Buildings and Classical Groups
Published on 6. November 2012
Book
Paperback/Softback
XII, 373 pages
978-94-010-6245-9 (ISBN)
Description
This book describes the structure of the classical groups, meaning general linear groups, symplectic groups, and orthogonal groups, both over general fields and in finer detail over p-adic fields. To this end, half of the text is a systematic development of the theory of buildings and BN-pairs, both spherical and affine, while the other half is illustration by and application to the classical groups. The viewpoint is that buildings are the fundamental objects, used to study groups which act upon them. Thus, to study a group, one discovers or con structs a building naturally associated to it, on which the group acts nicely. This discussion is intended to be intelligible after completion of a basic graduate course in algebra, so there are accounts of the necessary facts about geometric algebra, reflection groups, p-adic numbers (and other discrete val uation rings), and simplicial complexes and their geometric realizations. It is worth noting that it is the building-theoretic aspect, not the algebraic group aspect, which determines the nature of the basic representation theory of p-adic reductive groups.
More details
Edition
Softcover reprint of the original 1st ed. 1997
Language
English
Place of publication
Dordrecht
Netherlands
Target group
Professional and scholarly
Research
Product notice
Paperback (trade)
Illustrations
black & white illustrations
Dimensions
Height: 23.5 cm
Width: 15.5 cm
Thickness: 20 mm
Weight
593 gr
ISBN-13
978-94-010-6245-9 (9789401062459)
DOI
10.1007/978-94-011-5340-9
Schweitzer Classification
Other editions
Additional editions
Paul B. Garrett
Buildings and Classical Groups
Book
04/1997
1st Edition
CRC Press
€217.01
Article exhausted; check different version
Content
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.