Local Langlands correspondences

TL;DR

This paper reviews the properties of local Langlands correspondences, discusses known results, and outlines a strategy to prove finiteness and non-emptiness of L-packets, especially in positive characteristic cases.

Contribution

It provides a comprehensive review of canonical local Langlands correspondences and proposes a strategy to establish key properties for specific parametrizations in positive characteristic.

Findings

Reviewed properties of canonical local Langlands correspondences

Outlined a strategy to prove finiteness and non-emptiness of L-packets

Discussed results for specific groups and parametrizations

Abstract

The first part of this article is a review of the properties expected of any local Langlands correspondence that aims to be considered "canonical," and of known results that establish some or all of these properties for specific groups. In the absence of compatibility with a global correspondence it is not known in general that this list of desirable properties suffices to characterize the correspondence. The remainder of the article outlines elements of a strategy to prove that the $L$-packets attached to irreducible local parameters are finite and non-empty, in particular, for the specific parametrization constructed by Genestier and Lafforgue when $F$ is of positive characteristic.