The equivariant line bundle associated to a Borel character

Assume the Axiom of Choice (The Axiom of Choice). Let $G$ be the connected simply connected complex semisimple affine algebraic group with Borel subgroup $B=T⋉U$ and opposite unipotent subgroup $U^{-}$ of Complex semisimple algebraic group, Borel, and flag variety and Borel, opposite unipotent groups and root coordinates, and let $X=G\mathrm{/}B$ be the flag variety with its quotient structure and its $B$-torsor $G\to X$ of Zariski sections of Borel and minimal-parabolic orbit maps.

Characters of the Borel. By clause (iv) of Borel, opposite unipotent groups and root coordinates the restriction of characters is an isomorphism $X^{\ast }(B)\to X^{\ast }(T)$, so a character $\lambda \in X^{\ast }(T)$ of the maximal torus extends uniquely to a character of $B$, trivial on the unipotent radical $U$, which is also written $\lambda$; in these notes the group law of the character group $X^{\ast }(T)$ is written additively, so $-\lambda$ denotes the inverse character $b\mapsto \lambda (b{)}^{-1}$.

The line bundle. Let ${\mathbb{C}}_{-\lambda }$ be the one-dimensional $B$-module on which $b\in B$ acts by the character $-\lambda$, that is $b\cdot v=\lambda (b{)}^{-1}v$. Define the associated bundle ${\mathcal{L}}_{\lambda }=G{\times }^B{\mathbb{C}}_{-\lambda }=(G\times \mathbb{C})/\sim ,(gb,v)\sim (g,b\cdot v),g\in G,b\in B,v\in \mathbb{C},$ with the projection ${\mathcal{L}}_{\lambda }\to X$ induced by $(g,v)\mapsto gB$ and the left $G$-action $g^{\mathrm{\prime }}\cdot [g,v]=[g^{\mathrm{\prime }}g,v]$. The sign convention is fixed once and for all by this formula: the fibre of ${\mathcal{L}}_{\lambda }$ over a point $gB$ is $\{[g,v]:v\in \mathbb{C}\}\cong \mathbb{C}$, on which $B$ acts through $-\lambda$ when the point is $eB$, and the left action of $G$ commutes with this right $B$-action, so ${\mathcal{L}}_{\lambda }$ is a $G$-equivariant line bundle on $X$ with $H^0$ and all cohomological statements attached to it computed in this convention. In particular ${\mathcal{L}}_0={\mathcal{O}}_X$ is the structure sheaf, ${\mathcal{L}}_{\lambda }\otimes {\mathcal{L}}_{\mu }\cong {\mathcal{L}}_{\lambda +\mu }$ and ${\mathcal{L}}_{\lambda }^{\vee }\cong {\mathcal{L}}_{-\lambda }$ by the corresponding identities of one-dimensional $B$-modules.

Example: the rank-one case. For the simple root $\alpha$ with minimal parabolic $P_{\alpha }\supseteq B$ of Minimal parabolic from one negative simple root, the fibre of the projection $X=G\mathrm{/}B\to G\mathrm{/}{P}_{\alpha }$ at the point $P_{\alpha }$ is $P_{\alpha }\mathrm{/}B\cong {\mathbb{P}}^1$, and the restriction of ${\mathcal{L}}_{\lambda }$ to this projective line is computed in Flag line-bundle degree on a minimal-parabolic fiber; the sign in ${\mathbb{C}}_{-\lambda }$ is chosen so that this degree is the signed coroot pairing $⟨\lambda ,{\alpha }^{\vee }⟩$ for the identification of $P_{\alpha }\mathrm{/}B$ with ${\mathbb{P}}^1$ fixed there. The construction of the quotient ${\mathcal{L}}_{\lambda }$ above uses the local sections and local triviality of $G\to X$ supplied by Zariski sections of Borel and minimal-parabolic orbit maps. The Axiom of Choice is assumed in the first sentence and is inherited from the suppliers named above; no choice is made in the definition itself.