Topic Bruhat-Tits Buildings
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Bruhat-Tits Buildings

The Bruhat-Tits building \(\mathcal{B}(G, F)\) of a reductive group \(G\) over a non-archimedean local field \(F\) is the affine building attached to \(G(F)\). It is a complete CAT(0) metric space on which \(G(F)\) acts by simplicial isometries, and the canonical geometric object encoding the local arithmetic of \(G\).

Construction (split case)

Let \(G\) be a split connected reductive group over \(F\), with split maximal torus \(T\), cocharacter lattice \(X_* = X_*(T)\), coroots \(\Phi^\vee\), and extended affine Weyl group \(\tilde W = W \ltimes X_*\).

The building \(\mathcal{B}(G, F)\) is constructed as follows:

  1. The standard apartment \(\mathbb{A}\) is \(X_* \otimes \mathbb{R}\), tiled by alcoves under \(\tilde W\).
  2. \(G(F)\) acts on \(\mathbb{A}\) through the Iwahori-Weyl group \(\tilde W\), extending to an action on \(\mathbb{A}\) via affine transformations.
  3. The building is glued from copies of \(\mathbb{A}\) using \(G(F)\)-conjugacy: \(\mathcal{B}(G, F) = G(F) \times \mathbb{A} / \sim\), where \((g, a) \sim (g h, h^{-1} a)\) for \(h \in N_G(T)(F)\).

Equivalently: \(\mathcal{B}(G, F)\) is the building of the affine BN-pair \((G(F), I, N_G(T)(F), S_{\mathrm{aff}})\).

Properties

Theorem (Bruhat-Tits). \(\mathcal{B}(G, F)\) is:

  • a complete CAT(0) metric space (non-positively curved);
  • a simplicial complex of dimension equal to the split rank of \(G\);
  • acted on by \(G(F)\) properly and simplicially;
  • the building has apartments isomorphic to Euclidean space of dimension \(= \mathrm{rank}\, G\), tiled by alcoves.

The CAT(0) property gives a "centre" to any bounded subgroup: every bounded subgroup of \(G(F)\) has a fixed point in \(\mathcal{B}(G, F)\), the foundation of the Bruhat-Tits fixed-point theorem.

Construction (general case)

For \(G\) connected reductive over \(F\) but not split:

  1. Let \(L/F\) be a finite Galois extension splitting \(G\).
  2. Construct \(\mathcal{B}(G_L, L)\) for the split form over \(L\).
  3. Take Galois invariants: \[ \mathcal{B}(G, F) \;=\; \mathcal{B}(G_L, L)^{\mathrm{Gal}(L/F)}. \]
  4. The result is the affine building of \(G(F)\); it has dimension equal to the \(F\)-split rank of \(G\).

For an anisotropic group (e.g.\ \(\mathrm{SL}_1(D)\) for a division algebra \(D\)), the building is a single point.

Examples

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

\(\mathcal{B}(\mathrm{SL}_2, \mathbb{Q}_p)\) is the \((p+1)\)-regular tree:

  • vertices = homothety classes of \(\mathbb{Z}_p\)-lattices in \(\mathbb{Q}_p^2\);
  • edges connect lattices \(L \supseteq L'\) with \(L/L' = \mathbb{Z}/p\);
  • each vertex has \(p+1\) neighbours (= the \(p+1\) lines in \(\mathbb{F}_p^2\)).

This is the simplest non-trivial affine building, and the prototype for understanding \(p\)-adic representations of \(\mathrm{SL}_2\) and the local Langlands correspondence.

\(\mathrm{SL}_n(F)\)

\(\mathcal{B}(\mathrm{SL}_n, F)\) is \((n-1)\)-dimensional:

  • vertices = homothety classes of \(\mathcal{O}_F\)-lattices in \(F^n\);
  • chambers = chains \(L_0 \supseteq L_1 \supseteq \cdots \supseteq L_{n-1} \supsetneq pL_0\) of nested lattices.

Each apartment is \(\mathbb{R}^{n-1}\) tiled by the affine type \(\tilde A_{n-1}\) tessellation.

\(\mathrm{SO}_n(F)\), \(\mathrm{Sp}_{2n}(F)\)

Buildings of type \(\tilde B_n\), \(\tilde C_n\), \(\tilde D_n\); the combinatorics involve isotropic lattice flags.

Vertices and parahoric subgroups

Each vertex \(v \in \mathcal{B}(G, F)\) has a stabiliser \(\mathrm{Stab}_{G(F)}(v) = K_v\) called a parahoric subgroup. Different vertex types give different parahorics:

  • Hyperspecial vertices: stabilisers \(\cong G(\mathcal{O}_F)\) for a smooth reductive group scheme \(G/\mathcal{O}_F\); these exist iff \(G\) is unramified over \(F\).
  • Special vertices: maximal compact subgroups; always exist.
  • Generic vertices: stabilisers are smaller subgroups (parahorics with non-trivial pro-unipotent radical).

The Iwahori subgroup \(I\) is the stabiliser of a chamber (= an alcove = maximal simplex).

Reduction map

There is a building reduction map: for each parahoric \(P_v\), the quotient \(P_v / P_v^+\) (where \(P_v^+\) is the pro-unipotent radical) is a connected reductive group \(\bar P_v\) over the residue field. This gives the reductive quotient of the parahoric.

For a hyperspecial vertex: \(\bar P_v = G(\mathbb{F}_q)\).

Role in Langlands

  • Local Langlands: depth, level, and types in the Bushnell-Kutzko / Yu / Kim theory of supercuspidal representations are organised by the building's geometry.
  • Iwahori-Hecke algebra \(\mathcal{H}(I \backslash G(F)/I)\) is the algebra of \(I\)-bi-invariant functions; its representation theory is the geometric setup for Langlands at the Iwahori block.
  • Affine Grassmannian \(\mathrm{Gr}_G = G(F)/G(\mathcal{O}_F)\) is an ind-scheme version of the vertex set; geometric Satake realises \(\mathrm{Rep}(\hat G)\) on its perverse sheaves.
  • Mod \(p\) Langlands: the building's cell complex computes pro-\(p\) Iwahori-Hecke algebra cohomology.

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).
  • Brown, Buildings (1989).
  • Garrett, Buildings and Classical Groups (1997).
  • Yu, Bruhat-Tits theory: a new approach (preprint, 2002).

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