Integral line-bundle translation changes the infinitesimal character

In type $A_1$, $\rho$ is the fundamental weight and ${\mathcal{O}}_X(\rho )=\mathcal{O}(-1)$. Applying [F1] with $\kappa =0$ and $\eta =\rho$ sends $M=\mathcal{O}(-1)$ to $M\otimes {\mathcal{O}}_X(\rho )=\mathcal{O}(-2)$ at the new twist $\kappa +\eta =\rho$.

The Harish–Chandra polynomial from [F2] is $p(t)=t+\frac{1}{2}{t}^2$ in the $h$-coordinate. At $\kappa =0$, the evaluation point is $\kappa -\rho =-1$, so $p(-1)=-\frac{1}{2}$. At ${\kappa }^{\mathrm{\prime }}=\rho$, the evaluation point is $0$, so $p(0)=0$. Thus the central characters differ; also $⟨\rho ,{\alpha }^{\vee }⟩=1$, so the target parameter is non-antidominant.

Put $z=y\mathrm{/}x$ on $U_0$ and $w=x\mathrm{/}y$ on $U_1$. For $d=-2$, the Cech complex has $C^0=\mathbb{C}[z]e_0\oplus \mathbb{C}[w]e_1$ and $C^1=\mathbb{C}[z,z^{-1}]e_0$, with transition $e_1=z^{-2}{e}_0$. The image contains powers $z^n$ for $n\ge 0$ and $z^{-2-n}$ for $n\ge 0$, leaving the single quotient basis $z^{-1}$. Hence $H^0(\mathcal{O}(-2))=0$ and $\dim H^1(\mathcal{O}(-2))=1$ by [F3]. The nonzero identity of $M=\mathcal{O}(-2)$ is killed by global sections, so that functor is not faithful. For the standard chart inclusions $j_0,j_1,j_{01}$, put $E=j_{0\ast }(M{\mathrm{\mid }}_{U_0})\oplus j_{1\ast }(M{\mathrm{\mid }}_{U_1})$ and $F=j_{01\ast }(M{\mathrm{\mid }}_{U_0\cap U_1})$. Restriction and difference give a ${\mathcal{D}}_{\rho }$-linear exact sequence $0\to M\to E\to F\to 0$: differential operators act on direct images by restriction, the kernel consists of sections that glue, and on $U_i$ the other direct-image summand maps isomorphically onto $F{\mathrm{\mid }}_{U_i}$. These sheaves are quasi-coherent by [F5]: on $U_0=\mathrm{Spec}\mathbb{C}[z]$ their distinguished-open sections are the localizations of $\mathbb{C}[z]e_0$, $(\mathbb{C}[z]\oplus \mathbb{C}[z,z^{-1}])e_0$ and $\mathbb{C}[z,z^{-1}]e_0$, respectively; the same calculation holds with $w,e_1$ on $U_1$. The map $\mathrm{\Gamma }(E)\to \mathrm{\Gamma }(F)$ is exactly the computed Cech differential with nonzero cokernel. Thus global sections is not exact on quasi-coherent ${\mathcal{D}}_{\rho }$-modules. This computation makes no claim that line-bundle translation preserves global-section faithfulness.