Compatibility of the Fargues-Scholze and Gan-Takeda Local Langlands Correspondences

Abstract: In upcoming work, Fargues and Scholze construct a candidate for a general local Langlands correspondence, associating to a smooth irreducible representation of a connected reductive group $G/\mathbf{Q}_{p}$ a continuous semisimple Weil parameter, using the action of excursion operators on the moduli space of $G$-bundles on the Fargues-Fontaine curve. It is a natural question to ask whether this correspondence is compatible with known instances of the local Langlands correspondence after semi-simplification. For $G = \mathrm{GL}_{n}$, this compatibility is deduced from the fact that correspondence of Harris-Taylor is realized in the cohomology of the Lubin-Tate tower at infinite level, via its interpretation as a moduli space of mixed characteristic shtukas. For $G = \mathrm{GSp}_{4}$ or its inner form $\mathrm{GU}_{2}(D)$, there is a local Langlands correspondence constructed by Gan-Takeda and Gan-Tantono, respectively. We will discuss upcoming work related to proving compatibility in this case. Similar to the case of $\mathrm{GL}_{n}$, this involves realizing this local Langlands correspondence in the cohomology of the local Shimura varieties at infinite level associated with these groups. We do this by applying basic uniformization of these local Shimura varieties due to Shen, as well as results on Galois representations in the cohomology of the relevant global Shimura varieties due to Sorensen and Kret-Shin. After proving this compatibility, we employ various new ideas from the geometry of the Fargues Scholze correspondence to obtain a complete description of the $\rho$-isotypic part of the cohomology of this local Shimura variety at infinite level, where $\rho$ is a representation of $G$ with supercuspidal Gan-Takeda or Gan-Tantono parameter, thereby verifying the strongest form of the Kottwitz conjecture for these specific representations.

commutative algebraalgebraic geometrynumber theory

Audience: researchers in the topic

Recent Advances in Modern p-Adic Geometry (RAMpAGe)

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