The operators $f(t) \rightarrow f(t-a)$ and $f(t) \rightarrow e^{2\pi bt} f(t)$ on $L^2(\mathbb R)$ generate unitary representations of 
 the discrete Heisenberg group $H$ with central character $e^{2\pi abz}$.  What are the irreducible representations of $H$ with this central 
 character, and how can one synthesize the representation just described from them ? When $ab$ is rational, the answers are quite straight
 forward, but when $ab$ is irrational things are much more complicated. We shall describe results in both cases.