Paley–Wiener theorem for real reductive groups

E877881

mathematical theorem result in harmonic analysis

The Paley–Wiener theorem for real reductive groups is a fundamental result in harmonic analysis that characterizes the image of compactly supported smooth functions under the group Fourier transform in terms of holomorphic functions with specific growth and support conditions.

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All labels observed (6)

Label	Occurrences
Paley–Wiener theorem for K-finite functions	1
Paley–Wiener theorem for Schwartz space on real reductive groups	1
Paley–Wiener theorem for compact Lie groups	1
Paley–Wiener theorem for real reductive groups canonical	1
Paley–Wiener theorem for the Fourier transform on Lie groups	1
spherical Paley–Wiener theorem for real reductive groups	1

How this entity was disambiguated

This entity first appeared as the object of triple T10641506 — 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: Paley–Wiener theorem for real reductive groups
Context triple: [Plancherel theorem for real reductive groups, isRelatedTo, Paley–Wiener theorem for real reductive groups]

A. 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.
B. Harish-Chandra regularity theorem
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.
C. 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.
D. Harish-Chandra projection
The Harish-Chandra projection is a linear map from a universal enveloping algebra onto the symmetric algebra of a Cartan subalgebra that plays a central role in describing the center via the Harish-Chandra isomorphism.
E. 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.
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: Paley–Wiener theorem for real reductive groups
Target entity description: The Paley–Wiener theorem for real reductive groups is a fundamental result in harmonic analysis that characterizes the image of compactly supported smooth functions under the group Fourier transform in terms of holomorphic functions with specific growth and support conditions.

A. 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.
B. Harish-Chandra regularity theorem
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.
C. 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.
D. Harish-Chandra projection
The Harish-Chandra projection is a linear map from a universal enveloping algebra onto the symmetric algebra of a Cartan subalgebra that plays a central role in describing the center via the Harish-Chandra isomorphism.
E. 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.
F. None of above. chosen

Statements (49)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in harmonic analysis ⓘ
appliesTo	real reductive groups ⓘ
associatedWith	David A. Vogan Jr. ⓘ Elyahu P. Stein ⓘ Harish-Chandra ⓘ Michel Duflo ⓘ Nolan R. Wallach ⓘ linked to: Nolan Wallach Patrick Delorme ⓘ Paul Sally ⓘ
characterizes	image of compactly supported smooth functions under the group Fourier transform ⓘ support of a function via exponential type of its transform ⓘ
codomain	space of holomorphic functions on a suitable complexified parameter space ⓘ
concerns	Fourier transform on real reductive groups ⓘ Harish-Chandra transform ⓘ support properties of matrix coefficients ⓘ
describesAs	holomorphic functions on the complexified dual of a Cartan subalgebra ⓘ
domain	space of compactly supported smooth functions on a real reductive group ⓘ
field	harmonic analysis ⓘ noncommutative harmonic analysis ⓘ representation theory ⓘ
generalizes	Paley–Wiener theorem for compact Lie groups ⓘ linked to: Paley–Wiener theorem for real reductive groups classical Paley–Wiener theorem on ℝⁿ ⓘ linked to: Paley–Wiener theorem
gives	necessary and sufficient conditions for a holomorphic function to be a Fourier transform of a compactly supported smooth function ⓘ
hasVersion	Paley–Wiener theorem for K-finite functions ⓘ linked to: Paley–Wiener theorem for real reductive groups Paley–Wiener theorem for Schwartz space on real reductive groups ⓘ linked to: Paley–Wiener theorem for real reductive groups spherical Paley–Wiener theorem for real reductive groups ⓘ linked to: Paley–Wiener theorem for real reductive groups
imposesConditionOn	growth of holomorphic functions ⓘ support of holomorphic functions ⓘ
involves	Weyl group invariance conditions ⓘ discrete series representations ⓘ parabolic induction ⓘ tempered representations ⓘ
mathematicalSubjectClassification	22E30 ⓘ 43A85 ⓘ
relatedTo	Fourier inversion formula on real reductive groups ⓘ Harish-Chandra c-function ⓘ Plancherel theorem for real reductive groups ⓘ spherical Fourier transform ⓘ
usedIn	Langlands program ⓘ harmonic analysis on semisimple Lie groups ⓘ spectral decomposition of L²(G) ⓘ study of automorphic representations ⓘ trace formula ⓘ linked to: Arthur trace formula
usesConcept	Cartan subalgebra ⓘ linked to: Cartan subalgebras Cartan subgroup ⓘ linked to: Cartan subalgebras Harish-Chandra’s Plancherel theory ⓘ linked to: Plancherel theorem for real reductive groups Iwasawa decomposition ⓘ real reductive Lie group ⓘ

How these facts were elicited

Referenced by (6)

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

Plancherel theorem for real reductive groups → isRelatedTo → Paley–Wiener theorem for real reductive groups ⓘ

Paley–Wiener theorem for real reductive groups → generalizes → Paley–Wiener theorem for compact Lie groups ⓘ