Two Categorifications of the Local Langlands Correspondence for the Torus

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.