Sections of an associated line bundle as equivariant functions

By [F1] the map $\pi$ is a Zariski-locally trivial $B$-torsor, so $X$ has a finite open cover by charts $V_i$ on which $\pi$ admits regular sections ${\sigma }_i:V_i\to G$, with $\pi \circ {\sigma }_i=\mathrm{i}\mathrm{d}$; on an overlap $V_i\cap V_j$ the two sections satisfy ${\sigma }_j(x)={\sigma }_i(x)b_{ij}(x)$ for a morphism $b_{ij}:V_i\cap V_j\to B$, because both points lie in the same fibre, which is a right $B$-coset by [F3], and the two trivializations of the associated bundle over the overlap differ by $b_{ij}$, as [F1] records.

Let $s\in H^0(X,{\mathcal{L}}_{\lambda })$. Its pullback ${\pi }^{\ast }s$ is a regular section of ${\pi }^{\ast }{\mathcal{L}}_{\lambda }$ over $G$, and the pullback of an associated bundle along the torsor projection is canonically trivial: the map $\tau :G\times \mathbb{C}\to {\pi }^{\ast }{\mathcal{L}}_{\lambda }$, $\tau (g,v)=(g,[g,v])$, is an isomorphism over $G$, since it is bijective on the fibre ${\pi }^{-1}(\pi (g))=gB$ by the relation $[gb,v]=[g,\lambda (b{)}^{-1}v]$ of [F2] and is the identity trivialization over each chart of [F1]. Write ${\tau }^{-1}({\pi }^{\ast }s)(g)=(g,f(g))$ for a regular function $f\in \mathcal{O}(G)$. For $b\in B$ one has $s(\pi (gb))=s(\pi (g))$ by [F3], while $s(\pi (gb))=[gb,f(gb)]$ and $s(\pi (g))=[g,f(g)]=[gb,\lambda (b)f(g)]$ by [F2]; the second coordinate of the class over $gb$ is unique, so $f(gb)=\lambda (b)f(g)$. This defines the map from sections to functions.

Conversely, let $f\in \mathcal{O}(G)$ satisfy $f(gb)=\lambda (b)f(g)$ for all $g,b$, and define $s(x)=[g,f(g)]$ for any $g$ with $\pi (g)=x$. This is well-defined: every other representative of $x$ is $gb$ with $b\in B$ by [F3], and $[gb,f(gb)]=[gb,\lambda (b)f(g)]=[g,f(g)]$ by [F2]. It is a section: the projection sends $s(x)=[g,f(g)]$ to $gB=x$. It is regular: on a chart $V_i$ of step 1.1 the formula $s(x)=[{\sigma }_i(x),f({\sigma }_i(x))]$ exhibits $s$ as a composition of regular maps into the locally trivial bundle ${\mathcal{L}}_{\lambda }$, and regularity is local on the cover $\{V_i\}$, which is finite by [F1]; since $s$ is a section, the compatibility of the local formulae on overlaps is automatic from well-definedness.

The two constructions are inverse: a function $f$ recovered from $s$ satisfies $[g,f(g)]={\pi }^{\ast }s(g)=s(\pi (g))$, so reconstructing $s$ from $f$ returns the original section, and starting from $f$ the recovered function is $g\mapsto$ second coordinate of $[g,f(g)]$, which is $f(g)$. The identification is natural in $\lambda$: the canonical isomorphism ${\mathcal{L}}_{\lambda }\otimes {\mathcal{L}}_{\mu }\to {\mathcal{L}}_{\lambda +\mu }$ of [F2] sends $[g,u]\otimes [g,v]$ to $[g,uv]$, so on sections it corresponds to pointwise multiplication of the functions assigned to $\lambda$ and $\mu$. The dual bundle isomorphism is the fibrewise dual construction; evaluation ${\mathcal{L}}_{\lambda }\otimes {\mathcal{L}}_{-\lambda }\to {\mathcal{O}}_X$ corresponds to pointwise multiplication of functions with opposite $B$-equivariance. In particular, the dictionary does not identify a dual section with the pointwise reciprocal of an arbitrary section. These constructions are compatible with the trivialization $\tau$ used above.

For left translation: the equivariant structure acts on a section $s$ by $(g_0\cdot s)(x)=g_0\cdot s(g_0^{-1}x)$ and $g_0\cdot [g,v]=[g_{0}g,v]$ by [F2], so the function of $g_0\cdot s$ is $g\mapsto$ second coordinate of $g_0\cdot [g_0^{-1}g,f(g_0^{-1}g)]$, which is $f(g_0^{-1}g)$; this is the stated left translation action. For the right $B$-action on sections $(b\cdot s)(x)=s(x)\cdot b$, the associated function is $(b\cdot f)(g)=\lambda (b{)}^{-1}f(g)=f(g{b}^{-1})$ by [F2], so ${\mathrm{e}\mathrm{v}}_1(b\cdot f)=(b\cdot f)(1)=f(b^{-1})=\lambda (b{)}^{-1}f(1)=b\cdot ({\mathrm{e}\mathrm{v}}_{1}f)$ for the action $v\mapsto \lambda (b{)}^{-1}v$ on the fibre ${\mathbb{C}}_{-\lambda }$ at $eB$; hence ${\mathrm{e}\mathrm{v}}_1$ is $B$-equivariant into that fibre. Nothing here asserts that ${\mathrm{e}\mathrm{v}}_1$ is surjective: it is the zero map whenever the space of such functions is zero, and the present lemma only identifies that space with $H^0(X,{\mathcal{L}}_{\lambda })$.