Higher ramification and the local Langlands correspondence

Abstract

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 $\pi$ is an irreducible cuspidal representation of a general linear group ${\rm GL}_n(F)$ and $\sigma$ the corresponding irreducible representation of the Weil group $\mathcal{W}_F$ of $F$, the restriction of $\sigma$ to a ramification subgroup of $\mathcal{W}_F$ is determined by a truncation of the simple character $\theta_\pi$ contained in $\pi$, and conversely. Numerical aspects of the relation are governed by an Herbrand-like function $\Psi_\varTheta$ depending on the endo-class $\varTheta$ of $\theta_\pi$. We give a method for calculating $\Psi_\varTheta$ directly from $\varTheta$. Consequently, the ramification-theoretic structure of $\sigma$ can be predicted from the simple character $\theta_\pi$ alone.