ON FACTOR REPRESENTATIONS OF DISCRETE RATIONAL NILPOTENT GROUPS AND THE PLANCHEREL FORMULA

The purpose of this paper is to extend the Kirillov orbit picture of r epresentation theory for nilpotent Lie groups to discrete groups G(Q) defined over the rationals Q, following a program begun by Roger Howe. Let Ad be the coadjoint action of G(Q) on the Pontryagin dual g(Q) o f the Lie algebra of G(Q). It is shown that each coadjoint orbit closu re is a coset of the annihilator of an ideal of g(Q), that a certain i nduced representation canonically associated with an orbit closure is a traceable factor, and that there is an orbital integral formula whic h gives the trace. Finally, a Plancherel formula is proved.