Representation of finitely generated nilpotent groups

Abstract

A. I. Malcev has shown that finitely generated torsion free nilpotent groups imbed as lattices in nilpotent Lie groups, and hence their structure is similar to that of the Lie groups. Since A. A. Kirillov has classified the representations of nilpotent Lie groups and, in particular, shown that they are all monomial (induced from one dimensional representations of subgroups), one might conjecture that representations of finitely generated nilpotent groups were monomial. (A representation, here, is a weakly continuous unitary representation on separable Hubert space.) We prove a criterion for when a representations of finitely generated nilpotent groups are monomial. We will also show that representations induced from finite dimensional ones satisfy similar equivalence and irreducibility criteria to those deduced by Kirillov for nilpotent Lie groups.