Connections on parahoric torsors over curves

We define parahoric $\text{cG}$–torsors for certain Bruhat–Tits group scheme $\text{cG}$ on a smooth complex projective curve $X$ when the weights are real, and also define connections on them. We prove that a $\text{cG}$–torsor is given by a homomorphism from ${\pi }_1(X\setminus D)$ to a maximal compact subgroup of $G$, where $D\subset X$ is the parabolic divisor, if and only if the torsor is polystable.