Wave front sets of nilpotent Lie group representations

Abstract

Let G be a nilpotent, connected, simply connected Lie group with Lie algebra $g$, and π a unitary representation of G. In this article we prove that the wave front set of π coincides with the asymptotic cone of the orbital support of π, i.e. $\mathrm{WF}\left(\pi \right)=\mathrm{AC}\left({\bigcup }_{\sigma \in \mathrm{supp}\left(\pi \right)}{O}_{\sigma }\right)$, where $O_{\sigma }\subset i{g}^{⁎}$ is the coadjoint Kirillov orbit associated to the irreducible unitary representation $\sigma \in \hat{G}$.