Bernstein–Zelevinsky classification
E934436
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Bernstein–Zelevinsky classification canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11576263 — 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: Bernstein–Zelevinsky classification Context triple: [Joseph Bernstein, notableWork, Bernstein–Zelevinsky classification]
-
A.
Langlands classification
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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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.
Kazhdan–Lusztig theory
Kazhdan–Lusztig theory is a framework in representation theory and algebraic geometry that studies Hecke algebras and their bases via Kazhdan–Lusztig polynomials, with deep connections to the representation theory of Lie algebras and geometry of Schubert varieties.
-
D.
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.
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E.
Gelfand–Tsetlin basis
The Gelfand–Tsetlin basis is a canonical, combinatorially defined basis for representations of certain Lie algebras and groups, particularly used in the representation theory of GL(n) and related structures.
- 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: Bernstein–Zelevinsky classification Target entity description: 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.
-
A.
Langlands classification
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.
-
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.
Kazhdan–Lusztig theory
Kazhdan–Lusztig theory is a framework in representation theory and algebraic geometry that studies Hecke algebras and their bases via Kazhdan–Lusztig polynomials, with deep connections to the representation theory of Lie algebras and geometry of Schubert varieties.
-
D.
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.
-
E.
Gelfand–Tsetlin basis
The Gelfand–Tsetlin basis is a canonical, combinatorially defined basis for representations of certain Lie algebras and groups, particularly used in the representation theory of GL(n) and related structures.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Bernstein–Zelevinsky classification Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.