Langlands classification
E877880
The Langlands classification is a fundamental framework in representation theory that systematically describes all irreducible admissible representations of a real or p-adic reductive group in terms of data from its parabolic subgroups and their characters.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Langlands classification canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T10641438 — 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: Langlands classification Context triple: [Harish-Chandra character formula, relatedTo, Langlands classification]
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A.
Langlands program
The Langlands program is a far-reaching web of conjectures and theories in number theory and representation theory that seeks deep connections between Galois groups and automorphic forms, unifying many areas of modern mathematics.
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B.
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.
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C.
Deligne–Lusztig theory
Deligne–Lusztig theory is a framework in algebraic geometry and representation theory that constructs and studies representations of finite groups of Lie type using varieties defined over finite fields.
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D.
Langlands dual group
The Langlands dual group is an algebraic group constructed from a given reductive group by interchanging its root and coroot data, playing a central role in the Langlands program’s connections between number theory and representation theory.
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E.
Hecke theory
Hecke theory is a branch of number theory centered on Hecke operators and modular forms, providing powerful tools to study arithmetic properties of modular forms and related objects.
- 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: Langlands classification Target entity description: The Langlands classification is a fundamental framework in representation theory that systematically describes all irreducible admissible representations of a real or p-adic reductive group in terms of data from its parabolic subgroups and their characters.
-
A.
Langlands program
The Langlands program is a far-reaching web of conjectures and theories in number theory and representation theory that seeks deep connections between Galois groups and automorphic forms, unifying many areas of modern mathematics.
-
B.
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.
-
C.
Deligne–Lusztig theory
Deligne–Lusztig theory is a framework in algebraic geometry and representation theory that constructs and studies representations of finite groups of Lie type using varieties defined over finite fields.
-
D.
Langlands dual group
The Langlands dual group is an algebraic group constructed from a given reductive group by interchanging its root and coroot data, playing a central role in the Langlands program’s connections between number theory and representation theory.
-
E.
Hecke theory
Hecke theory is a branch of number theory centered on Hecke operators and modular forms, providing powerful tools to study arithmetic properties of modular forms and related objects.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
classification theorem
ⓘ
concept in representation theory ⓘ mathematical theory ⓘ |
| appliesTo |
p-adic reductive groups
ⓘ
real reductive groups ⓘ reductive groups ⓘ |
| assumes |
admissibility of representations
ⓘ
reductive group over a local field ⓘ |
| basedOn | parabolic induction from tempered representations twisted by characters ⓘ |
| characterizes | irreducible admissible representations up to isomorphism ⓘ |
| connectedTo | Plancherel formula for reductive groups NERFINISHED ⓘ |
| coreIdea |
parametrization of irreducible admissible representations by data from parabolic subgroups
ⓘ
realization of irreducible representations as unique irreducible quotients of standard induced modules ⓘ |
| describes | irreducible admissible representations ⓘ |
| developedBy | Robert Langlands NERFINISHED ⓘ |
| field |
harmonic analysis
ⓘ
number theory ⓘ representation theory ⓘ |
| formalism | classification via standard parabolic subgroups and their Levi components ⓘ |
| goal | systematic description of all irreducible admissible representations of a reductive group ⓘ |
| hasVersion |
archimedean Langlands classification
NERFINISHED
ⓘ
non-archimedean Langlands classification NERFINISHED ⓘ |
| historicalPeriod | late 20th century ⓘ |
| implies |
every irreducible admissible representation is a Langlands quotient
ⓘ
uniqueness of Langlands data for each irreducible admissible representation ⓘ |
| influenced |
development of the local Langlands conjectures
ⓘ
modern harmonic analysis on reductive groups ⓘ |
| output | parameter set for irreducible admissible representations ⓘ |
| relatedTo |
Bernstein decomposition
NERFINISHED
ⓘ
Harish-Chandra theory NERFINISHED ⓘ Langlands program NERFINISHED ⓘ local Langlands correspondence NERFINISHED ⓘ tempered dual ⓘ unitary dual ⓘ |
| scope |
admissible dual of a reductive group
ⓘ
local representations ⓘ |
| studiedIn |
automorphic forms
ⓘ
p-adic representation theory ⓘ representation theory of real Lie groups ⓘ |
| usesConcept |
Langlands data
NERFINISHED
ⓘ
Langlands parameters ⓘ Langlands quotient NERFINISHED ⓘ Levi subgroups ⓘ characters of Levi subgroups ⓘ parabolic induction ⓘ parabolic subgroups ⓘ standard modules ⓘ tempered representations ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Langlands classification Description of subject: The Langlands classification is a fundamental framework in representation theory that systematically describes all irreducible admissible representations of a real or p-adic reductive group in terms of data from its parabolic subgroups and their characters.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.