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 Hilbert 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.
