Wave Front Sets of Nilpotent Lie Group Representations

Abstract

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.

to appear in Journal of Functional Analysis