Beilinson–Bernstein localization theorem

E876104

mathematical theorem result in geometric representation theory

The Beilinson–Bernstein localization theorem is a fundamental result in geometric representation theory that realizes representations of semisimple Lie algebras as sheaves of differential operators on flag varieties, establishing an equivalence between algebraic and geometric categories.

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Observed surface forms (1)

Surface form	Occurrences
Beilinson–Bernstein localization for representations of semisimple Lie algebras	1

Statements (48)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in geometric representation theory ⓘ
appliesTo	complex semisimple Lie algebras ⓘ
asserts	global section functor is quasi-inverse for regular dominant central character ⓘ localization functor is an equivalence for regular dominant central character ⓘ
concerns	D-modules NERFINISHED ⓘ category O NERFINISHED ⓘ flag varieties ⓘ highest weight representations ⓘ semisimple Lie algebras ⓘ sheaves of differential operators ⓘ
context	Borel subalgebra ⓘ Cartan subalgebra NERFINISHED ⓘ Weyl group NERFINISHED ⓘ root system ⓘ
describes	global section functor from D-modules to modules ⓘ localization functor from modules to D-modules ⓘ
establishes	equivalence of categories ⓘ
field	algebraic geometry ⓘ geometric representation theory ⓘ representation theory ⓘ
generalizedBy	localization for singular characters ⓘ
hasVersionFor	real groups (via related localization techniques) ⓘ
influenced	Kazhdan–Lusztig theory NERFINISHED ⓘ geometric Langlands program NERFINISHED ⓘ study of category O via geometry ⓘ theory of perverse sheaves ⓘ
keyConcept	block decomposition by central character ⓘ equivariant D-modules ⓘ highest weight category ⓘ twisted sheaves of differential operators ⓘ
namedAfter	Alexander Beilinson NERFINISHED ⓘ Joseph Bernstein NERFINISHED ⓘ
originallyFormulatedFor	complex algebraic groups ⓘ
provides	geometric construction of irreducible highest weight modules ⓘ geometric realization of category O ⓘ
publishedIn	early 1980s ⓘ
relatedTo	Borel–Weil–Bott theorem NERFINISHED ⓘ Riemann–Hilbert correspondence (conceptually) NERFINISHED ⓘ
relates	coherent D-modules on flag varieties ⓘ representations of semisimple Lie algebras ⓘ
requires	regular dominant infinitesimal character ⓘ
uses	Harish-Chandra isomorphism NERFINISHED ⓘ algebraic D-modules ⓘ center of the universal enveloping algebra ⓘ flag variety of a semisimple algebraic group ⓘ universal enveloping algebra ⓘ
yearProved	late 1970s ⓘ

Referenced by (2)

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

Alexander Beilinson → knownFor → Beilinson–Bernstein localization theorem ⓘ

Alexander Beilinson → notableWork → Beilinson–Bernstein localization theorem ⓘ

this entity surface form: Beilinson–Bernstein localization for representations of semisimple Lie algebras