On the equivariant reduction of structure group of a principal bundle to a Levi subgroup

Let $M$ be an irreducible projective variety over an algebraically closed field $k$ of characteristic zero equipped with an action of a group $\mathrm{\Gamma }$. Let $E_G$ be a principal $G$–bundle over $M$, where $G$ is a connected reductive algebraic group over $k$, equipped with a lift of the action of $\mathrm{\Gamma }$ on $M$. We give conditions for $E_G$ to admit a $\mathrm{\Gamma }$–equivariant reduction of structure group to $H$, where $H\subset G$ is a Levi subgroup. We show that for $E_G$, there is a naturally associated conjugacy class of Levi subgroups of $G$. Given a Levi subgroup $H$ in this conjugacy class, $E_G$ admits a $\mathrm{\Gamma }$–equivariant reduction of structure group to $H$, and furthermore, such a reduction is unique up to an automorphism of $E_G$ that commutes with the action of $\mathrm{\Gamma }$.