Buildings and Classical Groups - Paul Garrett Introduction This book describes the structure of the classical groups meaning general linear groups symplectic groups and orthogonal groups both over general elds and in ner detail over padic elds To this end half of the text is a systematic development of the theory of buildings and BNpairs both spherical and ane 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 reection groups padic numbers and other discrete val uation rings and simplicial complexes and their geometric realizations It is worth noting that it is the buildingtheoretic aspect not the algebraic group aspect which determines the nature of the basic representation theory of padic reductive groups One important source of information for this and related material is the monumental treatise of BruhatTits which appeared in several parts widely separated in time This treatise concerned mostly application of the theory of ane buildings to padic groups of the theory of ane buildings One of the basic points made and an idea pervasive in the work is that buildings can be attached in an intrinsic manner to all padic reductive groups But this point is dicult to appreciate making this source not congenial to beginners pre suming as it does that the reader knows a great deal about algebraic groups and has a rm grasp of root systems and reection groups having presumably worked all the exercises in Bourbaki s Lie theory chapters IVVVI In contrast it is this author s opinion that rather than being corollaries of the theory of algebraic groups the mechanism by which a suitable action of a group upon a building illuminates the group structure is a fundamental thing itself Still much of the material of the present monograph can be found in or inferred from the following items F Bruhat and J Tits Groupes Reductifs sur un Corps Local I Donnees radicielles valuees Publ Math IHES pp F Bruhat and J Tits Groupes Reductifs sur un Corps Local II Schemas en groups existence d une donnee radicielle valuee ibid pp ii Garrett Introduction F Bruhat and J Tits Groupes Reductifs sur un Corps Local III Comple ments et applications a la cohomologie galoisienne J Fac Sci Univ Tokyo pp F Bruhat and J Tits Schemas en groupes et immeubles des groupes clas siques sur un corps local Bull Soc Math Fr pp I have beneted from the quite readable J Humphreys Re ection Groups and Coxeter Groups Camb Univ Press K Brown Buildings SpringerVerlag New York M Ronan Lectures on Buildings Academic Press Even though I do not refer to it in the text I have given as full a bibliog raphy as I can Due to my own motivations for studying buildings the bib liography also includes the representation theory of padic reductive groups especially items which illustrate the use of the ner structure of padic reduc tive groups discernible via buildingtheory By after the rst of the BruhatTits papers most of the issues seem to have been viewed as settled in principle For contrast one might see some papers of Hijikata which appeared during that period in which he studied p adic reductive groups both in a classical style and also in a style assimilating the IwahoriMatsumoto result H Hijikata Maximal compact subgroups of some padic classical groups mimeographed notes Yale University H Hijikata On arithmetic of padic Steinberg groups mimeographed notes Yale University H Hijikata On the structure of semisimple algebraic groups over valuation elds I Japan J Math vol no pp The third of these papers contains some very illuminating remarks about the state of the literature at that time Having made these acknowledgements I will simply try to tell a coherent story Garrett Contents iii Contents Coxeter groups Words lengths presentations of groups Coxeter groups systems diagrams Linear representation reections roots Roots and the length function More on roots and lengths Generalized reections Exchange Condition Deletion Condition The Bruhat order Special subgroups of Coxeter groups Seven important families Three spherical families Four ane families Complexes Chamber complexes The uniqueness lemma Foldings walls reections Coxeter complexes Characterization by foldings and walls Corollaries on foldings and halfapartments Buildings Apartments and buildings denitions Canonical retractions to apartments Apartments are Coxeter complexes Labels links maximal apartment system Convexity of apartments Spherical buildings BNpairs from buildings BNpairs denitions BNpairs from buildings Parabolic special subgroups Further BruhatTits decompositions Generalized BNpairs The spherical case Buildings from BNpairs Generic algebras and Hecke algebras Generic algebras Strict IwahoriHecke algebras Generalized IwahoriHecke algebras Geometric algebra GLn a prototype Bilinear and hermitian forms classical groups A Witttype theorem extending isometries Parabolics unipotent radicals Levi components Examples in coordinates Symplectic groups in coordinates Orthogonal groups Onn in coordinates Orthogonal groups Opq in coordinates Unitary groups in coordinates Construction for GLn Construciton of the spherical building for GLn Verication of the building axioms Action of GLn on the spherical building The spherical BNpair in GLn Analogous treatment of SLn The symmetric group as Coxeter group Spherical Construction for Isometry Groups Construction of spherical buildings for isometry groups Verication of the building axioms The action of the isometry group The spherical BNpair in isometry groups Analogues for similitude groups The Spherical Oriamme Complex The oriamme construction for SOnn Verication of the building axioms The action of SOnn The spherical BNpair in SOnn Analogues for GOnn Reections root systems Weyl groups Hyperplanes chambers walls Reection groups are Coxeter groups Root systems and nite reection groups Ane reection groups special vertices Ane Weyl groups Ane Coxeter complexes Tits cone model of Coxeter complexes Positivedeniteness the spherical case A lemma from PerronFrobenius Local niteness of Tits cones Denition of geometric realizations Criterion for aneness The canonical metric The seven innite families Ane buildings Ane buildings trees denitions The canonical metric Negative curvature inequality Contractibility Completeness BruhatTits xed 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 Congurations of three chambers Subsets of apartments strong isometries The spherical building at innity Sectors Bounded subsets of apartments Lemmas on isometries Subsets of apartments Congurations of chamber and sector Congurations of sector and three chambers Congurations of two sectors Geodesic rays The spherical building at innity Induced maps at innity Applications to groups Induced group actions at innity BNpairs parahorics and parabolics Translations and Levi components Filtration by sectors Levi decomposition Bruhat and Cartan decompositions Iwasawa decomposition Maximally strong transitivity Canonical translations Lattices padic numbers discrete valuations p-adic numbers Discrete valuations Hensel's Lemma Lattices Some topology Iwahori decomposition for GLn Construction for SLV Construction of the ane building for SLV Verication of the building axioms Action of SLV on the ane building The Iwahori subgroup B The maximal apartment system Construction of ane buildings for isometry groups Ane buildings for alternating spaces The double oriamme complex The ane single oriamme complex Verication of the building axioms Group actions on the buildings Iwahori subgroups The maximal apartment systems Index Bibliography