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I'm trying to find a proof/reference for a statement that I've seen quoted in some way or the other, but without reference.

The setting: let $P\longrightarrow M$ be a flat principal $G$-bundle over a compact Riemann surface, $\tilde M$ the universal cover of $M$. In addition, flat connections on $P$ are in bijection to representations of the fundamental group of $M$ into $G$ (we obtain this via holonomy).

The statement: For any $H$-reduction $(P_{H},\iota)$, where $H$ is a maximal compact subgroup of $G$, we also obtain a $\rho_{\omega}$-equivariant map $s:\tilde M\longrightarrow G/H$ associated to some flat connection $\omega$ on $P$. In addition, we can write $\iota^*\omega=A+\psi$, $A$ a connection on $P_H$ such that, locally, $A=s^{*}\nabla$. Here $\nabla$ is the LC-connection on $G/H$ with the metric that turns multiplication by a fix element of $G$ into isometries.

Thank you!

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Have you checked Sharpe's book? –  Jesse Madnick May 14 '14 at 18:17
    
I'm literally going through it right now .. the problem is, I have no prior knowledge of Cartan geometry other than what I can say from my principal bundle perspective where ever it coincides. –  David Hornshaw May 14 '14 at 18:52

1 Answer 1

Pages 6 and 7 of these notes by Peter Gothen might be of some help. You seem to be dealing with a setting that might lead to Corlette's theorem, so his original paper "Flat $G$-bundles with canonical metrics" may also be a good reading.

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These are the exact sources that lead me to formulating the question. They both, either implicitly or explicitly, leave the proof to the reader. I seem to be stuck doing this and I'd like to have a reference with an actual proof. –  David Hornshaw May 22 '14 at 12:05

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