Solution
= 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>[tensor-product space]. It is consequently a <unitary operator>, with
$$
D_x^\dagger|\alpha\rangle|y\rangle
=|\alpha\rangle|yx^\alpha\rangle.
$$