Harish-Chandra c-function
E876151
The Harish-Chandra c-function is a key analytic function in representation theory and harmonic analysis on semisimple Lie groups, encoding the Plancherel measure and asymptotic behavior of spherical functions.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Harish-Chandra c-function canonical | 3 |
How this entity was disambiguated
This entity first appeared as the object of triple T10641306 — 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 c-function Context triple: [Harish-Chandra, notableFor, Harish-Chandra c-function]
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A.
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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B.
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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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.
Representation Theory and Automorphic Functions
"Representation Theory and Automorphic Functions" is a seminal mathematical work by Israel Gelfand that develops the connections between representation theory of groups and the theory of automorphic forms, with deep applications in number theory and harmonic analysis.
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E.
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.
- 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 c-function Target entity description: The Harish-Chandra c-function is a key analytic function in representation theory and harmonic analysis on semisimple Lie groups, encoding the Plancherel measure and asymptotic behavior of spherical functions.
-
A.
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.
-
B.
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.
-
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.
Representation Theory and Automorphic Functions
"Representation Theory and Automorphic Functions" is a seminal mathematical work by Israel Gelfand that develops the connections between representation theory of groups and the theory of automorphic forms, with deep applications in number theory and harmonic analysis.
-
E.
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.
- F. None of above. chosen
Statements (43)
| Predicate | Object |
|---|---|
| instanceOf |
analytic function
ⓘ
object in representation theory ⓘ |
| appearsIn |
Harish-Chandra Plancherel formula
ⓘ
linked to:
Plancherel theorem for real reductive groups
Harish-Chandra’s work on the Plancherel formula ⓘ harmonic analysis on semisimple Lie groups and symmetric spaces literature ⓘ inversion formula for the spherical Fourier transform ⓘ |
| associatedWith |
Cartan decomposition of a semisimple Lie group
ⓘ
linked to:
Cartan decomposition
Iwasawa decomposition of a semisimple Lie group ⓘ
linked to:
Iwasawa decomposition
minimal parabolic subgroup ⓘ principal series representations ⓘ |
| dependsOn |
multiplicities of restricted roots
ⓘ
root system of the Lie algebra ⓘ |
| domain | complexified dual of a Cartan subalgebra ⓘ |
| encodes |
Plancherel measure for semisimple Lie groups
ⓘ
asymptotic behavior of spherical functions ⓘ |
| field |
harmonic analysis
ⓘ
representation theory ⓘ theory of semisimple Lie groups ⓘ |
| generalizationOf | Gamma function factors in rank-one cases ⓘ |
| hasRankOneForm | ratio of Gamma functions ⓘ |
| influenced | later developments in non-compact harmonic analysis ⓘ |
| namedAfter | Harish-Chandra ⓘ |
| property |
Weyl group invariant up to explicit factors
ⓘ
meromorphic in the spectral parameter ⓘ |
| relatedTo |
Harish-Chandra isomorphism
ⓘ
Plancherel theorem for semisimple Lie groups ⓘ
linked to:
Plancherel theorem for real reductive groups
c-function of Heckman–Opdam theory ⓘ spherical Fourier transform ⓘ spherical functions ⓘ zonal spherical functions ⓘ |
| role |
density factor in the Plancherel measure
ⓘ
normalizing factor for intertwining operators ⓘ normalizing factor for spherical functions ⓘ |
| satisfies | functional equations under Weyl group action ⓘ |
| specialCaseOf | Gindikin–Karpelevich c-function in p-adic theory ⓘ |
| usedIn |
decomposition of the regular representation
ⓘ
harmonic analysis on Riemannian symmetric spaces ⓘ harmonic analysis on real reductive groups ⓘ spectral decomposition of L^2(G/K) ⓘ |
| usedToCompute | L^2-norms of spherical functions ⓘ |
| usedToDefine | Plancherel density on the unitary dual ⓘ |
| usedToStudy |
tempered representations of semisimple Lie groups
ⓘ
unitary principal series ⓘ |
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Subject: Harish-Chandra c-function Description of subject: The Harish-Chandra c-function is a key analytic function in representation theory and harmonic analysis on semisimple Lie groups, encoding the Plancherel measure and asymptotic behavior of spherical functions.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.