Fargues-Scholze correspondence and endoscopic classification for special orthogonal and unitary groups

Abstract

Let $p$ be odd and let $K/\mathbb Q_p$ be unramified. For a special orthogonal group or a unitary group $G$ over $K$ that splits over an unramified extension, we prove that the Fargues-Scholze local Langlands correspondence agrees with the semisimplification of the classical correspondence for $G$ constructed in the work of Arthur and others. As applications, we construct an unambiguous local Langlands correspondence for even special orthogonal groups, deduce the strong Kottwitz conjecture and the eigensheaf conjecture of Fargues, and establish new torsion vanishing results for orthogonal and unitary Shimura varieties.