Let be a group and let be in the center of .
Why is for every irreducible representation of a scalair?
rho is just an arbitrary representation, thus is a homomorphism on a (complex) vector space , where GL(V) is the general lineair group.
I think it is easier then it looks:
Since , and , it follows directly from this corollary of Schur's Lemma:
PlanetMath: Schur's lemma