The Borel-Weil theorem

Given: The Axiom of Choice, the group $G$, its Borel $B$, the flag variety $X=G\mathrm{/}B$ of dimension $N=\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }$, a weight $\lambda \in X^{\ast }(T)$ and the equivariant line bundle ${\mathcal{L}}_{\lambda }$.

The function space $F_{\lambda }=\{f\in \mathcal{O}(G):f(gb)=\lambda (b)f(g)\}$ is identified with $H^0(X,{\mathcal{L}}_{\lambda })$; it is a finite-dimensional $\mathfrak{g}$-module for the derived left translation action $(x\cdot f)(g)=\frac{d}{dt}{\mid }_{0}f(\exp (-tx)g)$, and annihilation by every $x\in {\mathfrak{n}}^{-}$ is equivalent to invariance under left translation by the whole group $U^{-}$ (Sections of an associated line bundle as equivariant functions, The cohomology of a Borel-character line bundle is a rational G-module, A $U^{-}$-invariant section is determined on the big cell).

The space of $U^{-}$-invariant functions in $F_{\lambda }$ has dimension at most $1$ and, when nonzero, consists of the weight-$(-\lambda )$ line for the torus action; moreover $f$ is determined by $f(1)$ on this space (A $U^{-}$-invariant section is determined on the big cell).

For every dominant integral $\mu$ the subspace of ${\mathfrak{n}}^{-}$-invariants of the finite-dimensional irreducible module $L(\mu )$ equals its lowest weight space $L(\mu {)}_{w_0\mu }$ and is one-dimensional (The lowest weight space is the nilradical-invariant line).

Every finite-dimensional $\mathfrak{g}$-module is completely reducible, and the finite-dimensional irreducible modules are exactly the $L(\mu )$ with $\mu$ dominant integral; the dual $L(\lambda {)}^{\ast }$ of $L(\lambda )$ is irreducible of highest weight $-w_0\lambda$ (Weyl's complete reducibility theorem, Highest-weight classification, Highest weight of the dual representation).

If $\lambda$ is dominant integral, then there is a regular function $v\in \mathcal{O}(G)$ with $v(gb)=\lambda (b)v(g)$ and $v(1)=1$; in particular $F_{\lambda }\mathrm{\ne }0$ (The dominant Borel-Weil section extends from the big cell).

For $\mu =\lambda +\rho$ dominant and regular, there is a reduced expression $w_0=s_{i_N}\cdots s_{i_1}$ whose partial products $w_j=s_{i_j}\cdots s_{i_1}$ satisfy $N(w_j\mu )=j$ and end at the strictly antidominant weight $w_0\mu$; consequently, with ${\lambda }_j=w_j\cdot \lambda$, the simple reflection used at step $j$ satisfies $⟨{\lambda }_j,{\alpha }_{i_{j+1}}^{\vee }⟩\ge 0$ (A regular weight has a unique dominant dot translate).

If $⟨\nu ,{\alpha }^{\vee }⟩\ge -1$ for a simple root $\alpha$, then $H^i(X,{\mathcal{L}}_{\nu })\cong H^{i+1}(X,{\mathcal{L}}_{s_{\alpha }\cdot \nu })$ for all $i\ge 0$ (Rank-one cohomology shifts across a simple wall).

$X$ is smooth projective of pure dimension $N=\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }$ and ${\mathcal{L}}_{\lambda }$ is locally free, so $H^q(X,{\mathcal{L}}_{\lambda })=0$ for all $q>N$, and each $H^q$ is finite-dimensional (Serre duality for locally free sheaves on a smooth projective variety, A semisimple flag variety is smooth and projective, The cohomology of a Borel-character line bundle is a rational G-module).

A dominant weight pairs nonnegatively with every positive root. A weight $\nu$ is dominant if and only if $-w_0\nu$ is dominant: if $\nu$ is dominant then $w_0\nu$ is antidominant, so $-w_0\nu$ is dominant, and conversely if $-w_0\nu$ is dominant then $w_0(-w_0\nu )=-\nu$ is antidominant, so $\nu$ is dominant (Integral, dominant, and strictly dominant weights, Dot-Weyl facets and single-wall translation data, Weyl length equals inversion number).