Bernstein–Zelevinsky classification

E934436

classification theorem mathematical theory result in representation theory

The Bernstein–Zelevinsky classification is a foundational framework in representation theory that systematically describes irreducible smooth representations of general linear groups over non-archimedean local fields.

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Statements (47)

Predicate	Object
instanceOf	classification theorem ⓘ mathematical theory ⓘ result in representation theory ⓘ
appliesTo	GL(n,F) NERFINISHED ⓘ general linear groups over non-archimedean local fields ⓘ irreducible smooth representations ⓘ
assumes	non-archimedean local base field of characteristic 0 or positive characteristic ⓘ smooth complex representations ⓘ
characteristicFeature	description of composition series of parabolically induced representations ⓘ parametrization by multisegments of cuspidal representations ⓘ use of derivatives to analyze reducibility ⓘ
characterizes	irreducible representations as Langlands quotients of standard modules ⓘ irreducible representations via multisegments of cuspidal data ⓘ
clarifies	reducibility points of parabolic induction for GL(n,F) ⓘ
describes	irreducible smooth complex representations of GL(n,F) ⓘ structure of the unitary dual of GL(n,F) over non-archimedean local fields ⓘ
domain	non-archimedean local field ⓘ
field	p-adic representation theory ⓘ representation theory ⓘ
frameworkFor	computing characters of irreducible representations of GL(n,F) ⓘ studying unitary dual of GL(n,F) ⓘ understanding Hecke algebra modules attached to GL(n,F) ⓘ
generalizes	Zelevinsky classification for GL(n) over p-adic fields NERFINISHED ⓘ
hasImpactOn	representation theory of affine Hecke algebras ⓘ
influenced	classification of representations of p-adic reductive groups ⓘ development of the local Langlands program ⓘ
involves	irreducible essentially square-integrable representations ⓘ tempered representations ⓘ
namedAfter	Andrei Zelevinsky NERFINISHED ⓘ Joseph Bernstein NERFINISHED ⓘ
originallyFormulatedFor	GL(n) over p-adic fields NERFINISHED ⓘ
provides	combinatorial description of irreducible representations ⓘ parametrization of irreducible smooth representations of GL(n,F) ⓘ
relatedTo	Bernstein decomposition NERFINISHED ⓘ Langlands classification NERFINISHED ⓘ local Langlands correspondence NERFINISHED ⓘ
timePeriod	1970s ⓘ
usedIn	automorphic forms ⓘ harmonic analysis on p-adic groups ⓘ number theory ⓘ
usesConcept	Jacquet modules NERFINISHED ⓘ cuspidal representations ⓘ derivatives of representations ⓘ multisegments ⓘ parabolic induction ⓘ segments ⓘ supercuspidal representations ⓘ

Referenced by (1)

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Joseph Bernstein → notableWork → Bernstein–Zelevinsky classification ⓘ