Two minimal-parabolic projections for SL(3)

Root data and pairings. The flag manifold of $G=S{L}_3(\mathbb{C})$ has $\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }=3$ positive roots ${\alpha }_1,{\alpha }_2,{\alpha }_1+{\alpha }_2$ and $\dim X_B=3$; the simple coroots are the diagonal matrices $h_{{\alpha }_1}=\mathrm{diag}(1,-1,0)$ and $h_{{\alpha }_2}=\mathrm{diag}(0,1,-1)$, which act on the coordinate functionals by $⟨{\epsilon }_i,{\alpha }_j^{\vee }⟩={\delta }_{i,j}-{\delta }_{i,j+1}$. With ${\omega }_1={\epsilon }_1$ and ${\omega }_2={\epsilon }_1+{\epsilon }_2$ this gives the four pairings $⟨{\omega }_1,{\alpha }_1^{\vee }⟩=1,⟨{\omega }_1,{\alpha }_2^{\vee }⟩=0,⟨{\omega }_2,{\alpha }_1^{\vee }⟩=1-1=0,⟨{\omega }_2,{\alpha }_2^{\vee }⟩=0+1=1,$ that is $⟨{\omega }_i,{\alpha }_j^{\vee }⟩={\delta }_{ij}$, as required of the fundamental weights; also $\rho =\frac{1}{2}(2{\alpha }_1+2{\alpha }_2)={\alpha }_1+{\alpha }_2$, so $2\rho =2({\alpha }_1+{\alpha }_2)$ and $⟨2\rho ,{\alpha }_j^{\vee }⟩=2⟨{\alpha }_1+{\alpha }_2,{\alpha }_j^{\vee }⟩=2$ for $j=1,2$, using $⟨{\alpha }_1,{\alpha }_2^{\vee }⟩=⟨{\alpha }_2,{\alpha }_1^{\vee }⟩=-1$.

The flag variety as complete flags. The stabilizer in $G$ of the standard flag $E_1=\mathbb{C}{e}_1\subset E_2=\mathbb{C}{e}_1\oplus \mathbb{C}{e}_2$ is exactly the upper triangular subgroup $B$, and $G$ acts transitively on complete flags: given $0\mathrm{\ne }{v}_1\in L$ and $v_2\in H\setminus L$ one extends to a basis $v_1,v_2,v_3$ adapted to $L\subset H$, and after replacing $v_1$ by ${\lambda }^{-1}{v}_1$ for $\lambda =\det (v_1,v_2,v_3)$ the corresponding matrix lies in $S{L}_3$ and carries the standard flag to $L\subset H$; the replacement does not change $L$. Hence $G\mathrm{/}B\to \{L\subset H\}$, $gB\mapsto g(E_1\subset E_2)$, is a $G$-equivariant bijection.

The two parabolics are the two stabilizers. The subgroup of $G$ stabilising the plane $E_2$ consists of the block matrices $\left(\begin{array}{ccc}\ast & \ast & \ast \\ \ast & \ast & \ast \\ 0& 0& \ast \end{array}\right)$ of determinant one, a closed subgroup of dimension $6$ containing $B$ and the root group $U_{-{\alpha }_1}$; by [F1] the minimal parabolic $P_{{\alpha }_1}$ is an irreducible closed subgroup with $P_{{\alpha }_1}\mathrm{/}B\cong {\mathbb{P}}^1$, so $\dim P_{{\alpha }_1}=\dim B+1=8-3+1=6$, and a closed subgroup of the same dimension containing it equals it; hence $P_{{\alpha }_1}$ is the stabilizer of the plane $E_2$. The same count with the line $E_1$, whose stabilizer is the block subgroup of dimension $6$ containing $B$ and $U_{-{\alpha }_2}$, gives that $P_{{\alpha }_2}$ is the stabilizer of the line $E_1$. Consequently the projection $f_{{\alpha }_1}:X_B\to G\mathrm{/}{P}_{{\alpha }_1}$ forgets the line and keeps the plane, while $f_{{\alpha }_2}$ forgets the plane and keeps the line.

Both fibres are projective lines. Over a plane $H$ the fibre of $f_{{\alpha }_1}$ is the set of lines $L\subset H$, which is $\mathbb{P}H\cong {\mathbb{P}}^1$; over a line $L$ the fibre of $f_{{\alpha }_2}$ is the set of planes $H\supset L$, which corresponds to the set of lines in ${\mathbb{C}}^3\mathrm{/}L$, that is $\mathbb{P}({\mathbb{C}}^3\mathrm{/}L)\cong {\mathbb{P}}^1$. Both descriptions have exactly the two-chart shape $z\mapsto u_{-\alpha }(z)B$, $t\mapsto u_{\alpha }(t)n_{\alpha }B$ of [F1], with $z$ a coordinate on ${\mathbb{A}}^1={\mathbb{P}}^1\setminus \{\mathrm{\infty }\}$ and $t=z^{-1}$ on the other chart.

Fibre degrees. By [F2] the degree of ${\mathcal{L}}_{{\omega }_i}$ on the fibre of the projection attached to ${\alpha }_j$ is $⟨{\omega }_i,{\alpha }_j^{\vee }⟩={\delta }_{ij}$ by step 1.1; hence the fibre-degree pair of ${\mathcal{L}}_{{\omega }_1}$ is $(1,0)$ and that of ${\mathcal{L}}_{{\omega }_2}$ is $(0,1)$ on the two families of fibres. In particular ${\mathcal{L}}_{{\omega }_1}\otimes {\mathcal{L}}_{{\omega }_2}\cong {\mathcal{L}}_{{\omega }_1+{\omega }_2}={\mathcal{L}}_{\rho }$ has fibre-degree pair $(1,1)$, and ${\omega }_{G\mathrm{/}B}\cong {\mathcal{L}}_{-2\rho }$ has degree $⟨-2\rho ,{\alpha }_j^{\vee }⟩=-2$ on both families, matching the behaviour $\mathcal{O}(-2)$ of a canonical bundle on the fibres of a ${\mathbb{P}}^1$-bundle.

Wrap-up and the Axiom of Choice. The two projections of the statement are $f_{{\alpha }_1}$ and $f_{{\alpha }_2}$ of step 1.3, with fibers ${\mathbb{P}}^1$ by step 1.4 and fibre-degree pairs $(1,0)$, $(0,1)$ by step 2.1; the consistency check with ${\omega }_{G\mathrm{/}B}$ is step 2.1. The Axiom of Choice [A1] is assumed in the statement and inherited from the named suppliers [F1]-[F6]; no choice is made in the example itself, whose only selections are the finitely many standard data ${\alpha }_i,{\omega }_i$ and the standard flag.