This is probably really easy, but I just need someone to help me get mentally unstuck.  As part of a description of the <a href="http://www.valdostamuseum.org/hamsmith/McKay.html">McKay correspondence</a>, I want to show that if $G$ is a finite subgroup of $SU(2)$ and $V$ the corresponding 2-dimensional representation, then $\dim \text{Hom}(W, W \otimes V) = 0$ for any irreducible representation $W$ of $G$.  I suspect the result is true in slightly greater generality, but it clearly can't always be true.  Since $\dim \text{Hom}(W, W \otimes V) = \dim \text{Hom}(W \otimes W^{\ast}, V)$, the result is false if, for example, $V \simeq W \otimes W^{\ast}$ or is a direct summand thereof for some $W$.

So I am wondering when, for a given $G$ and $V$, it is always true that $\dim \text{Hom}(W, W \otimes V) = 0$ for all irreducible representations $W$.  One can easily reduce to the case that $V$ is irreducible.