Let $\rho : G \to GL(V)$ be an irreducible representation of a finite group. Schur's lemma says if $\pi:GL(V) \to GL(V)$ intertwines with $\rho$, that is, $\pi \rho(g) = \rho(g) \pi$ for every $g\in G$, then $\pi = \lambda I$ for some $\lambda \in \mathbb{C}$. 


Is there a similar lemma for $\rho = m\_1 \rho\_1 \oplus \ldots \oplus m\_k \rho\_k$, where $\rho\_1, \ldots, \rho\_k$ are different irreducible representations? The question is, given such $\rho$, if $\pi \rho = \rho \pi$, then what is the structure of $\pi$?



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. 


