The local Langlands correspondence conjecture for Mp(2n)

Let $\tilde{G}=\operatorname{Mp}(2n)$, let $\Phi_{\mathrm{bdd}}(\tilde{G})$ denote the set of bounded parameters, and let $C_{\phi,\mathrm{ss}}$ and $S_\phi$ be the semisimple centralizer and its component group attached to $\phi$. For $s\in C_{\phi,\mathrm{ss}}$, let $V_\phi^{s=-1}$ be the $(-1)$-eigenspace of $s$ in the representation associated with $\phi$, and define

$$\tilde{\epsilon}_\phi(s)=\epsilon\left(\frac{1}{2},V_\phi^{s=-1},\psi\right).$$

The local Langlands correspondence conjecture. There should be a local Langlands correspondence

$$\widetilde{\operatorname{Irr}}_{\rm temp}(\tilde{G})=\bigsqcup_{\phi\in\Phi_{\mathrm{bdd}}(\tilde{G})}\Pi_\phi$$

with character relations satisfying the stated hypothesis, such that the assignment $\pi\mapsto\langle\cdot,\pi\rangle:C_{\phi,\mathrm{ss}}\to\mathbb{C}^\times$ gives

$$\Pi_\phi\xrightarrow{1:1}\operatorname{Irr}(S_\phi)\cdot\tilde{\epsilon}_\phi,$$

a torsor under $\operatorname{Irr}(S_\phi)$. The conjecture is intended to describe tempered packets for the metaplectic group and their character relations; the source notes that the factor $\tilde{\epsilon}_\phi$ need not generally belong to $\operatorname{Irr}(S_\phi)$, although it does for discrete-series parameters, and that an analogous formulation for $A$-packets is possible.