Bruhat–Tits building

Bruhat–Tits building
Name	Bruhat–Tits building
Discipline	Mathematics
Subdiscipline	Algebraic groups, Number theory, Geometry
Introduced	1970s
Founders	François Bruhat, Jacques Tits
Related	BN-pair, Euclidean buildings, Tits building, p-adic groups, Reductive groups

Bruhat–Tits building. The Bruhat–Tits building is a combinatorial and geometric structure associated with reductive algebraic groups over non-archimedean local fields, introduced by François Bruhat and Jacques Tits in the 1960s and 1970s. It provides a simplicial complex and a metric space on which groups such as GL_n, SL_n, Sp_n, SO_n, and exceptional groups like E_8 act with strong transitivity properties, connecting Tits building theory, BN-pair axioms, and harmonic analysis on p-adic groups. The building underlies many advances in the study of automorphic forms, Langlands program, and the representation theory of adelic and local field groups.

Introduction

The construction of the Bruhat–Tits building synthesizes ideas from the works of François Bruhat and Jacques Tits with techniques influenced by Claude Chevalley, Jean-Pierre Serre, and developments in Weil group and Cartan decomposition theory. The building refines the earlier concept of the spherical Tits building for algebraic groups over algebraically closed fields, and adapts it to the arithmetic context of p-adic numbers, local fields, and valuation rings. Its combinatorial geometry encodes the structure of parahoric subgroups, facets corresponding to points fixed by compact subgroups such as Iwahori subgroups, and apartments modeled on affine Coxeter complexes like those arising in the study of Weyl groups and Coxeter groups.

Construction and Definitions

Bruhat–Tits buildings are constructed using data from a connected reductive algebraic group G over a non-archimedean local field K with valuation ring O_K, uniformizer, and residue field. One chooses a maximal torus T and studies the affine root system from the relative root datum developed by Serre and Chevalley. The building is assembled from affine spaces called apartments associated with conjugacy classes of T, with vertices representing equivalence classes of O_K-lattices in representation spaces for groups like GL_n(K), SL_n(K), and GSp_n(K). Parahoric subgroup classification uses the Bruhat decomposition, Iwahori–Hecke algebra, and the theory of Moy–Prasad filtrations to identify stabilizers of faces, while Langlands decomposition perspectives connect to Levi subgroups and parabolic subgroups studied by Bernstein and Casselman.

Apartments, Chambers, and Weyl Group Action

Each apartment in the building is an affine space modeled on the cocharacter lattice of a maximal torus and is tiled by alcoves (chambers) determined by affine roots linked to the affine Weyl group associated with the relative root system studied by Bourbaki and Kac. The chambers correspond to maximal compact subgroups like Iwahori subgroups and are permuted transitively by the normalizer of the torus, yielding an action of the extended affine Weyl group that mirrors the combinatorics of Dynkin diagrams and their affine extensions appearing in Kac–Moody algebra contexts. Galleries, retractions, and folding maps used by Tits and Rémy control geodesic combinatorics and relate to the Bruhat order familiar from Schubert variety theory and Borel subgroup orbit decompositions.

Metric and Geometric Properties

As a metric space, the Bruhat–Tits building is a complete CAT(0) space when endowed with a Euclidean metric on each apartment, linking the theory to works of Bridson and Haefliger. This nonpositive curvature property implies uniqueness of geodesics, fixed-point theorems for compact group actions such as those by Serre and Tits, and strong convexity properties used in rigidity results of Margulis and superrigidity phenomena studied by Zimmer. Spherical buildings at infinity correspond to asymptotic directions and relate to compactifications considered by Satake and Polyakov, while metric projections to apartments underpin reduction theory developed by Borel and Harish-Chandra.

Buildings for Reductive Groups over Local Fields

For classical groups like GL_n, SL_n, Sp_{2n}, and SO_{n}, explicit models of the building arise from lattice-theoretic descriptions tied to Hermitian forms and quadratic forms over division algebras studied by Dieudonné and Wedderburn. For exceptional groups such as G_2, F_4, E_6, E_7, and E_8, Bruhat–Tits theory requires delicate analysis of relative root data combined with descent theory of algebraic groups over ramified extensions as in the works of Tits and Bruhat. The classification of parahoric subgroup schemes over the valuation ring links to group scheme theory developed by Chevalley, Demazure, and SGA3 contributors.

Applications and Examples

Bruhat–Tits buildings are central in the study of representation theory of p-adic groups via the geometry of Bernstein center and the study of types by Bushnell and Kutzko, and they provide frameworks for analytic constructions in the theory of automorphic forms as in the work of Gelbart and Godement. They appear in the proof strategies for the local Langlands correspondence advanced by Harris, Taylor, Henniart, and Kazhdan, and in harmonic analysis on reductive groups by Moy, Prasad, and Howe. Buildings inform the study of discrete subgroups in Bruhat–Tits trees for SL_2 over local fields, with applications to arithmetic groups like SL_2(Z) analogues, and have been used in geometric group theory by Gromov and Bass–Serre theory.

Cohomology and Representation-Theoretic Connections

Cohomology of buildings, including compactly supported and L^2-cohomology, interacts with the representation theory of reductive p-adic groups through Hecke algebra modules and the theory of automorphic representations developed by Langlands and Arthur. Vanishing theorems and spectral sequences associated with the simplicial structure feed into the study of supercuspidal representations by Moy–Prasad and types by Bushnell–Kutzko, while the equivariant K-theory of buildings ties to index theory explored by Atiyah and Kasparov. The interplay between building cohomology and completed cohomology in the work of Emerton informs p-adic Banach space representations and the ongoing development of non-abelian Lubin–Tate and Drinfeld tower analyses.

Category:Buildings (group theory) Category:Algebraic groups