Kashiwara's singular parabolic kernel filtration

Choose once and for all the reduced expression and its simple-root sequence from [F1]. The explicit proof there constructs the Verma embeddings, their rational simple-parabolic descent, and the central-character projector proving global generation of the final term. Thus this finite chain is an actual local supplier, independent of $M$; it is not an unproved instruction to filter a localization functor. By [F2], each quotient belongs to $\ker p_{{\alpha }_j\ast }$.

Suppose the asserted filtration of $M$ exists. Every factor $Q_j$ satisfies $\mathrm{\Gamma }(X,Q_j)=\mathrm{\Gamma }(X_{{\alpha }_j},p_{{\alpha }_j\ast }{Q}_j)=0$. Applying exactness [F3] to $0\to M_j\to M_{j-1}\to Q_j\to 0$ gives $\mathrm{\Gamma }{M}_{j-1}=\mathrm{\Gamma }{M}_j$. Starting with $M_N=0$ proves $\mathrm{\Gamma }M=0$.

Now suppose $\mathrm{\Gamma }M=0$. The weight $\kappa -\rho$ is regular antidominant. As proved by the simple-reflection and dual-positive-root calculation in [F1], for every positive root $\beta$, $⟨\rho ,{\beta }^{\vee }⟩$ is a positive integer: subtracting it from a nonintegral complex pairing leaves a nonintegral pairing, while subtracting it from a permitted integral pairing, which is nonpositive, gives a strictly negative integer. Thus no positive integral or zero pairing occurs. By the integral twisting equivalence and [F4], ${\mathcal{O}}_X(-\rho )\otimes M$ is generated by finitely many global differential sections over ${\mathcal{D}}_{\kappa -\rho }$. Tensor back with ${\mathcal{O}}_X(\rho )$; the twist bimodule identity gives a surjection $E^n\twoheadrightarrow M$ for some finite $n$, with $E$ as in [F1]. Locally this identity is the equality of line-twisted differential operators from the PBW/twist construction in [F4].

Put $T={\mathcal{D}}_{\kappa }\mathrm{\Gamma }(X,E)\subseteq E$ and $Q=E\mathrm{/}T$. The submodule $T$ is the image of the global-section evaluation map and is quasi-coherent; the local filtered-noetherian PBW argument for a TDO makes it coherent as a submodule of the coherent module $E$. Every component $E\to M$ in step 1.3 kills $\mathrm{\Gamma }(X,E)$ because $\mathrm{\Gamma }M=0$, hence kills $T$, giving a surjection $Q^n\twoheadrightarrow M$. The globally generated submodule ${\mathcal{E}}_N\subseteq E$ from [F1] is contained in $T$: each of its global generators is a section of $E$. Therefore the images $Q_j$ of ${\mathcal{E}}_j$ in $Q$ give $Q=Q_0\supseteq \cdots \supseteq Q_N=0$. Their factors are coherent quotients of ${\mathcal{E}}_{j-1}\mathrm{/}{\mathcal{E}}_j$. Exactness of the corresponding $p_{{\alpha }_j\ast }$ in [F2] puts each factor in its kernel.

Take the direct sum of this chain $n$ times and its image under $Q^n\twoheadrightarrow M$. This gives the required finite coherent filtration of $M$ with the same root sequence; every factor is a coherent quotient of a finite sum of factors from step 2.1, and hence has zero corresponding direct image by [F2]. If the sequence is empty, [F1] gives $E={\mathcal{E}}_N$ globally generated, so $T=E$, $Q=0$ and $M=0$, exactly the length-zero filtration. The argument and step 1.2 prove both directions for all the stated complex parameters, with no unproved singular-kernel or derived-localization premise.