Representations of the Mautner group and cocyles of an irrational rotation
For \(\alpha\),\(\beta\in {\mathbb{R}}\) let \(M_{\alpha,\beta}\) be the Mautner group \({\mathbb{C}}\times {\mathbb{C}}\times {\mathbb{R}}\) with multiplication \[ (z,w,t)\quad (z',w',t'):=(z+\exp (2\pi i\alpha t)z',\quad w+\exp (2\pi i\beta t)w',\quad t+t'). \] It is shown that \(M_{\alpha,\beta}\cong M_{\alpha ',\beta '}\) iff \(\alpha /\beta =\alpha '/\beta '=:\theta\). Hence \(M_{\theta}:=M_{\alpha,\beta}\) if \(\alpha /\beta =\theta\). Let N be the subgroup \({\mathbb{C}}\times \{0\}\times \{0\}\) of \(M_{\theta}\). Then the irreducible representations of \(M_{\theta}\) acting non- trivially on N are induced from the stability group \(H={\mathbb{C}}\times {\mathbb{C}}\times {\mathbb{Z}}\) of N and the representation of H is defined via a representation of the ``discrete'' Mautner group \(D_{\theta}={\mathbb{C}}\times {\mathbb{Z}}.\) Let K be a Hilbert space, U the group of unitary operators on K and R: [0,1)\(\times {\mathbb{Z}}\to U\) a U-valued cocycle of irrational rotation - \(\theta\). A: [0,1)\(\to U\) is intertwining between R and R' if \(A(\cdot)R(\cdot,m)=R'(\cdot,m)A(\cdot -m\theta)\). The cocycles R and R' are \(\theta\)-cohomologous if an intertwining operator A exists. Measurable cocycles R of rotation -\(\theta\) describe the irreducible representations of \(D_{\theta}\) acting non-trivially on \({\mathbb{C}}\times \{0\}.\) In {\S} 3 a family of one-dimensional Lebesgue-measurable cocycles of irrational rotation -\(\theta\) is defined, the cohomology of which depends roughly spoken on the continued-fraction expansion of \(\theta\). Then the Mackey-Ramsay theory described above can be used to describe five- parameter families of unitary representations of \(M_{\theta}\). These methods are used to construct cocycles of higher dimensions.
scientific article; zbMATH DE number 4038110