Two Categorifications of the Local Langlands Correspondence for Tori
Ruide Fu

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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Algebra and Geometry · Geometry and complex manifolds · Algebraic Geometry and Number Theory
