Let $V$ be a finite-dimensional vector space and $\rho$ an irreducible abelian representation of $G$ on $V$. Is the centralizer of $\rho(G)$ in $End(V)$ necessarily a (commutative) field? (In particular, the commutativity is the only part that is not immediate.)
Motivation: it seems to me that a result of this sort is used in Serre's book on $\ell$-adic representations of elliptic curves. In particular, I am thinking about applications where $V$ is the Tate module of an abelian variety and $G$ is an absolute galois group of some base field, and the representation is given by the usual Galois action.