Ample translation maps split by central-character separation

Form the finite $G$-equivariant bundle $E(V_{\eta })=G{\times }^B{V}_{\eta }$. The map $[g,v]\mapsto (gB,gv)$ identifies it with the trivial bundle $X\times V_{\eta }$ with its diagonal $G$-action. Choose a weight basis in each weight space and order the weights with higher weights first. Positive root operators raise weights, so the resulting complete filtration is $B$-stable and its successive quotient lines have characters ${\eta }_j$ as in [F2]. Association to $G\to X$ is exact on this finite filtration: on every torsor chart it is tensoring a sequence of complex vector spaces with ${\mathcal{O}}_X$. Its quotient line bundles are ${\mathcal{O}}_X({\eta }_j)$, with highest line ${\mathcal{O}}_X(\eta )$. Tensoring this bundle filtration with arbitrary $M$ remains exact, since each quotient is locally free.

The coevaluation ${\mathcal{O}}_X\to L_{\eta }\otimes E(V_{\eta })$ is the highest line in this filtration; dually the evaluation $E(V_{\eta }{)}^{\ast }\to L_{\eta }$ is the quotient of weight $-\eta$. These are the duals of evaluation of the globally generated line bundle [F1], so they are respectively injective and surjective, locally split as vector-bundle maps. After tensoring with $M$ they give the maps in the Statement, with the same exact filtration positions. Each associated quotient of the first target is $M\otimes {\mathcal{O}}_X(-\eta +{\eta }_j)$, with central character ${\chi }_{\kappa -\eta +{\eta }_j}$ by [F4]. The evaluation source has quotients $M\otimes {\mathcal{O}}_X(-{\eta }_j)$ with characters ${\chi }_{\kappa -{\eta }_j}$; its target has character ${\chi }_{\kappa -\eta }$. All these filtrations are preserved by the diagonal $\mathfrak{g}$-action because their finite bundle factors are $G$-equivariant.

For the coevaluation, every quotient other than its image has ${\eta }_j\mathrm{\ne }\eta$. If its character were ${\chi }_{\kappa }$, equality of Harish–Chandra characters would give some $w\in W$ with $\kappa -w\kappa =\eta -{\eta }_j\in Q_{+}\setminus \{0\}$, contrary to [F3]. Thus the image is the sole filtration quotient with character ${\chi }_{\kappa }$. This argument applies to the actual filtered module involving $M$, not to a supposed tensor-character identity for an unfiltered arbitrary tensor product.

For the evaluation, if ${\eta }_j\mathrm{\ne }\eta$ had ${\chi }_{\kappa -{\eta }_j}={\chi }_{\kappa -\eta }$, then $\kappa -{\eta }_j=w(\kappa -\eta )$ for some $w$. Rearranging gives $\kappa -w^{-1}\kappa =\eta -w^{-1}{\eta }_j\in Q_{+}$, since $w^{-1}{\eta }_j$ is again a weight of $V_{\eta }$. If the right side is nonzero, antidominance [F3] rules it out; if it is zero, regularity [F3] forces $w=1$ and hence ${\eta }_j=\eta$, a contradiction. Thus its target is the sole quotient with the distinguished character.

For a sheaf of complex vector spaces with a finite filtration of exact central-character quotients, a product of powers of their finitely many maximal ideals in $Z(U(\mathfrak{g}))$ annihilates the sheaf. Powers of distinct maximal ideals are comaximal: expand $1=(u+v{)}^{a+b-1}$ with $u+v=1$, $u$ in the first ideal and $v$ in the second. Thus [F5] gives central polynomial projectors onto the corresponding generalized-character summands. These polynomials act on sections by the diagonal differential $U(\mathfrak{g})$-action, commute with restrictions and with $U(\mathfrak{g})$, and hence are morphisms of sheaves of complex vector spaces. They are natural in $M$. They are generally differential operators rather than ${\mathcal{O}}_X$-linear maps; that stronger linearity is neither asserted nor needed.

By step 3.1 the distinguished projector on the coevaluation target has image exactly the injected copy of $M$: it is identity on that copy, and its induced action on the filtered quotient is zero. Composing with the inverse of this identification gives a sheaf retraction. By step 3.2 the distinguished summand of the evaluation source maps isomorphically onto its target, since the kernel has no such character; invert that restriction to get its sheaf section. Step 3.3 supplies all asserted naturality and $U(\mathfrak{g})$-linearity. Applying sheaf cohomology to either split pair preserves its retraction or section, proving the final assertion.