Buildings - an introduction with view towards arithmetic

Transcription

1 Buildings - an introduction with view towards arithmetic Rudolf Scharlau Preliminary Version Contents 1 Buildings Combinatorial buildings Tits systems (BN-pairs) in a group The building of a semisimple algebraic group Buildings of spherical type Semisimple algebraic groups over arbitrary fields Affine buildings and semisimple algebraic groups over local fields Buildings of affine type The Bruhat-Tits building of a semisimple group over a local field The case of classical groups: Iwahori, Matsumoto, Hijikata An application: Vertex transitive S-arithmetic groups 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., g.l.b. A B, and all A := {X X A} are isomorphic to a full power set; 1

2 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: ϕ : induces an isomorphism of ordered sets A (ϕa) for all A ; d) Star of A in is St A = {B B A}; 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 St A are connected; j) numbering: a morphism type : P (I), I the set of types; equivalently, a partitioning of the vertex set X = X as X = i I X i 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 X, and x 0 < x 1 < < x d = x < < x n a maximal flag containing x, then d =: dim x, is independent of the choice of the flag. dim : X {0,..., n}, extending to Flag X P {0,..., n} 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 G i, i I a family of subgroups. Let (G, G i, i I) := J I G/G J and define a relation on (G, G i, i I) by (J, gg J ) (K, hg K ) J K and gg J hg K.

3 Rudolf Scharlau Buildings 3 a) (, ) is a numbered complex with vertex set {(i, xg i ) i I, x G}, chamber set {(I, xg I ) x G} = G/G I, and type function (i, xg i ) 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 G J : St (J, G J ) = (G J, G J i, i I J). d) Any numbered complex with a chamber transitive group G of special automorphisms is G-isomorphic to a complex of the shape (G, G i, i I). Notice that under the assumption J K, it actually follows from gg J hg K that gg J hg K. 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 = {s i i I} of generating involutions s.t. W = s i I (s i s j ) m ij = 1 if m ij. is a presentation of W, where m ij := ord(s i s j ). b) M = (m ij ) i,j I is called Coxeter matrix if m ij {2, 3,..., } if i j, m ij = m ji for all i, j I, m ii = 1 for all i I. c) Let (W, S) be a Coxeter system. The associated Coxeter-Tits complex is defined as the coset complex Σ(W, S) := (W, W s, s S), where W s := S {s}. 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: W (M) := i I (ij) m ij = 1 if m ij, together with the given generators i I forms a Coxeter system, and ord(ij) = m ij (i.e. M belongs to this system).

4 Rudolf Scharlau Buildings 4 b) The Coxeter-Tits complex of type M or belonging to M is defined as Σ(M) := (W (M), W i, i 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 Σ are Coxeter complexes. (S2) For any two A, B, there is a Σ Σ containing A and B. (S3) For any Σ, Σ Σ, A, B Σ Σ, there exists an isomorphism Σ Σ 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 apartments 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 S, w W, (T3) sbs B for all s 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 = T := nbn 1. n N

5 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 C, set B = G C, N = G Σ. 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, G i, i 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 W X := X W and G X = BW X B. a) The G X are subgroups of G, and G = G S = BW B. b) The mapping w BwB, W B\G/B is a bijection. c) For X, Y S, the mapping w G X wg Y, W G X \G/G Y induces a bijection W X \W/W Y = GX \G/G Y. The groups G X 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 G X, for some X S (namely X = X P := {s S s P }). b) G X = G Y only holds if X = Y. c) G Xi = G Xi holds for any family (X i ) 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 W as above. For s S, set W s := W S {s}, G s := G S {s}, where W X, G X, X S are as above. Consider the S-numbered complex := (G, G s, s S) = G/G X X S

6 Rudolf Scharlau Buildings 6 and its subcomplex Let Σ := {gσ g G}. a) (, Σ) is a thick building. Σ := {ng X n N, X S} b) G acts strongly transitively on. c) Σ = (W, W s, s S) canonically. In particular, W (Σ) = W. d) (W, S) is a Coxeter system. e) B is the stabilizer G B of the chamber B. The setwise stabilizer G Σ equals Ñ := N T where T is as in 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 complexes are parametrized by the following so called spherical Coxeter diagrams, where n is the number of nodes: A n n 1 B n = C n n 2 D n n 4 E 6 E 7

