Relative canonical weight for a minimal-parabolic flag projection

Product coordinates of the projection. By [F2] the morphisms ${\sigma }_B$ and ${\sigma }_{\alpha }$ are isomorphisms onto open charts, and by [F3] the multiplication $U_{\alpha }^{-}\times U_{-\alpha }\to U^{-}$, $(u,w)\mapsto uw$, is an isomorphism of varieties; put $U={\sigma }_B(U^{-})$ and $V={\sigma }_{\alpha }(U_{\alpha }^{-})$. For $u\in U_{\alpha }^{-}$ and $z\in \mathbb{C}$ one has $f({\sigma }_B(u{u}_{-\alpha }(z)))=u{u}_{-\alpha }(z)[v_{\alpha }]=u[v_{\alpha }]={\sigma }_{\alpha }(u),$ because $u_{-\alpha }(z)\in P_{\alpha }={\mathrm{Stab}}_G([v_{\alpha }])$ by [F1] and [F3]; in particular $f(U)\subseteq V$. Reading $U\cong U_{\alpha }^{-}\times {\mathbb{A}}^1$ and $V\cong U_{\alpha }^{-}$ as affine charts with coordinate rings $\mathcal{O}(U)=\mathbb{C}[x_{\beta },z]$ and $\mathcal{O}(V)=\mathbb{C}[x_{\beta }]$, the displayed computation says that for every $h\in \mathcal{O}(V)$ the regular functions $h\circ f{\mathrm{\mid }}_U$ and $h\circ {\mathrm{p}\mathrm{r}}_1$ on the reduced finite-type $\mathbb{C}$-variety $U_{\alpha }^{-}\times U_{-\alpha }$ agree at every closed point, hence are equal by [F12]; therefore $f{\mathrm{\mid }}_U$ corresponds to the first projection ${\mathrm{p}\mathrm{r}}_1$ and the comorphism $\mathcal{O}(V)\to \mathcal{O}(U)$ is the inclusion of the subring $\mathbb{C}[x_{\beta }]$ into $\mathbb{C}[x_{\beta },z]$.

The fibre and its two charts. Setting $u=1$ in step 1.1, the fibre meets $U$ in the $z$-chart ${\sigma }_B(U_{-\alpha })=\{u_{-\alpha }(z)[v_B]:z\in \mathbb{C}\}\cong {\mathbb{A}}^1$ with fibre coordinate $z$. The $s$-chart lies in the translate ${\mathrm{\ell }}_{n_{\alpha }}(U)=n_{\alpha }{\sigma }_B(U^{-})$: direct multiplication in $S{L}_2(\mathbb{C})$ gives $w^{-1}{u}_{+}(s)w=u_{-}(-s)$ and $u_{+}(s)w=u_{-}(z)\mathrm{diag}(z^{-1},z)u_{+}(-z)$ when $s=z^{-1}$, so applying the morphism ${\phi }_{\alpha }$ of [F5] gives $n_{\alpha }^{-1}{u}_{\alpha }(s)n_{\alpha }=u_{-\alpha }(-s),u_{\alpha }(s)n_{\alpha }=u_{-\alpha }(z){\alpha }^{\vee }(z^{-1})u_{\alpha }(-z)\in u_{-\alpha }(z)B,$ whence $u_{\alpha }(s)n_{\alpha }[v_B]=n_{\alpha }{u}_{-\alpha }(-s)[v_B]\in {\mathrm{\ell }}_{n_{\alpha }}(U)$, and the $s$-chart point coincides with the $z$-chart point of coordinate $z=s^{-1}$, i.e. ${\mathrm{\ell }}_{n_{\alpha }}^{-1}(u_{\alpha }(s)n_{\alpha }[v_B])=u_{-\alpha }(-s)[v_B]$. The two charts are each isomorphic to ${\mathbb{A}}^1$, contain respectively $eB$ ($z=0$) and $n_{\alpha }[v_B]$ ($s=0$), meet in $\{z\mathrm{\ne }0\}=\{s\mathrm{\ne }0\}$, and cover $F$ by [F1].

