The endoscopic character identity for even special orthogonal groups

Abstract: We establish the endoscopic character identity for bounded $A$-packets of non-quasi-split even special orthogonal groups, with respect to elliptic endoscopic triples. The proof reduces the non-quasi-split case to the quasi-split case and the real Adam--Johnson case by combining local-global compatibility principle with Arthur's multiplicity formula for non-quasi-split even special orthogonal groups established by Chen and Zou in arXiv:2103.07956. This result plays a key role in the author's work arXiv:2503.04623 on the compatibility between the Fargues--Scholze local Langlands correspondence and classical local Langlands correspondence for even special orthogonal groups.