Let $G$ be an abelian group, and let $E \to M$ be a $G$-equivariant complex vector bundle. If $E$ is irreducible, meaning it has no nontrivial $G$-equivariant subbundles, can anything special be said about $E$? (For instance, when $M$ is a point, the answer to this question is that $E$ must be $1$-dimensional.)
If it helps, I am mainly concerned with the case $G = S^1 \times \cdots \times S^1$ is a torus.
