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 strong property generalizing the higher ramification theorem of local class field theory. If $\pi$ is an irreducible cuspidal representation of a general linear group $G{L}_n(F)$ and $\sigma$ the corresponding irreducible representation of the Weil group $W_F$ of $F$, the restriction of $\sigma$ to a ramification subgroup of $W_F$ is determined by a truncation of the simple character ${\theta }_{\pi }$ contained in $\pi$, and conversely. Numerical aspects of the relation are governed by a Herbrand-like function ${\Psi }_{\Theta }$ depending on the endo-class $\Theta$ of ${\theta }_{\pi }$. We give a method for determining ${\Psi }_{\Theta }$. Consequently, the ramification-theoretic structure of $\sigma$ can be predicted from the simple character ${\theta }_{\pi }$ alone.

Revised version