Finite factor representations of 2-step nilpotent groups, and orbit theory

In this paper we describe factor representations of discrete 2-step nilpotent groups with 2-divisible center. We show that some standard theorems of the orbit theory are valid in the case of these groups. For countable 2-step nilpotent groups, we explain how to construct a factor representation starting from the orbit of the “coadjoint representation.” We also prove that every factor representation (more precisely, every trace) can be obtained by this construction, and prove a theorem on the decomposition of the factor representation restricted to a subgroup.