Apologies for the basic question, but my experience on math.stackexchange tells me that this will go unanswered there.

Background: In the principal block, the dual Verma modules (with highest weight $w\circ(-2\varrho)$, i.e. the shifted Weyl action of $w$ applied to the (shifted-)antidominant weight) under Beilinson-Drinfeld localization correspond to $(\iota_w)_\star\mathcal{O}_{X_w}$, where $\iota_w\colon X_w\to X$ is the locally closed embedding of the Bruhat cell into the flag variety and $f_\star$ denotes the D-module pushforward (given by the regular pushforward of $\mathcal{O}$-modules composed with tensoring by a transfer module). Evidently this should work for arbitrary (shifted-dominant and shifted-regular on the $\mathfrak{g}$-side of the story) weights $\lambda$ in general. I am defining the twisted D-sheaf (for arbitrary weights) $\mathcal{D}_X^\lambda$ as the universal enveloping algebroid modded out by the ideal generated by $\widetilde{\xi}-\widetilde{\lambda}(\widetilde{\xi})$ for $\widetilde\xi\in\widetilde{\mathfrak{b}}$, which for integral $\lambda$ corresponds to differential operators on the line bundle $\mathcal{L}(\lambda)$, which for $\mathfrak{sl}_2$ is $\mathcal{L}(n)=\mathcal{O}(-n)$; the exact signs/shifts that should appear here vary depending on your conventions.

Question: From following some example computations online (e.g. https://arxiv.org/pdf/2002.01540.pdf), it seems like this same sheaf, $(\iota_w)_\star\mathcal{O}_{X_w}$, considered as a *twisted* D-module over the twisted sheaf $\mathcal{D}_X^{-\lambda}$, should correspond to the dual Verma $M_{(ww_0)\circ\lambda}$ (here $\lambda$ is shifted-dominant). In the case of $\mathfrak{sl}_2$ it seems you can just sort of do the twisted D-action directly on this sheaf, but in general it is completely mysterious to me where the twisted D-action on $(\iota_w)_\star\mathcal{O}_{X_w}$ comes from, especially since the sheaf itself is not changing and there does not appear to be a twisted structure on $\mathcal{O}_{X_w}$ (indeed the symbol $\mathcal{D}_{X_w}^\lambda$ doesn't seem defined since there is no action of $G$ on $X_w$). So where does this twisted action come from? (In the case of integral $\lambda$ one has the equivalence of modules over $\mathcal{D}_X$ and $\mathcal{D}_X^\lambda$, but I do not think this is true in general as nonintegral blocks of category $\cal O$ can be ill-behaved.)