Endoscopic decomposition of elliptic Fargues-Scholze L-packets

Abstract: The main goal of this note is to show that the local L-packet of Fargues-Scholze [FS], corresponding to an elliptic L-parameter, has an endoscopic decomposition. Our argument is strongly motivated by a beautiful paper of Chenji Fu [Fu], where the stable case is proven. To put our results in a more general context, we also formulate a general endoscopic decomposition over complex numbers based on results of Arthur, and suggest a conjectural generalization of this decomposition over an arbitrary algebraically closed field of characteristic zero.