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

Question

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?

Answer 1

If you want to "take derivative" with respect to an element $Y$ of a Lie algebra, you want to take a Newton quotient as usual. That means that you pick a path $\gamma(t)\colon [0,\epsilon) \to G$ in your Lie group such that $\dot{\gamma}(0)=Y$. If you want to differentiate a function with respect to this, for example, you consider $$\frac{d}{dY}f(x)=\lim_{h\to 0}\frac{f(\gamma(h)\cdot x)-f(x)}{h}.$$ Differential geometers would say that this directional derivative defines a vector field, (since it gives a derivation on the sheaf of differentiable functions). However, if you're looking not at a function, but a section of some line bundle, you're in trouble: $f(\gamma(h)\cdot x)-f(x)$ makes no sense, since these are elements of different fibers of the line bundle $\mathcal{L}_{\gamma(h)\cdot x}$ and $\mathcal{L}_x$. The usual solution to this is to pick a connection (which again, people usually define as the rule for taking derivative of sections of your bundle), but because our vector field comes from a group, we have another solution. If $\mathcal{L}$ is equivariant, we can identify fibers using the action of the group; that is, we can look at $$\frac{d}{dY}\sigma(x)=\lim_{h\to 0}\frac{\gamma(h)^*\sigma(\gamma(h)\cdot x)-f(x)}{h}.$$ Here, $\gamma(h)^*$ means applying the isomorphism $\gamma(h)^*\colon \mathcal{L}_{\gamma(h)\cdot x}\to\mathcal{L}_x$ which comes from the equivariant structure. For functions (and the usual equivariant structure), this is just identifying $\mathbb{C}\cong \mathbb{C}$ by the identity, so this includes the formula above. This is what HTT are writing, but in a more abstract way (so for example, you can think about algebraic differential operators).

The filtration being preserved is just the fact that an element of the Lie algebra (something "first order") is sent to a derivation (also "first order").