Representation Theory of Finite Reductive GroupsAt the crossroads of representation theory, algebraic geometry and nite group theory,this book blends together many of the main concerns of modern algebra, synthesizingthe past 25 years of research, with full proofs of some of the most remarkableachievements in the area.Cabanes and Enguehard follow three main themes: rst, applications of etalecohomology, leading, via notions of twisted induction, unipotent characters andLusztigs approach to the Jordan decomposition of characters, to the proof of therecent Bonnaf eRouquier theorems. The second is a straightforward and simpliedaccount of the DipperJames theorems relating irreducible characters and modularrepresentations, while introducing modular Hecke and Schur algebras. The nal themeis local representation theory. One of the main results here is the authors version ofFongSrinivasan theorems showing the relations between twisted induction and blocksof modular representations.Throughout, the text is illustrated by many examples; background is provided byseveral introductory chapters on basic results, and appendices on algebraic geometryand derived categories. The result is an essential introduction for graduate students andreference for all algebraists.marccabanesis Charg e de Recherche at Universit e Paris 7mi chelenguehardis Professeur Em erite, Universit e Paris 8.New Mathematical MonographsEditorial BoardB ela Bollob as, University of MemphisWilliam Fulton, University of MichiganFrances Kirwan, Mathematical Institute, University of OxfordPeter Sarnak, Princeton UniversityBarry Simon, California Institute of TechnologyFor informationabout Cambridge UniversityPress mathematics publications visithttp://publishing.cambridge.org/stm/mathematics/Representation Theory of FiniteReductive GroupsMARCCABANES, MICHELENGUEHARDUniversit e Pariscaxniiociuxiviisir\iiissCambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,SoPauloCambridgeUniversityPressThe Edinburgh Building, Cambridge cn: :iu, UKFirst published in print format isnx-:,,;-c-,::-:,:;-;isnx-:,,;-c-,::-:o,oo-,CambridgeUniversityPress20042004Informationonthistitle:www.cambridge.org/9780521825177This publication is in copyright. Subject to statutory exception and to the provision ofrelevant collective licensing agreements, no reproduction of any part may take placewithout the written permission of Cambridge University Press.isnx-:cc-,::-:o,oo-isnx-:cc-,::-:,:;-:Cambridge University Press has no responsibility for the persistence or accuracy of uiisfor external or third-party internet websites referred to in this publication, and does notguarantee that any content on such websites is, or will remain, accurate or appropriate.PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYorkwww.cambridge.orghardbackeBook(NetLibrary)eBook(NetLibrary)hardbackContentsPreface page xiList of terminology xvPART I REPRESENTING FINITE BN-PAIRS 11 Cuspidality in finite groups 31.1 Subquotients and associated restrictions 41.2 Cuspidality and induction 61.3 Morphisms and an invariance theorem 81.4 Endomorphism algebras of induced cuspidal modules 111.5 Self-injective endomorphism rings and an equivalenceof categories 141.6 Structure of induced cuspidal modules and series 172 Finite BN-pairs 222.1 Coxeter groups and root systems 232.2 BN-pairs 272.3 Root subgroups 292.4 Levi decompositions 322.5 Other properties of split BN-pairs 353 Modular Hecke algebras for nite BN-pairs 413.1 Hecke algebras in transversal characteristics 423.2 Quotient root system and a presentation ofthe Hecke algebra 474 The modular duality functor and derived category 554.1 Homology 564.2 Fixed point coefcient system and cuspidality 594.3 The case of nite BN-pairs 63vvi Contents4.4 Duality functor as a derived equivalence 674.5 A theorem of Curtis type 695 Local methods for the transversal characteristics 745.1 Local methods and two main theoremsof Brauers 755.2 A model: blocks of symmetric groups 785.3 Principal series and the principal block 825.4 Hecke algebras and decomposition matrices 845.5 A proof of Brauers third Main Theorem 866 