Higher ramification and the local Langlands correspondence

TL;DR

This paper establishes a deep connection between the ramification properties of representations in the local Langlands correspondence over non-Archimedean fields and simple characters, extending classical ramification theorems.

Contribution

It introduces a new framework linking ramification subgroups to simple characters in the local Langlands correspondence, generalizing higher ramification theorems.

Findings

Ramification of Weil group representations is determined by simple characters.

A Herbrand-like function $ extPsi_ heta$ governs numerical relations.

Method provided to compute $ extPsi_ heta$ from endo-classes.

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.