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I am now reading Kevin Corlette's paper: Flat G-bundles with canonical metrics, JDG, 1988 Link

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).

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    $\begingroup$ I am not sure what do you mean by "this is true or not": Existence of an equivariant harmonic map? Then the answer is positive in much greater generality, including all reductive groups over local fields, where target spaces are symmetric spaces of nonpositive curvature as well as Euclidean buildings. One needs a mild assumption of reductivity of the representation, but that's all. $\endgroup$
    – Misha
    May 21, 2014 at 16:29

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I think Corlette chooses to work with complex semi-simple groups rather than reductive groups just in order to avoid discussing non-uniqueness which comes from the center of the group.

For example, for $\mathrm{GL}(n,\mathbb{C})$, multiplying a harmonic metric by a positive constant gives another one, and they correspond to different harmonic maps to $\mathrm{GL}(n,\mathbb{C})/U(n)$. But for $\mathrm{SL}(n,\mathbb{C})$, we only consider unimodular hermitian metrics (equivalently, equivariant maps to $\mathrm{SL}(n,\mathbb{C})/\mathrm{SU}(n)$), and there is a unique such metric which is harmonic.

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