I am now reading Kevin Corlette's paper: Flat G-bundles with canonical metrics, JDG, 1988 http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.jdg/1214442469

I think I do understand the core part of its argument, i.e. finding the harmonic metric via heat equation. However, it occurs to me that his argument also works for flat $GL(n,\mathbb{C})$-bundles (with reduction to $U(n)$). I would like to know whether this is true or not, and in addition, why did he work with $SL(n,\mathbb{C})$ (let us forget about the generalization to semisimple groups at this moment).