Answer by Venkataramana for Generalization of Schur's Lemma

This is an easy exercise. If $\rho _i$ are "different" (i.e. inequivalent) then by Schur's lemma, $Hom _G(\rho _i,\rho _i)={\mathbb C}I$ and $Hom _G(\rho _i, \rho _j)=0$. Hence the commutant of $G$ in $End(\rho)$ is easily seen to be the product

$$M_{m_1}({\mathbb C})\times \cdots \times M_{m_k}({\mathbb C}),$$ where $M_n({\mathbb C})$ is the algebra of $n\times n$ matrices with complex entries.