Spherical Buildings
A spherical building is a building whose apartments are sphere triangulations (= finite Coxeter complexes). They are the buildings of finite Coxeter type and are attached to reductive groups over arbitrary fields, encoding the Bruhat decomposition geometrically.
Definition
A simplicial complex \(\Delta\) is a spherical building if:
- It is a building (in the sense of Tits);
- Each apartment is a Coxeter complex \(\Sigma(W, S)\) for a finite Coxeter system \((W, S)\).
Equivalently: a spherical building has finite Weyl group, and each apartment is a triangulation of a sphere \(S^{r-1}\) (\(r\) = rank).
Construction from a reductive group
For \(G\) a connected reductive group over a field \(k\) with Borel \(B\), maximal torus \(T \subseteq B\), normaliser \(N = N_G(T)\):
The spherical building of \(G\) is the simplicial complex with:
- Vertices = maximal proper parabolic subgroups of \(G(k)\);
- Simplices = parabolic subgroups, ordered by reverse inclusion;
- Chambers = Borel subgroups of \(G(k)\).
The dimension is \(\mathrm{rank}(G) - 1\); the Weyl group is the finite Weyl group of \(G\); apartments correspond to conjugacy classes of maximal split tori.
Examples
\(\mathrm{GL}_n\)
The spherical building of \(\mathrm{GL}_n(k)\) is the flag complex of \(k^n\):
- Vertices = proper non-trivial subspaces of \(k^n\);
- Simplices = flags \(V_1 \subsetneq V_2 \subsetneq \cdots \subsetneq V_r\);
- Chambers = full flags \(V_1 \subsetneq \cdots \subsetneq V_{n-1}\).
Apartments correspond to direct-sum decompositions \(k^n = L_1 \oplus \cdots \oplus L_n\) into lines; each apartment is the boundary complex of a simplex.
\(\mathrm{SL}_n(k)\)
Same building as \(\mathrm{GL}_n(k)\) (since central isogenies don't change parabolic structure).
\(\mathrm{Sp}_{2n}(k)\)
The spherical building consists of totally isotropic subspaces of \(k^{2n}\) for the symplectic form, ordered by inclusion. Chambers = maximal isotropic flags. Type \(C_n\) building.
Finite buildings
For \(k = \mathbb{F}_q\): the spherical building of \(G(\mathbb{F}_q)\) is a finite simplicial complex; its cohomology gives the Steinberg representation of \(G(\mathbb{F}_q)\).
Solomon-Tits theorem
Theorem (Solomon, Tits). A spherical building \(\Delta\) of rank \(r\) (= dimension \(r-1\)) is homotopy-equivalent to a wedge of \((r-1)\)-spheres: \[ \Delta \;\simeq\; \bigvee_{i} S^{r-1}. \]
The number of spheres in the wedge is \(|G(\mathbb{F}_q)| / |B(\mathbb{F}_q)|\) in the finite case (= Steinberg's formula).
Tits classification
Theorem (Tits 1974). Irreducible spherical buildings of Tits rank \(\ge 3\) are classified: they are exactly the buildings of simple algebraic groups (or their close relatives, including twisted forms).
In rank 2, the buildings are generalized polygons (= bipartite graphs of girth \(2n\) and diameter \(n\)) and are classified only in the finite or Moufang case.
Tits's classification is a high-rank generalisation of the Fundamental Theorem of Projective Geometry.
Moufang property
A spherical building is Moufang if it has "many automorphisms" — specifically, root subgroups \(U_\alpha\) acting transitively on chambers opposite to a given chamber. All buildings of reductive groups are Moufang.
In Tits's classification: all irreducible thick spherical buildings of rank \(\ge 3\) are Moufang. Rank 2 (= generalized polygons) admits non-Moufang examples (e.g.\ "exotic" generalized hexagons).
Spherical building vs.\ affine building
For \(G\) over a local field \(F\): there are two buildings:
- Spherical building: parametrises parabolic subgroups of \(G(F)\); finite-dimensional, sphere triangulation.
- Affine building (Bruhat-Tits): parametrises parahoric subgroups; Euclidean tiling, infinite-dimensional in the sense that apartments are full Euclidean spaces.
The spherical building is the "boundary at infinity" of the affine building (in a precise visual sense).
Role in Langlands
- Steinberg representation: top cohomology of the spherical building of \(G(\mathbb{F}_q)\) gives the Steinberg representation, the prototype of "Whittaker-supported" representations.
- Parabolic induction: the spherical building organises parabolic-induction structure.
- Cohomological structure: spherical buildings appear in the computation of Borel-Serre boundary cohomology.
Mathlib
Not formalised.
References
- Tits, Buildings of Spherical Type and Finite BN-Pairs (LNM 386, 1974).
- Brown, Buildings (1989).
- Abramenko–Brown, Buildings: Theory and Applications (2008).