Endoscopic decomposition of elliptic Fargues-Scholze L-packets

TL;DR

This work proves that the local Fargues–Scholze L-packet associated to an elliptic L-parameter admits an endoscopic decomposition. It combines the Hansen–Kaletha–Weinstein character theory, Kaletha’s $z$-embeddings, and Arthur–Moeglin–Waldspurger endoscopy to transfer stability from elliptic elements to the whole group, yielding a canonical endoscopic splitting tied to $L$-parameters. The authors formulate and leverage the FS spectral action on the stack of $G$-bundles and Hecke operators to connect Langlands parameters with stable generalized functions, and they extend the endoscopic decomposition to arbitrary algebraically closed fields of characteristic zero via a recent KSV framework. An ancillary result is establishing independence from the field isomorphism $\iota$, aligning with motivic perspectives on the FS correspondence. Overall, the paper extends Fu's stable case to elliptic L-parameters and situates FS packets within Arthur-style endoscopy, with broad implications for endoscopic decompositions across bases and fields.

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 construct a general endoscopic decomposition over complex numbers based on results of Arthur, and a generalization of this decomposition over an arbitrary algebraically closed field of characteristic zero based on a recent work [KSV].

Theorems & Definitions (31)

• Theorem 6
• Proposition 1.9
• Lemma 2.4
• proof
• Proposition 2.6
• Proposition 2.8
• proof : Proof of Theorem \ref{['T:main']}
• Lemma 3.5
• proof
• Lemma 3.9