Harish-Chandra projection
E876153
The Harish-Chandra projection is a linear map from a universal enveloping algebra onto the symmetric algebra of a Cartan subalgebra that plays a central role in describing the center via the Harish-Chandra isomorphism.
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
algebra homomorphism
ⓘ
construction in representation theory ⓘ linear map ⓘ tool in Lie theory ⓘ |
| actsOn | universal enveloping algebra of a Cartan subalgebra via identification with its symmetric algebra ⓘ |
| appearsIn |
Harish-Chandra’s papers on representations of semisimple Lie groups
NERFINISHED
ⓘ
Harish-Chandra’s work on the Plancherel formula ⓘ |
| associatedWith |
Weyl group
NERFINISHED
ⓘ
center of the universal enveloping algebra ⓘ symmetric algebra of a Cartan subalgebra ⓘ |
| codomain | symmetric algebra of a Cartan subalgebra ⓘ |
| constructionStep | projection along nilpotent subalgebras in the triangular decomposition ⓘ |
| definedOn | universal enveloping algebra of a complex semisimple Lie algebra ⓘ |
| domain | universal enveloping algebra ⓘ |
| ensures | identification of the center with Weyl group invariant polynomials ⓘ |
| field |
Lie algebra representation theory
ⓘ
harmonic analysis on semisimple Lie groups ⓘ noncommutative algebra ⓘ |
| generalizationOf | projection of invariant differential operators to polynomial functions on a Cartan subalgebra ⓘ |
| maps | center of the universal enveloping algebra into Weyl group invariants in the symmetric algebra ⓘ |
| mathematicalContext |
category O of Bernstein–Gelfand–Gelfand
NERFINISHED
ⓘ
complex semisimple Lie algebras ⓘ real reductive Lie groups NERFINISHED ⓘ |
| namedAfter | Harish-Chandra NERFINISHED ⓘ |
| property |
W-equivariant with respect to the Weyl group action
ⓘ
depends on the choice of Cartan subalgebra and positive roots ⓘ restricts to an algebra isomorphism on the center ⓘ |
| relatedConcept |
Harish-Chandra homomorphism
NERFINISHED
ⓘ
Weyl group invariants ⓘ center of a universal enveloping algebra ⓘ central character ⓘ infinitesimal character ⓘ |
| relatedTo | Harish-Chandra isomorphism NERFINISHED ⓘ |
| roleIn |
construction of the Harish-Chandra isomorphism
ⓘ
description of the center of the universal enveloping algebra ⓘ |
| targetSubspace | Weyl group invariant subalgebra of the symmetric algebra ⓘ |
| technicalCondition | defined using PBW (Poincaré–Birkhoff–Witt) basis decomposition ⓘ |
| usedIn |
classification of central characters
ⓘ
description of infinitesimal characters of representations ⓘ harmonic analysis on Lie groups ⓘ representation theory of complex semisimple Lie algebras ⓘ representation theory of real reductive Lie groups ⓘ study of primitive ideals in enveloping algebras ⓘ |
| uses |
Cartan subalgebra
ⓘ
root space decomposition ⓘ triangular decomposition of a semisimple Lie algebra ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.