Ben-Zvi–Sakellaridis–Venkatesh reduced-dual period formula

Let $\hat{\Delta}_{\mathrm{red}}=(\hat{L},\hat{H}',1,\hat{\rho}_{\hat{H}',\hat{\iota}'})$, where

$$\hat{\rho}_{\hat{H}',\hat{\iota}'}=\hat{\rho}_{\hat{H}'}\oplus\bigoplus_{k\in\hat{I},\;k\;\mathrm{odd}}\hat{\rho}_k.$$

Let $\pi_L$ be a tempered automorphic representation of $L(\mathbb A)$ and let $\mathcal P_{\widehat{(\hat{\Delta}_{\mathrm{red}})}}$ be the associated period. Reduced-dual period formula. For any embedding $\nu:\pi_L\rightarrow L^2(L(k)\backslash L(\mathbb A))$, the period $\mathcal P_{\widehat{(\hat{\Delta}_{\mathrm{red}})}}(\phi)$, for $\phi\in\operatorname{Im}(\nu)$, is nonzero only if the Arthur parameter of $\pi_L$ factors through $\hat{H}'(\mathbb C)\subset\hat{L}(\mathbb C)$. In that case, $\pi_L$ is a lifting of a tempered Arthur packet $\Pi$ of $H'(\mathbb A)$, and one can choose $\nu$ so that

$$\frac{|\mathcal P_{\widehat{(\hat{\Delta}_{\mathrm{red}})}}(\phi)|^2}{\langle\phi,\phi\rangle}=\frac{L(1/2,\Pi,\rho_{\hat{H}'})\prod_{k\in\hat{I},\;k\;\mathrm{odd}}L(1/2,\Pi,\hat{\rho}_k)\cdot L(1,\Pi,\hat{\rho}_0)}{L(1,\Pi,\operatorname{Ad})^2}.$$

This is obtained in the source by applying the preceding BZSV period conjecture to the reduced dual quadruple; its general validity therefore remains open.