Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 324 / 1 / c / i / Solution

For each fixed $\alpha$, right multiplication by $x^{-\alpha }$ is a bijection of $G$, whose inverse is right multiplication by $x^{\alpha }$. Therefore $D_x$ permutes the displayed orthonormal basis of the tensor-product space. It is consequently a unitary operator, with