An Approach to the Characterization of the Local Langlands Correspondence

In this paper, we give a method for characterizing the local Langlands conjectures in the vein of Scholze’s alternate proof of the local Langlands conjecture for ${\mathrm{G}\mathrm{L}}_n$. More specifically, we show that if a local Langlands correspondence satisfies a Scholze–Shin equation, as in the paper of Scholze and Shin, in addition to the usual desiderata expected of such a correspondence then these properties uniquely characterize the correspondence.