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 straightforward, but when $ab$ is irrational things are much more complicated. We shall describe results in both cases.

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Last updated: 06 Mar 2020