Fargues-Scholze vs. classical local Langlands, and applications

For general reductive groups over a p-adic number field, Fargues and Scholze constructed a (semi-simplified) local Langlands with many good properties. On the other hand, classical local Langlands correspondences are known for classical groups via endoscopy theory and theta lifting. We review the construction of Fargues-Scholze and related geometric objects, and prove these two correspondences are compatible for all unramified special orthogonal and unitary groups. As an application, we prove torsion vanishing results for orthogonal Shimura varieties, extending results of Caraiani-Scholze, Koshikawa, Santos and Hamann-Lee, etc.