Compatibility of the Fargues–Scholze correspondence for unitary groups

We study unramified unitary and unitary similitude groups in an odd number of variables. Using work of the first and third named authors (Bertoloni Meli and Nguyen in J Reine Angew Math (Crelles J) 2023:1–68, 2021) on the Kottwitz Conjecture for the similitude groups, we show that the Fargues–Scholze local Langlands correspondence (Fargues and Scholze in Geometrization of the local Langlands correspondence, 2021. arXiv:2102.13459) agrees with the semi-simplification of the local Langlands correspondences constructed in Mok (Mem Am Math Soc 235(1108):248, 2015), Kaletha et al. (Endoscopic classification of representations: inner forms of unitary groups, 2014) and Bertoloni Meli and Nguyen (J Reine Angew Math (Crelles J) 2023:1–68, 2021) for the groups we consider. This compatibility result is then combined with the spectral action constructed by Fargues and Scholze (2021, Chapter X), to verify their categorical form of the local Langlands conjecture for supercuspidal L-parameters (Fargues and Scholze 2021, Conjecture X.2.2). We deduce Fargues’ conjecture (Fargues in Geometrization of the local Langlands correspondence: an overview, 2016. arXiv:1602.00999) and prove the strongest form of Kottwitz’s conjecture (Hansen et al. in On the Kottwitz conjecture for local Shimura varieties, 2021, Conjecture I.0.1. arXiv:1709.06651) for the groups we consider, even in the case of non minuscule \(\mu \) $\mu$ .