Exact twisted associated bundles and simple-parabolic descent

The relation $[gs,h]=[g,r(s)h]$ defines the associated sheaf by [F1]; rationality means every vector belongs to a finite-dimensional rational $S$-submodule, so the construction is the filtered union of the usual finite-rank associated bundles. On a torsor section it is precisely ${\mathcal{O}}_Y\otimes H$. A short exact sequence of twisted modules therefore becomes a short exact sequence of sheaves, since tensoring over $\mathbb{C}$ is exact. Equivariant morphisms give the same local maps on all charts.

Let $\beta (A)$ be the infinitesimal $G$-action on the associated sheaf. Define an ${\mathcal{O}}_Y$-linear action by $\gamma (A)[g,h]=[g,\pi (\mathrm{Ad}(g^{-1})A)h]\mathrm{.}$ The covariance hypothesis makes this independent of replacing $g$ by $gs$. Differentiating covariance gives $[\beta (A),\gamma (A^{\mathrm{\prime }})]=\gamma ([A,A^{\mathrm{\prime }}])$; the representation property gives $[\gamma (A),\gamma (A^{\mathrm{\prime }})]=\gamma ([A,A^{\mathrm{\prime }}])$. Consequently $a(A)=\beta (A)-\gamma (A)$ satisfies $[a(A),f]=A_Y(f),[a(A),a(A^{\mathrm{\prime }})]=a([A,A^{\mathrm{\prime }}])\mathrm{.}$ At the base point, an isotropy element $x\in \mathfrak{s}$ acts by $dr(x)-\pi (x)=\tau (x)$; transport gives the same isotropy relation at every point. Thus $a$ factors through ${\mathcal{A}}_{\tau }^S$. The displayed identities also show that maps in step 1.1 are differential-module maps, proving exactness of the differential functor.

The associated-bundle map for a finite $G$-module is $[g,h\otimes v]⟼[g,h]\otimes gv\mathrm{.}$ It respects the $S$ relation; its inverse sends $[g,h]\otimes w$ to $[g,h\otimes g^{-1}w]$. Under this map $\beta$ becomes ${\beta }_H\otimes 1+1\otimes dV$, because the $G$-equivariant structure on the trivial bundle has the diagonal $G$-action. Likewise $\gamma$ becomes ${\gamma }_H\otimes 1+1\otimes dV$: the identity $gdV(\mathrm{Ad}(g^{-1})A)g^{-1}=dV(A)$ gives its second summand. Subtraction cancels these finite-representation terms exactly. Thus $a=a_H\otimes 1$, with the Leibniz rule on function coefficients, and constant vectors of $V$ are horizontal. This proves the asserted identity as a differential-module identity, not merely as an equivariant bundle identity.

More generally fix an integral $B$-weight $\eta$ and put $\tau =\kappa +\rho$. On $M(-\tau +\eta )$ define the rational $B$-action by conjugating the $U(\mathfrak{g})$ factor and letting the highest vector have weight $\eta$. The infinitesimal action is $\pi +\tau$: commuting $x\in \mathfrak{b}$ past the enveloping factor leaves the scalar $(-\tau +\eta )(x)$ on the highest vector, whereas the rational action leaves $\eta (x)$. Its rationality follows from the finite-dimensional adjoint filtration of $U(\mathfrak{g})$ and the rational character $\eta$. The highest line gives a line subbundle ${\mathcal{O}}_X(\eta )$ of the associated sheaf. Apply the differential action to obtain ${\mathcal{D}}_{\kappa }{\otimes }_{{\mathcal{O}}_X}{\mathcal{O}}_X(\eta )\to {\mathsf{E}}_{\tau }^B(M(-\tau +\eta ))$. Locally on a torsor section, filter both sides by the degree of the negative-root PBW monomials. On the right, $\gamma$ supplies those monomials with sign $-1$ and $\beta$ preserves the associated filtration; hence the leading map sends a degree-$d$ tangent monomial to $(-1{)}^d$ times the corresponding Verma monomial. By [F2]–[F3] this is an isomorphism of symmetric tangent modules. Induction on degree therefore proves the filtered map is an isomorphism. The cases $\eta =0,\rho$ are the claimed formulas.

For twisted $(\mathfrak{g},P_{\alpha })$-module $H$ of twist $\kappa$, the character $\kappa$ is defined on ${\mathfrak{p}}_{\alpha }$ because its $\alpha$-coroot pairing is zero. The construction of steps 1.1 and 2.1 on $Y=G\mathrm{/}{P}_{\alpha }$ gives $N$ over ${\mathcal{A}}_{\kappa }^{P_{\alpha }}={\mathcal{D}}_{\kappa -\rho }^{\alpha }$. Restriction from $P_{\alpha }$ to $B$ identifies associated underlying sheaves $G{\times }^{B}H=p_{\alpha }^{\ast }(G{\times }^{P_{\alpha }}H)$. In a product chart its underlying module is ${\mathcal{O}}_{{\mathbb{P}}^1}⊠N$. For a vertical vector field, its $\beta$ and fiberwise $\gamma$ terms on the transported $H$ cancel, leaving only differentiation of the relative function coefficient. For base fields the same formula is the original $\beta -\gamma$ action on $N$. These are precisely the ordinary inverse image action in [F4]. They also prove agreement on overlaps, since both sides use the same $P_{\alpha }$ transition and chain rule. Thus ${\mathsf{E}}_{\kappa }^B(H)=p_{\alpha }^{†}N$.

Multiplying the restricted rational $B$-action by the integral character $\rho$ adds $\rho$ to its differential while leaving $\pi$ unchanged. The associated bundle is therefore tensored by ${\mathcal{O}}_X(\rho )$. By [F2] its differential twist becomes $\kappa +\rho$, so its TDO is ${\mathcal{D}}_{\kappa }$. Combine this with step 3.3 to obtain the displayed parabolic-descent identity. Choices of bases and of local torsor sections use only the inherited Axiom of Choice; all resulting constructions and isomorphisms are canonical.