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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Harish-Chandra projection canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10641406 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Harish-Chandra projection Context triple: [Harish-Chandra isomorphism, involves, Harish-Chandra projection]
-
A.
Harish-Chandra isomorphism
The Harish-Chandra isomorphism is a fundamental result in representation theory that identifies the center of the universal enveloping algebra of a semisimple Lie algebra with the algebra of Weyl group–invariant polynomials on a Cartan subalgebra.
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B.
Harish-Chandra character formula
The Harish-Chandra character formula is a fundamental result in representation theory that gives an explicit expression for the characters of irreducible admissible representations of real reductive Lie groups.
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C.
Plancherel theorem for real reductive groups
The Plancherel theorem for real reductive groups is a fundamental result in representation theory that describes how square-integrable functions on a real reductive Lie group decompose into irreducible unitary representations, generalizing Fourier analysis to this non-abelian setting.
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D.
Weil representation
The Weil representation is a fundamental projective unitary representation of symplectic groups (or their metaplectic covers) on spaces of functions, central to number theory, automorphic forms, and the theory of theta functions.
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E.
Borel–Weil theorem
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.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Harish-Chandra projection Target entity description: 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.
-
A.
Harish-Chandra isomorphism
The Harish-Chandra isomorphism is a fundamental result in representation theory that identifies the center of the universal enveloping algebra of a semisimple Lie algebra with the algebra of Weyl group–invariant polynomials on a Cartan subalgebra.
-
B.
Harish-Chandra character formula
The Harish-Chandra character formula is a fundamental result in representation theory that gives an explicit expression for the characters of irreducible admissible representations of real reductive Lie groups.
-
C.
Plancherel theorem for real reductive groups
The Plancherel theorem for real reductive groups is a fundamental result in representation theory that describes how square-integrable functions on a real reductive Lie group decompose into irreducible unitary representations, generalizing Fourier analysis to this non-abelian setting.
-
D.
Weil representation
The Weil representation is a fundamental projective unitary representation of symplectic groups (or their metaplectic covers) on spaces of functions, central to number theory, automorphic forms, and the theory of theta functions.
-
E.
Borel–Weil theorem
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.
- F. None of above. chosen
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
ⓘ
linked to:
Harish-Chandra theory
Harish-Chandra’s work on the Plancherel formula ⓘ |
| associatedWith |
Weyl group
ⓘ
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
ⓘ
complex semisimple Lie algebras ⓘ real reductive Lie groups ⓘ
linked to:
reductive Lie group
|
| namedAfter | Harish-Chandra ⓘ |
| 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
ⓘ
linked to:
Harish-Chandra isomorphism
Weyl group invariants ⓘ center of a universal enveloping algebra ⓘ central character ⓘ infinitesimal character ⓘ |
| relatedTo | Harish-Chandra isomorphism ⓘ |
| 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 ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Harish-Chandra projection Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.