Flag line-bundle degree on a minimal-parabolic fiber

The fibre and its two frames. By [F1] the fibre of $f$ over $[v_{\alpha }]$ is $F=P_{\alpha }[v_B]=P_{\alpha }B\mathrm{/}B$, and by [F2] it is covered by the two chart maps $z\mapsto u_{-\alpha }(z)[v_B]$ and $s\mapsto u_{\alpha }(s)n_{\alpha }[v_B]$; these are injective, agree exactly at $s=z^{-1}$ with $z\mathrm{\ne }0$, and their images are complementary in the sense that the first contains $B$ but not $n_{\alpha }B$ and the second contains $n_{\alpha }B$ but not $B$, so together they cover $F$. For a point $x$ of the first chart define $e_0(x)=[u_{-\alpha }(z(x)),1]$, where $z(x)$ is the unique preimage of $x$, and for a point $x$ of the second chart define $e_{\mathrm{\infty }}(x)=[u_{\alpha }(s(x))n_{\alpha },1]$. Since $u_{-\alpha }(z)\in P_{\alpha }$ and $u_{\alpha }(s)n_{\alpha }\in P_{\alpha }$ the classes lie in the fibre of ${\mathcal{L}}_{\lambda }$ at $x$, and by [F10] that fibre is $\{[g,v]:v\in \mathbb{C}\}\cong \mathbb{C}$ with the second coordinate $v=1\mathrm{\ne }0$, so $e_0$ and $e_{\mathrm{\infty }}$ are nowhere-vanishing sections and therefore frames trivializing ${\mathcal{L}}_{\lambda }{\mathrm{\mid }}_F$ over the two charts.

Matrix identity and change of lift. For $z\in {\mathbb{C}}^{\times }$ put $s=z^{-1}$. Direct multiplication in $S{L}_2(\mathbb{C})$ gives $u_{+}(s)w=\left(\begin{array}{cc}s& -1\\ 1& 0\end{array}\right)=\left(\begin{array}{cc}1& 0\\ z& 1\end{array}\right)\left(\begin{array}{cc}{z}^{-1}& -1\\ 0& z\end{array}\right)=u_{-}(z)\mathrm{diag}(z^{-1},z)u_{+}(-z),$ where the middle factorisation uses $\mathrm{diag}(z^{-1},z)u_{+}(-z)=\left(\begin{array}{cc}{z}^{-1}& -1\\ 0& z\end{array}\right)$. Applying the morphism ${\phi }_{\alpha }$ of [F3] to this product identity and using its values on the standard unipotent subgroups, on $w$ and on the diagonal torus gives, in $G$, $u_{\alpha }(s)n_{\alpha }=u_{-\alpha }(z){\alpha }^{\vee }(z^{-1})u_{\alpha }(-z)(s=z^{-1})\mathrm{.}$ The right-hand factor $b_z:={\alpha }^{\vee }(z^{-1})u_{\alpha }(-z)$ lies in $B$, because ${\alpha }^{\vee }(z^{-1})\in T$, $u_{\alpha }(-z)\in U$ and $B=T⋉U$ by [F4].

Character value of the change of lift. Every character of $B$ is trivial on the unipotent radical by [F4], so $\lambda (u_{\alpha }(-z))=1$, while [F5] identifies the symbol ${\alpha }^{\vee }$ with the coroot of $\alpha$ and [F3] gives $\lambda ({\alpha }^{\vee }(z^{-1}))=(z^{-1}{)}^{⟨\lambda ,{\alpha }^{\vee }⟩}$. Hence with $m:=⟨\lambda ,{\alpha }^{\vee }⟩$ one has $(-\lambda )(b_z)=\lambda (b_z{)}^{-1}=((z^{-1}{)}^m{)}^{-1}=z^m\mathrm{.}$

Change of frame. On the overlap, step 1.2 exhibits the same point of $F$ with the two lifts $u_{\alpha }(s)n_{\alpha }$ and $u_{-\alpha }(z)$, and the equivalence relation of [F10] applied to $b_z$ gives $e_{\mathrm{\infty }}(x)=[u_{\alpha }(s)n_{\alpha },1]=[u_{-\alpha }(z)b_z,1]=[u_{-\alpha }(z),b_z\cdot 1]=(-\lambda )(b_z)[u_{-\alpha }(z),1]=z^m{e}_0(x)\mathrm{.}$

