A fixed singular Verma chain with globally generated final bundle

Write $\kappa =a+ib$ in the real root space. Choose a real number $t$ outside the following finite exceptional set: for each root, exclude a value making $⟨a+tb,{\beta }^{\vee }⟩=0$ unless both pairings already vanish; for each Weyl element, exclude a value making $w(a+tb)=a+tb$ unless $wa=a$ and $wb=b$. Each nonidentical linear equation excludes at most one value. Then the real vector $c=a+tb$ has stabilizer $W_c=W_a\cap W_b=W_{\kappa }$, and its perpendicular roots are exactly ${\mathrm{\Phi }}_{\kappa }$. The real closed-chamber stabilizer theorem [F1] therefore identifies $W_{\kappa }$ with the group generated by these root reflections. The assumed simple system gives $W_{\kappa }=W_J$. Reduced-word positivity in [F1] gives $m_j=⟨{\mu }_{j-1},{\alpha }_j^{\vee }⟩\in {\mathbb{Z}}_{>0}$: the inverse prefix sends ${\alpha }_j$ to a positive root. For completeness, $s_i$ permutes the positive roots other than ${\alpha }_i$, so $s_i\rho =\rho -{\alpha }_i$ and $⟨\rho ,{\alpha }_i^{\vee }⟩=1$. Each positive coroot is a nonzero nonnegative integral sum of simple coroots (apply the positive-root statement to the dual root system), giving the asserted positive integral pairing. The final ${\mu }_N=w_J\rho$ pairs strictly negatively with every positive coroot of ${\mathrm{\Phi }}_{\kappa }$.

Choose root generators $e_{\alpha },f_{\alpha },h_{\alpha }$ with $[e_{\alpha },f_{\alpha }]=h_{\alpha }$. In $H_{j-1}$ its highest vector $v$ has ${\alpha }_j$-weight $m_j-1$, because $⟨\kappa ,{\alpha }_j^{\vee }⟩=0$ and $⟨\rho ,{\alpha }_j^{\vee }⟩=1$. The identity $e_{\alpha }{f}_{\alpha }^{m}v=m(a-m+1)f_{\alpha }^{m-1}v$ for a highest vector of weight $a$, proved by commuting $e_{\alpha }$ past the $m$ factors, shows that $f_{{\alpha }_j}^{m_j}v$ is killed by $e_{{\alpha }_j}$. Every other simple positive generator commutes with $f_{{\alpha }_j}$ and kills $v$. Thus this is a highest vector of weight $-\kappa -\rho +{\mu }_j$, defining $H_j\to H_{j-1}$. On the negative-root PBW model it is multiplication on the right by $f_{{\alpha }_j}^{m_j}$. The associated graded is multiplication by a nonzero monomial in the polynomial algebra $S({\mathfrak{n}}^{-})$, hence injective; taking the highest nonzero degree proves injectivity of the original map. Twisting the rational highest weight adds $\kappa +\rho$, so the same map is $B$-equivariant. This gives the fixed chain.

In the highest-weight module $V$ of [F3] put $V^{\mathrm{\prime }}={⨁}_{{\mu }_N-\xi \mathrm{\notin }{Q}_{+}}{V}_{\xi },V^{\mathrm{\prime }\mathrm{\prime }}=V\mathrm{/}{V}^{\mathrm{\prime }}\mathrm{.}$ This is a $B$-submodule: raising a weight outside the downset of ${\mu }_N$ cannot enter that downset, since subtracting a positive root from an element of $Q_{+}$ stays incompatible with the assumed outside condition in the reverse direction. More explicitly, if ${\mu }_N-(\xi +\beta )\in Q_{+}$ for $\beta >0$, then ${\mu }_N-\xi \in Q_{+}$, a contradiction. All weights of $V^{\mathrm{\prime }\mathrm{\prime }}$ satisfy ${\mu }_N-\xi \in Q_{+}$. Its one-dimensional extremal weight line $V_{{\mu }_N}$ is a $B$-submodule because raising it leaves this downset.