Simple modules in the natural characteristic 886.1 Modular Hecke algebra associated with aSylow p-subgroup 886.2 Some modules in characteristic p 936.3 Alperins weight conjecture in characteristic p 956.4 The p-blocks 97PART II DELIGNELUSZTIG VARIETIES, RATIONALSERIES, AND MORITA EQUIVALENCES 1017 Finite reductive groups and DeligneLusztig varieties 1037.1 Reductive groups and Langs theorem 1047.2 Varieties dened by the Lang map 1057.3 DeligneLusztig varieties 1097.4 DeligneLusztig varieties are quasi-afne 1148 Characters of finite reductive groups 1188.1 Reductive groups, isogenies 1198.2 Some exact sequences and groups in duality 1228.3 Twisted induction 1258.4 Lusztigs series 1279 Blocks of nite reductive groups and rational series 1319.1 Blocks and characters 1329.2 Blocks and rational series 1339.3 Morita equivalence and ordinary characters 13610 Jordan decomposition as a Morita equivalence:the main reductions 14110.1 The condition iR jF = 0 14210.2 A rst reduction 14410.3 More notation: smooth compactications 146Contents vii10.4 Ramication and generation 14910.5 A second reduction 15011 Jordan decomposition as a Morita equivalence: sheaves 15511.1 Ramication in DeligneLusztig varieties 15611.2 Coroot lattices associated with intervals 16211.3 DeligneLusztig varieties associated with intervals 16511.4 Application: some mapping cones 16812 Jordan decomposition as a Morita equivalence: modules 17312.1 Generating perfect complexes 17412.2 The case of modules induced by DeligneLusztig varieties 17612.3 Varieties of minimal dimension inducing asimple module 17712.4 Disjunction of series 181PART III UNIPOTENT CHARACTERS ANDUNIPOTENT BLOCKS 18713 Levi subgroups and polynomial orders 18913.1 Polynomial orders of F-stable tori 18913.2 Good primes 19313.3 Centralizers of -subgroups and some Levisubgroups 19414 Unipotent characters as a basic set 19914.1 Dual conjugacy classes for -elements 19914.2 Basic sets in the case of connected center 20115 Jordan decomposition of characters 20515.1 From non-connected center to connected center anddual morphism 20615.2 Jordan decomposition of characters 20916 On conjugacy classes in type D 21916.1 Notation; some power series 22016.2 Orthogonal groups 22116.3 Special orthogonal groups and their derived subgroup;Clifford groups 22716.4 Spin2n(F) 23516.5 Non-semi-simple groups, conformal groups 23916.6 Group with connected center and derived groupSpin2n (F); conjugacy classes 245viii Contents16.7 Group with connected center and derived group Spin2n (F);Jordan decomposition of characters 24816.8 Last computation, y1, y2, y425017 Standard isomorphisms for unipotent blocks 25917.1 The set of unipotent blocks 26017.2 -series and non-connected center 26117.3 A ring isomorphism 264PART IV DECOMPOSITION NUMBERS AND q-SCHURALGEBRAS 26918 Some integral Hecke algebras 27118.1 Hecke algebras and sign ideals 27218.2 Hecke algebras of type A 27518.3 Hecke algebras of type BC, Hoefsmits matrices andJucysMurphy elements 27918.4 Hecke algebras of type BC: some computations 28118.5 Hecke algebras of type BC: a Morita equivalence 28518.6 Cyclic Clifford theory and decomposition numbers 28819 Decomposition numbers and q-Schur algebras: generallinear groups 29719.1 Hom functors and decomposition numbers 29819.2 Cuspidal simple modules and GelfandGraev lattices 30119.3 Simple modules and decomposition matrices for unipotentblocks 30519.4 Modular Harish-Chandra series 30920 Decomposition numbers and q-Schur algebras:linear primes 31820.1 Finite classical groups and linear primes 31920.2 Hecke algebras 32220.3 Type BC 32620.4 Type D 328PART V UNIPOTENTBLOCKS ANDTWISTEDINDUCTION 33121 Local methods; twisted induction for blocks 33321.1 Connected subpairs in nite reductive groups 33321.2 Twisted induction for blocks 33421.3 A bad prime 341Contents ix22 Unipotent blocks and generalized Harish-Chandra theory 34522.1 Local subgroups in nite reductive groups, -elements and tori 34622.2 The theorem 35022.3 Self-centralizing subpairs 35222.4 The defect groups 35423 Local structure and ring structure of unipotent blocks 36023.1 Non-unipotent