$G$ is a finite group with a subgroup $H$. Let $\rho_1:G \to GL(V)$ and $\rho_2:H \to GL(U)$ be representations. $Z=\mathbb{C}[G]^H$, i.e., $Z$ is the centralizer of $H$ in $\mathbb{C}[G]$.
How do I show the representation $$\phi:Z \to End(Hom_H(U,V))$$ given by $\phi_g:f \to \rho_1(g) \circ f$ is irreducible?
If I am not mistaken, $Z$ might be infinite and $Hom_H(U,V)$ might be infinite dimensional.