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Beilinson-Bernstein localization: $\mathfrak{g}$ action on $G$-equivariant sheaf

More generally, is there some appropriate notion encapsulating this idea of "differentiating" the equivariant structure of a sheaf? Edit: Let me try to make my first question more precise. … The equivariant structure on $\mathcal{L}$ is then just a fancy way of saying that we have a $G$-action on the total space $L$ such that $\pi$ is $G$-linear. …