Bruhat–Tits theory

E921613

mathematical theory theory in arithmetic geometry theory in representation theory

Bruhat–Tits theory is a framework in arithmetic geometry and representation theory that studies reductive algebraic groups over non-archimedean local fields via associated geometric objects called Bruhat–Tits buildings.

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Statements (48)

Predicate	Object
instanceOf	mathematical theory ⓘ theory in arithmetic geometry ⓘ theory in representation theory ⓘ
appliesTo	reductive groups over non-archimedean local fields ⓘ reductive groups over p-adic fields ⓘ
assumes	non-archimedean local field with discrete valuation ⓘ
basedOn	Tits theory of buildings NERFINISHED ⓘ structure theory of reductive algebraic groups ⓘ
characteristicOf	non-archimedean geometry ⓘ
developedIn	1970s ⓘ
field	algebraic groups ⓘ arithmetic geometry ⓘ p-adic groups ⓘ representation theory ⓘ
hasApplication	construction of local factors in the Langlands correspondence ⓘ study of reduction of algebraic groups modulo the maximal ideal of the valuation ring ⓘ
hasPart	classification of parahoric subgroups ⓘ construction of buildings from valued root data ⓘ description of special and hyperspecial vertices ⓘ
influenced	Moy–Prasad theory NERFINISHED ⓘ theory of buildings in geometric group theory ⓘ
mainSubject	reductive algebraic groups over non-archimedean local fields ⓘ
namedAfter	François Bruhat NERFINISHED ⓘ Jacques Tits NERFINISHED ⓘ
provides	classification of reductive groups over non-archimedean local fields up to isomorphism ⓘ construction of Bruhat–Tits buildings attached to reductive groups ⓘ description of maximal compact subgroups via vertices in buildings ⓘ description of parahoric subgroups via facets in buildings ⓘ structure theory for p-adic groups ⓘ
relates	algebraic structure of a reductive group ⓘ geometry of its associated building ⓘ
studies	Moy–Prasad filtrations (via its later developments) ⓘ filtrations of parahoric subgroups ⓘ group schemes over valuation rings ⓘ integral models of reductive groups ⓘ
usedIn	construction of types and Hecke algebras for p-adic groups ⓘ harmonic analysis on p-adic groups ⓘ local aspects of the Langlands program ⓘ representation theory of p-adic groups ⓘ study of automorphic forms ⓘ
usesConcept	Bruhat–Tits buildings NERFINISHED ⓘ affine buildings ⓘ apartment ⓘ chamber ⓘ maximal compact subgroups ⓘ parahoric subgroups ⓘ root data ⓘ valuations of root data ⓘ

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Deligne–Lusztig theory → relatesTo → Bruhat–Tits theory ⓘ