Two Categorifications of the Local Langlands Correspondence for the Torus

Author: Fu, Ruide

Year: 2025

Degree: Dissertation (Ph.D.)

Advisor: Zhu, Xinwen

Committee Members: Flach, Matthias; Mantovan, Elena; Fu, Yu; Zhu, Xinwen

Option: Mathematics

DOI: 10.7907/3fcs-9c06

Abstract

The stack of local Langlands parameters is a Picard stack when the relevant reductive group is a torus. 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. This second categorification relates to the previous correspondence in the sense that taking the categorical trace construction allows one to reproduce the previous result.

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