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.

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