The global relative trace formula conjecture for spherical varieties

Let $\mathfrak X$ be the global spherical period space associated with $X$, let $\mathcal L_k$ be the global Langlands group, and let ${^LG_X}\to {^LG}$ be the dual-group map. Let $\operatorname{RTF}_{\mathfrak X}$ denote the relative trace formula distribution. Global relative trace formula conjecture. There is a decomposition

$$\operatorname{RTF}_{\mathfrak X}=\int J_\phi^{\operatorname{aut}}\,\mu_X^{\operatorname{aut}}(\phi),$$

where the measure is on global Arthur parameters factoring through

$$\mathcal L_k\times\operatorname{SL}_2\xrightarrow{\phi}{^LG_X}\times\operatorname{SL}_2\xrightarrow{}{^LG},$$

with $4\phi4$ lying over the identity on the $4\operatorname{SL}_24$ projections, and $J_\phi^{\operatorname{aut}}$ is a sum of relative characters for automorphic representations in the corresponding Arthur packet. When $\mathfrak X$ is stable, the restriction to the most tempered Arthur type, away from the poles of $L_X(\phi|_{\mathcal L_k})$, equals

$$\frac{1}{|S_\phi|}\prod_v' J_{\phi_v}\cdot\mu_{{^LG_X}}(\phi),$$

where $S_\phi$ is the stabilizer of $\phi$ in $\check G_X$ and the factors are the local Plancherel relative characters from the relative local Langlands conjecture. This conjecture connects the global relative trace formula with the local Plancherel formula and generalizes Ichino–Ikeda-type factorizations; it remains open in the stated generality.