Plancherel theorem for real reductive groups

E250731

mathematical theorem result in harmonic analysis result in representation theory

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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All labels observed (6)

Label	Occurrences
Plancherel formula for real reductive groups	1
Plancherel theorem	1
Plancherel theorem for SL2(R)	1
Plancherel theorem for SU(1,1)	1
Plancherel theorem for real rank one groups	1
Plancherel theorem for real reductive groups canonical	1

How this entity was disambiguated

This entity first appeared as the object of triple T2267716 — 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 theorem for real reductive groups
Context triple: [Harish-Chandra, notableWork, Plancherel theorem for real reductive groups]

A. 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.
B. L’intégration dans les groupes topologiques et ses applications
L’intégration dans les groupes topologiques et ses applications is a foundational mathematical monograph by André Weil that develops the theory of integration on topological groups and explores its far-reaching applications in analysis and number theory.
C. The Classical Groups: Their Invariants and Representations
The Classical Groups: Their Invariants and Representations is a foundational mathematical monograph by Hermann Weyl that systematically develops the theory of classical Lie groups, their invariants, and their representation theory.
D. Adeles and Algebraic Groups
"Adeles and Algebraic Groups" is a foundational mathematical work by André Weil that develops the theory of adeles and its deep connections with algebraic groups and number theory.
E. Selberg trace formula
The Selberg trace formula is a fundamental result in analytic number theory and spectral theory that relates lengths of closed geodesics on a Riemannian manifold to the spectrum of its Laplace operator, serving as a non-abelian analogue of the Poisson summation formula.
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 theorem for real reductive groups
Target entity description: 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.

A. 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.
B. L’intégration dans les groupes topologiques et ses applications
L’intégration dans les groupes topologiques et ses applications is a foundational mathematical monograph by André Weil that develops the theory of integration on topological groups and explores its far-reaching applications in analysis and number theory.
C. The Classical Groups: Their Invariants and Representations
The Classical Groups: Their Invariants and Representations is a foundational mathematical monograph by Hermann Weyl that systematically develops the theory of classical Lie groups, their invariants, and their representation theory.
D. Adeles and Algebraic Groups
"Adeles and Algebraic Groups" is a foundational mathematical work by André Weil that develops the theory of adeles and its deep connections with algebraic groups and number theory.
E. Selberg trace formula
The Selberg trace formula is a fundamental result in analytic number theory and spectral theory that relates lengths of closed geodesics on a Riemannian manifold to the spectrum of its Laplace operator, serving as a non-abelian analogue of the Poisson summation formula.
F. None of above. chosen

Statements (49)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in harmonic analysis ⓘ result in representation theory ⓘ
appliesTo	real reductive Lie groups ⓘ real reductive groups ⓘ
characterizes	tempered representations as those occurring in the Plancherel decomposition ⓘ
describes	decomposition of L2(G) into irreducible unitary representations ⓘ spectral decomposition of the left regular representation ⓘ unitary dual of a real reductive group up to Plancherel measure zero ⓘ
ensures	Parseval-type identity for matrix coefficients of unitary representations ⓘ existence of an isometric isomorphism between L2(G) and a direct integral over the unitary dual ⓘ
field	Lie theory ⓘ harmonic analysis ⓘ representation theory ⓘ
generalizes	Fourier analysis on the real line ⓘ Fourier transform on Euclidean space ⓘ Plancherel theorem for locally compact abelian groups ⓘ
hasSpecialCase	Plancherel theorem for real reductive groups self-linksurface differs ⓘ surface form: Plancherel theorem for SL2(R) Plancherel theorem for real reductive groups self-linksurface differs ⓘ surface form: Plancherel theorem for SU(1,1) Plancherel theorem for real reductive groups self-linksurface differs ⓘ surface form: Plancherel theorem for real rank one groups
involves	Harish-Chandra c-function ⓘ Harish-Chandra character formula ⓘ surface form: Harish-Chandra characters Langlands classification ⓘ Plancherel measure ⓘ Weyl group ⓘ discrete series representations ⓘ irreducible unitary representations ⓘ parabolic induction ⓘ principal series representations ⓘ tempered representations ⓘ unitary dual of a group ⓘ
isBasedOn	Harish-Chandra’s theory of Schwartz space on real reductive groups ⓘ Harish-Chandra regularity theorem ⓘ surface form: Harish-Chandra’s theory of characters Harish-Chandra’s theory of the Fourier transform on real reductive groups ⓘ
isImportantFor	automorphic forms ⓘ harmonic analysis on semisimple Lie groups ⓘ non-abelian harmonic analysis ⓘ Arthur trace formula ⓘ surface form: the Arthur trace formula Selberg trace formula ⓘ surface form: the Selberg trace formula the theory of unitary representations ⓘ
isRelatedTo	Fourier inversion formula on real reductive groups ⓘ Paley–Wiener theorem for real reductive groups ⓘ
requires	Cartan decomposition ⓘ Iwasawa decomposition ⓘ structure theory of real reductive Lie groups ⓘ
states	L2(G) decomposes as a direct integral of irreducible unitary representations ⓘ the left regular representation is unitarily equivalent to a direct integral of irreducibles with multiplicities given by Plancherel measure ⓘ
wasDevelopedBy	Harish-Chandra ⓘ
wasDevelopedIn	mid 20th century ⓘ

How these facts were elicited

Referenced by (6)

Full triples — surface form annotated when it differs from this entity's canonical label.

Harish-Chandra → notableWork → Plancherel theorem for real reductive groups ⓘ

Riemann–Lebesgue lemma → relatedTo → Plancherel theorem for real reductive groups ⓘ

this entity surface form: Plancherel theorem

Harish-Chandra character formula → relatedTo → Plancherel theorem for real reductive groups ⓘ

this entity surface form: Plancherel formula for real reductive groups

Plancherel theorem for real reductive groups → hasSpecialCase → Plancherel theorem for real reductive groups self-linksurface differs ⓘ

this entity surface form: Plancherel theorem for SL2(R)

Plancherel theorem for real reductive groups → hasSpecialCase → Plancherel theorem for real reductive groups self-linksurface differs ⓘ

this entity surface form: Plancherel theorem for SU(1,1)

Plancherel theorem for real reductive groups → hasSpecialCase → Plancherel theorem for real reductive groups self-linksurface differs ⓘ

this entity surface form: Plancherel theorem for real rank one groups

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Referenced by (6)

How this entity was disambiguated

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