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 $GL_n(F)$ 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.