Compatibility of the Fargues–Scholze and Gan–Takeda local Langlands

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 $\pi$ of G ( L ), Fargues and Scholze recently attached a semisimple L -parameter to such a $\pi$ , 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 and Scholze, and Hansen, Kaletha and Weinstein showed that the correspondence is compatible with the correspondence of Harris, and Taylor and 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 and Takeda, and Gan and 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 with $\mathrm{GSp}_{4}$ , using basic uniformization of abelian-type Shimura varieties due to Shen, combined with various global results of Kret and Shin, and Sorensen on Galois representations in the cohomology of global Shimura varieties associated with 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. This allows us 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.