In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, in the field of representation theory

Representation theory

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...

, the Borel–Weil theorem, named after Armand Borel

Armand Borel

Armand Borel was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993...

 and André Weil

André Weil

André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...

, provides a concrete model for irreducible representations of compact Lie groups and complex semisimple Lie groups. These representations are realized in the spaces of global sections of holomorphic line bundles on the flag manifold

Flag manifold

In mathematics, a generalized flag variety is a homogeneous space whose points are flags in a finite-dimensional vector space V over a field F. When F is the real or complex numbers, a generalized flag variety is a smooth or complex manifold, called a real or complex flag manifold...

 of the group. Its generalization to higher cohomology spaces is called the Borel–Weil–Bott theorem.

Statement of the 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

Connected space

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...

 complex semisimple Lie group, B its Borel subgroup

Borel subgroup

In the theory of algebraic groups, a Borel subgroup of an algebraic group G is a maximal Zariski closed and connected solvable algebraic subgroup.For example, in the group GLn ,...

, and X=G/B the flag variety. In this picture, X is a complex manifold and a nonsingular algebraic G-variety. The flag variety can also be described as a compact homogeneous space

Homogeneous space

In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...

 K/T, where T=K∩B is a (compact) Cartan subgroup

Cartan subgroup

In mathematics, a Cartan subgroup of a Lie group or algebraic group G is one of the subgroups whose Lie algebrais a Cartan subalgebra. The dimension of a Cartan subgroup, and therefore of a Cartan subalgebra, is the rank of G.-Conventions:...

 of K. An integral weight λ
determines a G-equivariant holomorphic line bundle L_λ 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 an 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.

Concrete description

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


for all g∈G and b∈B.

The action of G on these sections is given by

for g,h∈G.

Example

Let G be the complex special linear group

Special linear group

In mathematics, the special linear group of degree n over a field F is the set of n×n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion....

 SL(2,C), with a Borel subgroup consisting of upper triangular matrices with determinant one. Integral weights for G may be identified with integer

Integer

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s, with dominant weights corresponding to nonnegative integers, and the corresponding characters χ_n of B have the form

The flag variety G/B may be identified with the complex projective line P¹ with homogeneous coordinates X, Y and the space of the global sections of the line bundle L_n 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 2i−n, and the highest weight vector Xⁿ has weight n.

History

The theorem dates back to the early 1950s and can be found in and .

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