7 Rudolf Scharlau Buildings 7 E 8 F 4 H 3 5 H 4 5 I 2 (m) m m 2 Figure 3.1. The irreducible spherical Coxeter diagrams Definition 2.1 A building is called spherical or of spherical type if its Weyl group is finite. All spherical buildings of rank at least 3 have been classified by Jacques Tits [Tits74]. They all admit a strongly transitive automorphism group, that is, they belong to a Tits system in a group. Also, they are all of algebraic origin ; see in particular Theorem 2.2 below. An easy special case: the buildings of Coxeter type A n are precisely the flag complexes of projective spaces of dimension n. If the division ring belonging to that space is infinite-dimensional over is center, this building is easy, but not of the type described in Semisimple algebraic groups over arbitrary fields We use a few facts from the structure theory of algebraic groups: K a (global) field, K s its separable closure G a connected semisimple (reductive) algebraic group over K S a maximal K-split torus of G T a maximal torus of G, defined over K, s.t. T S N G (S) =: N the normalizer of S in G Z G (S) the centralizer of S in G analogously for T W K = W K (G, S) := N G (S)(K)/Z G (S)(K) the relative Weyl group of G W = W (G, T ) := N G (T )(K s )/Z G (T )(K s ) the absolute Weyl group of G X (T ) := Hom(T, Ks ) the character group Φ X (T ) the root system (of T (K s ) acting on Lie G) Φ a root basis (another use of Delta ) 0 Φ 0 := {α Φ α S = 1} K Φ K X (S) relative K-roots Φ 0 Φ Φ K canonically B a minimal K-parabolic subgroup of G with G S

8 Rudolf Scharlau Buildings 8 Theorem 2.2 (The Relative Building) Let G be reductive over an arbitrary field K, and B G, N G as above. Then (B(K), N(K)) is a Tits system in G(K). Naturally, the building (G(K), P α (K), α K ) according to the last theorem and Theorem 1.7 is called the building of G over K. Definition 2.2 (Split amd quasi-split groups) a) A reductive group over K is called split if S = T in the above notation, i.e. if it contains a maximal torus which splits over K. b) A reductive group over K is called quasi-split if the following two equivalent conditions hold: def:quasisplit (i) Z G (S) = T. (ii) G contains a Borel subgroup defined over K. So, the split groups are those that look like over the algebraic closure (or like reductive complex Lie groups). For a natural quasi-split example, see the group 2 D n,n 1 described below. Definition 2.3 (K-index, Tits diagram of a reductive group ) The K-index of the reductive group G over K consists of the following data: i) The Dynkin diagram of G(K s ), with set of nodes ; ii) the distinguished subset 0 ; iii) a Γ-action on by diagram automorphisms, leaving 0 invariant. A group is called of inner type it the Γ-action is trivial, otherwiese, it is of outer type. Notice that 0 = for quasi-split groups. Over finite fields, all reductive groups are quasi-split, according to a classical theorem of Steinberg. Proposition 2.3 The Dynkin diagram (type) of the relative building of a reductive group G depends only on its index. The nodes of the relative Coxeter diagram are in one-to-one correspondence with the orbits of Γ on 0 as above. The m αβ can be computed in terms of numbers of roots of certain sub-root systems of ; see [Tits65, Scha95]. When drawing the diagrams, the anisotropic roots, i.e. elements of 0, are represented by solid nodes, whereas the isotropic roots, giving the K-roots, a represented by hollow nodes. As an example where a non-trivial

9 Rudolf Scharlau Buildings 9 Γ-action is possible, we choose the absolute type D n. For the following examples, see e.g. [Tits65], pp. 56/7. They work the same way for all n 4. Example: (Some groups of absolute type D n.) 1. 2 D 4,3 quasi-split, index 3 non-trivial Γ-action 2. 2 D 4,2 non-split, index 2 non-trivial Γ-action 3. 1 D 4,2 non-split, index 2 trivial Γ-action The explicit description is as follows. Let us assume for simplicity that the characteristic of K is 2. All three groups are special orthogonal groups SO(f, K) (or spin groups) of a quadratic form of rank 8 over K, the index as stated is the Witt-index of the form, and also the K-rank of the group. The relative Dynkin diagram is B 3, B 2, B 2, respectively. By scaling, we can assume that the anisotropic part f 0 of f, which is of rank 2, 4, 4 respectively, represents 1. So in case 1, f 0 is the norm form of a quadratic extension, over which the form becomes hyperbolic and the group splits. Cases 2 and 3 are distinguished as follows: In case 2, f 0 is a form of discriminant 1. So it is the norm form of a non-split quaternion algebra, and thus f 0 splits (becomes hyperbolic) over a quadratic extension. In case 3, the discriminant of f 0 is not 1. Then there is a unique quadratic extension over which f 0 is as in case 2. But f 0 does not become hyperbolic over any quadratic extension. Notice that case 3 does not occur over local fields. 3 Affine buildings and semisimple algebraic groups over local fields 3.1 Buildings of affine type Proposition 3.1 Let M be an irreducible Coxeter diagram of rank n + 1. The following are equivalent. (i) M is the Coxeter diagram of a discrete cocompact reflection group on euclidean n-space. (ii) The canonical bilinear form of M is positive semi-definite with onedimensional radical.

