Buildings - an introduction with view towards arithmetic Rudolf Scharlau Preliminary Version Contents 1 Buildings 1 1.1 Combinatorial buildings . 1 1.2 Tits systems (BN-pairs) in a group . 4 2 The building of a semisimple algebraic group 6 2.1 Buildings of spherical type . 6 2.2 Semisimple algebraic groups over arbitrary fields . 7 3 Affine buildings and semisimple algebraic groups over local fields 9 3.1 Buildings of affine type . 9 3.2 The Bruhat-Tits building of a semisimple group over a local field . 11 3.3 The case of classical groups: Iwahori, Matsumoto, Hijikata . 11 3.4 An application: Vertex transitive S-arithmetic groups . 12 1 Buildings This section is based on my book chapter [Scha95]; the classical reference is [Tits74]. 1.1 Combinatorial buildings Definition 1.1 (Numbered complex) Basic vocabulary: a) (simplicial) complex: a partially ordered set (∆; ⊆) s.t. 9 ;, 9 g.l.b. A \ B, and all A∗ := fX 2 ∆ j X ⊆ Ag are isomorphic to a full power set; 1 Rudolf Scharlau Buildings 2 b) Simplex: an element of ∆, \A is a face of B" if A ⊆ B; vertex a minimal nonempty face; c) morphism: ' : ∆ ! ∆0 induces an isomorphism of ordered sets A∗ ! ('A)∗ for all A 2 ∆; d) Star of A in ∆ is St∆ A = fB 2 ∆ j B ⊇ Ag; e) Rank of A: cardinality of its vertex set; f) Chamber: a maximal simplex; ∆ is pure, if all chambers have the same rank, the rank of ∆; g) Panel: chamber of corank / codimension 1; h) Gallery: chain of succesively \adjacent" chambers; connected; i) strongly connected, if all StA are connected; j) numbering: a morphism type : ∆ ! P (I), I the set of types; equiv- S alently, a partitioning of the vertex set X = X∆ as X = i2I Xi s.t. every chamber contains one vertex of each type. Example 1.1 (Flag complex of a poset) a) For a poset (partially ordered set) (X; ≤), the set of flags Flag(X; ≤) is a simplicial complex with vertex set X. b) (X; ≤) is pure if any two maximal flags have the same finite cardinality n + 1 (or \length" n). Assuming this, let x 2 X, and x0 < x1 < ··· < xd = x < ··· < xn a maximal flag containing x, then d =: dim x, is independent of the choice of the flag. dim : X ! f0; : : : ; ng; extending to Flag X ! P f0; : : : ; ng makes Flag(X; ≤) into a numbered complex. Special case: the proper subspaces of a projective space (or a vector space). Proposition 1.2 (Chamber transitive complexes) Let G be any group, and Gi; i 2 I a family of subgroups. Let a ∆(G; Gi; i 2 I) := G=GJ J⊆I and define a relation ≤ on ∆(G; Gi; i 2 I) by (J; gGJ ) ≤ (K; hGK ) () J ⊆ K and gGJ \ hGK 6= ;: Rudolf Scharlau Buildings 3 a) (∆; ≤) is a numbered complex with vertex set f(i; xGi) j i 2 I; x 2 Gg, chamber set f(I; xGI ) j x 2 Gg =∼ G=GI , and type function (i; xGi) 7! i. b) G operates type preservingly and chamber transitively on ∆ by left translations. c) The star in ∆ of a typical simplex of type J equals the corresponding complex for the group GJ : J J J[i St∆ (J; G ) = ∆ (G ;G ; i 2 I r J): d) Any numbered complex with a chamber transitive group G of special automorphisms is G-isomorphic to a complex of the shape ∆ (G; Gi; i 2 I). Notice that under the assumption J ⊆ K, it actually follows from gGJ \ hGK 6= ; that gGJ ⊇ hGK . So ≤ is indeed a partial ordering. Definition 1.2 (Coxeter group, - system, Coxeter-Tits complex) a) A Coxeter system is a group W together with a set S = fsi j i 2 Ig of generating involutions s.t. mij W = hsi 2 I j (sisj) = 1 if mij 6= 1i: is a presentation of W , where mij := ord(sisj). b) M = (mij)i;j2I is called Coxeter matrix if mij 2 f2; 3;:::; 1g if i 6= j; mij = mji for all i; j 2 I; mii = 1 for all i 2 I: c) Let (W; S) be a Coxeter system. The associated Coxeter-Tits com- plex is defined as the coset complex s s Σ(W; S) := ∆(W; W ; s 2 S); where W := hS r fsgi: Proposition 1.3 (Existence of Coxeter groups) Let M be a Coxeter matrix (Coxeter diagram) over I. a) The group W (M) defined by generators and relations as follows: mij W (M) := hi 2 I j (ij) = 1 if mij 6= 1i; together with the given generators i 2 I forms a Coxeter system, and ord(ij) = mij (i.e. M belongs to this system). Rudolf Scharlau Buildings 4 b) The Coxeter-Tits complex of type M or belonging to M is de- fined as