The classical Shimura correspondence lifts automorphic representations on the double cover of $SL_2$ to automorphic representations on $PGL_2$. Here we take key steps towards establishing a relative trace formula that would give a new global Shimura lift, from the triple cover of $SL_3$ to $PGL_3$, 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 $\Theta_{ SO_8}$ for split $SO_8({ \mathbb{ A}})$, consistent with a 2001 conjecture of Bump, Friedberg and Ginzburg. In this paper, we first analyze a global distribution on $PGL_3({ \mathbb{ A}})$ 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 $SL_3({ \mathbb{ A}})$. We then match the corresponding local orbital integrals for the unit elements of the spherical Hecke algebras; that is, we establish the Fundamental Lemma.
Categories: math.NT math.RT