Connections on Parahoric Torsors over Curves

Abstract

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