Geometric approach to the explicit local Langlands correspondence

TL;DR

This paper introduces a geometric method to explicitly describe Langlands parameters for supercuspidal representations of GL(n) over non-archimedean fields, using cohomological comparisons in Lubin-Tate spaces.

Contribution

It presents a novel geometric approach to explicitly determine Langlands parameters, focusing on the cohomology of Lubin-Tate spaces and their reductions.

Findings

Explicit descriptions for depth 0 supercuspidal representations

Explicit descriptions for simple supercuspidal representations

Establishment of a geometric framework for Langlands correspondence

Abstract

We propose a geometric strategy of giving explicit description of the Langlands parameter of an irreducible supercuspidal representation of GL(n) over a non-archimedean local field. The key is to compare the cohomology of an affinoid in the Lubin-Tate space at infinite level and that of the reduction of its formal model. As examples, we treat the cases of depth 0 supercuspidal representations and simple supercuspidal representations.