Q:

commuting algebra of an irreducible representation

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.  

A:

As is implicit in the question, the fact that the centralizer is a division algebra is automatic, by the usual Schur's Lemma argument. 
Call this division algebra $D$; then $V$ is isomorphic to $D^n$ for some $n$, and by the double centralizer theorem, the image of the group ring of $G$ is $M_n(D)$ (acting on the right, so really I should write $M_n(D^{op})$).   But this image is a commutative ring (by the assumption that the rep'n is abelian), so $n = 1$ and $D$ actually is a commutative field.