Globalization of supercuspidal representations over function fields and applications

TL;DR

This paper develops a method to globalize supercuspidal representations over function fields, maintaining ramification control and distinction properties, and applies it to establish stability of gamma-factors and the local Langlands correspondence for classical groups.

Contribution

It introduces a new globalization technique for supercuspidal representations over function fields that preserves key properties and enables applications to Langlands program results.

Findings

Globalization of supercuspidal representations with ramification control

Stability of Langlands-Shahidi gamma-factors

Validation of local Langlands correspondence for classical groups

Abstract

Let H be a connected reductive group defined over a non-archimedean local field F of characteristic p>0. Using Poincar\'e series, we globalize supercuspidal representations of H(F) in such a way that we have control over ramification at all other places, and such that the notion of distinction with respect to a unipotent subgroup (indeed more general subgroups) is preserved. In combination with the work of Vincent Lafforgue on the global Langlands correspondence, we present some applications, such as the stability of Langlands-Shahidi \gamma-factors and the local Langlands correspondence for classical groups.