The identification with the standard projective line. Identify $F$ with ${\mathbb{P}}_{\mathbb{C}}^1$ as fixed in the statement by sending the $z$-chart to $U_0$ with $t=z$ and the $s$-chart to $U_{\mathrm{\infty }}$ with $u=s$; the gluing relation $s=z^{-1}$ of the fibre's overlap matches $tu=1$, and by step 1.1 the two charts cover both sides, so this is an isomorphism of varieties using the quotient variety structure supplied by [F2]. Under this identification step 2.1 says that the frames $e_0,e_{\mathrm{\infty }}$ of ${\mathcal{L}}_{\lambda }{\mathrm{\mid }}_F$ are related by $e_{\mathrm{\infty }}=t^m{e}_0$ on the overlap, which is exactly the prescription by which [F7] glues the invertible sheaf $\mathcal{O}(m)$ from its two chart trivializations. Both ${\mathcal{L}}_{\lambda }{\mathrm{\mid }}_F$ and $\mathcal{O}(m)$ therefore admit trivializations on $U_0$ and $U_{\mathrm{\infty }}$ whose induced overlap identifications agree (both are multiplication by $t^{-m}$), and the uniqueness part of the gluing theorem [F9] gives an isomorphism ${\mathcal{L}}_{\lambda }{\mathrm{\mid }}_F\cong \mathcal{O}(m)$ compatible with these trivializations.

Degree. By [F8] the twist index of an invertible sheaf on ${\mathbb{P}}^1$ that is isomorphic to a twist is well defined, so step 3.1 computes the degree of ${\mathcal{L}}_{\lambda }{\mathrm{\mid }}_F$ under the fixed identification to be $m=⟨\lambda ,{\alpha }^{\vee }⟩$. The computation covers positive, zero and negative $m$: for every $m\in \mathbb{Z}$ the transition $z^m$ is a unit on the overlap $z\mathrm{\ne }0$.

The root cases. By [F3] and [F6], $⟨\alpha ,{\alpha }^{\vee }⟩=\alpha (h_{\alpha })=2$, because $[h_{\alpha },e_{\alpha }]=2e_{\alpha }$ with $e_{\alpha }\in {\mathfrak{g}}_{\alpha }$ and ${\mathfrak{g}}_{\alpha }$ consists of the vectors with $[H,x]=\alpha (H)x$; substituting $\lambda =\alpha$ and $\lambda =-\alpha$ in step 4.1 gives ${\mathcal{L}}_{\alpha }{\mathrm{\mid }}_F\cong \mathcal{O}(2)$ of degree $2$ and ${\mathcal{L}}_{-\alpha }{\mathrm{\mid }}_F\cong \mathcal{O}(-2)$ of degree $-2$. For $\lambda =0$ the same computation gives $m=0$ and ${\mathcal{L}}_0{\mathrm{\mid }}_F\cong \mathcal{O}(0)$, consistent with ${\mathcal{L}}_0={\mathcal{O}}_{X_B}$; and a nonzero character with $⟨\lambda ,{\alpha }^{\vee }⟩=0$ has degree $0$, so the degree records exactly the restriction of $\lambda$ to the coroot torus ${\alpha }^{\vee }({\mathbb{C}}^{\times })$.

Conclusion. Steps 1.1-2.1 trivialize the restriction and compute the change of frame, step 3.1 identifies it with the standard twist $\mathcal{O}(m)$, and steps 4.1-5.1 extract the degree $m=⟨\lambda ,{\alpha }^{\vee }⟩$ with the special cases $\lambda =\pm \alpha$. The Axiom of Choice [A1] is assumed in the statement and is inherited through the orbit-quotient, minimal-parabolic and flag-torsor suppliers behind [F1], [F2] and [F10]; the argument itself makes no further choice, the only data fixed being the two chart coordinates and the single matrix identity of step 1.2. The quotient, two-chart fibre and associated bundle used here are supplied by [F1], [F2] and [F10].