Bruhat–Tits Theory: A New Approach

Contents
Tables
Introduction
Goals
A Brief Overview of the Theory
Our Approach
A Summary of Each Chapter
On the Logical Structure of the Exposition
Different Paths through the Book
Acknowledgements
PART ONE: BACKGROUND AND REVIEW
1. Affine Root Systems and Abstract Buildings
1.1 Metric Spaces
1.2 Affine Spaces
1.3 Affine Root Systems
1.4 Tits Systems
1.5 Abstract Buildings
1.6 The Monoid ~R
2. Algebraic Groups
2.1 Henselian Fields
2.2 Bounded Subgroups of Reductive Groups
2.3 Fields of Dimension ≼ 1
2.4 Affine Group Schemes over Perfect Fields
(a) Homomorphisms and Kernels
(b) Unipotent Groups
(c) The Reductive Quotient and Conjugacy Results
2.5 Tori
(a) Induced Tori
(b) The Valuation Homomorphism
(c) The Maximal Bounded Subgroup
(d) The Iwahori Subgroup
(e) The Lie Algebra
2.6 Reductive Groups
(a) Basic Notation
(b) The Absolute and Relative Root Datum
(c) The Relative Root System of a Tits Index
(d) The Valuation Homomorphism and the Subgroup G(k)¹
(e) The Subgroups G(k)♯ and G(k)⁰
2.7 The Group SU₃
2.8 Separable Quadratic Extensions of Local Fields
2.9 Chevalley Systems
(a) Pinnings
(b) The Split Case
(c) The Quasi-split Case
(d) Commutation Relations
(e) Simply Connected Cover
2.10 Integral Models
2.11 Group Scheme Actions and the Dynamic Method
PART TWO: BRUHAT–TITS THEORY
3. Examples: Quasi-split Simple Groups of Rank 1
3.1 The Example of SL₂
(a) The Standard Apartment
(b) The Affine Roots
(c) The affineWeyl group
(d) The Building
(e) The Moy–Prasad Filtration Subgroups
(f) The Apartment as an Affine Space
3.2 The Example of SU₃
(a) The Filtration of the Maximal Torus
(b) The Filtrations of the Root Subgroups
(c) The Standard Apartment and the Affine Roots
(d) The AffineWeyl Group
(e) Unshifted Filtrations
(f) The Building
4. Overview and Summary of Bruhat–Tits Theory
4.1 Axiomatization of Bruhat–Tits Theory
4.2 Metric
4.3 The Enlarged Building
4.4 Uniqueness of the Apartment and the Building
5. Bruhat, Cartan, and Iwasawa Decompositions
5.1 The (affine) Bruhat Decomposition
5.2 The Cartan Decomposition
5.3 The Iwasawa Decomposition
5.4 The Intersection of Cartan and Iwasawa Double Cosets
6. The Apartment
6.1 The Apartment of a Quasi-split Reductive Group
(a) Split Semi-simple Groups
(b) Quasi-split Semi-simple Groups
(c) Quasi-split Reductive Groups
6.2 Affine Reflections and Uniqueness of Valuations
6.3 Affine Roots and Affine Root Groups
6.4 The Affine Root System of a Quasi-split Group
(a) Split Groups
(b) Quasi-split Groups
6.5 Change of Valuation
6.6 The AffineWeyl Group
6.7 Projection to a Levi Subgroup
7. The Bruhat–Tits Building for a Valuation of the Root Datum
7.1 Commutator Computations
7.2 A Filtration of Z(k)
7.3 Concave Functions
7.4 Parahoric Subgroups
7.5 The Iwahori–Tits System
7.6 The (Reduced) Building
7.7 Disconnected Parahoric Subgroups
7.8 The Iwahori–Weyl Group
7.9 Change of Base Field and Automorphisms
(a) Change of Base Field
(b) Automorphisms
7.10 Passage to Completion
7.11 Absolutely Special Points
8. Integral Models
8.1 Preliminaries
8.2 General Properties of Smooth Models of G
8.3 Parahoric Integral Models
8.4 The Structure of the Special Fiber of G⁰_Ω
8.5 Integral Models Associated to Concave Functions
8.6 Passage to Completion
9. Unramified Descent
9.1 Preliminaries
9.2 Statement of the Main Result
9.3 The Building and its Apartments
9.4 The Affine Root System
9.5 Completion of the Proof of the Main Result
9.6 Valuation of Root Datum
9.7 Levi Subgroups
9.8 Concave Function Groups
9.9 Special, Superspecial, and Hyperspecial Points
9.10 Residually Split and Residually Quasi-split Groups
