Compatibility of the Fargues-Scholze and Gan-Takeda Local Langlands
Linus Hamann
TL;DR
The paper proves that Fargues–Scholze’s general local Langlands candidate is compatible with the Gan–Takeda (GSp$_4$) and Gan–Tantono (GU$_2(D)$) correspondences after semisimplification. It treats non-supercuspidal parameters uniformly by leveraging FS’s compatibility with parabolic induction, and handles supercuspidal/mixed cases via a combination of Shen’s basic uniformization, the spectral action in FS, and global trace-formula arguments to realize strong transfers and multiplicity-one results. A key technical advance is identifying the $W_L$-action on cohomology of local Shimura varieties with standard $L$-parameters and describing the middle-degree cohomology for supercuspidal packets, yielding a strong form of the Kottwitz conjecture in this setting. The work also derives explicit descriptions of the Galois representations in the cohomology of global Shimura varieties and demonstrates compatibility for the endoscopic and stable cases, with applications to Sp$_4$ and SU$_2(D)$ and potential extensions to non-minuscule shtuka spaces. Overall, the results bridge the geometric–spectral framework of FS with the classical Langlands program for GSp$_4$ and its inner forms, clarifying the interplay between local parameters and global Galois actions in this important case.
Abstract
Given a prime $p$, a finite extension $L/\mathbb{Q}_{p}$, a connected $p$-adic reductive group $G/L$, and a smooth irreducible representation $π$ of $G(L)$, Fargues-Scholze recently attached a semisimple Weil parameter to such $π$, giving a general candidate for the local Langlands correspondence. It is natural to ask whether this construction is compatible with known instances of the correspondence after semisimplification. For $G = \mathrm{GL}_{n}$ and its inner forms, Fargues-Scholze and Hansen-Kaletha-Weinstein showed that the correspondence is compatible with the correspondence of Harris-Taylor/Henniart. We verify a similar compatibility for $G = \mathrm{GSp}_{4}$ and its unique non-split inner form $G = \mathrm{GU}_{2}(D)$, where $D$ is the quaternion division algebra over $L$, assuming that $L/\mathbb{Q}_{p}$ is unramified and $p > 2$. In this case, the local Langlands correspondence has been constructed by Gan-Takeda and Gan-Tantono. Analogous to the case of $\mathrm{GL}_{n}$ and its inner forms, this compatibility is proven by describing the Weil group action on the cohomology of a local Shimura variety associated to $\mathrm{GSp}_{4}$, using basic uniformization of abelian type Shimura varieties due to Shen, combined with various global results of Kret-Shin and Sorensen on Galois representations in the cohomology of global Shimura varieties associated to inner forms of $\mathrm{GSp}_{4}$ over a totally real field. After showing the parameters are the same, we apply some ideas from the geometry of the Fargues-Scholze construction explored recently by Hansen, to give a more precise description of the cohomology of this local Shimura variety, verifying a strong form of the Kottwitz conjecture in the process.