The \(\sigma\)-regular representation of \({\mathbb{Z}}\times {\mathbb{Z}}\) (Q1084521)

Let \(\sigma\) be a cocycle on \(G={\mathbb{Z}}\times {\mathbb{Z}}\), and denote by \(R^{\sigma}\) the \(\sigma\)-regular representation of G. The main result of the paper is that, if \(\sigma^ k\) is not a coboundary for \(k\neq 0\), then \(R^{\sigma}\) is cohomologous to a direct integral of irreducible induced representations. Via Fourier transform, \(R^{\sigma}\) is lifted to a projective representation \(\hat R^{\sigma}\) of G on \(L^ 2(\hat G)=L^ 2({\mathbb{T}}^ 2)\). Then every matrix \(M\in SL(2, {\mathbb{R}})\) acts on \(\hat R^{\sigma}\), and therefore also on \(\chi\cdot \hat R^{\sigma}\) for every character \(\chi\) of G: denote by U the image \(M\cdot (\chi \cdot \hat R^{\sigma})\). As a preliminary step for his main result, the author obtains a direct integral decomposition for the projective representation U, and studies irreducibility and unitary equivalence of the components.