Langlands classification

E877880

classification theorem concept in representation theory mathematical theory

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.

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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 ⓘ

Referenced by (2)

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

Harish-Chandra character formula → relatedTo → Langlands classification ⓘ

Plancherel theorem for real reductive groups → involves → Langlands classification ⓘ