Solution
= Solution
For $z\in Z$, the <eigenspaces> of $\rho(z)$ are invariant under $G$, since every representing matrix commutes with $\rho(z)$. Irreducibility forces one such <eigenspace> to be all of $W$, so $\rho(z)=\chi(z)I$. The multiplication law gives $\chi(zz')=\chi(z)\chi(z')$, making $\chi$ a one-dimensional, hence irreducible, <group representation> of $Z$. Therefore
$$
\boxed{\operatorname{Res}^G_ZW=(\dim W)\chi}.
$$