Endoscopic Decomposition of Elliptic Fargues–Scholze L-Packets

The main goal of this note is to show that the local L-packets of Fargues–Scholze [FS21], corresponding to elliptic L-parameters, admit an endoscopic decomposition. Our argument is strongly motivated by a beautiful paper of Chenji Fu [Fu25], where the stable case is proven. To put our results in a more general context, we also construct a general endoscopic decomposition over ℂ based on results of Arthur, and a generalization of this decomposition over arbitrary algebraically closed fields of characteristic zero based on a recent work [KSV25].