Buildings and Classical Groups

Buildings are highly structured, geometric objects, primarily used in the finer study of the groups that act upon them. In Buildings and Classical Groups, the author develops the basic theory of buildings and BN-pairs, with a focus on the results needed to apply it to the representation theory of p-adic groups. In particular, he addresses spherical and affine buildings, and the "spherical building at infinity" attached to an affine building. He also covers in detail many otherwise apocryphal results. Classical matrix groups play a prominent role in this study, not only as vehicles to illustrate general results but as primary objects of interest. The author introduces and completely develops terminology and results relevant to classical groups. He also emphasizes the importance of the reflection, or Coxeter groups and develops from scratch everything about reflection groups needed for this study of buildings. In addressing the more elementary spherical constructions, the background pertaining to classical groups includes basic results about quadratic forms, alternating forms, and hermitian forms on vector spaces, plus a description of parabolic subgroups as stabilizers of flags of subspaces. The text then moves on to a detailed study of the subtler, less commonly treated affine case, where the background concerns p-adic numbers, more general discrete valuation rings, and lattices in vector spaces over ultrametric fields. Buildings and Classical Groups provides essential background material for specialists in several fields, particularly mathematicians interested in automorphic forms, representation theory, p-adic groups, number theory, algebraic groups, and Lie theory. No other available source provides such a complete and detailed treatment.

Table of Contents:
Coxeter Groups Words, Lengths, Presentations of Groups Coxeter Groups, Systems, Diagrams Linear Representation, Reflections, Roots Roots and the Length Function More on Roots and Lengths Generalized Reflections Exchange Condition, Deletion Condition The Bruhat Order Special Subgroups of Coxeter Groups Seven Important Families Three Spherical Families Four Affine Families Complexes Chamber Complexes The Uniqueness Lemma Foldings, Walls, Reflections Coxeter Complexes Characterization by Foldings and Walls Corollaries on Foldings and Half-Apartments Buildings Apartments and Buildings: Definitions Canonical Retractions to Apartments Apartments are Coxeter Complexes Labels, Links, Maximal Apartment System Convexity of Apartments Spherical Buildings BN-Pairs from Buildings GN-Pairs: Definitions BN-Pairs from Buildings Parabolic (Special) Subgroups Further Bruhat-Tits Decompositions Generalized BN-Pairs The Spherical Case Buildings from BN Pairs Generic Algebras and Hecke Algebras Generic Algebras Strict Iwahori-Hecke Algebras Generalized Iwahori-Hecke Algebras Geometric Algebra GL(n)-A Prototype Bilinear and Hermitian Forms: Classical Groups A Witt-Type Theorem: Extending Isometries Parabolics, Unipotent Radicals, Levi Components Examples in Coordinates Symplectic Groups in Coordinates Orthogonal Groups O(n,n) in Coordinates Orthogonal Groups O(p,q) in Coordinates Unitary Groups in Coordinates Construction for GL(n) Construction of the Spherical Building for GL(n) Verification of the Building Axioms Action of GL(n) on the Spherical Building The Spherical BN-Pair in GL(n) Analogous Treatment of SL(n) The Symmetric Group as Coxeter Group Spherical Construction for Isometry Groups Construction of Spherical Buildings for Isometry Groups Verification of the Building Axioms The Action of the Isometry Group The Spherical BN-Pair in Isometry Groups Analogues for Similitude Groups The Spherical Oriflamme Complex The Oriflamme Construction for SO(n,n) Verification of the Building Axioms The Action of SO(n,n) The Spherical BN-Pair in SO(n,n) Analogues for GO(n,n) Reflections, Root Systems, Weyl Groups Hyperplanes, Chambers, Walls Reflection Groups are Coxeter Groups Root Systems and Finite Reflection Groups Affine Reflection Groups, Special Vertices Affine Weyl Groups Affine Coxeter Complexes Tits' Cone Model of Coxeter Complexes Positive-Definiteness: The Spherical Case A Lemma from Perron-Frobenius Local Finiteness of Tits' Cones Definition of Geometric Realizations Criterion for Affineness The Canonical Metric The Seven Infinite Families Affine Buildings Affine Buildings, Trees: Definitions The Canonical Metric Negative Curvature Inequality Contractibility Completeness Bruhat-Tits Fixed Point Theorem Conjugacy Classes of Maximal Compact Subgroups Special Vertices, Good Compact Subgroups Finer Combinatorial Geometry Minimal Galleries and Reduced Galleries Characterizing Apartments Existence of Prescribed Galleries Configurations of Three Chambers] Subsets of Apartments, Strong Isometries The Spherical Building at Infinity Sectors Bounded Subsets of Apartments Lemmas on Isometries Subsets of Apartments Configurations of Chamber and Sector Configurations of Sector and Three Chambers Configurations of Two Sectors Geodesic Rays The Spherical Building at Infinity Induced Maps at Infinity Applications to Groups Induced Group Actions at Infinity BN-Pairs, Parahorics and Parabolic Translations and Levi Components Filtration by Sectors: Levi Decomposition Bruhat and Cartan Decompositions Iwasawa Decomposition Maximally Strong Transitivity Canonical Translation Lattices, p-adic Numbers, Discrete Valuations p-adic Numbers Discrete Valuations Hensel's Lemma Lattices Some Topology Iwahori Decomposition for GL(n) Construction for SL(V) Construction of the Affine Building for SL(V) Verification of the Building Axioms Action of SL(V) on the Affine Building The Iwahori Subgroup 'B' The Maximal Apartment System Construction of Affine Buildings for Isometry Groups Affine Buildings for Alternating Spaces The Double Oriflamme Complex The (Affine) Single Oriflamme Complex Verification of the Building Axioms Group Actions on the Buildings Iwahori Subgroups The Maximal Apartment Systems Index Bibliography