Bott-Borel-Weil Theorem

The Bott-Borel-Weil theorem extends the Borel-Weil theorem by using sheaf cohomology in all degrees, not just degree zero. While Borel-Weil realizes irreducible representations as global sections of line bundles over flag varieties, Bott's extension shows that higher cohomology groups also carry representation-theoretic meaning: each irreducible representation appears exactly once, in a specific cohomological degree determined by the Weyl group element that relates the weight to the dominant chamber.

This cohomological perspective is essential for infinite-dimensional representation theory, where global sections may be zero but higher cohomology is not. The theorem provides a unified geometric framework that connects the finite-dimensional representations of compact groups, the discrete series representations of real semisimple groups, and the unitary representations of loop groups and Kac-Moody algebras. In each case, the representation is realized as the cohomology of a sheaf over an appropriate homogeneous space, and the Bott-Borel-Weil machinery gives explicit formulas for characters and multiplicities.

The Bott-Borel-Weil theorem reveals that representation theory is a branch of algebraic geometry in disguise. The fact that every irreducible representation has a unique geometric 'home' in the cohomology of a flag variety suggests that the classification of representations is not an algebraic accident but a topological necessity.