The relative cotangent sheaf on the big cell. Since $f(U)\subseteq V$, restriction of relative differentials to the open subscheme $U\subseteq X_B$ over $V\subseteq X_{\alpha }$ gives ${\mathrm{\Omega }}_f^1{\mathrm{\mid }}_U={\mathrm{\Omega }}_{U\mathrm{/}V}^1$ with the same universal derivation by [F8]; by step 1.1 and [F8] its global sections are the Kähler module ${\mathrm{\Omega }}_{\mathbb{C}[x_{\beta },z]\mathrm{/}\mathbb{C}[x_{\beta }]}$ of a polynomial extension in the single variable $z$, which is free of rank one with basis $\mathrm{d}z$ by [F9]. Hence ${\mathrm{\Omega }}_f^1$ is free of rank one on $U$ with frame $\mathrm{d}z$. Over the chart $V$ the base-change isomorphism of [F10] applied to ${\mathbb{A}}^1\to \mathrm{Spec}\mathbb{C}$ and $U_{\alpha }^{-}\to \mathrm{Spec}\mathbb{C}$ identifies ${\mathrm{\Omega }}_{U\mathrm{/}V}^1$ with the pullback of ${\mathrm{\Omega }}_{{\mathbb{A}}^1\mathrm{/}\mathbb{C}}$ along the second projection; the fibre of that projection over the point $u=1$ is $\{1\}\times {\mathbb{A}}^1\cong {\mathbb{A}}^1$, and restricting the pullback to it returns ${\mathrm{\Omega }}_{{\mathbb{A}}^1\mathrm{/}\mathbb{C}}$ on the nose with frame $\mathrm{d}z$. So the fibre chart $F\cap U$ carries the cotangent sheaf of the affine line with frame $\mathrm{d}z$, the restriction of the frame of ${\mathrm{\Omega }}_f^1$ over $U$.

The canonical equivariant structure and invertibility. For $g\in G$ the left translations on $X_B$ and $X_{\alpha }$ are automorphisms satisfying $f\circ {\mathrm{\ell }}_g={\mathrm{\ell }}_g\circ f$ by [F1]; thus they form an automorphism of the arrow $f$, with the base also translated. On affine charts the universal relative derivation sends a function $a$ to $\mathrm{d}a$ modulo differentials pulled back from the base. Since ${\mathrm{\ell }}_g^{\ast }$ takes base functions to base functions, its differential induces a canonical ${\mathcal{O}}_{X_B}$-linear isomorphism ${\mathrm{\ell }}_g^{\ast }{\mathrm{\Omega }}_f^1\to {\mathrm{\Omega }}_f^1$; the inverse comes from $g^{-1}$ and the cocycle law from the chain rule of [F11]. These algebraic isomorphisms give the $G$-equivariant structure, with $g$ acting on local forms by pullback along ${\mathrm{\ell }}_{g^{-1}}$. Moreover the freeness of rank one proved on $U$ in step 2.2 transports along the isomorphisms to each open chart ${\mathrm{\ell }}_g(U)=g{\sigma }_B(U^{-})$, and these charts cover $X_B$ because $X_B=G\cdot [v_B]\subseteq G\cdot (U^{-}[v_B])={\bigcup }_{g\in G}{\mathrm{\ell }}_g(U)$ by the transitivity of [F2]. Hence ${\mathrm{\Omega }}_f^1$ is an invertible sheaf of rank one, ${\omega }_{(G\mathrm{/}B)\mathrm{/}(G\mathrm{/}{P}_{\alpha })}=\det {\mathrm{\Omega }}_f^1={\mathrm{\Omega }}_f^1$ is a $G$-equivariant line bundle on $X_B$, and its fibre at every point is one-dimensional.

