On the cubic Shimura lift to $\mathrm{PGL}(3)$: The fundamental lemma

Abstract

The classical Shimura correspondence lifts automorphic representations on the double cover of ${\mathrm{SL}}_2$ to automorphic representations on ${\mathrm{PGL}}_2$. Here we take key steps towards establishing a relative trace formula that would give a new global Shimura lift, from the triple cover of ${\mathrm{SL}}_3$ to ${\mathrm{PGL}}_3$, 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 ${\Theta }_{{\mathrm{SO}}_8}$ for split ${\mathrm{SO}}_8(\mathbb{A})$, consistent with a conjecture of Bump, Friedberg and Ginzburg (2001). In this paper, we first analyze a global distribution on ${\mathrm{PGL}}_3(\mathbb{A})$ 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 ${\mathrm{SL}}_3(\mathbb{A})$. We then match the corresponding local orbital integrals for the unit elements of the spherical Hecke algebras; that is, we establish the fundamental lemma.