Bruhat-Tits Apartments
A Bruhat-Tits apartment is the basic Euclidean piece of the affine building of a reductive group over a local field. It is a real vector space tiled by alcoves under the action of the extended affine Weyl group, and copies of it glue to form the full Bruhat-Tits building.
Definition
Let \(G\) be a connected reductive group over a non-archimedean local field \(F\) with maximal \(F\)-split torus \(S \subseteq G\). The standard Bruhat-Tits apartment is the real vector space \[ \mathbb{A}(G, S) \;:=\; X_*(S) \otimes_\mathbb{Z} \mathbb{R}, \] together with the action of the extended affine Weyl group \[ \tilde W \;=\; W(G, S) \;\ltimes\; X_*(S), \] where \(W(G, S) = N_G(S)/Z_G(S)\) is the relative Weyl group, and \(X_*(S)\) acts by translations.
Structure
\(\mathbb{A}(G, S)\) is tiled by alcoves — fundamental domains for \(\tilde W\) — and facets of lower dimensions:
- Alcoves (top-dimensional facets) = chambers of the affine Coxeter complex.
- Vertices (0-dimensional facets) = corners of alcoves.
- Walls = boundaries of alcoves = hyperplanes defined by affine roots \(\psi = \alpha + n\) for \(\alpha\) a relative root and \(n \in \mathbb{Z}\).
Affine root system on \(\mathbb{A}\)
The set of affine roots \[ \Psi \;=\; \{\alpha + n : \alpha \in \Phi(G, S), \;n \in \mathbb{Z}\} \] is the valued root system of \((G, F)\). Each \(\psi \in \Psi\) gives an affine hyperplane \(\ker(\psi) \subseteq \mathbb{A}\).
For non-quasi-split or unramified \(G\): the affine roots may have non-integer translates (e.g.\ \(\alpha + n/2\) for ramified cases).
Construction from valued root data
Let \(K = \widehat{F^{\mathrm{unr}}}\) and let \(\sigma\) be Frobenius. The apartment is constructed via the valued root datum of \((G, S)\): the additional data of:
- A valuation \(\omega : K^\times \to \mathbb{Q}\) (normalised so \(\omega(\pi) = 1\) for a uniformiser \(\pi\));
- For each root \(\alpha\) and root subgroup \(U_\alpha\): a filtration \(U_\alpha = \cup_n U_{\alpha, n}\) compatible with \(\omega\).
The apartment is the unique smooth affine space realising this filtration combinatorics.
Examples
\(\mathrm{SL}_2(\mathbb{Q}_p)\)
\(X_*(S) = \mathbb{Z}\); apartment = \(\mathbb{R}\) (1-dimensional). Affine roots: \(\pm \varepsilon + n\) for \(n \in \mathbb{Z}\), giving walls at \(\frac{1}{2}\mathbb{Z}\). Alcoves: intervals \([n, n+\frac{1}{2}]\).
The full Bruhat-Tits building is the \((p+1)\)-regular tree, with each apartment being a "line" in the tree.
\(\mathrm{SL}_n(F)\)
Apartment = \(\mathbb{R}^{n-1}\) tiled by the affine type-\(\tilde A_{n-1}\) tessellation; alcoves are \((n-1)\)-simplices.
\(\mathrm{Sp}_{2n}(F)\)
Apartment = \(\mathbb{R}^n\); type \(\tilde C_n\) tessellation.
Bruhat-Tits theory
Theorem (Bruhat-Tits). Each connected reductive \(G\) over \(F\) has a unique (up to isomorphism) affine building \(\mathcal{B}(G, F)\) constructed by gluing copies of \(\mathbb{A}(G, S)\) (apartments) along \(G(F)\)-translates.
The building has the CAT(0) property (non-positive curvature), making it a complete metric space on which \(G(F)\) acts properly by isometries.
Galois descent
For \(G\) not necessarily split, but quasi-split:
- Construct the apartment for \(G_K\) over \(K = \widehat{F^{\mathrm{unr}}}\) (split).
- Take \(\mathrm{Gal}(K/F)\)-invariants: \(\mathbb{A}(G, S)^{\mathrm{Gal}} = \mathbb{A}(G_F, S_F)\).
For anisotropic \(G\) (no \(F\)-split torus): the apartment is a single point, and the building is a single point — but the parahoric theory is non-trivial via the residue-field reductive structure.
Role in Langlands
- Iwahori-Hecke algebra: arises from the \(G(F)\)-action on the apartment via the affine Weyl group.
- Moy-Prasad filtration: defined point-by-point on the apartment; gives the depth filtration of \(G(F)\) used in supercuspidal parametrisation.
- \(L^2\)-cohomology: of the building computes pro-\(p\) Iwahori cohomology, foundational for mod \(p\) Langlands.
References
- Bruhat–Tits, Groupes réductifs sur un corps local I, II (Publ.\ IHES 1972, 1984).
- Tits, Reductive groups over local fields (Proc.\ AMS Symp.\ 1979).
- Yu, Bruhat-Tits theory: a new approach (preprint, 2002).