Converse theorems and the local Langlands correspondence in families

TL;DR

This paper establishes a descent criterion for families of smooth representations of GL_n over p-adic fields, linking gamma factors to the local Langlands correspondence and describing the Bernstein center via Galois theory.

Contribution

It introduces a new descent criterion for families of representations and proves conjectures connecting the Bernstein center with Galois theory and the local Langlands correspondence.

Findings

Proved a descent criterion for families of smooth representations.

Established a complete description of the integral Bernstein center.

Confirmed the conjectural 'local Langlands correspondence in families'.

Abstract

We prove a descent criterion for certain families of smooth representations of GL_n(F) (F a p-adic field) in terms of the gamma factors of pairs constructed in previous work of the second author. We then use this descent criterion, together with a theory of gamma factors for families of representations of the Weil group W_F (developed previously by both authors), to prove a series of conjectures, due to the first author, that give a complete description of the integral Bernstein center in terms of Galois theory and the local Langlands correspondence. An immediate consequence is the conjectural "local Langlands correspondence in families" of Emerton and Helm.