Put $\alpha ={\alpha }_j$, $m=m_j$ and $C_j=H_{j-1}\mathrm{/}{H}_j$. Let ${\mathfrak{u}}_{\alpha }^{-}$ be the sum of negative root spaces other than ${\mathfrak{g}}_{-\alpha }$. It is a Lie subalgebra stable under the $\alpha$-Levi. Ordering its basis before $f_{\alpha }$ in PBW gives $C_j\cong U({\mathfrak{u}}_{\alpha }^{-})\otimes \mathrm{span}(v,f_{\alpha }v,\dots ,f_{\alpha }^{m-1}v)\mathrm{.}$ Indeed $\mathrm{im}{H}_j$ has exactly the basis with final $f_{\alpha }$ exponent at least $m$. The last factor is the $m$-dimensional simple $\alpha$-module. Positive root spaces other than ${\mathfrak{g}}_{\alpha }$ kill this factor: commute them past powers of $f_{\alpha }$; all resulting roots stay positive and different from $\alpha$, or vanish. Therefore this basis identifies $C_j$ with the parabolic induction from that finite-dimensional ${\mathfrak{p}}_{\alpha }$-module. In particular $\pi (e_{\alpha })$ and $\pi (f_{\alpha })$ are locally nilpotent on $C_j$: the adjoint $\alpha$-action on each finite PBW degree of $U({\mathfrak{u}}_{\alpha }^{-})$ is locally finite, and its tensor with the last factor is a sum of finite-dimensional ${\mathfrak{s}\mathfrak{l}}_2$ modules.

Suppose a weight $\xi$ of $V^{\mathrm{\prime }\mathrm{\prime }}$ has ${\chi }_{-\kappa +\xi }={\chi }_{-\kappa +{\mu }_N}$. By [F4] and the $W$-invariance of the sign change, $\kappa -\xi \in W(\kappa -{\mu }_N)=W(\kappa -\rho )$, where the last equality uses $w_J\kappa =\kappa$. Choose $w$ with $w(\kappa -\xi )=\kappa -\rho$. Then $\kappa -w\kappa =\rho -w\xi \in Q_{+},$ since $w\xi$ is a weight of $V$ and lies below $\rho$ by [F3]. By [F2] the difference is zero, so $w\kappa =\kappa$ and $w\in W_{\kappa }=W_J$. Thus $\xi =w^{-1}\rho \in W_J{\mu }_N$. For a strictly antidominant weight ${\mu }_N$ in this subsystem, $v{\mu }_N-{\mu }_N\in Q_{+}$ for every $v\in W_J$: telescope a reduced word, whose simple-root pairings with ${\mu }_N$ are negative and whose prefix roots are positive by [F1]. Consequently $\xi -{\mu }_N\in Q_{+}$, whereas ${\mu }_N-\xi \in Q_{+}$ by step 2.2. The pointed root cone forces $\xi ={\mu }_N$. Thus all other weights of $V^{\mathrm{\prime }\mathrm{\prime }}$ have central characters different from the distinguished one. This is separation for arbitrary complex antidominant $\kappa$, using no real-chamber restriction.

Remove the integral character $\rho$ from the rational $B$-action on $C_j$. Its infinitesimal action becomes $\pi +\kappa$, and all torus weights are integral, beginning with ${\mu }_{j-1}-\rho$. This action extends to a rational $P_{\alpha }$-action. Explicitly keep the torus weight action and the existing positive root groups, and define the additional negative root group by $x_{-\alpha }(t)=\exp (t\pi (f_{\alpha }))$. This exponential is a finite sum on each vector by step 3.1. On finite PBW-degree subspaces, the Levi ${\mathrm{S}\mathrm{L}}_2$ relations hold by [F6]; the torus action agrees because its $\alpha$-weight is that of $\pi$, as $\kappa (h_{\alpha })=0$. The other torus characters are integral and hence rational. The non-Levi positive nilradical acts locally nilpotently and its conjugation relations with the Levi follow by exponentiating the Lie brackets; the finite Baker–Campbell–Hausdorff sums agree with its root-group factorization [F6]. These actions are locally finite: commuting a positive non-Levi root generator through a PBW word never increases its ${\mathfrak{u}}_{\alpha }^{-}$ degree, and the Levi preserves that degree. Each bounded-degree tensor the last finite factor is finite-dimensional and stable under ${\mathfrak{p}}_{\alpha }$. Hence all the displayed matrix coefficients are regular and glue to the rational parabolic action. Covariance with $\pi (\mathfrak{g})$ follows on the root generators by the same bracket/exponential calculation. This is a twisted $(\mathfrak{g},P_{\alpha })$-module of twist $\kappa$.

