# Connections on parahoric torsors over curves

## TL;DR

This paper introduces parahoric torsors with real weights on complex curves, defines connections on them, and establishes a correspondence between polystability and certain fundamental group homomorphisms.

## Contribution

It extends the theory of torsors by defining parahoric $ ext{G}$-torsors with real weights and characterizes polystability via fundamental group representations.

## Key findings

- Polystable parahoric torsors correspond to homomorphisms from the fundamental group to maximal compact subgroups.
- Defines connections on parahoric torsors with real weights.
- Establishes a criterion linking polystability to fundamental group representations.

## Abstract

We define parahoric $\cG$--torsors for certain Bruhat--Tits group scheme $\cG$ on a smooth complex projective curve $X$ when the weights are real, and also define connections on them. We prove that a $\cG$--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.