Lie group or for its compact form . Let be a connected complex semisimple Lie group, a Borel subgroup of , and the flag variety. In this scenario, is a complex manifold and a nonsingular algebraic . The flag variety can also be described as a compact homogeneous space , where is a (compact) Cartan subgroup of . An integral weight determines a holomorphic line bundle on and the group acts on its space of global sections, The Borel–Weil theorem states that if is a dominant integral weight then this representation is a holomorphic irreducible highest weight representation of with highest weight . Its restriction to is an irreducible unitary representation of with highest weight , and each irreducible unitary representations of is obtained in this way for a unique value of . (A holomorphic representation of a complex Lie group is one for which the corresponding Lie algebra representation is complex linear.) Concrete description The weight gives rise to a character (one-dimensional representation) of the Borel subgroup , which is denoted . Holomorphic sections of the holomorphic line bundle over may be described more concretely as holomorphic maps for all and . The action of on these sections is given by for . Example Let be the complex special linear group , with a Borel subgroup consisting of upper triangular matrices with determinant one. Integral weights for may be identified with integers, with dominant weights corresponding to nonnegative integers, and the corresponding characters of have the form The flag variety may be identified with the complex projective line with homogeneous coordinates and the space of the global sections of the line bundle is identified with the space of homogeneous polynomials of degree on . For , this space has dimension and forms an irreducible representation under the standard action of on the polynomial algebra . Weight vectors are given by monomials of weights , and the highest weight vector has weight . See also Theorem of the highest weight Notes References . . (reprinted by Dover) A Proof of the Borel–Weil–Bott Theorem, by Jacob Lurie. Retrieved on Jul. 13, 2014. . . . . Reprint of the 1986 original. Further reading Representation theory of Lie groups Theorems in representation theory