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 ${\mathrm{PGL}}_3$.

108 pages. Minor revisions to wording and presentation, mathematical results unchanged. Comments are welcome!