The global numerical conjecture for polarized hyperspherical Hamiltonian varieties

Let $M=T^*X$ and $\check{M}=T^*\check{X}$ be a pair of dual polarized hyperspherical Hamiltonian varieties, where $X$ is a $G$-spherical variety and $\check{X}$ is the corresponding $\check{G}$-spherical variety. Let $\pi$ be a tempered automorphic representation of $G(\mathbb{A})$ with $L$-parameter $\phi$. Suppose that the fixed point set

$$\check{X}^{\phi}=\{x_1,\dots,x_r\}$$

is finite. For a suitably normalized spherical vector $f\in\pi$, The global numerical conjecture. one has

$$\mathcal{P}_X(f)\sim \sum_{i=1}^rL(0,(T_{x_i}\check{X})^{\talloblong}).$$

Here $\sim$ suppresses the global constants and local factors at ramified places occurring in a precise period formula. This is the expected numerical relation between the automorphic period attached to $X$ and special values of $L$-functions associated with the fixed points of the Langlands parameter on the dual variety; the precise global period formula, including its constants and ramified local factors, remains to be established in this setting.