Plancherel measure
E876156
The Plancherel measure is a canonical measure on the unitary dual of a group that describes how the regular representation decomposes into irreducible unitary representations in harmonic analysis.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Plancherel measure canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T10641474 — 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: Plancherel measure Context triple: [Plancherel theorem for real reductive groups, involves, Plancherel measure]
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A.
Haar measure
Haar measure is a fundamental concept in harmonic analysis and topological group theory, providing a translation-invariant way to assign measures to subsets of locally compact groups.
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B.
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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C.
Wiener measure
Wiener measure is the canonical probability measure on the space of continuous paths that models standard Brownian motion in probability theory and stochastic analysis.
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D.
Schwartz–Bruhat space
The Schwartz–Bruhat space is a function space of rapidly decreasing smooth (or locally constant with compact support, in the non-Archimedean case) test functions on a locally compact abelian group, fundamental in harmonic analysis and number theory.
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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: Plancherel measure Target entity description: The Plancherel measure is a canonical measure on the unitary dual of a group that describes how the regular representation decomposes into irreducible unitary representations in harmonic analysis.
-
A.
Haar measure
Haar measure is a fundamental concept in harmonic analysis and topological group theory, providing a translation-invariant way to assign measures to subsets of locally compact groups.
-
B.
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.
-
C.
Wiener measure
Wiener measure is the canonical probability measure on the space of continuous paths that models standard Brownian motion in probability theory and stochastic analysis.
-
D.
Schwartz–Bruhat space
The Schwartz–Bruhat space is a function space of rapidly decreasing smooth (or locally constant with compact support, in the non-Archimedean case) test functions on a locally compact abelian group, fundamental in harmonic analysis and number theory.
-
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 (46)
| Predicate | Object |
|---|---|
| instanceOf |
canonical measure
ⓘ
mathematical concept ⓘ measure in harmonic analysis ⓘ |
| appearsIn |
Langlands program
ⓘ
representation theory of real reductive groups ⓘ |
| appliesTo |
locally compact groups
ⓘ
unitary dual of a group ⓘ |
| associatedWith | Haar measure on a locally compact group ⓘ |
| characterizedBy |
preservation of L2 norm under Fourier transform
ⓘ
unitary isomorphism between L2 of the group and L2 of the unitary dual ⓘ |
| codomain | nonnegative real numbers ⓘ |
| context | noncommutative generalization of Fourier inversion ⓘ |
| definedFor |
non-unimodular locally compact groups with modifications
ⓘ
unimodular locally compact groups ⓘ |
| dependsOn | choice of Haar measure up to normalization ⓘ |
| describes |
decomposition into irreducible unitary representations
ⓘ
decomposition of the left regular representation ⓘ decomposition of the right regular representation ⓘ |
| domain | unitary dual of a locally compact group ⓘ |
| ensures |
Plancherel formula for L2 functions on the group
ⓘ
orthogonality relations for matrix coefficients ⓘ |
| field |
abstract harmonic analysis
ⓘ
harmonic analysis ⓘ representation theory ⓘ |
| guarantees | isometry between L2 of the group and direct integral of Hilbert spaces over the unitary dual ⓘ |
| namedAfter | Michel Plancherel ⓘ |
| property |
Borel measure on the unitary dual
ⓘ
sigma-finite measure ⓘ uniqueness up to measure-zero sets ⓘ |
| relatedConcept |
continuous spectrum of a representation
ⓘ
discrete series representation ⓘ tempered representation ⓘ unitary dual ⓘ |
| relatedTo |
Fourier transform on groups
ⓘ
irreducible unitary representation ⓘ regular representation of a group ⓘ |
| role | weights irreducible unitary representations in the decomposition of L2 of the group ⓘ |
| specialCase | Lebesgue measure on the dual group of a locally compact abelian group ⓘ |
| specialCaseOf | spectral measure of a unitary representation ⓘ |
| usedFor |
harmonic analysis on finite groups via counting measure analogue
ⓘ
harmonic analysis on p-adic groups ⓘ harmonic analysis on semisimple Lie groups ⓘ spectral decomposition of convolution operators ⓘ |
| usedIn |
Fourier analysis on groups
ⓘ
Plancherel theorem ⓘ noncommutative harmonic analysis ⓘ |
How these facts were elicited
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Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Plancherel measure Description of subject: The Plancherel measure is a canonical measure on the unitary dual of a group that describes how the regular representation decomposes into irreducible unitary representations in harmonic analysis.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.