Two Categorifications of the Local Langlands Correspondence for Tori

TL;DR

This paper explicitly determines the Picard dual of the stack of local Langlands parameters for a torus and demonstrates that the Fourier-Mukai transform provides a categorical local Langlands correspondence, confirming a conjecture of X. Zhu.

Contribution

It introduces a geometric categorification of the local Langlands correspondence for tori using Fourier-Mukai transforms and explicitly describes the Picard dual of the parameter stack.

Findings

Explicit determination of the Picard dual of the parameter stack

Establishment of the integral categorical local Langlands correspondence for tori

Confirmation of X. Zhu's conjecture

Abstract

The stack of local Langlands parameters for a torus is a Picard stack. In this article, we explicitly determine its Picard dual and show that the Fourier-Mukai transform gives rise to the integral categorical local Langlands correspondence for the torus. This is the categorification of the local Langlands correspondence and answers a conjecture of X. Zhu. Moreover, we establish a geometric version of this correspondence, whose categorical trace reproduces the previous result.