Induce ${\mathbb{C}}_{-\kappa -\rho }\otimes V^{\mathrm{\prime }\mathrm{\prime }}$ from $\mathfrak{b}$. By exactness of induction, its distinguished weight line induces the submodule $H_N$. The quotient has a finite Verma filtration with one factor $M(-\kappa -\rho +\xi )$ for each other weight, counted with multiplicity: order the finitely many weight spaces from higher to lower and refine within each torus weight space; the positive nilradical raises weights and the torus acts diagonally. Choose a central element $z$ whose value $c_0$ on the distinguished character differs from its values $c_i$ on all the other characters. Such $z$ exists: each forbidden equality is a proper linear hyperplane in the center, and a finite union of proper linear subspaces cannot cover a complex vector space (restrict to the finite span of elements individually separating the finitely many pairs). Let $g(t)={\prod }_i(t-c_i)$ with repetition along the quotient filtration. The product $g(z)$ annihilates that quotient, because each factor kills its corresponding successive Verma factor; all factors commute. On $H_N$ it is the nonzero scalar $g(c_0)$. Therefore $g(z)\mathrm{/}g(c_0)$ maps the induced module onto $H_N$ and is the identity on $H_N$, giving a projection. It intertwines the rational twisted $B$-action because $z$ is invariant under the adjoint action of connected $B$. This is an actual module projection; no ${\mathcal{O}}_X$-linearity is asserted for central operators on arbitrary differential modules.

Apply [F5] to the exact chain in step 2.1. The PBW identity gives ${\mathcal{E}}_0={\mathcal{D}}_{\kappa }\otimes {\mathcal{O}}_X(\rho )$. Descent of step 4.1 gives ${\mathcal{E}}_{j-1}\mathrm{/}{\mathcal{E}}_j={\mathcal{O}}_X(\rho )\otimes p_{\alpha }^{†}{N}_j$, with $N_j={\mathsf{E}}_{\kappa }^{P_{\alpha }}(C_j)$. These modules are coherent: ${\mathcal{E}}_0$ is locally a rank-one free differential module, and the PBW filtered-noetherian argument makes its differential submodules ${\mathcal{E}}_j$ and quotients coherent. Locally the quotient has the form $\mathcal{O}(-1)⊠N_j$; finite differential generators on the two affine fiber charts involve finitely many coefficients in $N_j$. Their base-differential span must be $N_j$, since vertical differentiation creates no new coefficients. The partial TDO is locally noetherian by the same PBW argument, so $N_j$ is coherent.

Composing induction of $V\twoheadrightarrow V^{\mathrm{\prime }\mathrm{\prime }}$ with the projection of step 4.2 gives a surjection of twisted modules $M(-\kappa -\rho )\otimes V\twoheadrightarrow H_N\mathrm{.}$ Here the tensor identity is the map $u\otimes (c\otimes v)\mapsto \sum (u_{(1)}\otimes c)\otimes u_{(2)}v$, with inverse obtained by inserting the antipode in the second enveloping factor. The coproduct and antipode from [F7] check on generators that both maps respect the $\mathfrak{b}$ relations, are mutual inverses, and commute with the twisted rational $B$-actions. Apply the exact functor [F5]. Its PBW and tensor identities give ${\mathcal{D}}_{\kappa }{\otimes }_{\mathbb{C}}V\twoheadrightarrow {\mathcal{E}}_N$ with differential action only on ${\mathcal{D}}_{\kappa }$. The sections $1\otimes v_i$ for a finite basis of $V$ are global differential generators. This proves finite global generation and completes all chain assertions, including the empty-word case. Choice enters through the inherited highest-weight and Weyl suppliers and the permitted bases.