Bernstein center in representation theory

E934437

center of a category commutative algebra mathematical object

The Bernstein center in representation theory is a commutative algebra that acts as the center of the category of smooth representations of a p-adic reductive group, playing a key role in decomposing and classifying these representations.

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Statements (49)

Predicate	Object
instanceOf	center of a category ⓘ commutative algebra ⓘ mathematical object ⓘ
actsBy	natural endomorphisms on every smooth representation ⓘ scalars on irreducible smooth representations via central characters ⓘ
actsOn	category of smooth representations of a p-adic reductive group ⓘ
alsoKnownAs	Bernstein center NERFINISHED ⓘ Bernstein’s center NERFINISHED ⓘ
analogy	Harish-Chandra’s Schwartz algebra center for real reductive groups NERFINISHED ⓘ
appearsIn	Bushnell–Kutzko theory of types NERFINISHED ⓘ theory of types for p-adic groups ⓘ
constructedAs	algebra of natural transformations from the identity functor to itself ⓘ endomorphism ring of the identity functor on the category of smooth representations ⓘ
context	smooth complex representations of p-adic reductive groups ⓘ smooth representations over algebraically closed fields of characteristic 0 ⓘ
definedFor	locally compact totally disconnected groups under suitable hypotheses ⓘ p-adic reductive groups ⓘ
dependsOn	the coefficient field of representations ⓘ the underlying p-adic reductive group ⓘ
field	representation theory ⓘ
generalizationOf	center of the group algebra in the finite group case ⓘ
geometricRealization	algebra of regular functions on the Bernstein variety (Bernstein spectrum) in many cases ⓘ
hasComponent	idempotents projecting to individual Bernstein blocks ⓘ
introducedBy	Joseph Bernstein NERFINISHED ⓘ
is	center of the category of smooth representations of a p-adic reductive group ⓘ commutative algebra of endomorphisms of the identity functor on the category of smooth representations ⓘ
property	commutative ⓘ functorial in the group under suitable morphisms ⓘ idempotents correspond to Bernstein components ⓘ
relatedTo	Bernstein decomposition NERFINISHED ⓘ Bernstein spectrum ⓘ Hecke algebras attached to p-adic groups ⓘ Langlands classification NERFINISHED ⓘ cuspidal representations ⓘ inertial equivalence classes of cuspidal data ⓘ local Langlands correspondence NERFINISHED ⓘ parabolic induction ⓘ tempered representations ⓘ
role	classifies central characters of smooth representations ⓘ controls decomposition of the category of smooth representations into Bernstein blocks ⓘ parametrizes the block decomposition of the category of smooth representations ⓘ provides spectral decomposition of the category of smooth representations ⓘ
studiedIn	automorphic forms ⓘ p-adic harmonic analysis ⓘ
typicalCoefficientField	algebraically closed fields of characteristic 0 GENERATED ⓘ complex numbers GENERATED ⓘ
usedFor	classification of irreducible smooth representations of p-adic reductive groups ⓘ definition of Bernstein components (blocks) of the category of smooth representations ⓘ localization of the category of smooth representations ⓘ

Referenced by (1)

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

Joseph Bernstein → notableWork → Bernstein center in representation theory ⓘ