in [BH,Proposition,§1.4]. To deal with the case r = 0, we proceed as follows.…”

“…5] we have B ∩ A = B ∩ A where the overline denotes closure in X. If r > 0 then, by the first statement in [BH,Proposition,§1.4], we have…”