Ikeda–Yamana multiplicity conjecture for CAP representations of the metaplectic group

Let $F$ be a totally real number field, let $n$ be odd, and let $\pi\simeq\otimes'_v\pi_v$ be an irreducible cuspidal automorphic representation of $\mathrm{PGL}_2(\mathbb A)$ such that $\pi_v\simeq D_{2\kappa_v}$ for every archimedean place $v\in\mathfrak S_\infty$. For each place $v$, let $\pi_v^{\epsilon_v}$ denote the member of the local packet indexed by $\epsilon_v\in\{+,-\}$, and let $\varepsilon(\frac12,\pi)$ be the global root number. The metaplectic multiplicity conjecture. The representation $\otimes'_v\mathrm{Mp}_n^{\psi_v}(\pi_v^{\epsilon_v})$ has multiplicity

$$\frac{1}{2}\left(1+\varepsilon\left(\frac{1}{2},\pi\right)\prod_v\epsilon_v\right)$$

in ${\mathscr A}_\mathrm{cusp}(\mathrm{Mp}(W_n))$. This predicts precisely which members of the global packet occur cuspidally and with what multiplicity; the statement is presented in the paper as a conjectural multiplicity formula, while related cases are discussed through explicit constructions.