Langlands dual group
E870220
algebraic group
concept in number theory
concept in representation theory
object in the Langlands program
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Langlands dual group canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10550238 — 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 dual group Context triple: [Robert Langlands, notableIdea, Langlands dual group]
-
A.
Weyl group
A Weyl group is a finite reflection group associated with a root system that encodes the symmetries of Lie algebras and Lie groups in representation theory and geometry.
-
B.
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.
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C.
Weil–Deligne group
The Weil–Deligne group is an extension of the Weil group by a copy of the additive group that encodes both arithmetic and monodromy data, playing a central role in the local Langlands correspondence and the study of l-adic Galois representations.
-
D.
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.
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E.
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.
- 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 dual group Target entity description: 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.
-
A.
Weyl group
A Weyl group is a finite reflection group associated with a root system that encodes the symmetries of Lie algebras and Lie groups in representation theory and geometry.
-
B.
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.
-
C.
Weil–Deligne group
The Weil–Deligne group is an extension of the Weil group by a copy of the additive group that encodes both arithmetic and monodromy data, playing a central role in the local Langlands correspondence and the study of l-adic Galois representations.
-
D.
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.
-
E.
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.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic group
ⓘ
concept in number theory ⓘ concept in representation theory ⓘ object in the Langlands program ⓘ |
| appearsIn |
Tannakian formalism
ⓘ
geometric Satake equivalence NERFINISHED ⓘ unramified local Langlands correspondence ⓘ |
| centralRoleIn |
classification of automorphic representations
ⓘ
parameterization of L-parameters ⓘ |
| constructedBy |
interchanging roots and coroots
ⓘ
interchanging weight lattice and coweight lattice ⓘ |
| context |
arithmetic geometry
ⓘ
harmonic analysis on reductive groups ⓘ |
| correspondsTo |
Langlands parameters
NERFINISHED
ⓘ
original group via dual root datum ⓘ |
| definedFor | reductive algebraic group ⓘ |
| definedFrom |
character lattice of a maximal torus
ⓘ
cocharacter lattice of a maximal torus ⓘ coroot system of the original group ⓘ root system of the original group ⓘ |
| definedOver | algebraically closed field of characteristic zero ⓘ |
| example |
dual of GL_n is GL_n
ⓘ
dual of PGL_n is SL_n ⓘ dual of SL_n is PGL_n ⓘ dual of SO_{2n+1} is Sp_{2n} ⓘ dual of SO_{2n} is SO_{2n} ⓘ dual of Sp_{2n} is SO_{2n+1} ⓘ |
| hasComponent |
dual Borel subgroup
ⓘ
dual maximal torus ⓘ |
| hasInput | root datum of a reductive group ⓘ |
| hasOutput | connected reductive algebraic group ⓘ |
| introducedBy | Robert Langlands NERFINISHED ⓘ |
| property |
root datum dual to that of the original group
ⓘ
same Weyl group as the original group ⓘ |
| relatedTo |
Galois representations
NERFINISHED
ⓘ
Hecke eigenvalues ⓘ L-group NERFINISHED ⓘ Satake isomorphism NERFINISHED ⓘ Weil group NERFINISHED ⓘ automorphic forms ⓘ |
| usedIn |
Langlands correspondence
NERFINISHED
ⓘ
automorphic representation theory ⓘ geometric Langlands program NERFINISHED ⓘ global Langlands correspondence NERFINISHED ⓘ local Langlands correspondence NERFINISHED ⓘ |
| usedTo |
define L-functions via representations
ⓘ
describe unramified representations of p-adic groups ⓘ formulate functoriality conjectures ⓘ parametrize Hecke eigen-sheaves in geometric Langlands ⓘ |
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 dual group Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.