Canonical weight of a flag variety

The big-cell chart and the torus action. By [F4] the map ${\sigma }_B:U^{-}\to V\subset X$ is an isomorphism onto an open affine chart containing $eB$, and by [F2] the chart carries the coordinates $z_{\beta }$ ($\beta \in {\mathrm{\Phi }}^{+}$) of $U^{-}$. For $t\in T$ and $u\in U^{-}$ one has ${\mathrm{\ell }}_t({\sigma }_B(u))=tu[v_B]=(tu{t}^{-1})[v_B]={\sigma }_B(tu{t}^{-1})$, because $t^{-1}$ fixes $[v_B]$ by [F1]; hence $V$ is $T$-stable and $t$ acts on the chart by conjugation of $U^{-}$, which by [F3] scales the coordinate $z_{\beta }$ by $\beta (t{)}^{-1}$. Consequently the comorphism of the action satisfies ${\mathrm{\ell }}_{t^{-1}}^{\ast }(z_{\beta })=\beta (t)z_{\beta }$ for every $\beta \in {\mathrm{\Phi }}^{+}$.

The cotangent space at $eB$. By [F6] the ${\mathcal{O}}_X$-module ${\mathrm{\Omega }}_{X\mathrm{/}\mathbb{C}}^1$ restricted to the affine chart $V=\mathrm{Spec}\mathbb{C}[z_{\beta }]$ corresponds to the Kähler differential module ${\mathrm{\Omega }}_{\mathbb{C}[z_{\beta }]\mathrm{/}\mathbb{C}}$, which is free with basis the differentials $\mathrm{d}{z}_{\beta }$. Its fibre at the origin $eB$, the maximal ideal $(z_{\beta })$, is therefore the $\mathbb{C}$-vector space $({\mathrm{\Omega }}_{X\mathrm{/}\mathbb{C}}^1{)}_{eB}={⨁}_{\beta \in {\mathrm{\Phi }}^{+}}\mathbb{C}\mathrm{d}{z}_{\beta }\mathrm{.}$ The generator $\mathrm{d}{z}_{\beta }$ corresponds to the universal derivation of the coordinate $z_{\beta }$, so by the naturality of $\mathrm{d}$ under the chart automorphisms [F6] the left action $\omega \mapsto ({\mathrm{\ell }}_{g^{-1}}{)}^{\ast }\omega$ of $G$ on differential forms satisfies ${\mathrm{\ell }}_{b^{-1}}^{\ast }(\mathrm{d}{z}_{\beta })=\mathrm{d}({\mathrm{\ell }}_{b^{-1}}^{\ast }(z_{\beta }))$ for $b\in B$; on the torus $T$ this is $\mathrm{d}(\beta (t)z_{\beta })=\beta (t)\mathrm{d}{z}_{\beta }$ by step 1.1. Hence the cotangent space at $eB$ is a $B$-representation of dimension $\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }$, whose $T$-weights are the positive roots $\beta$.

The fibre of the canonical bundle. Since ${\omega }_X={\bigwedge }^{\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }}{\mathrm{\Omega }}_{X\mathrm{/}\mathbb{C}}^1$ by [F5] and the chart module is free, forming the top exterior power commutes with taking the fibre at $eB$: the fibre $({\omega }_X{)}_{eB}$ is the top exterior power of the cotangent space of step 2.1, one-dimensional and spanned by the class of the wedge product $\mathrm{d}{z}_{{\beta }_1}\wedge \cdots \wedge \mathrm{d}{z}_{{\beta }_{\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }}},$ with ${\beta }_1,\dots ,{\beta }_{\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }}$ an enumeration of ${\mathrm{\Phi }}^{+}$. The action of $t\in T$ multiplies each factor $\mathrm{d}{z}_{\beta }$ by $\beta (t)$, so the $T$-weight of this generator is ${\prod }_{\beta \in {\mathrm{\Phi }}^{+}}\beta (t)=2\rho (t)$ by [F7]. The fibre is a one-dimensional algebraic $B$-representation, hence given by a character of $B$; by [F2] every character of $B$ is trivial on $U$ and is determined by its restriction to $T$, so the $B$-module $({\omega }_X{)}_{eB}$ is exactly the one-dimensional module ${\mathbb{C}}_{2\rho }$ on which $b$ acts by $\lambda (b{)}^{-1}$ for the character $\lambda =-2\rho$ of $B$.

The equivariant structure on ${\omega }_X$. For $g\in G$ the left translation ${\mathrm{\ell }}_g$ is an isomorphism $X\to X$, and the canonical isomorphisms ${\mathrm{\ell }}_g^{\ast }{\omega }_X\cong {\omega }_X$ of [F5] compose compatibly because the differential satisfies the chain rule and identity [F6]; using them in the form $({\mathrm{\ell }}_{g^{-1}}{)}^{\ast }$ defines a left action of $G$ on the total space of ${\omega }_X$ covering the action on $X$. Thus ${\omega }_X$ is a $G$-equivariant algebraic line bundle on $X$ whose induced $B$-action on the fibre at the fixed point $eB$ is the one computed in step 3.1, and the fibre functor of [F9] assigns to ${\omega }_X$ the one-dimensional $B$-module ${\mathbb{C}}_{2\rho }$.

Conclusion. By [F8] the fibre of ${\mathcal{L}}_{-2\rho }$ at $eB$ is the $B$-module ${\mathbb{C}}_{-(-2\rho )}={\mathbb{C}}_{2\rho }$, the same one attached to ${\omega }_X$ in step 4.1; the fibre functor of [F9] is an equivalence, so it reflects isomorphisms and there is a (necessarily $G$-equivariant) isomorphism ${\omega }_X\cong {\mathcal{L}}_{-2\rho }$. In particular the canonical bundle is ${\mathcal{L}}_{-2\rho }$, and no choice of a different sign or identification enters: the identification is forced by the fibre characters.

Axiom-of-choice bookkeeping. The Axiom of Choice [A1] is assumed in the statement and is inherited through the quotient, torsor and cohomological suppliers behind [F1], [F4], [F5] and [F9]; the computation itself uses only the fixed chart, the finitely many root coordinates $z_{\beta }$ and the conjugating tori, and makes no further choice. The open chart and quotient structure used in steps 1.1–5.1 are supplied by [F1] and [F4], and the equivariant bundle identification by [F9].