Parabolic Lie algebroid connections on parabolic principal bundles over curves

Let $X$ be a compact connected Riemann surface and $S\subset X$ a finite subset. We consider parabolic principal $G$–bundles ${\mathcal{E}}_G$ on $X$ with parabolic structure on $S$, where $G$ is a connected complex reductive affine algebraic group. Let $P\subset G$ be a parabolic subgroup and ${\mathcal{E}}_P\subset {\mathcal{E}}_G$ a reduction of structure group of ${\mathcal{E}}_G$ to $P$. We give a criterion for the existence of a parabolic Lie algebroid connection on ${\mathcal{E}}_P$ for any given parabolic Lie algebroid on $(X,S)$ whose anchor map is not surjective. More precisely, ${\mathcal{E}}_P$ admits a parabolic Lie algebroid connection if the reduction ${\mathcal{E}}_P\subset {\mathcal{E}}_G$ is parabolically infinitesimally rigid. In particular, the Harder–Narasimhan reduction of ${\mathcal{E}}_G$ admits a parabolic Lie algebroid connection.