On factor representations of discrete rational nilpotent groups and the Plancherel formula (Q1318190)

The purpose of this paper is to extend the Kirillov orbit picture of representation theory for nilpotent Lie groups to discrete groups \(G_ \mathbb{Q}\) defined over the rationals \(\mathbb{Q}\), following a program begun by Roger Howe. Let \(Ad^*\) be the coadjoint action of \(G_ \mathbb{Q}\) on the Pontryagin dual \(\widehat{{\mathfrak g}_ \mathbb{Q}}\) of the Lie algebra of \(G_ \mathbb{Q}\). It is shown that each coadjoint orbit closure is a coset of the annihilator of an ideal of \({\mathfrak g}_ \mathbb{Q}\), that a certain induced representation canonically associated with an orbit closure is a traceable factor, and that there is an orbital integral formula which gives the trace. Finally, a Plancherel formula is proved.