Harish-Chandra regularity theorem
E876154
The Harish-Chandra regularity theorem is a fundamental result in representation theory that asserts characters of irreducible admissible representations of real reductive Lie groups are given by real-analytic, locally integrable functions on the group.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Harish-Chandra regularity theorem canonical | 1 |
| Harish-Chandra’s theory of characters | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10641423 — 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 regularity theorem Context triple: [Harish-Chandra character formula, uses, Harish-Chandra regularity theorem]
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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.
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.
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E.
Peter–Weyl theorem
The Peter–Weyl theorem is a fundamental result in representation theory and harmonic analysis that decomposes square-integrable functions on a compact topological group into a direct sum of finite-dimensional irreducible unitary representations.
- 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 regularity theorem Target entity description: The Harish-Chandra regularity theorem is a fundamental result in representation theory that asserts characters of irreducible admissible representations of real reductive Lie groups are given by real-analytic, locally integrable functions on the group.
-
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.
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.
-
E.
Peter–Weyl theorem
The Peter–Weyl theorem is a fundamental result in representation theory and harmonic analysis that decomposes square-integrable functions on a compact topological group into a direct sum of finite-dimensional irreducible unitary representations.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem in representation theory ⓘ |
| appliesTo |
irreducible admissible representations
ⓘ
real reductive Lie groups ⓘ |
| asserts |
characters of irreducible admissible representations of real reductive Lie groups are given by real-analytic functions on the group
ⓘ
characters of irreducible admissible representations of real reductive Lie groups are locally integrable functions on the group ⓘ distribution characters of irreducible admissible representations are given by locally integrable functions on the group ⓘ distribution characters of irreducible admissible representations are represented by real-analytic functions on the regular set ⓘ |
| characterType |
global character regularity
ⓘ
local regularity on regular semisimple set ⓘ |
| concerns | characters of representations ⓘ |
| context |
admissible representations of real reductive Lie groups
ⓘ
unitary representation theory of real reductive Lie groups ⓘ |
| ensures |
distribution characters are smooth on the regular semisimple set
ⓘ
singularities of characters are controlled and mild ⓘ |
| field |
Lie theory
ⓘ
harmonic analysis ⓘ representation theory ⓘ |
| hasConsequence |
characters can be studied using analytic methods on Lie groups
ⓘ
characters determine irreducible admissible representations up to equivalence ⓘ |
| holdsFor |
connected real reductive Lie groups
ⓘ
finite-length admissible representations ⓘ |
| implies |
characters are real-analytic on the set of regular elements
ⓘ
characters extend as locally integrable functions on the whole group ⓘ characters of irreducible admissible representations are class functions ⓘ |
| involves |
Harish-Chandra’s Schwartz space
ⓘ
center of the universal enveloping algebra ⓘ infinitesimal character ⓘ invariant eigendistributions ⓘ universal enveloping algebra of a Lie algebra ⓘ |
| isPartOf |
Harish-Chandra’s program on harmonic analysis on semisimple Lie groups
ⓘ
linked to:
Harish-Chandra theory
|
| namedAfter | Harish-Chandra ⓘ |
| refines | the description of characters as conjugation-invariant distributions ⓘ |
| relatedTo |
Harish-Chandra’s Plancherel theorem
ⓘ
linked to:
Plancherel theorem for real reductive groups
Harish-Chandra’s character formula ⓘ
linked to:
Harish-Chandra character formula
Harish-Chandra’s subquotient theorem ⓘ |
| strengthens | the fact that characters are distributions ⓘ |
| usedFor |
analysis of the unitary dual of real reductive Lie groups
ⓘ
classification of irreducible admissible representations ⓘ harmonic analysis on real reductive Lie groups ⓘ study of trace formulas ⓘ |
| usesConcept |
Harish-Chandra’s theory of characters
ⓘ
conjugation-invariant distributions ⓘ distribution characters ⓘ locally integrable functions ⓘ real-analytic functions ⓘ regular elements of a Lie group ⓘ |
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Subject: Harish-Chandra regularity theorem Description of subject: The Harish-Chandra regularity theorem is a fundamental result in representation theory that asserts characters of irreducible admissible representations of real reductive Lie groups are given by real-analytic, locally integrable functions on the group.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Harish-Chandra regularity theorem