Equivariant line bundle-divisor correspondence?
In reading about equivariant bundles, I've become a bit confused about how the usual line bundle-divisor correspondence $c_1:\text{Pic}(X)\xrightarrow{\cong}A^1(X)$ works in the equivariant setting. Let $X$ be a smooth complex variety with $G$-action, and let $\mathcal L$ denote a $G$-equivariant line bundle. I've read that there is an equivariant first Chern class $c_1^G$ defined by$$ c_1^G(\mathcal L)=c_1(EG\times_G\mathcal L)\in H^2(EG\times_GX)=H^2_G(X) $$which classifies equivariant line bundles, and am wondering if this leads to a concrete equivalence between equivariant line bundles on $X$ and some equivariant version of the Weil class group (with effective divisors replaced by $G$-invariant divisors).
For instance, in my current reading I came across an example (slightly simplified) where $\mu_n\cong\mathbb Z/n\mathbb Z$ acts on $\mathbb C$ by multiplication by roots of unity, and an equivariant line bundle $\mathcal L$ is defined on $\mathbb C$ by writing $\mathcal L=O(p)$ where $p$ is the origin (the point fixed under the action). How can I understand the action of $\mu_n$ on this line bundle?
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