The $T$-weight of the fibre at $eB$. The torus $T$ normalizes $U^{-}$, $U_{\alpha }^{-}$ and $U_{-\alpha }$ by [F3], so ${\mathrm{\ell }}_t$ preserves the chart $U$ for every $t\in T$, and by [F3] and the conjugation formula of [F4] its action in the coordinates of step 1.1 is ${\mathrm{\ell }}_t({\sigma }_B(u{u}_{-\alpha }(z_0)))=\left(tu{t}^{-1}\right)(t{u}_{-\alpha }(z_0)t^{-1})[v_B]={\sigma }_B(tu{t}^{-1}\cdot u_{-\alpha }(\alpha (t{)}^{-1}{z}_0)),$ because $t[v_B]=[v_B]$. Hence the coordinate function $z$ satisfies ${\mathrm{\ell }}_{t^{-1}}^{\ast }(z)=\alpha (t)z$ on $U$, and taking differentials, as is legitimate for the pullback of forms under a morphism and compatible with the universal derivation by [F11], the frame $\mathrm{d}z$ of step 2.2 satisfies $t\cdot \mathrm{d}z={\mathrm{\ell }}_{t^{-1}}^{\ast }(\mathrm{d}z)=\mathrm{d}({\mathrm{\ell }}_{t^{-1}}^{\ast }(z))=\alpha (t)\mathrm{d}z\mathrm{.}$ Evaluating at the $T$-fixed point $eB$, where $z=0$, this says that $T$ acts on the one-dimensional fibre $({\mathrm{\Omega }}_f^1{)}_{eB}$ by the character $\alpha \in X^{\ast }(T)$: the fibre weight is the root $\alpha$.

The $B$-character and ${\mathrm{\Omega }}_f^1\cong {\mathcal{L}}_{-\alpha }$. By [F2] the stabilizer of $[v_B]$ is $B$, so the $G$-equivariant structure of step 3.1 restricts to an action of $B$ on the one-dimensional fibre $({\mathrm{\Omega }}_f^1{)}_{eB}$; this action is algebraic and linear, hence given by a character $\chi \in X^{\ast }(B)$ of [F7]. Step 3.2 computes $\chi {\mathrm{\mid }}_T=\alpha {\mathrm{\mid }}_T$, and by [F7] every character of $B$ is trivial on $U$ and the restriction $X^{\ast }(B)\to X^{\ast }(T)$ is an isomorphism, so $\chi$ is the unique character of $B$ extending the root $\alpha$; in particular $\chi (b)=\alpha (b)$ for all $b\in B$. By [F6] the fibre of ${\mathcal{L}}_{-\alpha }$ at $eB$ is the $B$-module on which $b$ acts by $(-\alpha )(b{)}^{-1}=\alpha (b)$, that is, by the same character $\chi$. The fibre functor of [F7] is an equivalence and therefore reflects isomorphism classes: two $G$-equivariant line bundles on $X_B$ whose fibres at $eB$ are isomorphic as $B$-modules are $G$-equivariantly isomorphic, and the isomorphism is unique up to a scalar; hence ${\mathrm{\Omega }}_f^1\cong {\mathcal{L}}_{-\alpha }$ as $G$-equivariant line bundles.

