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.
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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.