characters in unipotent blocks 36123.2 Control subgroups 36323.3 (q 1)-blocks and abelian defect conjecture 366APPENDICES 373Appendix 1 Derived categories and derived functors 374A1.1 Abelian categories 374A1.2 Complexes and standard constructions 375A1.3 The mapping cone 375A1.4 Homology 376A1.5 The homotopic category 376A1.6 Derived categories 377A1.7 Cones and distinguished triangles 378A1.8 Derived functors 379A1.9 Composition of derived functors 379A1.10 Exact sequences of functors 380A1.11 Bi-functors 380A1.12 Module categories 381A1.13 Sheaves on topological spaces 382A1.14 Locally constant sheaves and the fundamental group 384A1.15 Derived operations on sheaves 385Appendix 2 Varieties and schemes 389A2.1 Afne F-varieties 389A2.2 Locally ringed spaces and F-varieties 390A2.3 Tangent sheaf, smoothness 392A2.4 Linear algebraic groups and reductive groups 393A2.5 Rational structures on afne varieties 395A2.6 Morphisms and quotients 395A2.7 Schemes 397A2.8 Coherent sheaves 399A2.9 Vector bundles 400A2.10 A criterion of quasi-afnity 401x ContentsAppendix 3 Etale cohomology 404A3.1 The etale topology 404A3.2 Sheaves for the etale topology 405A3.3 Basic operations on sheaves 406A3.4 Homology and derived functors 407A3.5 Base change for a proper morphism 408A3.6 Homology and direct images with compact support 408A3.7 Finiteness of cohomology 409A3.8 Coefcients 409A3.9 The open-closed situation 410A3.10 Higher direct images and stalks 411A3.11 Projection and K unneth formulae 411A3.12 Poincar eVerdier duality and twisted inverse images 412A3.13 Purity 413A3.14 Finite group actions and constant sheaves 414A3.15 Finite group actions and projectivity 414A3.16 Locally constant sheaves and the fundamental group 415A3.17 Tame ramication along a divisor withnormal crossings 417A3.18 Tame ramication and direct images 418References 422Index 431PrefaceThis book is an introduction to the study of representations of a special classof nite groups, called nite reductive groups. These are the groups of rationalpoints over a nite eld in reductive groups. According to the classication ofnite simple groups, the alternating groups and the nite reductive groups yieldall nite non-abelian simple groups, apart from 26 sporadic groups.Representation theory, when applied to a given nite group G, traditionallyrefers to the program of study dened by R. Brauer. Once the ordinary charac-ters of Gare determined, this consists of expressing the Brauer characters aslinear combinations of ordinary characters, thus providing the decompositionmatrix and Cartan matrix of group algebras of the form k[G] where k is somealgebraically closed eld of prime characteristic . One may add to the abovea whole array of problems:r blocks of k[G] and induced partitions of characters,r relations with -subgroups,r computation of invariants controlling the isomorphism type of these blocks,r checking of niteness conjectures on blocks,r study of certain indecomposable modules,r further information about the category k[G]mod and its derived categoryD(k[G]).In the case of nite reductive groups, only parts of this program have beencompleted but, importantly, more specic questions or conjectures have arisen.For this reason, the present book may not match Brauers programon all points.It will generally follow the directions suggested by the results obtained duringthe last 25 years in this area.Before describing the content of the book, we shall outline very briey thehistory of the subject.xixii PrefaceThe subject. Finite simple groups are organized in three stages of mountingcomplexity, plus the 26 sporadic groups. First are the cyclic groups of primeorder. Second are the symmetric groups (or, better, their derived subgroups)whose representation theory has been fairly well known since the 1930s. Thenthere are the nite reductive groups, each associated with a power q of a primep, a dimension n, and a geometry in dimension n taken in a list similar to the onefor Dynkin ADE diagrams. A little before this classication was complete,DeligneLusztigs paper [DeLu76] on representations of nite reductive groupsappeared. It