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In mathematics, the Borel–Weil–Bott theorem is a basic result in the representation theory of Lie groups, showing how a family of representations can be obtained from holomorphic sections of certain complex vector bundles, and, more generally, from higher sheaf cohomology groups associated to such bundles. It is built on the earlier Borel–Weil theorem of Armand Borel and André Weil, dealing just with the space of sections (the zeroth cohomology group), the extension to higher cohomology groups being provided by Raoul Bott. One can equivalently, through Serre's GAGA, view this as a result in complex algebraic geometry in the Zariski topology.

Formulation


Given an integral weight λ, one of two cases occur:

The theorem states that in the first case, we have

and in the second case, we have

Example


Positive characteristic


Borel–Weil theorem


The Borel–Weil theorem 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. The Borel–Weil–Bott theorem is its generalization to higher cohomology spaces. The theorem dates back to the early 1950s and can be found in Serre & 1951-4 and Tits (1955).

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 = KB 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,

The Borel–Weil theorem states that if λ is a dominant integral weight then this representation is a holomorphic irreducible highest weight representation of G with highest weight λ. Its restriction to K is an irreducible unitary representation of K with highest weight λ, and each irreducible unitary representations of K is obtained in this way for a unique value of λ. (A holomorphic representation of a complex Lie group is one for which the corresponding Lie algebra representation is complex linear.)

The weight λ gives rise to a character (one-dimensional representation) of the Borel subgroup B, which is denoted χλ. Holomorphic sections of the holomorphic line bundle Lλ over G/B may be described more concretely as holomorphic maps

for all gG and bB.

The action of G on these sections is given by

for g, hG.

Let G be the complex special linear group SL(2, C), with a Borel subgroup consisting of upper triangular matrices with determinant one. Integral weights for G may be identified with integers, with dominant weights corresponding to nonnegative integers, and the corresponding characters χn of B have the form

The flag variety G/B may be identified with the complex projective line CP1 with homogeneous coordinates X, Y and the space of the global sections of the line bundle Ln is identified with the space of homogeneous polynomials of degree n on C2. For n ≥ 0, this space has dimension n + 1 and forms an irreducible representation under the standard action of G on the polynomial algebra C[X, Y]. Weight vectors are given by monomials

of weights 2in, and the highest weight vector Xn has weight n.

See also


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