The singular simple-parabolic kernel on the flag variety

Cover $X_{\alpha }$ by the product charts of [F1]. On each chart [F2] proves $R^q{p}_{\alpha \ast }M=0$ for every $q>0$, including arbitrary base affine opens. Higher direct image is a sheaf on the base, so vanishing on this open cover proves assertion (1). In particular the underlying sheaf direct-image functor is exact on coherent ${\mathcal{D}}_{\kappa }$-modules: its long exact sequence has zero $R^1$ term for every coherent kernel.

Suppose $p_{\alpha \ast }M=0$. On a product chart, [F2] gives the canonical differential-module isomorphism $M=\mathcal{O}(-1)⊠N$, with $N=p_{\alpha \ast }({\mathcal{O}}_X(-\rho )\otimes M)$ in that chart's partial TDO convention. The inverse image in [F1] has relative structure-sheaf factor, so this is precisely the displayed form ${\mathcal{O}}_X(\rho )\otimes p_{\alpha }^{†}N$. The proof of [F2] makes its evaluation natural in all commuting base operators; on overlaps it is intertwined by the chain rule and the partial TDO cocycle in [F1]. Thus the base actions glue on the stated direct image and the local isomorphisms glue to the asserted global one. Coherence of $N$ is part of [F2], not an inference from ${\mathcal{O}}_X$-coherence of $M$.

Conversely, the displayed form has local relative factor $\mathcal{O}(-1)⊠N$ by [F1], whose direct image is zero by [F2]; hence $p_{\alpha \ast }M=0$ globally. Tensoring this form by ${\mathcal{O}}_X(-\rho )$ and applying $p_{\alpha \ast }$ recovers $N$, since $p_{\alpha \ast }{p}_{\alpha }^{†}N=N$. The same identity proves uniqueness up to the recovered canonical isomorphism. This proves both directions and all higher-image assertions; no rank-one classification is left as a source-only premise.