PBW and integral line-bundle twisting for flag TDOs

Work on a Zariski open set where $T_X$ and ${\mathfrak{b}}_X$ are free and the anchor splits as an ${\mathcal{O}}_X$-module map. Combine tangent lifts $x_1,\dots ,x_d$ and a frame $b_1,\dots ,b_s$ of ${\mathfrak{b}}_X$ into one frame $e_1,\dots ,e_n$ of the full action algebroid, ordered with the tangent lifts first and the isotropy frame last. For every frame generator use the relations $e_{a}f=f{e}_a+a(e_a)(f)$ and, for $b>a$, $e_b{e}_a=e_a{e}_b+[e_b,e_a]$ to move functions left and order all generators; write each bracket in the chosen frame with its local function coefficients. Reordering terminates by lexicographic induction on generator degree, the number of generator/function inversions (a generator to the left of a coefficient), and the inversion count of the frame-generator word. A commutator correction lowers generator degree; the leading function move lowers the second count, and an out-of-order generator swap lowers the third. The only overlap types are $e_a(fg)$, $e_b{e}_{a}f$, and $e_c{e}_b{e}_a$ for arbitrary frame indices, including isotropy indices. Their reductions agree by the derivation rule for the anchor, the anchor identity $a([e_b,e_a])=[a(e_b),a(e_a)]$, and Jacobi for the action-algebroid bracket [F2]. A terminating rewriting system with every such overlap resolved has unique normal forms: induction on maximum remaining reduction length joins any two first reductions at their common overlap, after which induction identifies the normal forms. Hence the ordered monomials in all $e_a$, with coefficients on the left, form a local ${\mathcal{O}}_X$-basis, and the indicated ordering puts every tangent-lift monomial before isotropy monomials. Their leading symbols are the corresponding symmetric monomials, proving $\mathrm{gr}{\mathcal{U}}_X(\mathfrak{g})\cong {\mathrm{Sym}}_{{\mathcal{O}}_X}({\mathcal{O}}_X\otimes \mathfrak{g})$.

On a chart pulled back to $G$, the isotropy character is the constant $B$-character $\kappa +\rho$. It kills $[\mathfrak{b},\mathfrak{b}]$ and is invariant under $B$-conjugation; equivariance therefore gives ${\chi }^X([x,\xi ])=a(x)({\chi }^X(\xi ))$ for every local algebroid section $x$ and isotropy section $\xi$. In the ordered forms of step 1.1, use $b_a={\chi }^X(b_a)$ to eliminate each isotropy generator. The overlaps of this elimination with two isotropy generators agree because ${\chi }^X$ kills their bracket; the overlaps with a tangent lift agree by the displayed invariance identity; and overlaps with functions agree because $a(b_a)=0$. Thus the quotient has the ordered tangent-lift monomials as a local basis, and its associated graded is ${\mathrm{Sym}}_{{\mathcal{O}}_X}({\mathcal{O}}_X\otimes \mathfrak{g})\mathrm{/}({\mathfrak{b}}_X)\cong {\mathrm{Sym}}_{{\mathcal{O}}_X}{T}_X\mathrm{.}$ The degree-zero part is ${\mathcal{O}}_X$, and the first-order commutator is the anchor inherited from ${\mathcal{U}}_X(\mathfrak{g})$. This proves the TDO axioms and the two-sidedness of the defining ideal.

Suppose $\kappa +\rho$ is integral and put $L={\mathcal{O}}_X(\kappa +\rho )$. The infinitesimal G-linearization of $L$ gives a filtered algebra map ${\mathcal{U}}_X(\mathfrak{g})\to \mathrm{Diff}(L)$. Its isotropy action on each fiber is the character $\kappa +\rho$, so the map kills the ideal in step 2.1 and descends to ${\mathcal{D}}_{\kappa }\to \mathrm{Diff}(L)$. On associated graded it is the identity map $\mathrm{Sym}{T}_X\to \mathrm{Sym}{T}_X$ because both maps send a vector field to its principal symbol. A filtered map with associated-graded isomorphism is an isomorphism: lift any section order by order for surjectivity, and take the leading symbol of a kernel section for injectivity. Thus ${\mathcal{D}}_{\kappa }\cong \mathrm{Diff}(L)$. Taking $\kappa =-\rho$ gives $L={\mathcal{O}}_X$ and the asserted untwisted identity.

For integral $\eta$, write $L_{\eta }={\mathcal{O}}_X(\eta )$. Locally choose a frame $e$ of $L_{\eta }$. Tensoring a ${\mathcal{D}}_{\kappa }$-module with $e$ identifies its local differential-operator action with the action for the twist $\kappa +\eta$. If $e^{\mathrm{\prime }}=ue$ on an overlap, with $u$ an invertible regular function, the two local actions are related by conjugation $D\mapsto u^{-1}Du$; this is exactly the transition rule for the TDO parameter shifted by $\eta$. These local equivalences glue. Tensoring by $L_{-\eta }$ reverses the shift and composes with the first functor to the identity.