On the cubic Shimura lift to PGL (3): The fundamental lemma

The classical Shimura correspondence lifts automorphic representations on the double cover of SL₂ to automorphic representations on PGL₂. Here we take key steps towards establishing a relative trace formula that would give a new global Shimura lift, from the triple cover of SL₃ to PGL₃, and also characterize the image of the lift. The characterization would be through the non-vanishing of a certain global period involving a function in the space of the automorphic minimal representation ₒ₎䃘 for split SO₈ (A), consistent with a conjecture of Bump, Friedberg and Ginzburg (2001). In this paper, we first analyze a global distribution on PGL₃ (A) involving this period and show that it is a sum of factorizable orbital integrals. The same is true for the Kuznetsov distribution attached to the triple cover of SL₃ (A). We then match the corresponding local orbital integrals for the unit elements of the spherical Hecke algebras; that is, we establish the fundamental lemma.