Borel-Weil-Bott Theorem

The Borel-Weil-Bott theorem provides a geometric realization of irreducible representations as cohomology of line bundles on flag varieties.

TheoremBorel-Weil Theorem

Let $G$ be a complex semisimple Lie group with Borel subgroup $B$ and maximal torus $T \subset B$ . For a dominant integral weight $\lambda$ , let $\mathcal{L}_\lambda$ be the line bundle on $G/B$ associated to the character $\lambda: B \to \mathbb{C}^*$ .

Then: $H^0(G/B, \mathcal{L}_\lambda) \cong L(\lambda)$ is the irreducible representation with highest weight $\lambda$ , and all higher cohomology vanishes: $H^i(G/B, \mathcal{L}_\lambda) = 0$ for $i > 0$ .

This theorem realizes representations geometrically: sections of line bundles are representations, and geometric properties of $G/B$ translate to representation-theoretic properties.

TheoremBott's Extension (Borel-Weil-Bott)

For any weight $\lambda$ (not necessarily dominant), define $w \cdot \lambda = w(\lambda + \rho) - \rho$ (the "dot" action of the Weyl group).

If $w \cdot \lambda$ is dominant for some unique $w \in W$ , then: $H^{\ell(w)}(G/B, \mathcal{L}_\lambda) \cong L(w \cdot \lambda)$ where $\ell(w)$ is the length of $w$ in the Weyl group. All other cohomology groups vanish.

If $w \cdot \lambda$ is on a wall (non-dominant for all $w$ ), then all cohomology vanishes.

Example$SL_2$ and $\mathbb{P}^1$

For $G = SL_2(\mathbb{C})$ , the flag variety is $G/B \cong \mathbb{P}^1$ .

Line bundles are $\mathcal{O}(n)$ for $n \in \mathbb{Z}$ :

$n \geq 0$ (dominant): $H^0(\mathbb{P}^1, \mathcal{O}(n)) =$ homogeneous polynomials of degree $n$ , dimension $n+1$ . This is $L(n)$ .
$n = -1$ : No cohomology
$n \leq -2$ : $H^1(\mathbb{P}^1, \mathcal{O}(n)) \cong L(-n-2)$ (dimension $-n-1$ )

TheoremGeometric Interpretation

Flag varieties: $G/B$ parametrizes Borel subalgebras (or maximal flags in $\mathbb{C}^n$ ). For $SL_n$ , it's the variety of complete flags.

Line bundles: $\mathcal{L}_\lambda$ corresponds to taking the $\lambda$ -weight space of the flag. Sections are functions satisfying transformation laws.

Cohomology: Higher cohomology measures obstructions to extending sections. The vanishing (for dominant weights) reflects complete reducibility.

Remark

The Borel-Weil-Bott theorem is fundamental for:

Geometric representation theory: Representations via geometry (D-modules, perverse sheaves, geometric Langlands)
Localization: Atiyah-Bott localization computes characters via fixed points
Quantum cohomology: q-deformations and mirror symmetry
Categorification: Derived categories of coherent sheaves on flag varieties

This geometric perspective connects representation theory to:

Algebraic geometry (Schubert calculus, intersection theory)
Differential geometry (K-theory, index theory)
Mathematical physics (gauge theory, conformal field theory)