I have a few elementary questions related to Beilinson-Bernstein localization.

Let $G$ be a semisimple algebraic group over $\mathbb{C}$ with Lie algebra $\mathfrak{g}$. Consider the setup of Beilinson-Bernstein localization: we have a sheaf of (twisted) differential operators $$\mathscr{D}(\mathcal{L})=\mathcal{L}\otimes\mathscr{D}_X\otimes\mathcal{L}^\vee$$ acting on a locally free $G$-equivariant sheaf $\mathcal{L}$ on a smooth variety $X$. We would like to build a homomorphism $\Phi:\mathfrak{g}\to\Gamma(X,\mathscr{D}(\mathcal{L}))$ using the equivariance condition $\varphi:p_X^*\mathcal{L}\xrightarrow{\sim}\sigma^*\mathcal{L}$, where $\sigma:G\times X\to X$ and $p_X:G\times X\to X$ are the $G$-action on $X$ and the projection, respectively. HTT, in "D-Modules, Perverse Sheaves, and Representation Theory" give the following homomorphism for the algebraic category. For $s\in\Gamma(X,\mathcal{L}),a\in\mathfrak{g}$, they write $$\varphi^{-1}(\sigma^*(\Phi(a)s))=(a\otimes 1)\cdot \varphi^{-1}(\sigma^*s).$$ Here $\mathfrak{g}$ is acting as a right-invariant vector field on $G$, since $\Gamma(G\times X,p_2^*\mathcal{L})\cong\mathbb{C}[G]\otimes_\mathbb{C}\Gamma(X,\mathcal{L}).$

I'm not sure what is meant here by $\sigma^*s$ (unless we are working instead with the underlying bundles), and more importantly, what the intuitive picture behind this formulation is. Moreover, [HTT] uses this formula to claim a number of facts, e.g. that $\Phi$ is filtration-preserving (on $\mathcal{U}\mathfrak{g}$ and $\mathscr{D}(\mathcal{L})$) or that $\Phi$ is equivariant, which I don't know how to obtain from such a formal definition. A worked-out/elementary reference to this construction or an explanation would be much appreciated.

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. I don't really understand/see how to compute with the formula above. Is there a more straightforward definition? In particular, does the following work? Replace $\mathcal{L}$ with its underlying vector bundle $\pi:L\to X$. 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. For any fixed $l\in L$ we have $\sigma_l:G\to L$ taking $g\mapsto g\cdot l$. Passing to tangent spaces and varying $l$ yields a map $\mathfrak{g}\to\Gamma(L,TL)$. Does this give a map $\mathfrak{g}\to\Gamma(X,\mathscr{D}(\mathcal{L}))$? If so, is this construction any easier to work with?