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. $\mathrm{WF}(\pi)=\mathrm{AC}(\bigcup_{\sigma\in \mathrm{supp}(\pi)}\mathcal O_\sigma)$, where $\mathcal O_\sigma\subset i\mathfrak g^\ast$ is the coadjoint orbit associated to the irreducible unitary representation $\sigma\in \hat{G}$ by Kirillov.