Partial-flag twist and local product decomposition

The derived algebra $[{\mathfrak{p}}_{\alpha },{\mathfrak{p}}_{\alpha }]$ is spanned by positive-root spaces and the rank-one subalgebra $⟨{\mathfrak{g}}_{\alpha },{\mathfrak{g}}_{-\alpha },{\alpha }^{\vee }⟩$. The functional $\kappa$ vanishes on the root spaces by its definition and on ${\alpha }^{\vee }$ by the assumed zero pairing. It therefore kills the derived algebra, so it is a Lie algebra character of ${\mathfrak{p}}_{\alpha }$. Its transport to the isotropy bundle on $Y$ is well-defined because a character is invariant under conjugation by the connected group $P_{\alpha }$. Applying the PBW quotient construction of [F3] to the action algebroid of $G$ on $Y$ defines ${\mathcal{D}}_{\kappa -\rho }^{\alpha }$ with associated graded $\mathrm{Sym}{T}_Y$.

Choose a Zariski open $U\subseteq Y$ with a section $s:U\to G$ of the $P_{\alpha }$-torsor, as in [F2]. For a principal $H$-bundle $Q\to Z$, write its Atiyah algebroid as $T_Q\mathrm{/}H$, the bundle of $H$-invariant vector fields with its anchor and bracket. The infinitesimal left $G$-action identifies the action algebroid of $G$ on $G\mathrm{/}B$ with this Atiyah algebroid for $G\to X$: on each fiber the left fundamental fields identify $\mathfrak{g}$ with $T_{g}G$, and the sign convention is the one in the displayed action-algebroid bracket. The section identifies $G{\mathrm{\mid }}_U$ with $U\times P_{\alpha }$, so the Borel torsor is $U\times P_{\alpha }\to U\times (P_{\alpha }\mathrm{/}B)$. Its Atiyah algebroid splits as $T_U\oplus {\mathrm{pr}}_2^{\ast }\mathrm{At}(P_{\alpha }\to P_{\alpha }\mathrm{/}B)$, with zero cross-brackets because the two factors act on separate coordinates. The action algebroid of $G$ on $Y=G\mathrm{/}{P}_{\alpha }$ similarly becomes the Atiyah algebroid of $G\to Y$, locally $T_U\oplus (U\times {\mathfrak{p}}_{\alpha })$; quotienting its isotropy by $\kappa {\mathrm{\mid }}_{{\mathfrak{p}}_{\alpha }}$ leaves ${\mathrm{Diff}}_U$, which is the local form of ${\mathcal{D}}_{\kappa -\rho }^{\alpha }$. In the vertical factor $\mathrm{At}(P_{\alpha }\to P_{\alpha }\mathrm{/}B)$, the rank-one isotropy character is zero because $\kappa$ vanishes on the root spaces and on ${\alpha }^{\vee }$; the central isotropy acts trivially on $P_{\alpha }\mathrm{/}B$ and its scalar relation adds no relative differential operator. Thus the vertical quotient is the rank-one TDO with parameter $-{\rho }_{{\mathfrak{s}\mathfrak{l}}_2}$, which by [F3] is $\mathrm{Diff}({\mathcal{O}}_{{\mathbb{P}}^1})$. The PBW normal forms of the direct-product algebroid therefore identify ${\mathcal{D}}_{\kappa -\rho }{\mathrm{\mid }}_{p_{\alpha }^{-1}(U)}\cong {\mathcal{D}}_{{\mathbb{P}}^1}⊠{\mathcal{D}}_{\kappa -\rho }^{\alpha }{\mathrm{\mid }}_U\mathrm{.}$ The line bundle ${\mathcal{O}}_X(\rho )$ restricts to degree $-1$ on the fiber because its associated character pairs to $1$ with ${\alpha }^{\vee }$; hence its restriction is ${\mathcal{O}}_{{\mathbb{P}}^1}(-1)$.

On overlaps, write the torsor sections as $s_j=s_i{p}_{ij}$ for a regular map $p_{ij}:U_i\cap U_j\to P_{\alpha }$. The change of product trivialization is $(u,qB)\mapsto (u,p_{ij}(u)qB)$. On the Atiyah algebroid it is the gauge transformation of the product principal $B$-bundle: its base component includes the derivative of $p_{ij}$ in ${\mathfrak{p}}_{\alpha }$, and its fiber component is the differential of left multiplication by $p_{ij}(u)$ on $P_{\alpha }\mathrm{/}B$. The quotient of the base component by $\kappa {\mathrm{\mid }}_{{\mathfrak{p}}_{\alpha }}$ is exactly the transition of ${\mathcal{D}}_{\kappa -\rho }^{\alpha }$; the fiber component acts by the chain rule on relative operators. The isotropy character is preserved because it is a Lie-algebra character invariant under $\mathrm{Ad}(P_{\alpha })$, and the equality $p_{ik}=p_{ij}{p}_{jk}$ makes the gauge maps compose, giving the cocycle identity. Thus the local product TDO identifications agree on overlaps. The local ordinary inverse-image modules ${\mathcal{O}}_{{\mathbb{P}}^1}⊠N{\mathrm{\mid }}_U$ glue to $p_{\alpha }^{†}N$: in product coordinates a vertical field acts by ${\mathrm{\partial }}_z(f\otimes n)={\mathrm{\partial }}_z(f)\otimes n$, and a base operator acts by its derivative on $f$ plus its given action on $n$. The chain rule is precisely the algebroid transition just checked, so this is a module for the pulled-back TDO, not its regular left module. Finally [F3] identifies ${\mathcal{O}}_X(\rho )\otimes p_{\alpha }^{†}N$ as a ${\mathcal{D}}_{\kappa }$-module because tensoring by ${\mathcal{O}}_X(\rho )$ shifts the parameter from $\kappa -\rho$ to $\kappa$.