Wave Front Sets of Nilpotent Lie Group Representations

Let $G$ be a nilpotent, connected, simply connected Lie group with Lie algebra $\mathfrak{g}$, and $\pi$ a unitary representation of $G$. The goal is to prove that the wave front set of $\pi$ coincides with the asymptotic cone of the orbital support of $\pi$, i.e. $$\mathrm{W}\mathrm{F}$(\pi)=$\mathrm{A}\mathrm{C}$(\bigcup_{\sigma\in $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}$(\pi)}\mathcal O_\sigma)$,where$\mathcal O_\sigma\subset i\mathfrak g^\ast$ is the coadjoint orbit associated to the irreducible unitary representation $\sigma \in \widehat{G}$ by Kirillov.