On the Cubic Shimura lift to $PGL(3)$: The Fundamental Lemma

The classical Shimura correspondence lifts automorphic representations on the double cover of $S{L}_2$ to automorphic representations on $PG{L}_2$. Here we take key steps towards establishing a relative trace formula that would give a new global Shimura lift, from the triple cover of $S{L}_3$ to $PG{L}_3$, and also characterize the image of the lift. The characterization would be through the nonvanishing of a certain global period involving a function in the space of the automorphic minimal representation ${\mathrm{\Theta }}_{S{O}_8}$ for split $S{O}_8(\mathbb{A})$, consistent with a 2001 conjecture of Bump, Friedberg and Ginzburg. In this paper, we first analyze a global distribution on $PG{L}_3(\mathbb{A})$ 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 $S{L}_3(\mathbb{A})$. We then match the corresponding local orbital integrals for the unit elements of the spherical Hecke algebras; that is, we establish the Fundamental Lemma.