Global sections of twisted differential operators on the flag variety

Let ${\varphi }_{\kappa }:U(\mathfrak{g})\to \mathrm{\Gamma }(X,{\mathcal{D}}_{\kappa })$ be the infinitesimal-action map. Use the fiber $Q={\mathcal{D}}_{\kappa ,eB}\mathrm{/}{\mathcal{D}}_{\kappa ,eB}{\mathfrak{m}}_{eB}$ for the right ${\mathcal{O}}_X$-module structure, and write $\delta$ for the class of $1$. This is a left $U(\mathfrak{g})$-module. Choose local coordinates $t_j$ at $eB$ and a local differential-operator trivialization. For $b\in \mathfrak{b}$, its operator is ${\sum }_j{f}_j{\mathrm{\partial }}_{t_j}+c_b$, with $f_j(eB)=0$ and $c_b(eB)=(\kappa +\rho )(b)$ by the isotropy relation. Moving coefficients to the right gives $b\delta =(c_b(eB)-{\sum }_j{\mathrm{\partial }}_{t_j}{f}_j(eB))\delta$. The anchor action on functions is the negative of the tangent action on points, so its linear trace on $\mathfrak{g}\mathrm{/}\mathfrak{b}$ is $2\rho (b)$, since the tangent weights are the negative roots; equivalently this is the canonical character in [F4]. Thus $b\delta =(\kappa -\rho )(b)\delta$, and ${\mathfrak{n}}^{+}\delta =0$. Negative-root vector fields span the tangent space at $eB$; PBW symbols [F1] show that their ordered monomials on $\delta$ span $Q$ by induction on order. Hence $Q$ is a cyclic highest-weight module of weight $\kappa -\rho$, so every central $z$ acts by $\mathrm{pr}(z)(\kappa -\rho )={\chi }_{\kappa }(z)$ by [F5]. Equivariance gives the same scalar on every closed right fiber. Only Lie algebra characters were used, so no nonintegral line bundle is required.

Fix an order bound $m$. The space $U(\mathfrak{g}{)}_{\le m}$ is finite-dimensional by PBW, and $F_m{\mathcal{D}}_{\kappa }$ is locally free of finite rank as a right ${\mathcal{O}}_X$-module by the same PBW normal forms [F1]. For $z\in \ker {\chi }_{\kappa }$, the right ${\mathcal{O}}_X$-linear multiplication map $U(\mathfrak{g}{)}_{\le m}\otimes {\mathcal{O}}_X\to F_{m+d(z)}{\mathcal{D}}_{\kappa }$ given by $u\otimes f\mapsto {\varphi }_{\kappa }(u(z-{\chi }_{\kappa }(z)))f$ is zero on every closed fiber by step 1.1. In local frames this is a finite matrix of regular functions vanishing at every closed point; $X$ is reduced and of finite type over $\mathbb{C}$, so each entry is zero. Thus ${\varphi }_{\kappa }$ kills the whole two-sided central ideal and factors through a filtered map ${\bar{\varphi }}_{\kappa }:A_{\kappa }\to \mathrm{\Gamma }(X,{\mathcal{D}}_{\kappa })$. This finite-rank argument does not apply Nakayama to an infinitely generated module.

The associated-graded map fits into $\mathbb{C}[N]⟶\mathrm{gr}{A}_{\kappa }⟶\mathrm{gr}\mathrm{\Gamma }(X,{\mathcal{D}}_{\kappa })\hookrightarrow \mathrm{\Gamma }(X,\mathrm{gr}{\mathcal{D}}_{\kappa })=\mathrm{\Gamma }(T^{\ast }X,\mathcal{O})\mathrm{.}$ The last inclusion follows by applying the left-exact functor $\mathrm{\Gamma }$ to each order-filtration sequence. The composite is pullback by the Springer moment map: for $x\in \mathfrak{g}$, its linear function on $N$ pulls back to the principal symbol of the vector field $x_X$, namely $\xi \mapsto \xi (x_X)$ on $T^{\ast }X$. Linear functions generate $\mathbb{C}[N]$ as a quotient of $S(\mathfrak{g})$, so the two algebra maps agree on all generators. By [F3] the composite is an isomorphism.

Since the composite in [step 3.1] is an isomorphism and its right-hand map is injective, both intermediate associated-graded maps are isomorphisms. A filtered map with associated-graded isomorphism is an isomorphism: lift a target symbol and subtract a lift to lower the order until surjectivity holds; for injectivity, a nonzero kernel element would have a nonzero leading symbol in the graded kernel. Hence ${\bar{\varphi }}_{\kappa }$ is the asserted filtered algebra isomorphism. The argument uses no order condition on $\kappa$.