10 Rudolf Scharlau Buildings 10 (iii) M occurs in the list of irreducible affine (or euclidean) diagrams given in the following list: Ã n a circuit of length n + 1 n 2 B n n 2 C n n 2 D n n 4 Ẽ 6 Ẽ 7 Ẽ 8 F4 G 2 6 Figure 6.1. The irreducible affine Coxeter diagrams The affine building of a vector space over a field with discrete valuation. K denotes a field equipped with a discrete valuation v : K Z o := {α K v(α) 0} its valuation ring p := {α K v(α) > 0} its maximal ideal π o p a prime element in o k := o/p the residue class field of v. V denotes an vector space of dimension n + 1 over K

11 Rudolf Scharlau Buildings 11 L is a lattice on V, i.e. a finitely generated (and thus free) o-module which generates the vector space V. [L] := {αl α K } = {π m L m Z} denotes the similarity class of L. Two lattice classes [L] and [M] are incident if πl M L for appropriate representatives L, M. The flag complex aff (V ) of this incidence relation on the set X V of all lattice classes is called the affine building of V. Proposition 3.2 a) aff (V ) is a numbered complex of rank n + 1. b) aff (V ) belongs to the Coxeter diagram à n. c) For any vertex [L] X V, the star of [L] in aff (V ) is canonically isomorphic to the ordinary building (L/πL) of the k-vector space L/πL. d) aff (V ) is a building. Of course, a) is contained in d), but it is listed separately, and d) is listed last, since its proof is slightly more difficult than the rest. The type function t : X V Z/nZ can be described as follows: Fix one lattice L 0. Write an arbitrary lattice L as L = gl 0 for an element g GL(V ). The valuation of the determinant v(det(g)) =: t(l) does not depend on the choice of g. Furthermore, t(l) changes only modulo n + 1 if L is replaced by a similar lattice λl. A typical chamber [L 0 ], [L 1 ],..., [L n ] is the following: fix a basis e 1,..., e n+1 of V, and set L i := oe oe i + pe i pe n+1. A typical apartment of aff (V ) is obtained by considering one basis and all its transforms by monomial matrices, and defining chambers as above. 3.2 The Bruhat-Tits building of a semisimple group over a local field To be written up. 3.3 The case of classical groups: Iwahori, Matsumoto, Hijikata The following theorem was first proved/published by Iwahori and Matsumoto in 1964 for Chevalley groups over local fields. Shortly afterwards

12 Rudolf Scharlau Buildings 12 it was generalized by Hijikata, first to Steinberg (quasi-split) groups, and then to all classical groups as stated. Notice that symplectic groups and orthogonal groups in characteristic 2 are included, but orthogonal groups are excluded by the extra condition it the residue characteristic is 2. Recall that an integral lattice L on V is P-elementary if PL # L, where L # denotes the dual lattice. In other words, all elementary divisors of L in L # are O or P. Theorem 3.3 (H. Hijikata, 1964, Yale lecture notes) Let K be a local field, D a finite-dimensional division algebra over K with valuation ring O = O D, involution x x and (V, h) an ε-hermitian space of Witt index r over D. Let G = U(D, h) be the its (special) unitary group. Assume further that the hermitian form is residually trace-valued, i.e. with tr ε : D D, x x + εx. h(v ) I = tr ε (I) for all ideals I O, a) There are exactly r+1 classes of P-elementary lattices on V. They are the vertices of an affine building with a strongly-transitive G-action. b) The stabilizers in G of the lattices from a) represent all conjugacy classes of maximal subgroups of G which admit an invariant lattice. c) If K is locally compact, the groups from b) are precisely the maximal compact subgroups of G. Decompose V as V = V V an where V is hyperbolic with hyperbolic basis e 1,..., e r, f 1,..., f r. Let us consider the case V an {0}, let L an be the unique O-maximal lattice on V an. The lattices L s, s = 0,..., r of the theorem are now as follows: πe 1,..., πe s, e s+1,..., e r, f 1,..., f r L an, s = 0, 1,..., r, where as usual π is a fixed prime element of O. The maximality of the stablizers of the L i is proved by Hijikata in an indirect way: He verifies the axioms of a Tits system 1.4 in the group G, and then he refers to the theorem 1.6 about parablic subgroups. 3.4 An application: Vertex transitive S-arithmetic groups In the 1980s, certain classes of (locally) finite incidence geometries belonging to an affine Coxeter diagram had been studied, whose universal cover (in an appropriate sense) turned out to be an affine building, together with a chamber-transitive action of a discrete group. A survey of these works is

