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 (5)
| Label | Occurrences |
|---|---|
| Langlands classification canonical | 5 |
| Langlands data | 1 |
| Langlands quotient | 1 |
| archimedean Langlands classification | 1 |
| non-archimedean Langlands classification | 1 |
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]
-
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
- 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
ⓘ
linked to:
Plancherel theorem for real reductive groups
|
| 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 ⓘ |
| 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
ⓘ
linked to:
Langlands classification
non-archimedean Langlands classification ⓘ
linked to:
Langlands classification
|
| 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
ⓘ
Harish-Chandra theory ⓘ Langlands program ⓘ local Langlands correspondence ⓘ 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
ⓘ
linked to:
Langlands classification
Langlands parameters ⓘ Langlands quotient ⓘ
linked to:
Langlands classification
Levi subgroups ⓘ characters of Levi subgroups ⓘ parabolic induction ⓘ parabolic subgroups ⓘ standard modules ⓘ tempered representations ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
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: 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 (9)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Langlands classification
linked to: Langlands classification
linked to: Langlands classification
linked to: Langlands classification