Holomorphic Lie algebroid connections on holomorphic principal bundles on compact Riemann surfaces

For a $\mathrm{\Gamma }$–equivariant holomorphic Lie algebroid $(V,\phi )$, on a compact Riemann surface $X$ equipped with an action of a finite group $\mathrm{\Gamma }$, we investigate the equivariant holomorphic Lie algebroid connections on holomorphic principal $G$–bundles over $X$, where $G$ is a connected affine complex reductive group. If $(V,\phi )$ is nonsplit, then it is proved that every holomorphic principal $G$–bundle admits an equivariant holomorphic Lie algebroid connection. If $(V,\phi )$ is split, then it is proved that the following four statements are equivalent: An equivariant principal $G$–bundle $E_G$ admits an equivariant holomorphic Lie algebroid connection. The equivariant principal $G$–bundle $E_G$ admits an equivariant holomorphic connection. The principal $G$–bundle $E_G$ admits a holomorphic connection. For every triple $(P,L(P),\chi )$, where $L(P)$ is a Levi subgroup of a parabolic subgroup $P\subset G$ and $\chi$ is a holomorphic character of $L(P)$, and every $\mathrm{\Gamma }$–equivariant holomorphic reduction of structure group $E_{L(P)}$ of $E_G$ to $L(P)$, the degree of the line bundle over $X$ associated to $E_{L(P)}$ for $\chi$ is zero. The correspondence between $\mathrm{\Gamma }$–equivariant principal $G$–bundles over $X$ and parabolic $G$–bundles on $X\mathrm{/}\mathrm{\Gamma }$ translates the above result to the context of parabolic $G$–bundles.