Let X be a smooth complex projective variety equipped with an action of a linear algebraic group G over \(\mathbb {C}\) . Let D be a reduced effective divisor on X that is invariant under the G–action on X. Let \(s_D\) be the canonical section of \(\mathcal {O}_X(D)\) vanishing along D. Given a positive integer r, consider the stack \(\mathfrak {X}:= \mathfrak {X}_{(\mathcal {O}_X(D),\, s_D,\, r)}\) of r-th roots of \((\mathcal {O}_X, s_D)\) together with the natural morphism \(\pi : \mathfrak {X} \rightarrow X\) . 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 \(\mathfrak {X}\) such that \(\pi \) becomes G–equivariant, and the tautological invertible sheaf \(\mathscr {M}\) on \(\mathfrak {X}\) admits a linearization of this G–action. Finally, we define the notions of G–equivariant logarithmic connections on \(\mathfrak {X}\) 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 \(\mathfrak {X}\) and the category of G–equivariant parabolic connections on X with rational parabolic weights along D.
