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

Write $x=g^{\eta }$. Applying $D_x$ and then changing the summation variable from $k$ to $\ell =k-\alpha \eta$ givesHence this state is an eigenvector of $D_x$ with eigenvalue $\boxed{{\omega }^{\alpha \beta \eta }}$.