Harish-Chandra regularity theorem

E876154

mathematical theorem theorem in representation theory

The Harish-Chandra regularity theorem is a fundamental result in representation theory that asserts characters of irreducible admissible representations of real reductive Lie groups are given by real-analytic, locally integrable functions on the group.

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Observed surface forms (1)

Surface form	Occurrences
Harish-Chandra’s theory of characters	1

Statements (47)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem in representation theory ⓘ
appliesTo	irreducible admissible representations ⓘ real reductive Lie groups ⓘ
asserts	characters of irreducible admissible representations of real reductive Lie groups are given by real-analytic functions on the group ⓘ characters of irreducible admissible representations of real reductive Lie groups are locally integrable functions on the group ⓘ distribution characters of irreducible admissible representations are given by locally integrable functions on the group ⓘ distribution characters of irreducible admissible representations are represented by real-analytic functions on the regular set ⓘ
characterType	global character regularity ⓘ local regularity on regular semisimple set ⓘ
concerns	characters of representations ⓘ
context	admissible representations of real reductive Lie groups ⓘ unitary representation theory of real reductive Lie groups ⓘ
ensures	distribution characters are smooth on the regular semisimple set ⓘ singularities of characters are controlled and mild ⓘ
field	Lie theory ⓘ harmonic analysis ⓘ representation theory ⓘ
hasConsequence	characters can be studied using analytic methods on Lie groups ⓘ characters determine irreducible admissible representations up to equivalence ⓘ
holdsFor	connected real reductive Lie groups ⓘ finite-length admissible representations ⓘ
implies	characters are real-analytic on the set of regular elements ⓘ characters extend as locally integrable functions on the whole group ⓘ characters of irreducible admissible representations are class functions ⓘ
involves	Harish-Chandra’s Schwartz space NERFINISHED ⓘ center of the universal enveloping algebra ⓘ infinitesimal character ⓘ invariant eigendistributions ⓘ universal enveloping algebra of a Lie algebra ⓘ
isPartOf	Harish-Chandra’s program on harmonic analysis on semisimple Lie groups NERFINISHED ⓘ
namedAfter	Harish-Chandra NERFINISHED ⓘ
refines	the description of characters as conjugation-invariant distributions ⓘ
relatedTo	Harish-Chandra’s Plancherel theorem NERFINISHED ⓘ Harish-Chandra’s character formula NERFINISHED ⓘ Harish-Chandra’s subquotient theorem NERFINISHED ⓘ
strengthens	the fact that characters are distributions ⓘ
usedFor	analysis of the unitary dual of real reductive Lie groups ⓘ classification of irreducible admissible representations ⓘ harmonic analysis on real reductive Lie groups ⓘ study of trace formulas ⓘ
usesConcept	Harish-Chandra’s theory of characters ⓘ conjugation-invariant distributions ⓘ distribution characters ⓘ locally integrable functions ⓘ real-analytic functions ⓘ regular elements of a Lie group ⓘ

Referenced by (2)

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

Harish-Chandra character formula → uses → Harish-Chandra regularity theorem ⓘ

Plancherel theorem for real reductive groups → isBasedOn → Harish-Chandra regularity theorem ⓘ

this entity surface form: Harish-Chandra’s theory of characters