Topic Tits Buildings
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Tits Buildings

A (Tits) building is a simplicial complex equipped with a system of apartments — each apartment a Coxeter complex — and a "highly homogeneous" geometric structure. Buildings provide a uniform geometric model for reductive groups (spherical buildings) and \(p\)-adic reductive groups (affine = Bruhat-Tits buildings).

Axiomatic definition

A building is a simplicial complex \(\mathcal{B}\) equipped with a non-empty system \(\mathcal{A}\) of subcomplexes (called apartments) satisfying:

(B1) Each apartment \(A \in \mathcal{A}\) is isomorphic to the Coxeter complex \(\Sigma(W, S)\) for some fixed Coxeter system \((W, S)\);

(B2) For any two simplices \(\sigma, \tau \in \mathcal{B}\), there is an apartment \(A \in \mathcal{A}\) containing both;

(B3) For two apartments \(A, A'\) containing simplices \(\sigma, \tau\), there is an isomorphism \(A \to A'\) fixing \(\sigma \cap A \cap A'\) and \(\tau \cap A \cap A'\) pointwise.

\(\mathcal{B}\) is called of type \((W, S)\) and inherits a Weyl distance \(\delta : \text{chambers} \times \text{chambers} \to W\).

Equivalent formulations

  • W-metric. A pair (set of chambers, function \(\delta\) to \(W\)) with axioms (W1)-(W4) (Tits).
  • Combinatorial. A chamber complex satisfying connectedness and uniqueness-of-residues axioms.
  • From a BN-pair. Given a Tits system \((G, B, N, S)\), the building has cosets of standard parabolic subgroups as simplices, with \(G\) acting.

Spherical buildings

For a connected reductive group \(G\) over a field \(k\), the spherical building \(\Delta(G, k)\) is:

  • vertices = maximal proper parabolic subgroups of \(G(k)\) (modulo conjugation);
  • chambers = Borel subgroups;
  • simplices = parabolics, ordered by reverse inclusion.

The associated Coxeter group is the finite Weyl group of \(G\); apartments are sphere triangulations.

Example: \(\mathrm{GL}_n\)

\(\Delta(\mathrm{GL}_n, k)\) is the simplicial complex of proper non-trivial flags in \(k^n\). Vertices = proper subspaces; simplices = flags. Chambers = full (maximal) flags.

This is the prototypical building of type \(A_{n-1}\).

Affine (Bruhat-Tits) buildings

For \(G\) reductive over a non-archimedean local field \(F\), the affine building \(\mathcal{B}(G, F)\) is built using the affine BN-pair \((G(F), I, N, S_{\mathrm{aff}})\) with \(I\) the Iwahori subgroup:

  • vertices = maximal parahoric subgroups;
  • chambers = Iwahori subgroups;
  • apartments = Euclidean Coxeter complexes for the affine Weyl group.

\(\mathcal{B}(G, F)\) carries a natural CAT(0) metric (Bruhat-Tits), making it a complete metric space with non-positive curvature.

Example: \(\mathrm{SL}_2(\mathbb{Q}_p)\)

\(\mathcal{B}(\mathrm{SL}_2, \mathbb{Q}_p)\) is the \((p+1)\)-regular tree: an infinite tree where every vertex has \(p+1\) neighbours. This is the simplest non-trivial affine building and the geometric model for many \(p\)-adic constructions.

Solomon-Tits theorem

Theorem (Solomon-Tits). A spherical building \(\Delta\) of rank \(r\) is homotopy-equivalent to a wedge of \(r\)-spheres: \[ \Delta \simeq \bigvee S^r. \]

In particular, \(\widetilde{H}_r(\Delta, \mathbb{Z})\) is free of high rank — the Steinberg representation appears here, foundational to representation theory of finite reductive groups.

Tits' classification

Theorem (Tits, 1974). Irreducible spherical buildings of rank \(\ge 3\) are classified: they all come from BN-pairs of simple algebraic groups (or twisted versions). Rank-2 buildings include generalized polygons (\(n\)-gons), classified for finite case but not in general.

This is the high-rank generalisation of the Fundamental Theorem of Projective Geometry, and the proof uses careful "Moufang"-style axiomatic refinement.

Role in Langlands

  • Local Langlands: the affine building of \(G\) over a local field is the natural geometric setting for the Bushnell-Kutzko theory of types and the parameterisation of supercuspidals.
  • Iwahori-Hecke algebras: \(\mathcal{H}(I \backslash G/I)\) is the convolution algebra of \(I\)-bi-invariant functions; its category of modules is equivalent to a category of perverse sheaves on the affine flag variety.
  • Geometric Satake: the building's vertices correspond to lattices, and the geometry of the affine Grassmannian \(\mathrm{Gr}_G\) projects to the building structure.
  • Mod \(p\) Langlands: the Galois cohomology of buildings (via Borel-Serre compactification) realises pro-\(p\) Iwahori-Hecke algebras.

Mathlib

Not formalised; depends on Coxeter group infrastructure and étale descent.

References

  • Tits, Buildings of Spherical Type and Finite BN-Pairs (LNM 386, 1974).
  • Brown, Buildings (1989).
  • Bruhat–Tits, Groupes réductifs sur un corps local I, II (Publ.\ IHES 1972, 1984).
  • Abramenko–Brown, Buildings: Theory and Applications (2008).
  • Tits, A local approach to buildings (1980).

Prerequisite Topics