Comparison of the analytic and algebraic categorical local Langlands correspondences

We prove a folklore conjecture identifying the two known candidates for the automorphic side of the categorical local Langlands correspondence, allowing the passage of ideas and results from one side to the other. Precisely, for G a connected reductive group, we construct an equivalence between the derived category of etale sheaves on the algebraic stack Isoc_{G} of G-isocrystals constructed by Zhu and the derived category of étale sheaves on the analytic moduli stack of G-bundles Bun_{G} on the Fargues–Fontaine curve constructed by Fargues–Scholze. To a (very crude) first approximation, this is accomplished by considering an explicit geometric object, denoted Bun_{G}^{mer}, which defines a correspondence between the analytification of the algebraic object Isoc_{G} and the analytic object Bun_{G}, and then pushing and pulling along this correspondence. The resulting functor can be roughly thought of as “nearby cycles” between the generic and special fiber of the formal scheme (or rather its generalization to kimberlites in the sense of Gleason) Bun_{G}^{mer}. In usual formal/adic geometry, we know that such nearby cycles functors allow us to compare cohomology on the rigid generic fiber and special fiber of the formal scheme via showing that the formal scheme is henselian along the analytic locus coming from the rigid generic fiber. We prove our functor is an equivalence by verifying such henselianity properties hold inside the space Bun_{G}^{mer}. In particular, under our functor, this henselianity property allows us to compare a natural excision filtration (or semi-orthogonal decomposition) on the category attached to Isoc_{G} with an “exotic” one on the category Bun_{G} coming from the existence of certain exceptional adjoints to the usual six operations. This reduces us to showing our functor is an equivalence on the induced functor on the graded pieces of this filtration, where it is easily checked to be true. This is joint work with Ian Gleason, Joao Lourenco, Alexander Ivanov, and Konrad Zou.