Restriction to the fibre and its degree. On the $z$-chart the frame $\mathrm{d}z$ of ${\mathrm{\Omega }}_f^1$ restricts to the fibre as computed in step 2.2. On the $s$-chart, inside the translate ${\mathrm{\ell }}_{n_{\alpha }}(U)$, the transported frame is $\mathrm{d}{z}^{\mathrm{\prime }}$ with $z^{\mathrm{\prime }}=z\circ {\mathrm{\ell }}_{n_{\alpha }}^{-1}$, since pulling a differential form back along ${\mathrm{\ell }}_{n_{\alpha }}^{-1}$ and applying $\mathrm{d}$ to the pulled-back coordinate gives $\mathrm{d}(z^{\mathrm{\prime }})={\mathrm{\ell }}_{n_{\alpha }^{-1}}^{\ast }(\mathrm{d}z)$ by [F11]; by step 2.1 one has $z^{\mathrm{\prime }}(u_{\alpha }(s)n_{\alpha }[v_B])=z(u_{-\alpha }(-s)[v_B])=-s$, so on the overlap $z^{\mathrm{\prime }}=-z^{-1}$ and hence $\mathrm{d}{z}^{\mathrm{\prime }}=\mathrm{d}(-z^{-1})=z^{-2}\mathrm{d}z$, using that $z$ is a unit on the overlap and that $\mathrm{d}(z^{-1})=-z^{-2}\mathrm{d}z$ holds for the universal derivation localized there by [F8]. Under the identification of $F$ with the two-affine projective line of [F13] given by $U_0\mapsto \{z\}$, $t=z$ and $U_{\mathrm{\infty }}\mapsto \{s\}$, $u=s$, the frames $e_0=\mathrm{d}z$ and $e_{\mathrm{\infty }}=\mathrm{d}{z}^{\mathrm{\prime }}$ satisfy $e_{\mathrm{\infty }}=t^{-2}{e}_0$ on the overlap, which is exactly the gluing prescription defining $\mathcal{O}(-2)$ in [F13]; by the uniqueness of gluing [F14] and the well-definedness of the twist index [F13], ${\mathrm{\Omega }}_f^1{\mathrm{\mid }}_F\cong {\mathcal{O}}_{{\mathbb{P}}^1}(-2)$, of degree $-2$. The computation is confirmed by the equivariant description: by step 4.1 and [F6], ${\mathrm{\Omega }}_f^1{\mathrm{\mid }}_F\cong {\mathcal{L}}_{-\alpha }{\mathrm{\mid }}_F\cong \mathcal{O}(⟨-\alpha ,{\alpha }^{\vee }⟩)=\mathcal{O}(-2)$ has degree $⟨-\alpha ,{\alpha }^{\vee }⟩=-2$ since $⟨\alpha ,{\alpha }^{\vee }⟩=\alpha (h_{\alpha })=2$; the two routes give the same frame transition up to the fixed identifications, so no sign ambiguity remains. For a general fibre $gF=f^{-1}(g[v_{\alpha }])={\mathrm{\ell }}_g(F)$ the translation identifies ${\mathrm{\Omega }}_f^1{\mathrm{\mid }}_{gF}$ with the pullback of ${\mathrm{\Omega }}_f^1{\mathrm{\mid }}_F$ along an isomorphism of projective lines, so by the invariance of the twist index under isomorphism [F13] the restriction to every fibre is again $\mathcal{O}(-2)$ of degree $-2$.

Conclusion and choice bookkeeping. Steps 2.2 and 3.1 show that ${\mathrm{\Omega }}_f^1$ is an invertible sheaf of rank one with a canonical $G$-equivariant structure, so the relative canonical bundle ${\omega }_{(G\mathrm{/}B)\mathrm{/}(G\mathrm{/}{P}_{\alpha })}=\det {\mathrm{\Omega }}_f^1={\mathrm{\Omega }}_f^1$ is a $G$-equivariant line bundle; steps 3.2 and 4.1 compute its fibre character at $eB$ as the root $\alpha$ and conclude the $G$-equivariant isomorphism ${\omega }_{(G\mathrm{/}B)\mathrm{/}(G\mathrm{/}{P}_{\alpha })}\cong {\mathcal{L}}_{-\alpha }$, whose fibre at $eB$ is ${\mathbb{C}}_{\alpha }$; step 5.1 computes the restriction to every fibre as $\mathcal{O}(-2)$ of degree $⟨-\alpha ,{\alpha }^{\vee }⟩=-2$, in agreement with the degree of ${\mathcal{L}}_{-\alpha }$ on the fibre. In the degenerate rank-one case $\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }=1$, i.e. $G=P_{\alpha }$, the base $X_{\alpha }$ is a single point, $U_{\alpha }^{-}=1$, the chart $V$ is the whole base, and step 1.1 is the projection ${\mathbb{A}}^1\to \mathrm{Spec}\mathbb{C}$ on the affine chart $U$ of $X_B\cong {\mathbb{P}}^1$; steps 2.2–5.1 apply verbatim with ${\mathrm{\Omega }}_f^1={\omega }_{X_B}$, the canonical bundle of the projective line. The Axiom of Choice [A1] is assumed in the statement and is inherited through the quotient, torsor, representation-theoretic and differential suppliers behind [F1], [F2], [F7] and [F8]; the proof itself fixes only the two charts of the flag varieties, the root coordinates $z$ and $x_{\beta }$, and the finitely many root data, making no further choice. The quotient and chart claims used here are supplied by [F1] and [F2], while [F6] and [F7] supply the equivariant line bundle comparison.