Local Langlands correspondence, local factors, and zeta integrals in analytic families

TL;DR

This paper develops a framework for understanding how the local Langlands correspondence varies in characteristic-zero families, enabling interpolation of local factors and zeta integrals, with applications to eigenvarieties.

Contribution

It establishes existence, uniqueness, and recognition theorems for the local Langlands correspondence in families, extending prior work to characteristic-zero settings.

Findings

Existence and uniqueness of the correspondence in families.

Recognition criteria for identifying Langlands correspondents.

Interpolation of local zeta integrals and L- and epsilon-factors.

Abstract

We study the variation of the local Langlands correspondence for ${\rm GL}_{n}$ in characteristic-zero families. We establish an existence and uniqueness theorem for a correspondence in families, as well as a recognition theorem for when a given pair of Galois- and reductive-group- representations can be identified as local Langlands correspondents. The results, which can be used to study local-global compatibility questions along eigenvarieties, are largely analogous to those of Emerton, Helm, and Moss on the local Langlands correspondence over local rings of mixed characteristic. We apply the theory to the interpolation of local zeta integrals and of $L$- and $\varepsilon$-factors.