Component-group formula for the Gan–Gross–Prasad constant

Let $G_n=\operatorname{GSpin}_n$ and $G_{n+1}=\operatorname{GSpin}_{n+1}$, and let $\pi_n$ and $\pi_{n+1}$ be cuspidal automorphic representations with conjectural global Langlands parameters $\phi_n$ and $\phi_{n+1}$. For a parameter $\phi$, let $\mathcal{S}_\phi$ and $\mathcal{S}_{\phi,\mathrm{sc}}$ be the component groups defined from the centralizer of the image of $\phi$ in the dual group and its simply connected cover. Component-group conjecture. If $\pi_{n+1}$ and $\pi_n$ appear with multiplicity one in the discrete spectrum, then

$$2^\beta=4|\mathcal{S}_{\phi_n}||\mathcal{S}_{\phi_{n+1}}|=\frac{1}{2}|\mathcal{S}_{\phi_n,\mathrm{sc}}||\mathcal{S}_{\phi_{n+1},\mathrm{sc}}|.$$

Here $2^\beta$ is the constant occurring in the global Gan–Gross–Prasad period formula. The conjecture predicts that this normalization constant is determined by global Langlands component groups; the existence of the relevant global parameters is itself conjectural.