Prove that there exists a basis of $V$ in which the matrices of $u$ and $h$ are block diagonal and the diagonal blocks are respectively of the form
$$J_r \quad \text{and} \quad D_{r,a} = \operatorname{diag}(\zeta^a, \zeta^{a+1}, \ldots, \zeta^{a+r-1})$$
for $r \in \mathbb{N}^*$ and $a \in \{0, \ldots, N-1\}$ suitable.