Global Endoscopic Lifts from Pgl3 to G2 Wee Teck Gan and Gordan Savin

Let π be a cuspidal automorphic representation of PGL3(A) (where A is the adele ring of F ). The functoriality conjecture of Langlands predicts that, via the inclusion γ of dual groups, there is a global L-packet L(π) of G2 associated to π. Indeed, for each place v of F , there should be a local L-packet L(πv) of G2(Fv). It is expected that L(πv) contains a unique generic representation σ(πv). Hence, the global L-packet should contain a distinguished element σ(π) = ⊗vσ(πv). It is the unique (abstractly) generic representation in L(π) and the conjecture of Langlands predicts that it is cuspidal automorphic. The lifting π 7→ σ(π) is the endoscopic lift of the title. In order to prove the existence of this lift, two major problems need to be solved. Firstly, the distinguished element σ(πv) of the local L-packet L(πv) needs to be defined, and secondly the occurence of the representation σ(π) in the space of cusp forms on G2 needs to be proved.