13 Rudolf Scharlau Buildings 13 given in the paper [Kan90] by William M. Kantor. The following proposition, taken from [Scha09] makes precise a certain relation between class numbers and chamber transitivity of discrete groups as indicated on p. 40 of Kantor s above-mentioned paper. We consider the following objects: k a totally real algebraic number field o, p, k p, o p as usual G GL(V ) simply connected semisimple, almost simple over k anisotropic at the infinite places p a fixed finite place of k s.t. rk kp G 2 k p := o/po the residue field at p := (G(k p )) the Bruhat-Tits building of G(k p ). L a lattice in V s.t. o p L =: L p defines a vertex of 0 = (G( kp )) the residue (star, link) of L in. Γ := G(o[ 1 p ]) a {p}-arithmetic discrete subgroup of G(k p) Γ 0 := G(o) the finite stabilizer of L in G(k). Proposition 3.4 Under the above assumptions, the following properties of the lattice L (resp. the arithmetic groups Γ, Γ 0 ) are equivalent: a) Γ acts chamber transitively on. b) (i) Γ 0 acts chamber transitively on 0, (ii) h G (L) = 1. For the proof, see [Scha09]. It is completely analogous to the derivation of the neighbour method from strong approximation. As a consequence of the above result, discrete chamber transitive groups on affine buildings are very rare. A full classification has been announced by Kantor, Liebler and Tits in [KLT87]. Shortly after the appearance of [KLT87], a proof of the finiteness result based on the computation of covolumes of S-arithmetic groups has been given by Borel and Prasad in [BoPr89] and [Pra89]. References [BoPr89] A. Borel, G. Prasad: Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups. Publ. Math. I.H.E.S. 69 (1989), ; Addendum, ibid. 71 (1990), [BoTi65] A. Borel, J. Tits: Groupes réductifs. Publ. Math. I.H.E.S. 27 (1965), [BoTi72] A. Borel, J. Tits: Compléments à l article: Groupes réductifs. Publ. Math. I.H.E.S. 41 (1972),

14 Rudolf Scharlau Buildings 14 [Boulder65] A. Borel, G.D. Mostow (eds.): Algebraic Groups and Discontinuous Subgroups, Symposium on Algebraic Groups, July 5 August 6, 1965, Boulder Co. ; Proceedings of Symposia in Pure Mathematics, AMS, 1966 [BrTi72] F. Bruhat, J. Tits: Groupes réductifs sur un corps local. I.: Données radicielles valuées. Publ. Math. I.H.E.S. 41 (1972), [BrTi84] F. Bruhat, J. Tits: Schémas en groupes et immeubles des groupes classiques sur un corps local. Bull. Soc. Math France 112 (1984), [BrTi87a] F. Bruhat, J. Tits: Schémas en groupes et immeubles des groupes classiques sur un corps local. II. Groupes unitaires. Bull. Soc. Math France 115 (1987), [BrTi87b] F. Bruhat, J. Tits: Groupes algébriques sur un corps local Chapitre III. Compléments et applications à la cohomologie galoisienne. J. Fac. Sci. Univ. Tokyo 34 (1987), [Hij64a] H. Hijikata: On the arithmetic of p-adic Steinberg groups. Mimeographed Notes, Yale University, [Hij64b] H. Hijikata: Maximal compact subgroups of some p-adic classical groups. Mimeographed Notes, Yale University, [IwMa65] N. Iwahori, H. Matsumoto: On some Bruhat decomposition and the structure of the Hecke ring of p-adic Chevalley groups. Publ. Math. I.H.E.S. 25 (1965), [KLT87] W.M. Kantor, R. Liebler, J. Tits: On discrete chamber-transitive automorphism groups of affine buildings, Notices of the AMS 16 (1987), [Kan90] W.M. Kantor: Finite geometries via algebraic affine buildings, pp in: Finite Geometries, Buildings and Related Topics (Eds. W. M. Kantor et al.), Oxford University Press, Oxford 1990 [Kne65] M. Kneser: Galois-Kohomologie halbeinfacher algebraischer Gruppen über p-adischen Körpern I. and II. Math. Z.88 (1965), 40 47, 89 (1965), [Pra89] G. Prasad: Volumes of S-arithmetic quotients of semi-simple groups, Publ. Math., Inst. Hautes Étud. Sci. 69 (1989), [Scha95] R. Scharlau: Buildings. In: F. Buekenhout (ed.), Handbook of Incidence Geometry, Amsterdam: North-Holland, (1995). [Scha09] R. Scharlau: Martin Kneser s Work on Quadratic Forms and Algebraic Groups. In: R. Baeza et al. (eds.), Quadratic Forms - Algebra, Arithmetic and Geometry, Contemp. Math. 493, (2009).

15 Rudolf Scharlau Buildings 15 [Tits65] J. Tits: Classification of algebraic semisimple groups. in: [Boulder65] [Tits74] J. Tits: Buildings of spherical type and finite BN-pairs. Springer Lecture Notes in Mathematics 386 (1974). [Tits79] J. Tits: Reductive groups over local fields. In: Proc. A.M.S. Symp. Pure Math. 33, Part I, (1979).