Flag-variety twisted differential operators and their modules

Assume the Axiom of Choice as inherited from the published group and sheaf suppliers. Let $G$ be a connected simply connected complex semisimple affine algebraic group, $B\supseteq T$ a Borel subgroup and maximal torus, $X=G\mathrm{/}B$, $\mathfrak{g}=\mathrm{Lie}G$, $\mathfrak{h}=\mathrm{Lie}T$, and ${\mathrm{\Phi }}^{+}$ the positive roots selected by $B$. Write $\rho =\frac{1}{2}{\sum }_{\alpha \in {\mathrm{\Phi }}^{+}}\alpha$. All schemes are over $\mathbb{C}$. For an integral weight $\eta$, ${\mathcal{O}}_X(\eta )$ denotes the $G$-equivariant line bundle $G{\times }^B{\mathbb{C}}_{\eta }$ with fiber character $\eta$ at $eB$.

The infinitesimal $G$-action gives an anchor $a_X:{\mathcal{L}}_X:={\mathcal{O}}_X{\otimes }_{\mathbb{C}}\mathfrak{g}⟶T_X,f\otimes x⟼f{x}_X\mathrm{.}$ Give ${\mathcal{L}}_X$ the action-algebroid bracket $[f\otimes x,g\otimes y]=fg\otimes [x,y]+f{x}_X(g)\otimes y-g{y}_X(f)\otimes x\mathrm{.}$ Its kernel ${\mathfrak{b}}_X=\ker a_X$ is the isotropy Lie-algebra bundle; at $gB$ its fiber is $\mathrm{Ad}(g)\mathfrak{b}$.

For any $\kappa \in {\mathfrak{h}}^{\ast }$, the character $\kappa +\rho$ of $\mathfrak{b}$ (zero on ${\mathfrak{n}}^{+}$) transports to an ${\mathcal{O}}_X$-linear map ${\chi }_{\kappa +\rho }^X:{\mathfrak{b}}_X\to {\mathcal{O}}_X$. The transport is independent of the representative $g$ of $gB$, since a character is invariant under the adjoint action of $B$. The action-algebroid enveloping sheaf ${\mathcal{U}}_X(\mathfrak{g})$ is generated by ${\mathcal{O}}_X$ and ${\mathcal{L}}_X$ with the displayed bracket and $[x,f]=x_X(f),x\in {\mathcal{L}}_X,f\in {\mathcal{O}}_X\mathrm{.}$ Define the sheaf of twisted differential operators ${\mathcal{D}}_{\kappa }:={\mathcal{U}}_X(\mathfrak{g})\mathrm{/}(\xi -{\chi }_{\kappa +\rho }^X(\xi ):\xi \in {\mathfrak{b}}_X)\mathrm{.}$ Its order filtration assigns order zero to ${\mathcal{O}}_X$ and order at most one to the image of ${\mathcal{L}}_X$. A TDO means a filtered ${\mathcal{O}}_X$-ring with $F_0={\mathcal{O}}_X$, associated graded ${\mathrm{Sym}}_{{\mathcal{O}}_X}{T}_X$, and first-order commutator equal to the anchor; the well-definedness and PBW properties of ${\mathcal{D}}_{\kappa }$ are proved in PBW and integral line-bundle twisting for flag TDOs ↗. A left ${\mathcal{D}}_{\kappa }$-module below is required to be quasi-coherent as an ${\mathcal{O}}_X$-module.

For $z\in Z(U(\mathfrak{g}))$, let $\mathrm{pr}(z)\in U(\mathfrak{h})$ be the Harish–Chandra projection along the chosen triangular decomposition, and put ${\chi }_{\kappa }(z)=\mathrm{pr}(z)(\kappa -\rho )\mathrm{.}$ This is the shifted central character: the Harish–Chandra isomorphism makes it $W$-invariant in $\kappa$. Define $A_{\kappa }=U(\mathfrak{g})\mathrm{/}U(\mathfrak{g})\ker {\chi }_{\kappa }\mathrm{.}$ The regularity convention is antidominant when $⟨\kappa ,{\alpha }^{\vee }⟩\mathrm{\notin }{\mathbb{Z}}_{>0}$ for each positive root $\alpha$, and regular when $⟨\kappa ,{\alpha }^{\vee }⟩\mathrm{\ne }0$ for every root. For complex weights, ${\mathbb{Z}}_{>0}$ means the positive real integers; no order on arbitrary complex numbers is used.

For integral $\kappa$, the proved identification is ${\mathcal{D}}_{\kappa }=\mathrm{Diff}({\mathcal{O}}_X(\kappa +\rho ))$. In particular ${\mathcal{D}}_{-\rho }=\mathrm{Diff}({\mathcal{O}}_X)$. The construction uses the isotropy sheaf ${\mathfrak{b}}_X$; quotienting $U(\mathfrak{g})$ by the fixed subalgebra $\mathfrak{b}$ would not define an algebra. With the convention of The equivariant line bundle associated to a Borel character, ${\mathcal{O}}_X(\eta )={\mathcal{L}}_{-\eta }$.