Buildings and Classical Groups

1 Coxeter Groups.- 1.1 Words, lengths, presentations of groups.- 1.2 Coxeter groups, systems, diagrams.- 1.3 Reflections, roots.- 1.4 Roots and the length function.- 1.5 More on roots and lengths.- 1.6 Generalized reflections.- 1.7 Exchange, Deletion conditions.- 1.8 The Bruhat order.- 1.9 Special subgroups of Coxeter groups.- 2 Seven Infinite Families.- 2.1 Three spherical families.- 2.2 Four affine families.- 3 Chamber Complexes.- 3.1 Chamber complexes.- 3.2 The uniqueness lemma.- 3.3 Foldings, walls, reflections.- 3.4 Coxeter complexes.- 3.5 Characterization by foldings and walls.- 3.6 Corollaries on foldings.- 4 Buildings.- 4.1 Apartments and buildings: definitions.- 4.2 Canonical retractions to apartments.- 4.3 Apartments are Coxeter complexes.- 4.4 Labels, links.- 4.5 Convexity of apartments.- 4.6 Spherical buildings.- 5 BN-pairs from Buildings.- 5.1 BN-pairs: definitions.- 5.2 BN-pairs from buildings.- 5.3 Parabolic (special) subgroups.- 5.4 Further Bruhat-Tits decompositions.- 5.5 Generalized BN-pairs.- 5.6 The spherical case.- 5.7 Buildings from BN-pairs.- 6 Generic and Hecke Algebras.- 6.1 Generic algebras.- 6.2 Iwahori-Hecke algebras.- 6.3 Generalized Iwahori-Hecke algebras.- 7 Geometric Algebra.- 7.1 GL(n) (a prototype).- 7.2 Bilinear and hermitian forms.- 7.3 Extending isometries.- 7.4 Parabolics.- 8 Examples in Coordinates.- 8.1 Symplectic groups.- 8.2 Orthogonal groups O(n,n).- 8.3 Orthogonal groups O(p,q).- 8.4 Unitary groups in coordinates.- 9 Spherical Construction for GL(n).- 9.1 Construction.- 9.2 Verification of the building axioms.- 9.3 Action of GL(n) on the building.- 9.4 The spherical BN-pair in GL(n).- 9.5 Analogous treatment of SL(n).- 9.6 Symmetric groups as Coxeter groups.- 10 Spherical Construction for Isometry Groups.- 10.1 Constructions.- 10.2 Verification of the building axioms.- 10.3 The action of the isometry group.- 10.4 The spherical BN-pair.- 10.5 Analogues for similitude groups.- 11 Spherical Oriflamme Complex.- 11.1 Oriflamme construction for SO(n,n).- 11.2 Verification of the building axioms.- 11.3 The action of SO(n,n).- 11.4 The spherical BN-pair in SO(n,n).- 11.5 Analogues for GO(n,n).- 12 Reflections, Root Systems and Weyl Groups.- 12.1 Hyperplanes, chambers, walls.- 12.2 Reflection groups are Coxeter groups.- 12.3 Finite reflection groups.- 12.4 Affine reflection groups.- 12.5 Affine Weyl groups.- 13 Affine Coxeter Complexes.- 13.1 Tits' cone model of Coxeter complexes.- 13.2 Positive-definite (spherical) case.- 13.3 A lemma from Perron-Frobenius.- 13.4 Local finiteness of Tits' cones.- 13.5 Definition of geometric realizations.- 13.6 Criterion for affineness.- 13.7 The canonical metric.- 13.8 The seven infinite families.- 14 Affine Buildings.- 14.1 Affine buildings, trees: definitions.- 14.2 Canonical metrics on affine buildings.- 14.3 Negative curvature inequality.- 14.4 Contractibility.- 14.5 Completeness.- 14.6 Bruhat-Tits fixed-point theorem.- 14.7 Maximal compact subgroups.- 14.8 Special vertices, compact subgroups.- 15 Combinatorial Geometry.- 15.1 Minimal and reduced galleries.- 15.2 Characterizing apartments.- 15.3 Existence of prescribed galleries.- 15.4 Configurations of three chambers.- 15.5 Subsets of apartments.- 16 Spherical Building at Infinity.- 16.1 Sectors.- 16.2 Bounded subsets of apartments.- 16.3 Lemmas on isometries.- 16.4 Subsets of apartments.- 16.5 Configurations of chamber and sector.- 16.6 Sector and three chambers.- 16.7 Configurations of two sectors.- 16.8 Geodesic rays.- 16.9 The spherical building at infinity.- 16.10 Induced maps at infinity.- 17 Applications to Groups.- 17.1 Induced group actions at infinity.- 17.2 BN-pairs, parahorics and parabolics.- 17.3 Translations and Levi components.- 17.4 Levi filtration by sectors.- 17.5 Bruhat and Cartan decompositions.- 17.6 Iwasawa decomposition.- 17.7 Maximally strong transitivity.- 17.8 Canonical translations.- 18 Lattices, p-adic Numbers, Discrete Valuations.- 18.1 p-adic numbers.- 18.2 Discrete valuations.- 18.3 Hensel's Lemma.- 18.4 Lattices.- 18.5 Some topology.- 18.6 Iwahori decomposition for GL(n,k).- 19 Affine Building for SL(n).- 19.1 Construction.- 19.2 Verification of the building axioms.- 19.3 The action of SL(V).- 19.4 The Iwahori subgroup 'B'.- 19.5 The maximal apartment system.- 20 Affine Buildings for Isometry Groups.- 20.1 Affine buildings for alternating spaces.- 20.2 The double oriflamme complex.- 20.3 The (affine) single oriflamme complex.- 20.4 Verification of the building axioms.- 20.5 Group actions on the buildings.- 20.6 Iwahori subgroups.- 20.7 The maximal apartment systems.