Kazhdan–Lusztig theory

E503515

framework in algebraic geometry framework in representation theory mathematical theory

Kazhdan–Lusztig theory is a framework in representation theory and algebraic geometry that studies Hecke algebras and their bases via Kazhdan–Lusztig polynomials, with deep connections to the representation theory of Lie algebras and geometry of Schubert varieties.

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Statements (49)

Predicate	Object
instanceOf	framework in algebraic geometry ⓘ framework in representation theory ⓘ mathematical theory ⓘ
aimsToDescribe	characters of irreducible highest weight modules ⓘ composition multiplicities in category O ⓘ
appliesTo	affine Lie algebras NERFINISHED ⓘ modular representation theory ⓘ quantum groups ⓘ representation theory of reductive algebraic groups ⓘ representation theory of semisimple Lie algebras ⓘ
connects	combinatorics of Coxeter groups ⓘ geometry of Schubert varieties ⓘ representation theory of Lie algebras ⓘ
defines	Kazhdan–Lusztig basis NERFINISHED ⓘ Kazhdan–Lusztig polynomial NERFINISHED ⓘ
developedIn	late 1970s ⓘ
field	algebra ⓘ algebraic geometry ⓘ representation theory ⓘ
hasPart	Kazhdan–Lusztig basis NERFINISHED ⓘ Kazhdan–Lusztig conjecture NERFINISHED ⓘ Kazhdan–Lusztig polynomials NERFINISHED ⓘ R-polynomials NERFINISHED ⓘ canonical basis of Hecke algebra ⓘ character formulas for simple modules ⓘ
introducedBy	David Kazhdan NERFINISHED ⓘ George Lusztig NERFINISHED ⓘ
mainObjectOfStudy	Hecke algebras NERFINISHED ⓘ Kazhdan–Lusztig polynomials NERFINISHED ⓘ Schubert varieties NERFINISHED ⓘ
namedAfter	David Kazhdan NERFINISHED ⓘ George Lusztig NERFINISHED ⓘ
relatedTo	Borel–Weil–Bott theorem NERFINISHED ⓘ Langlands program NERFINISHED ⓘ Schubert calculus NERFINISHED ⓘ Soergel bimodules NERFINISHED ⓘ Springer correspondence NERFINISHED ⓘ Weyl groups NERFINISHED ⓘ canonical bases ⓘ flag varieties ⓘ geometric representation theory ⓘ
uses	Bruhat order NERFINISHED ⓘ Coxeter group NERFINISHED ⓘ D-modules ⓘ Hecke algebra NERFINISHED ⓘ Verma modules NERFINISHED ⓘ category O ⓘ intersection cohomology ⓘ perverse sheaves ⓘ

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Weyl group → usedIn → Kazhdan–Lusztig theory ⓘ