Springer correspondence

E921612

mathematical correspondence result in geometric representation theory theorem in representation theory

The Springer correspondence is a fundamental result in geometric representation theory that links representations of Weyl groups to the geometry of nilpotent orbits in Lie algebras via the cohomology of Springer fibers.

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Statements (49)

Predicate	Object
instanceOf	mathematical correspondence ⓘ result in geometric representation theory ⓘ theorem in representation theory ⓘ
aimsTo	classify irreducible representations of Weyl groups geometrically ⓘ
appliesTo	complex semisimple Lie algebras ⓘ reductive algebraic groups ⓘ
characterizedBy	Weyl group action on the cohomology of Springer fibers ⓘ
codomain	pairs of nilpotent orbits and local systems ⓘ
constructs	Weyl group action on cohomology of Springer fibers ⓘ
context	complex algebraic geometry ⓘ equivariant derived categories ⓘ ℓ-adic cohomology ⓘ
describes	irreducible representations of Weyl groups ⓘ topology of Springer fibers ⓘ
developedBy	T. A. Springer NERFINISHED ⓘ
domain	Weyl group of a reductive group ⓘ
field	Lie theory ⓘ algebraic geometry ⓘ algebraic groups ⓘ geometric representation theory ⓘ representation theory ⓘ
generalizationOf	classical representation theory of symmetric groups ⓘ
hasVariant	generalized Springer correspondence NERFINISHED ⓘ
influenced	Kazhdan–Lusztig theory NERFINISHED ⓘ character sheaves ⓘ geometric Langlands program NERFINISHED ⓘ modular representation theory of finite groups of Lie type ⓘ
involves	Borel subgroups NERFINISHED ⓘ Grothendieck simultaneous resolution NERFINISHED ⓘ flag varieties ⓘ nilpotent cone ⓘ
is	a bijection up to certain equivalences ⓘ
namedAfter	T. A. Springer NERFINISHED ⓘ
refinedBy	David Kazhdan NERFINISHED ⓘ George Lusztig NERFINISHED ⓘ
relatedConcept	Springer fiber NERFINISHED ⓘ Weyl group NERFINISHED ⓘ flag variety ⓘ nilpotent orbit ⓘ
relates	cohomology of Springer fibers ⓘ geometry of nilpotent orbits ⓘ representations of Weyl groups ⓘ
timePeriod	second half of the 20th century ⓘ
uses	Springer fibers NERFINISHED ⓘ Weyl groups NERFINISHED ⓘ equivariant cohomology ⓘ intersection cohomology ⓘ nilpotent orbits ⓘ perverse sheaves ⓘ

Referenced by (1)

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

Deligne–Lusztig theory → relatesTo → Springer correspondence ⓘ