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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.

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  • $\begingroup$ Why you think it must be a field? $\endgroup$ – Bombyx mori Mar 4 '13 at 5:22
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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.

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What is an abelian representation? If $\rho$ is an irreducible representation then $\text{End}_G(V)$ is a division algebra over the base field $k$ by Schur's lemma. It isn't necessarily a commutative division algebra; for example, if $k = \mathbb{R}$ then one of the possible options is the quaternions $\mathbb{H}$.

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    $\begingroup$ What is the example here? (G =? , V= ?). An abelian representation is one whose image is abelian. The examples I'm thinking about have G = the absolute Galois group of a p-adic field, V = the Tate module of an abelian variety. $\endgroup$ – Tony Mar 4 '13 at 18:06

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