Facets in Bruhat-Tits Buildings
A facet in a Bruhat-Tits building is a (relatively open) simplex of the underlying simplicial complex. Facets correspond bijectively to conjugacy classes of parahoric subgroups in \(G(F)\) and are the geometric organisation of the local representation theory.
Definition
For a Bruhat-Tits building \(\mathcal{B}(G, F)\):
A facet is the relative interior of a simplex of \(\mathcal{B}(G, F)\). Facets are classified by dimension:
- Vertices: 0-dim facets (corners of alcoves).
- Edges: 1-dim facets.
- Faces of intermediate dim: \(k\)-dim facets for \(0 \le k < r\), where \(r = \dim \mathcal{B}\).
- Alcoves (chambers): top-dimensional facets, \(r\)-dim.
Classification
The set of facets in \(\mathcal{B}(G, F)\), modulo \(G(F)\)-action, is in bijection with the subsets \(I \subseteq S_{\mathrm{aff}}\) of the simple reflections of the affine Weyl group: \[ \{G(F)\text{-orbits of facets}\} \;\cong\; \{\text{subsets } I \subsetneq S_{\mathrm{aff}}\}. \]
For each \(I\), the corresponding facet's stabiliser in \(G(F)\) is a standard parahoric subgroup \(P_I\); see parahoric subgroups.
Facet types
| Facet | Subset \(I \subseteq S_{\mathrm{aff}}\) | Stabiliser \(P_I\) |
|---|---|---|
| Alcove (chamber) | \(I = \emptyset\) | Iwahori \(I\) |
| Vertex \(v\) | \(I = S_{\mathrm{aff}} \setminus \{s_v\}\) | maximal compact \(K_v\) |
| Hyperspecial vertex | \(I\) maximally large, "all but one" | hyperspecial \(K_v\) |
| General facet | other \(I\) | intermediate parahoric |
Vertices and parahoric stabilisers
Each vertex \(v \in \mathcal{B}(G, F)\) has a stabiliser \(\mathrm{Stab}_{G(F)}(v)\). The connected component (in the Bruhat- Tits sense) is the parahoric \(\mathcal{P}_v\); the full stabiliser \(K_v\) may be slightly larger (differs by a finite quotient).
For hyperspecial vertices: \(\mathcal{P}_v = K_v\) is a maximal compact subgroup naturally isomorphic to \(G(\mathcal{O}_F)\) for a smooth reductive \(\mathcal{O}_F\)-model of \(G\); requires \(G\) to be unramified.
Examples
\(\mathrm{SL}_2(\mathbb{Q}_p)\) tree
\(\mathcal{B}(\mathrm{SL}_2, \mathbb{Q}_p)\) is the \((p+1)\)-regular tree:
- Vertices: 2 conjugacy classes (one for each hyperspecial class).
- Edges = alcoves: 1 conjugacy class.
- The two vertex classes give two conjugate copies of \(\mathrm{SL}_2(\mathbb{Z}_p)\) — corresponding to the two hyperspecial vertices in the apartment.
\(\mathrm{SL}_n(F)\)
Vertices = homothety classes of lattices in \(F^n\); facets = chains of lattices.
Number of vertex conjugacy classes: \(n\) (corresponding to "type" of lattice, indexed by determinant valuation mod \(n\)). Only one is hyperspecial; the others are special but not hyperspecial.
Operations on facets
- Star of a facet \(F\): set of facets containing \(F\) in their closure. The star encodes the local structure of the building.
- Link of a facet \(F\): boundary of the star, a spherical sub-building isomorphic to the spherical building of the residue reductive group at \(F\).
The link structure gives the reduction of representation theory: representations of \(G(F)\) at depth determined by \(F\) reduce to representations of the residue-field reductive group via the link.
Galois descent for non-split groups
For \(G\) non-split: facets of \(\mathcal{B}(G, F)\) are obtained as \(\mathrm{Gal}(K/F)\)-fixed sub-facets of \(\mathcal{B}(G_K, K)\) (\(K\) the splitting field). Some absolute facets may not descend; the rest parametrise the \(G(F)\)-conjugacy classes.
Role in Langlands
- Parahoric / Iwahori structure: facets ↔ parahoric subgroups ↔ reduction-mod-\(p\) reductive groups.
- Bushnell-Kutzko types: defined on facets of the building, with the choice of facet controlling the depth.
- Local Langlands at depth zero: representations of \(G(F)\) with invariants under a parahoric pro-unipotent radical reduce to representations of the residue-field reductive group.
References
- Bruhat–Tits, Groupes réductifs sur un corps local I, II (1972, 1984).
- Tits, Reductive groups over local fields (Proc.\ AMS Symp.\ 1979).
- Yu, Bruhat-Tits theory: a new approach (preprint, 2002).