  Given a simply connected nilpotent Lie group having unitary irreducible
representations that are square-integrable modulo the center (SI/Z), we develop
a notion of periodization on the group Fourier transform side, and use this
notion to give a characterization of shift-invariant spaces in $L^2(N)$ in
terms of range functions. We apply these results to study the structure of
frame and Reisz families for shift-invariant spaces. We illustrate these
results for the Heisenberg group as well as for other groups with SI/Z
representations.
