Borel–Weil theorem

E504917

mathematical theorem theorem in representation theory

The Borel–Weil theorem is a fundamental result in representation theory that realizes irreducible representations of compact Lie groups as spaces of holomorphic sections of line bundles over their flag manifolds.

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Observed surface forms (2)

Surface form	Occurrences
Borel–Bott–Weil theorem	1
Borel–Weil–Bott theorem	1

Statements (48)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem in representation theory ⓘ
appliesTo	compact Lie groups ⓘ complex semisimple Lie groups ⓘ
assumes	compactness of the Lie group in its classical form ⓘ
characterizes	irreducible representations by highest weights ⓘ
codomain	holomorphic sections of line bundles ⓘ
concerns	compact connected Lie groups ⓘ complex semisimple algebraic groups ⓘ irreducible finite-dimensional representations ⓘ
constructionMethod	global holomorphic sections of an equivariant line bundle ⓘ
constructs	irreducible representation from a dominant weight ⓘ
context	complex analytic geometry on homogeneous spaces ⓘ representation theory of compact connected Lie groups ⓘ
describes	irreducible representations of compact Lie groups ⓘ
domain	representation theory of Lie algebras ⓘ representation theory of Lie groups ⓘ
field	Lie theory ⓘ algebraic geometry ⓘ representation theory ⓘ
generalizedBy	Borel–Weil–Bott theorem NERFINISHED ⓘ
hasVariant	algebraic version for complex reductive groups ⓘ
implies	existence of all irreducible finite-dimensional representations ⓘ uniqueness of irreducible representation for each dominant integral weight ⓘ
namedAfter	André Weil NERFINISHED ⓘ Armand Borel NERFINISHED ⓘ
realizesAs	spaces of holomorphic sections of line bundles ⓘ
relatedTo	Borel–Weil–Bott theorem NERFINISHED ⓘ Bott–Borel–Weil theory NERFINISHED ⓘ Peter–Weyl theorem NERFINISHED ⓘ highest weight classification ⓘ
relates	Lie group representations and line bundles on flag varieties ⓘ representation theory and complex geometry ⓘ
toolFor	classification of irreducible representations of compact Lie groups ⓘ geometric representation theory ⓘ
typicalDomainObject	maximal torus of a compact Lie group ⓘ weight lattice of a Lie group ⓘ
typicalGeometricObject	complex flag manifold ⓘ projective homogeneous variety ⓘ
usedIn	modern geometric representation theory ⓘ theory of automorphic forms ⓘ
usesConcept	Borel subgroup NERFINISHED ⓘ dominant integral weights ⓘ flag manifolds ⓘ highest weight theory ⓘ holomorphic line bundles ⓘ
usesObject	flag variety G/B NERFINISHED ⓘ homogeneous space of a Lie group ⓘ

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Raoul Bott → knownFor → Borel–Weil theorem ⓘ

this entity surface form: Borel–Bott–Weil theorem

Weyl character formula → relatedTo → Borel–Weil theorem ⓘ

this entity surface form: Borel–Weil–Bott theorem