At the singular SL2 twist global sections is not conservative

The twist parameter is $\kappa =0$, so $⟨\kappa ,{\alpha }^{\vee }⟩=0$: it is antidominant and singular under the definitions in Flag-variety twisted differential operators and their modules. The line bundle $\mathcal{O}(-1)$ is nonzero and is a ${\mathcal{D}}_0$-module by [F1].

Let $z=y\mathrm{/}x$ on $U_0$ and $w=x\mathrm{/}y$ on $U_1$, with frames related by $e_1=z^{-1}{e}_0$. The Cech complex has $C^0=\mathbb{C}[z]e_0\oplus \mathbb{C}[w]e_1,C^1=\mathbb{C}[z,z^{-1}]e_0,$ and differential $(f_0,f_1)\mapsto z^{-1}{f}_1(1\mathrm{/}z)-f_0(z)$. A global section would identify a polynomial in nonnegative powers of $z$ with a sum of strictly negative powers, so $H^0=0$. In degree one the image contains all nonnegative powers from $f_0$ and all negative powers from $z^{-1}{f}_1(1\mathrm{/}z)$; these span the Laurent polynomials, so $H^1=0$.

By [F3], $p_{\alpha }:X\to G\mathrm{/}{P}_{\alpha }$ is the map to a point. Hence $p_{\alpha \ast }\mathcal{O}(-1)=H^0(X,\mathcal{O}(-1))=0$, while $\mathcal{O}(-1)\mathrm{\ne }0$. The global-sections kernel and the parabolic-pushforward kernel are therefore nonzero, contradicting the refuted assertion. This is the singular Serre-kernel branch, not a counterexample to exactness: the antidominant global-sections functor remains exact by Antidominant twists are D-affine and regular ones are conservative.