The one-step geometric wavefront set speculation for positive-real representations

Let $\pi(s,u,\tau)\in {\rm Irr}_\iota(\overline{G})^I$ be such that $s$ is positive real. Write $d^{(n)}_{BV,G}$ for the relevant Barbasch–Vogan duality map and $\mathcal{O}_u^\vee$ for the nilpotent orbit associated with $\pi(s,u,\tau)$. Let $\operatorname{Im}(d^{(n)}_{BV,G})$ denote its image, ordered by the orbit closure order.

One-step geometric wavefront set speculation. Either

$${\rm WF}^{\rm geo}(\operatorname{AZ}(\pi(s,u,\tau)))=d^{(n)}_{BV,G}(\mathcal{O}_u^\vee),$$

or ${\rm WF}^{\rm geo}(\operatorname{AZ}(\pi(s,u,\tau)))$ is equal to a maximal orbit in

$$\left\{\mathcal{O}\in\operatorname{Im}(d^{(n)}_{BV,G}):\mathcal{O}<d^{(n)}_{BV,G}(\mathcal{O}_u^\vee)\right\}.$$

Thus the geometric wavefront set is at most one step smaller than $d^{(n)}_{BV,G}(\mathcal{O}_u^\vee)$ within the image of $d^{(n)}_{BV,G}$. The speculation is motivated by numerical evidence and follows a discussion of examples where the geometric wavefront set can be smaller than the Barbasch–Vogan prediction; its general validity remains open.