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.
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
canonical measure
ⓘ
mathematical concept ⓘ measure in harmonic analysis ⓘ |
| appearsIn |
Langlands program
NERFINISHED
ⓘ
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 NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ noncommutative harmonic analysis ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.