In B. Anchouche and I.Biswas's paper " Einstein-Hermitian connections on polystable principal bundles over a compact K\"ahler manifold" Page 209 there states that:
A principal $G$-bundle $E_G$ over a compact K\"ahler manifold is called stable if for every maximal parabolic subgroup $P\subset G$ and every reduction of structure group $\sigma :U\rightarrow E_G /P$ over some open subset $U$ with $co\dim(M\setminus U)\geq 2 ,$ there holds $\deg \sigma^*(T_{\text{rel}})>0,$ where $T_{\text{rel}}$ is the relative tangent bundle for the natural projection $E_G /P\rightarrow U.$
However, when I read Kevin Corlette's well-known work "Flat G-bundles with Canonical metrics" Page362, there is another definition of flat stable principal $G$-bundle
"we call a flat bundle stable if the image of its holonomy homomorphism is not contained in a proper parabolic subgroup of $G$" where $G$ is a complex semisimple algebraic group. Maybe this is a explicit result.
Do the above two definitions coincide for flat principal $G$ bundles? I appreciate any help. Thank you very much!
