Higher ramification and the local Langlands correspondence

Let F be a non-Archimedean locally compact field. We show that the local Langlands correspondence over F has a property generalizing the higher ramification theorem of local class field theory. If π is an irreducible cuspidal representation of a general linear group GLn(F) and σ the corresponding irreducible representation of the Weil group WF of F, the restriction of σ to a ramification subgroup of W_F is determined by a truncation of the simple character θ_π contained in π, and conversely. Numerical aspects of the relation are governed by an Herbrand-like function ψ_Θ depending on the endo-class Θ of θ_π. We give a method for calculating ψ_Θ directly from Θ. Consequently, the ramification-theoretic structure of σ can be predicted from the simple character θ_π alone.