Borel–Weil theorem
In mathematics, in the field of representation theory, the Borel–Weil theorem, named after Armand Borel and André Weil, provides a concrete model for irreducible representations of compact Lie groups and irreducible holomorphic representations of complex semisimple Lie groups. These representations are realized in the spaces of global sections of holomorphic line bundles on the flag manifold of the group. Its generalization to higher cohomology spaces is called the Borel–Weil–Bott theorem.
The theorem can be stated either for a complex semisimple Lie group G or for its compact form K. Let G be a connected complex semisimple Lie group, B a Borel subgroup of G, and X = G/B the flag variety. In this scenario, X is a complex manifold and a nonsingular algebraic G-variety. The flag variety can also be described as a compact homogeneous space K/T, where T = K ∩ B is a (compact) Cartan subgroup of K. An integral weight λ determines a G-equivariant holomorphic line bundle Lλ on X and the group G acts on its space of global sections,
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