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–Bernstein 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. [Romanov - Four examples of Beilinson-Bernstein localization](https://arxiv.org/abs/2002.01540)), 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 sometimes 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.) The possibilities to me are that either the structure sheaves $\mathcal{O}_{X_w}$ somehow have some hidden `twisted' structure with respect to $\lambda$, or that the pushforward $f_\star$ has some twisted variant (in which case I would have to redefine the side-switching modules, the transfer modules, etc. etc., all over again for the twisted case), or that somehow any $\mathcal{D}_X$-module can be granted a $\mathcal{D}_X^\lambda$-module for general $\lambda$. Or maybe some combination of the above.

Unnecessary example (just for added detail): 

For $\mathfrak{sl}_2$, $X=\mathbb{P}^1$. My choice of coordinates is such that $D(x)$ contains the point $z=y/x=0$ corresponding to the standard Borel. Let $\iota$ denote the closed embedding of the point $z=0$ (the Bruhat cell for $w=1$) and $\jmath$ denote the open embedding of $D(y)$ (the Bruhat cell for $w=s$). Then $\iota_\star\mathcal{O}_{z=0}$ should correspond to the antidominant Verma, and indeed its global sections is $\mathbb{C}[\partial_z]\delta_0$. The trivial central character case is clear to me, so let's take the twist. By checking locally the action of $h=2z\partial_z+n$ (here I am taking differential operators on $\mathcal{L}(n)=\mathcal{O}_{\mathbb{P}^1}(-n)$, where $-n$ being shifted-dominant and shifted-regular implies $n\le 0$), we see $h\cdot\partial_z^k\delta_0=(2z\partial_z+n)\partial_z^k\delta_0=2(\partial_z^{k+1}z-(k+1)\partial_z^k)\delta_0+n\partial_z^k\delta_0=(n-2-2k)\partial_z^k\delta_0$, so that e.g. the `highest-weight vector' $\partial_z^0$ has weight $n-2$. As the nonpositive integer $n$ varies this gives the integral antidominant Vermas (the $e,f$ actions can be checked similarly). So in this case the twisted action still works *on the nose*, no modifications necessary.

However, things no longer work so nicely for the dominant Verma module, corresponding to $\mathbb{D}_{\mathbb P^1}\jmath_\star \mathcal{O}_{D(y)}$. This sheaf (I believe) has global sections $\mathbb{C}[\partial_z]$, considered first as a (right-)submodule of $\mathbb{C}[z,\partial_z]/\partial_z z\mathbb C[z,\partial_z]$ and then side-switched to a left module (note well that the thing we quotient out by is a right subobject). Let again $\lambda=n$ label the differential operator side, so that $-n$ is shifted-dominant and shifted-regular, so that $n\le 0$. Checking again the action of $h=2z\partial_z+n$, we see $h\cdot \partial_z^k=\partial_z^k\cdot(-\partial_z\cdot 2z+n)=-2\partial_z(z\partial_z^k+k\partial_z^{k-1})+n\partial_z^k=(n-2k)\partial_z^k$. By looking at $k=0$ we see this is *not* the dominant Verma, as $n$ is nonpositive. So whatever the action is, it is no longer the naive one. One runs into other problems when one tries to compute $e\cdot \partial_z^0$ also, since $e=z^2\partial_z+nz$ and $z$ doesn't make sense in $\mathbb{C}[\partial_z]$ (this is not a problem for $n=0$). So what is going on here?