unitarily invariant subspaces

Question

If $U_1$ is the group of unit norm complex scalars and $V$ is a subspace of complex variable polynomials. If $f(uz)\in V$ for $u\in U_1$ and $f\in V$, why $f(Uz)\in V$ for $f\in V$ and $U\in U(\mathbb{C}^n)$. {It was written this is true since elements of $U_1$ can commute with elements of $U(\mathbb{C}^n)$ but I can not understand why?!}