Borel characters classify equivariant flag line bundles

The fibre functor. For a $G$-equivariant line bundle $\mathcal{L}$ on $X$, the point $eB$ is fixed by $B$, so the action of $G$ on $\mathcal{L}$ restricts to an action of $B$ on the one-dimensional vector space ${\mathcal{L}}_{eB}$; this action is a morphism $B\times {\mathcal{L}}_{eB}\to {\mathcal{L}}_{eB}$ of varieties and is linear on each fibre, hence makes ${\mathcal{L}}_{eB}$ a one-dimensional algebraic $B$-representation ${\mathbb{C}}_{\chi }$ for a character $\chi \in X^{\ast }(B)$. A $G$-equivariant morphism $\phi :\mathcal{L}\to \mathcal{M}$ of line bundles induces a $B$-equivariant linear map ${\phi }_{eB}:{\mathcal{L}}_{eB}\to {\mathcal{M}}_{eB}$, so $\mathrm{\Phi }$ is a functor from $G$-equivariant line bundles to one-dimensional algebraic $B$-representations.

Faithfulness and fullness. Let $\phi :\mathcal{L}\to \mathcal{M}$ be $G$-equivariant with ${\phi }_{eB}=0$. For a point $x=gB$ and $v\in {\mathcal{L}}_x$, choose $w\in {\mathcal{L}}_{eB}$ with $v=g\cdot w$, possible because $G$ acts transitively on $X$ and the action map is an isomorphism on fibres of a $G$-equivariant line bundle; then ${\phi }_x(v)=g\cdot {\phi }_{eB}(w)=0$, so $\phi =0$. Hence $\mathrm{\Phi }$ is faithful. Conversely let $\psi :{\mathcal{L}}_{eB}\to {\mathcal{M}}_{eB}$ be any $B$-equivariant linear map. Define ${\phi }_x(g\cdot w)=g\cdot \psi (w)$ for $x=gB$; this is well defined because an ambiguity $g\mapsto gb$ changes the fibre coordinate by the $B$-action and $\psi$ commutes with that action. To see regularity, use each Zariski-local section $s:V\to G$ of [F1]: the action identifies $\mathcal{L}{\mathrm{\mid }}_V$ and $\mathcal{M}{\mathrm{\mid }}_V$ with $V\times {\mathcal{L}}_{eB}$ and $V\times {\mathcal{M}}_{eB}$, and in these trivializations $\phi {\mathrm{\mid }}_V={\mathrm{id}}_V\times \psi$ is a morphism. The formulas agree on overlaps by $B$-equivariance, so they glue to a $G$-equivariant morphism of line bundles. Thus the induced map ${\mathrm{Hom}}_G(\mathcal{L},\mathcal{M})\to {\mathrm{Hom}}_B({\mathcal{L}}_{eB},{\mathcal{M}}_{eB})$ is bijective.

Essential surjectivity. Let $\mathcal{L}$ be a $G$-equivariant line bundle and ${\mathbb{C}}_{\chi }={\mathcal{L}}_{eB}$ its fibre at $eB$ as in step 1.1. The evaluation morphism $G\times {\mathbb{C}}_{\chi }⟶\mathcal{L},(g,v)⟼g\cdot v,$ is $B$-equivariant for the right $B$-action $(g,v)\cdot b=(gb,b^{-1}\cdot v)$ on the product, because $gb\cdot (b^{-1}\cdot v)=g\cdot v$; it therefore descends to a morphism $G{\times }^B{\mathbb{C}}_{\chi }\to \mathcal{L}$ of line bundles over $X=G\mathrm{/}B$, and this morphism is $G$-equivariant for the left action $g^{\mathrm{\prime }}[g,v]=[g^{\mathrm{\prime }}g,v]$. On the fibre over each point it is the linear isomorphism $g\cdot (-):{\mathcal{L}}_{eB}\to {\mathcal{L}}_{gB}$, so it is an isomorphism of line bundles. Hence $\mathcal{L}$ is $G$-equivariantly isomorphic to an associated bundle of a one-dimensional $B$-representation, and this evaluation supplies the quasi-inverse comparison. Conversely, for every character $\chi$ of $B$, put $\lambda =-\chi {\mathrm{\mid }}_T$. By [F2] and [F3], ${\mathcal{L}}_{\lambda }=G{\times }^B{\mathbb{C}}_{\chi }$ has fibre ${\mathbb{C}}_{\chi }$ at $eB$, which proves essential surjectivity. The construction is Zariski-locally trivial by [F1].

Classification. Steps 2.1 and 2.2 show that $\mathrm{\Phi }$ is an equivalence of groupoids. A one-dimensional algebraic representation of $B$ is determined up to isomorphism by its character, and distinct characters give non-isomorphic representations, so the isomorphism classes of the targets of $\mathrm{\Phi }$ are in bijection with $X^{\ast }(B)$; equivalently $[\mathcal{L}]\mapsto \chi$ where ${\mathcal{L}}_{eB}\cong {\mathbb{C}}_{\chi }$ is a bijection onto $X^{\ast }(B)$. By [F2] the restriction map $X^{\ast }(B)\to X^{\ast }(T)$ is an isomorphism, so the classes are also in bijection with $X^{\ast }(T)$; the sign convention of [F3] makes the fibre of ${\mathcal{L}}_{\lambda }$ over $eB$ the module ${\mathbb{C}}_{-\lambda }$, so under this bijection the class of ${\mathcal{L}}_{\lambda }$ corresponds to $-\lambda$.

Conclusion. Clauses (i) and (ii) rest on the fibre functor of step 1.1, its full faithfulness in step 2.1, essential surjectivity in step 2.2 and the character computation in step 3.1. The Axiom of Choice [A1] is assumed in the statement and is inherited through the quotient and torsor structure [F1] and the associated-bundle construction [F3]; the proof itself makes no choice, the constructions being canonical and the only cover used being the fixed finite torsor chart cover of [F1]. The quotient and local triviality used at steps 1.1 and 2.2 are supplied by [F1], and the associated bundle and its sign convention by [F3].