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

TL;DR

This paper verifies the compatibility of the Fargues-Scholze local Langlands construction with the known Gan-Takeda correspondence for GSp4 and its inner form, using Shimura varieties and Galois representations.

Contribution

It extends compatibility results of the Fargues-Scholze construction to GSp4 and its inner form, employing Shimura variety cohomology and global Galois representations.

Findings

Confirmed compatibility of local Langlands parameters for GSp4 and its inner form.

Described the Weil group action on Shimura variety cohomology.

Verified a strong form of the Kottwitz conjecture in this context.

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-Scholze recently attached a semisimple Weil parameter to such $\pi$, 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-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.