Σ(M) := ∆(W (M);W i; i 2 I) Definition 1.3 (Building) A building is a complex ∆ together with a family Σ of subcomplexes such that the following holds: (S1) The elements Σ 2 Σ are Coxeter complexes. (S2) For any two A; B 2 ∆, there is a Σ 2 Σ containing A and B. (S3) For any Σ; Σ0 2 Σ; A; B 2 Σ\Σ0, there exists an isomorphism Σ ! Σ0 which is the identity on A; B and all their faces. The elements of Σ are called apartments, and any Σ subject to (S1), (S2), (S3) is called a system of apartments. It follows from the axioms that the type functions on the different apart- ments are compatible, and therefore any building is a numbered complex in a natural way. 1.2 Tits systems (BN-pairs) in a group We say that a group G acts strongly transitively on a building ∆ if it acts transitively on the pairs (C; Σ), where C is a chamber of ∆ and Σ an apartment containing C. Definition 1.4 (Tits system, BN-pair) Let G be a group and B and N subgroups of G, set T := B \ N. The pair (B; N) is called a Tits system or a BN-pair in G if the following axioms hold: (T0) T is normal in N. (T1) B [ N generates G. There exists a system S of involutions generating W := N=T such that (T2) sBw ⊆ BwB [ BswB for all s 2 S; w 2 W , (T3) sBs 6= B for all s 2 S. One also says that (G; B; N) is a Tits system, W is called its Weyl group. A system (G; B; N) of groups satisfying (T0), (T1), (T2) is called saturated if \ T = Te := nBn−1: n2N Rudolf Scharlau Buildings 5 Theorem 1.4 (The Tits system of a strongly transitive building) Let (∆; Σ) be a building and G a group acting strongly transitively on (∆; Σ). Fix a chamber C and an apartment Σ 3 C, set B = GC , N = GΣ. a) The system of subgroups (B; N) in G has the properties (T0), (T1), (T2). If ∆ is thick, then (T3) holds as well, i.e. (B; N) is a Tits system. b) (G; B; N) is saturated. c) The restriction mapping N ! W (Σ) induces an isomorphism of Weyl groups W = N=(B \ N) ! W (Σ): d) If ∆ is thick, then (∆; Σ) is canonically isomorphic to the building ∆(G; Gi; i 2 I) belonging to (G; B; N), where I = I(∆). Proposition 1.5 (Bruhat decomposition) Assume that (G; B; N) satisfies (T0), (T1), (T2). For X ⊆ S, set WX := hXi ⊆ W and GX = BWX B. a) The GX are subgroups of G, and G = GS = BW B. b) The mapping w 7! BwB, W ! BnG=B is a bijection. c) For X; Y ⊆ S, the mapping w 7! GX wGY , W ! GX nG=GY induces a bijection ∼ WX nW=WY = GX nG=GY : The groups GX are called parabolic subgroups of G (with respect to the specified Tits system). Theorem 1.6 (Parabolic subgroups) Let (G; B; N) be a Tits system. a) Any subgroup P ⊆ G containing B is parabolic, i.e. equal to GX , for some X ⊆ S (namely X = XP := fs 2 S j s ⊆ P g). b) GX = GY only holds if X = Y . c) G\Xi = \GXi holds for any family (Xi) of subsets of S. Theorem 1.7 (The building belonging to a Tits system) Let (B; N) be a Tits system in the group G, with Weyl group W and generating system s s S ⊆ W as above. For s 2 S, set W := WSrfsg, G := GSrfsg, where WX , GX , X ⊆ S are as above. Consider the S-numbered complex s [ ∆ := ∆(G; G ; s 2 S) = G=GX X⊆S Rudolf Scharlau Buildings 6 and its subcomplex Σ := fnGX j n 2 N; X ⊆ Sg Let Σ := fgΣ j g 2 Gg. a) (∆; Σ) is a thick building. b) G acts strongly transitively on ∆. c) Σ =∼ ∆(W; W s; s 2 S) canonically. In particular, W (Σ) =∼ W . d) (W; S) is a Coxeter system. e) B is the stabilizer GB of the chamber B. The setwise stabilizer GΣ equals Ne := NTe where Te is as in 4.3.9. The difficult part of this theorem is part a). The other statements are easy. Notice that part e) is trivial. We have stated it explicitly to set in evidence that a saturated system (G; B; N) can be recovered in a canonical way from its associated complex. 2 The building of a semisimple algebraic group 2.1 Buildings of spherical type Theorem 2.1 (H.S.M. Coxeter) The finite irreducible Coxeter-Tits com- plexes are parametrized by the following so called spherical Coxeter diagrams, where n is the number of nodes: An ◦ ◦ ◦ · · · ◦ ◦ ◦ n ≥ 1 Bn = Cn ◦ ◦ ◦ · · · ◦ ◦ ◦ n ≥ 2 "◦ " Dn ◦ ◦ ◦ · · · ◦ ◦b n ≥ 4 b◦ ◦ E6 ◦ ◦ ◦ ◦ ◦ ◦ E7 ◦ ◦ ◦ ◦ ◦ ◦ Rudolf Scharlau Buildings 7 ◦ E8 ◦ ◦ ◦ ◦ ◦ ◦ ◦ F4 ◦ ◦ ◦ ◦ 5 H3 ◦ ◦ ◦ 5 H4 ◦ ◦ ◦ ◦ m I2(m) ◦ ◦ m ≥ 2 Figure 3.1.