The local Gross–Prasad multiplicity-one and epsilon-dichotomy conjectures

Let $(G,H,\xi)$ be a Gross–Prasad triple attached to an admissible pair $(W,V)$ over $F$. For a generic local $L$-parameter $\varphi$ of $G$, let $\Pi_{\mathrm{rel}}^{\mathrm{Vogan}}(\varphi)$ be the relevant Vogan $L$-packet, let $m(\pi)=\dim \operatorname{Hom}_{H(\mathbb{R})}(\pi,\xi)$, and let $\chi_\pi$ be the character of the component group $\mathcal{S}_\varphi$ attached to $\pi$. Let $\chi_\varphi$ be the distinguished character defined by the local root-number formula in the source. Gross–Prasad's local conjecture. There exists a unique member $\pi_\varphi\in \Pi_{\mathrm{rel}}^{\mathrm{Vogan}}(\varphi)$ such that $m(\pi_\varphi)=1$. After fixing the specified Whittaker datum, its attached character satisfies

$$\chi_{\pi_\varphi}=\chi_\varphi.$$

This conjecture refines multiplicity one by identifying the unique distinguished representation in a relevant Vogan packet through epsilon factors. The paper studies it over archimedean local fields; the surrounding discussion records multiplicity one as known, while the packet-level characterization is the conjectural refinement.