The wavefront set upper bound conjecture for Brylinski–Deligne covers

Let $G$ be a reductive group over a local field $F$, and let $\overline{G}^{(n)}$ be a Brylinski–Deligne cover of $G$. Write $\operatorname{Irr}_{\rm gen}(\overline{G}^{(n)})$ for the irreducible genuine representations, let $\phi_\pi:{\rm WD}_F\to{}^L\overline{G}$ be the L-parameter of $\pi$, and let $\mathcal{O}(\phi_\pi)\in\mathcal{N}(\overline{G}^\vee)$ be the nilpotent orbit associated with $\phi_\pi(1,e_\alpha)$. Let $\operatorname{AZ}(\pi)$ denote the Aubert–Zelevinsky dual, and let

$$d_{BV,G}^{(n)}:\mathcal{N}(\overline{G}^\vee)\longrightarrow\mathcal{N}(\mathbf{G})$$

be covering Barbasch–Vogan duality. The wavefront set upper bound conjecture. For every $\pi\in\operatorname{Irr}_{\rm gen}(\overline{G}^{(n)})$,

$$\operatorname{WF}^{\rm geo}(\operatorname{AZ}(\pi))\leqslant d_{BV,G}^{(n)}(\mathcal{O}(\phi_\pi)),$$

and, if $\phi_\pi$ is tempered, equality is achieved by some representation in the $L$-packet. This generalizes the corresponding conjectures for linear groups to covering groups. The upper bound and its tempered-packet equality are the expected relationship between hypothetical local Langlands parameters and geometric wavefront sets; the paper presents evidence, but does not establish the conjecture in general.