Rank-one cohomology shifts across a simple wall

Given: The Axiom of Choice, the group $G$, its Borel $B$, the flag variety $X=G\mathrm{/}B$, a simple root $\alpha$ with minimal parabolic $P_{\alpha }$ and projection $f:X\to X_{\alpha }=G\mathrm{/}{P}_{\alpha }$, a weight $\lambda$, and the number $n=⟨\lambda ,{\alpha }^{\vee }⟩$.

The projection $f:X_B\to X_{\alpha }$, $g[v_B]\mapsto g[v_{\alpha }]$, is a surjective morphism which is a Zariski-locally trivial fibre bundle with fibre $P_{\alpha }\mathrm{/}B\cong {\mathbb{P}}^1$, trivialized over the single $G$-translates of the open torsor chart and covering $X_{\alpha }$ by finitely many of them; left translation by $G$ permutes these charts (A minimal-parabolic flag projection is a projective-line bundle, Minimal parabolic from one negative simple root).

Under the fixed identification of the fibre $F=P_{\alpha }[v_B]$ with ${\mathbb{P}}^1$, the restriction ${\mathcal{L}}_{\lambda }{\mathrm{\mid }}_F$ is isomorphic to ${\mathcal{O}}_{{\mathbb{P}}^1}(n)$; in particular the fibre degree of ${\mathcal{L}}_{\lambda }$ for $f$ is the constant integer $n=⟨\lambda ,{\alpha }^{\vee }⟩$ (Flag line-bundle degree on a minimal-parabolic fiber).

The relative canonical line bundle of $f$ satisfies ${\omega }_{X_B\mathrm{/}{X}_{\alpha }}\cong {\mathcal{L}}_{-\alpha }$, $G$-equivariantly, and its fibre degree is $-2$ (Relative canonical weight for a minimal-parabolic flag projection).

The canonical identifications ${\mathcal{L}}_{\lambda }\otimes {\mathcal{L}}_{\mu }\cong {\mathcal{L}}_{\lambda +\mu }$ and ${\mathcal{L}}_{\lambda }^{\vee }\cong {\mathcal{L}}_{-\lambda }$ are $G$-equivariant (The equivariant line bundle associated to a Borel character).

Let $\pi :E\to S$ be a Zariski locally trivial ${\mathbb{P}}^1$-bundle of complex schemes and $L$ an invertible sheaf of constant geometric fibre degree $n\ge -1$, with relative canonical bundle $K_{\pi }={\omega }_{E\mathrm{/}S}$. After the invariant apolarity normalization of the relative cohomology computation, there is for every $i\ge 0$ an isomorphism $H^i(E,L)\cong H^{i+1}(E,L\otimes K_{\pi }^{\otimes (n+1)})$, natural in $(E\mathrm{/}S,L)$ and under restriction of $S$; when $n=-1$ both sides of the underlying relative isomorphism are handled by the same statement with the zero sheaf identification (Relative projective-line cohomology shift, Relative projective-line cohomology and apolarity, Leray spectral sequence for sheaf cohomology).

For a smooth projective complex scheme $X$ of pure dimension $N$ and a locally free sheaf $E$, the groups $H^q(X,E)$ vanish outside $0\le q\le N$; here $\dim X=\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }=N$ (Serre duality for locally free sheaves on a smooth projective variety, A semisimple flag variety is smooth and projective).

The equivariant structure induces for every $i\ge 0$ a linear action of $G$ on $H^i(X,{\mathcal{L}}_{\lambda })$, and every natural isomorphism of equivariant bundles induces a $G$-equivariant map on cohomology (The cohomology of a Borel-character line bundle is a rational G-module).

The dot action is $w\cdot \lambda =w(\lambda +\rho )-\rho$, the simple reflection acts by $s_{\alpha }(\mu )=\mu -⟨\mu ,{\alpha }^{\vee }⟩\alpha$ and is an involution, and $⟨\rho ,{\alpha }^{\vee }⟩=1$ (Dot-Weyl facets and single-wall translation data, Root reflections and the Weyl group action, The Weyl vector in fundamental coordinates).