Artin–Deligne duality conjecture for Arthur packets

Let $G$ be a quasi-split connected reductive group over a local or global field $F$. For a parameter $\mathfrak{\psi}$, let $mathit{Pi}_{\mathfrak{\psi}}(G)^A$ and $mathit{Pi}_{\mathfrak{\psi}}(G)^D$ denote the Arthur packets defined using Artin's and Deligne's conventions, respectively; let $\mathfrak{w}$ be a Whittaker datum, $\mathcal{S}_{\mathfrak{\psi}}$ the component group, and $\widehat C$ the relevant duality automorphism. Artin–Deligne duality conjecture. Locally,

$$mathit{Pi}_{\mathfrak{\psi}}(G)^D=(mathit{Pi}_{\mathfrak{\psi}}(G)^A)^\vee=mathit{Pi}_{\mathfrak{\psi}^\vee}(G)^A,$$

and, for $mathit{pi}\inmathit{Pi}_{\mathfrak{\psi}}(G)^D$ and $x\inmathcal{S}_{\mathfrak{\psi}}$,

$$\langle\mathit{pi},x\rangle^D_{\mathfrak{w},\mathfrak{\psi}}=\langle\mathit{pi}^\vee,x^{-1}\rangle^A_{\mathfrak{w}^{-1},\mathfrak{\psi}}=\langle\mathit{pi},\widehat C^{-1}(x)\rangle^A_{\mathfrak{w},\mathfrak{\psi}^\vee}.$$

Globally,

$$L^2_{\mathfrak{\psi}}(G)^D=(L^2_{\mathfrak{\psi}}(G)^A)^\vee=L^2_{\mathfrak{\psi}^\vee}(G)^A, \qquad m(\mathit{pi}^\vee,\mathfrak{\psi}^\vee)=m(\mathit{pi},\mathfrak{\psi}).$$

This conjecture predicts compatibility between Artin and Deligne normalizations, contragredient representations, dual parameters, packet pairings, and global multiplicities. The source gives no resolution status.