The sl2 singular weight has no cohomology

Given: The Axiom of Choice, $G=S{L}_2(\mathbb{C})$, its upper triangular Borel, the flag variety $X=G\mathrm{/}B\cong {\mathbb{P}}^1$, the fundamental weight ${\omega }_1$ and Weyl vector $\rho ={\omega }_1$, the simple root $\alpha$ with reflection $s$, and the bundle ${\mathcal{L}}_{-{\omega }_1}$.

In rank one $\alpha =2{\omega }_1$ and $\rho =\alpha \mathrm{/}2={\omega }_1$. Thus for $\lambda =-{\omega }_1$ one has $\lambda +\rho =0$, $s\cdot \lambda =s(0)-\rho =-{\omega }_1$, and $⟨\lambda ,{\alpha }^{\vee }⟩=-1$. The shifted weight is therefore not regular (Fundamental weights for a chosen simple root system, The Weyl vector, Rank-one cohomology shifts across a simple wall).

For $G=S{L}_2$ the unique simple root $\alpha$ has $P_{\alpha }=G$, so the fibre $F=P_{\alpha }\mathrm{/}B$ is all of $X=G\mathrm{/}B$, and under the fixed identification of the fibre with the two-affine projective line ${\mathbb{P}}^1$ the restriction ${\mathcal{L}}_{\lambda }{\mathrm{\mid }}_F$ is isomorphic to ${\mathcal{O}}_{{\mathbb{P}}^1}(⟨\lambda ,{\alpha }^{\vee }⟩)$; in particular ${\mathcal{L}}_{-{\omega }_1}\cong \mathcal{O}(-1)$ (A minimal-parabolic flag projection is a projective-line bundle, Flag line-bundle degree on a minimal-parabolic fiber, Two-affine projective line and its twists).

On ${\mathbb{P}}^1$ over $\mathbb{C}$ one has $H^q(\mathcal{O}(d))=0$ unless $q=0$ or $q=1$; $H^0(\mathcal{O}(d))=0$ for $d<0$, and $H^1(\mathcal{O}(d))$ is nonzero exactly for $d\le -2$. In particular at $d=-1$ both $H^0$ and $H^1$ vanish (Cohomology of O(d) on projective space).

The wall-crossing isomorphism: since the pairing is $-1$, $H^i(X,{\mathcal{L}}_{-{\omega }_1})\cong H^{i+1}(X,{\mathcal{L}}_{s\cdot (-{\omega }_1)})=H^{i+1}(X,{\mathcal{L}}_{-{\omega }_1})$ for all $i\ge 0$; and all cohomology vanishes by the singular-weight lemma (Rank-one cohomology shifts across a simple wall, Singular dot weights have zero line-bundle cohomology, The Borel-Weil-Bott theorem).

$X$ is smooth projective of dimension $1$, so $H^q(X,{\mathcal{L}}_{-{\omega }_1})=0$ for $q>1$ (Serre duality for locally free sheaves on a smooth projective variety).