# Borel–Weil–Bott theorem

# Borel–Weil–Bott theorem

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

`LetGbe asemisimpleLie group oralgebraic groupover, and fix amaximal torusTalong with aBorel subgroupBwhich containsT. Letλbe anintegral weightofT;λdefines in a natural way a one-dimensional representation`

*C*_{λ}ofB, by pulling back the representation on*T*=*B*/*U*, whereUis theunipotent radicalofB. Since we can think of the projection map*G*→*G*/*B*as aprincipalB-bundle, for each*C*_{λ}we get anassociated fiber bundle*L*_{−λ}on*G*/*B*(note the sign), which is obviously aline bundle. Identifying*L*_{λ}with itssheafof holomorphic sections, we consider thesheaf cohomologygroups. SinceGacts on the total space of the bundleby bundle automorphisms, this action naturally gives aG-module structure on these groups; and the Borel–Weil–Bott theorem gives an explicit description of these groups asG-modules.`We first need to describe theWeyl groupaction centered at. For any integral weightλandwin the Weyl groupW, we set, whereρdenotes the half-sum of positive roots ofG. It is straightforward to check that this defines a group action, although this action is`

*not*linear, unlike the usual Weyl group action. Also, a weightμis said to be*dominant*iffor all simple rootsα. Letℓdenote thelength functiononW.Given an integral weight λ, one of two cases occur:

There is no such that is dominant, equivalently, there exists a nonidentity such that ; or

There is a

*unique*such that is dominant.

The theorem states that in the first case, we have

- for alli;

and in the second case, we have

- for all, while

- is the dual of the irreducible highest-weight representation ofGwith highest weight.

`It is worth noting that case (1) above occurs if and only iffor some positive rootβ. Also, we obtain the classical`

**Borel–Weil theorem**as a special case of this theorem by takingλto be dominant andwto be the identity element.Example

`For example, consider`

*G*=SL_{2}(**C**), for which*G*/*B*is theRiemann sphere, an integral weight is specified simply by an integern, and*ρ*= 1. The line bundle*L*_{n}is, whosesectionsare thehomogeneous polynomialsof degreen(i.e. the*binary forms*). As a representation ofG, the sections can be written asSym^{n}(**C**^{2})*, and is canonically isomorphic toSym^{n}(**C**^{2}).`This gives us at a stroke the representation theory of:is the standard representation, andis itsnthsymmetric power. We even have a unified description of the action of the Lie algebra, derived from its realization as vector fields on the Riemann sphere: ifH,X,Yare the standard generators of, then`

Positive characteristic

`One also has a weaker form of this theorem in positive characteristic. Namely, letGbe a semisimple algebraic group over analgebraically closed fieldof characteristic. Then it remains true thatfor alliifλis a weight such thatis non-dominant for allas long asλis "close to zero".`

^{[1]}This is known as theKempf vanishing theorem. However, the other statements of the theorem do not remain valid in this setting.`More explicitly, letλbe a dominant integral weight; then it is still true thatfor all, but it is no longer true that thisG-module is simple in general, although it does contain the unique highest weight module of highest weightλas aG-submodule. Ifλis an arbitrary integral weight, it is in fact a large unsolved problem in representation theory to describe the cohomology modulesin general. Unlike over, Mumford gave an example showing that it need not be the case for a fixedλthat these modules are all zero except in a single degreei.`

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).

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 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 *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 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.)

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 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 *χ**n* of *B* have the form

The flag variety *G*/*B* may be identified with the complex projective line **CP**1 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 **C***2*. 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 2*i* − *n*, and the highest weight vector *X**n* has weight *n*.

See also

Theorem of the highest weight

## References

*Representations of algebraic groups*(second ed.). American Mathematical Society. ISBN 978-0-8218-3527-2.