I am reading a paper that says $L$ is a flat complex $G$-line bundle over $M$ with holonomy $\alpha$. Here $G$ is an abelian Lie group and $\alpha$ is a character of $G$. I have two questions:
- If the bundle is flat then isn't its holonomy trivial?
- I'm a little familiar with holonomy being a group so whats it mean for the holonomy to be a character? The bundle is constructed as the associated line bundle to a principal $G$ bundle using the representation $\alpha$. So is saying that the holonomy is $\alpha$ just repeating this fact? If so, how does it connect to the other definition of holonomy?
Any references where this stuff is talked about in some detail would be highly appreciated.