Changing the line-bundle sign changes the Borel-Weil section space

Assume the Axiom of Choice (The Axiom of Choice). Let $G=S{L}_2(\mathbb{C})$ and consider the two bundles attached to $m{\omega }_1\in X^{\ast }(T)$: the library convention ${\mathcal{L}}_{m{\omega }_1}=G{\times }^B{\mathbb{C}}_{-m{\omega }_1}\cong \mathcal{O}(m)$ of The equivariant line bundle associated to a Borel character and the opposite convention ${\tilde{\mathcal{L}}}_{m{\omega }_1}=G{\times }^B{\mathbb{C}}_{+m{\omega }_1}\cong {\mathcal{L}}_{-m{\omega }_1}\cong \mathcal{O}(-m)$. For $m>0$ the reader who silently replaces ${\mathbb{C}}_{-\lambda }$ by ${\mathbb{C}}_{+\lambda }$ gets $H^0(X,{\tilde{\mathcal{L}}}_{m{\omega }_1})=H^0(X,\mathcal{O}(-m))=0\text{instead of}{H}^0(X,{\mathcal{L}}_{m{\omega }_1})\cong L(m{\omega }_1{)}^{\ast }\mathrm{\ne }0,$ so the opposite convention gives zero sections where the library convention gives a nonzero irreducible module. The line bundle is dualized by the sign change, but its space of sections is not obtained by dualizing the original space of sections. The two conventions are not interchangeable.

Given: The Axiom of Choice, $G=S{L}_2(\mathbb{C})$ with upper triangular Borel and flag variety $X=G\mathrm{/}B\cong {\mathbb{P}}^1$, an integer $m>0$, the fundamental weight ${\omega }_1$, the library bundles ${\mathcal{L}}_{m{\omega }_1}=G{\times }^B{\mathbb{C}}_{-m{\omega }_1}$, and the opposite bundles ${\tilde{\mathcal{L}}}_{m{\omega }_1}=G{\times }^B{\mathbb{C}}_{+m{\omega }_1}$.