Equivariant Parabolic connections and stack of roots

Let X be a smooth complex projective variety equipped with an action of a linear algebraic group G over C. Let D be a reduced effective divisor on X that is invariant under the G--action on X. Let sD be the canonical section of OX(D) vanishing along D. Given a positive integer r, consider the stack X := X(OX(D),\, sD,\, r) of r-th roots of (OX, sD) together with the natural morphism π : X → X. Under the assumption that G has no non-trivial characters, we show that the G--action on X naturally lifts to a G--action on X such that π become G--equivariant, and the tautological invertible sheaf M on X admits a linearization of this G--action. Finally, we define the notions of G--equivariant logarithmic connections on X and G--equivariant parabolic connections on X with rational parabolic weights along D, and establish an equivalence between the category of G--equivariant logarithmic connections on X and the category of G--equivariant parabolic connections on X with rational parabolic weights along D.