Lie algebroid connection and Harder-Narasimhan reduction

Take a holomorphic Lie algebroid $(V,\phi )$ on a compact connected Riemann surface $X$ such that the anchor map $\phi$ is not surjective. Let $P$ be a parabolic subgroup of a complex reductive affine algebraic group $G$ and $E_P\subset E_G$ a holomorphic reduction of structure group, to $P$, of a holomorphic principal $G$–bundle $E_G$ on $X$. We prove that $E_P$ admits a holomorphic Lie algebroid connection for $(V,\phi )$ if the reduction $E_P$ is infinitesimally rigid. If $E_P$ is the Harder–Narasimhan reduction of $E_G$, then it is shown that $E_P$ admits a holomorphic Lie algebroid connection for $(V,\phi )$. In particular, for any point $x_0\in X$, the Harder–Narasimhan reduction $E_P$ admits a logarithmic connection that is nonsingular on the complement $X\setminus \{x_0\}$.