Endoscopic description of the local Langlands correspondence for $\mathrm{G}_2$

Abstract

We prove that the $L$-packets for $p$-adic $\mathrm{G}_2$ constructed by Gan and Savin satisfy the endoscopic character identities. In the course of the proof, we also prove twisted endoscopic character identities for functorial lifts from $\mathrm{G}_2$ to $\mathrm{PGSO}_8$ with triality. As a byproduct, we also prove a global multiplicity formula for the discrete automorphic spectrum of $\mathrm{G}_2$ coming from the cuspidal automorphic spectrum of $\mathrm{PGL}_3$.