9.11 Restriction of Scalars
PART THREE: ADDITIONAL DEVELOPMENTS
10 Residue Field f of Dimension ≼ 1
10.1 Conjugacy of Special Tori
10.2 Superspecial Points
10.3 Anisotropic Groups
10.4 Fixed Points of Large Subgroups of Tori
10.5 Existence of Anisotropic Tori
10.6 Cohomological Results
10.7 Classification of Connected Reductive k-Groups
(a) (¹A_{n−1}, ¹A_{n−1})
(b) (¹A_{n−1}, ²A_{n−1})
(c) (²Aₙ, ²Aₙ)
(d) (Bₙ, Bₙ), n ≽ 3
(e) (Cₙ, Cₙ), n ≽ 2
(f) (¹Dₙ, ¹Dₙ), n > 4
(g) (¹Dₙ, ²Dₙ), n > 4
(h) (²Dₙ, ²Dₙ), n > 4
(i) D₄
(j) (¹E₆, ¹E₆)
(k) (¹E₆, ²E₆)
(l) E₇
11. Component Groups of Integral Models
11.1 The Kottwitz Homomorphism for Tori
11.2 The Component Groups of T^{ft} and T^{lft}
11.3 The Algebraic Fundamental Group
11.4 z-Extensions
11.5 The Kottwitz Homomorphism for Reductive Groups
11.6 The Component Groups of Parahoric Integral Models
11.7 The Case of dim(f) ≼ 1
12. Finite Group Actions and Tamely Ramified Descent
12.1 Preliminaries
12.2 Certain Group Schemes Associated to H and G
12.3 A Reduction
12.4 Apartments of B
12.5 The Polyhedral Structure on B
12.6 Identification of Parahoric Subgroups
12.7 The Main Theorem
12.8 The Case of a Finite Cyclic Group
12.9 Tamely Ramified Descent
13. Moy–Prasad Filtrations
13.1 Filtrations of Tori
13.2 Filtrations of Parahoric Subgroups
13.3 Filtrations of the Lie Algebra and its Dual
13.4 Optimal Points
13.5 The Moy–Prasad Isomorphism
13.6 Semi-stability
13.7 G-Domains in the Lie Algebra g
13.8 Vanishing of Cohomology
14. Functorial Properties
14.1 Quotient Maps
14.2 Embeddings: Isometric Properties
14.3 Embeddings: Factorization through a Levi Subgroup
14.4 Embeddings of Apartments
14.5 Adapted Points: Definition and Properties
14.6 Embeddings of Buildings via Adapted Points
14.7 The Space of Embeddings and Galois Descent
14.8 Existence of Adapted Points
14.9 Uniqueness of Admissible Embeddings
15. The Buildings of Classical Groups via Lattice Chains
15.1 The Special and General Linear Groups
15.2 Symplectic, Orthogonal, and Unitary groups
PART FOUR: APPLICATIONS
16. Classification of Maximal Unramified Tori (d’après DeBacker)
17. Classification of Tamely Ramified Maximal Tori
18. The Volume Formula
18.1 Remarks on Arithmetic Subgroups
18.2 Notations, Conventions and Preliminaries
18.3 Tamagawa Forms on Quasi-split Groups
18.4 Volumes of Parahoric Subgroups
18.5 Covolumes of Principal S-Arithmetic Subgroups
18.6 Euler–Poincaré characteristic of S-arithmetic subgroups.
18.7 Bounds for the Class Number of Simply Connected Groups
18.8 The Discriminant Quotient Formula for Global Fields
PART FIVE: APPENDICES
A. Operations on Integral Models
A.1 Base Change
A.2 Schematic Closure
A.3 Weil Restriction of Scalars
A.4 The Greenberg Functor
(a) Review of Witt Vectors
(b) Some Module Schemes and Ring Schemes
(c) Definition and Properties of the Greenberg Functor
(d) Beyond the Affine Case
(e) Applications
A.5 Dilatation
A.6 Smoothening
A.7 Schematic Subgroups
A.8 Reductive Models
B. Integral Models of Tori
B.1 Preliminaries
B.2 Split Tori
B.3 Induced Tori
B.4 The Standard Model
B.5 The Standard Filtration
B.6 Weakly induced tori
B.7 The ft-Néron Model
B.8 The Néron Mapping Properties and the lft-Néron Model
B.9 The pro-unipotent radical
B.10 The Minimal Congruent Filtration
C. Integral Models of Root Groups
C.1 Introduction
C.2 Integral Models for Filtration Subgroups of Ga
C.3 Integral Models for Filtration Subgroups of R⁰_{L/K} Ga
C.4 Integral Models for Filtration Subgroups of U_{L/K}
C.5 Summary
References
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Index of Symbols
General Index
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