Unitary matrix that commutes with a diagonal matrix

Question

Let $D$ be a diagonal matrix and let $U$ be a unitary matrix such that $U$ commutes with $D$, i.e. \begin{equation} UD = DU \qquad \text{or} \qquad UDU^\dagger = D. \end{equation}

What properties must $U$ have?

Answer 1

It's easy. We may assume (up to an orthonormal change of basis -in fact, a permutaton-) that $D=diag(\lambda_1I_{i_1},\cdots,\lambda_kI_{i_k})$ where the $(\lambda_i)$ are distinct.

Then $UD=DU$ IFF $U=diag(U_1,\cdots,U_k)$ where the $(U_i)$ are unitary.