introduced to the subject the powerful methods of etale cohomol-ogy, primarily devised in the 1960s and 1970s by Grothendieck and his teamin their re-foundation of algebraic geometry and proof of the Weil conjectures.DeligneLusztigs paper set the framework in which most subsequent studiesof representations over the complex eld of nite reductive groups took place,mainly by Lusztig himself [Lu84] with contributions by Asai, Shoji and others.The modular study of these representations was initiated by the papers ofFongSrinivasan [FoSr82], [FoSr89], giving the partition of complex charac-ters induced by the blocks of the group algebras over a eld of characteristic ( ,= the characteristic p of the eld of denition of the nite reductive group).Meanwhile, Dipper [Dip85ab] produced striking results about the decompo-sition numbers (relating irreducible representations over the complex eld andover nite elds of order a) for nite linear groups GLn( pb), emphasizing ther ole played by analogues in characteristicof concepts previously studied onlyover the complex eld, such as Hecke algebras and cuspidal representations.These works opened a new eld of research with numerous contributions byteams in Paris (Brou e, Michel, Puig, Rouquier, and the present authors) andGermany (Dipper, Geck, Hiss, Malle), producing several new results on blocksof modular representations, DeligneLusztigvarieties, non-connectedreductivegroups, and giving new impulse to adjacent (or larger) elds such as derivedcategoriesfornitegrouprepresentations,cyclotomicHeckealgebras,non-connected reductive groups, quasi-hereditary rings or braid groups. Dipperswork was fully rewritten and generalized in a series of papers by James andhimself, linking with James study of modular representations of symmetricgroups, thusgeneralizingthelattertoHeckealgebrasoftypeAorB, andintroducing the so-called q-Schur algebra.In1988and1994, Brou epublishedaset ofconjecturespostulatingthatmost correspondencesinLusztigtheoryshouldbeconsequencesofMoritaor derived equivalences of integral group algebras. One of them was recentlyproved by Bonnaf eRouquier in [BoRo03]. It asserts that the so-called Jordandecompositionofcharacters([Lu84];see[DiMi91]13.23)isinducedbyaMoritaequivalencebetweengroupalgebrasoveran-adiccoefcient ring.Preface xiiiTheir proof consists of a clever use and generalization of DeligneLusztigsmost signicant results, inparticularavanishingtheoremfor etalesheavessupplemented by the construction of Galois coverings for certain subvarietiesin the smooth compactication of DeligneLusztig varieties.The book. Our aimis to gather the main theorems around Bonnaf eRouquierscontributionandtheaccount ofDeligneLusztigsmethodsthat it requires.This makes a core of six chapters (712). After establishing the main algebraic-geometricpropertiesoftherelevant varieties, weexpoundDeligneLusztigtheory and Bonnaf eRouquier theorems. The methods are a balanced mix ofmodule theory and sheaf theory. We use systematically the notions and methodsof derived categories.In contrast to this high-ying sophistication, our Part I gathers most of theproperties that can be proved by forgetting about algebraic groups and work-ing within the framework of nite BN-pairs, or Tits systems, a frameworkcommon to nite, algebraic or p-adic reductive groups. (There are not evenBN-pairs in Chapter 1 but nite groups possessing a set of subquotients sat-isfying certain properties.) This, however, allows us to prove several substan-tial results, such as the determination of simple modules in natural character-istic([Ri69],[Gre78],[Tin79],[Tin80]),theresultsabouttheindependenceof Harish-Chandra induction in relation to parabolic subgroups in transversalcharacteristics ([HowLeh94], [DipDu93]), or the theorem asserting that AlvisCurtisDeligneLusztig duality of characters induces an auto-equivalence ofthe derived category (transversal characteristics again, [CaRi01]). Chapter 5 onblocks is a model of what will be done in Part V, while Chapter 4 gives a avorof sheaf theory and derived categories, topics that are at the heart of Part II.ApartfrominPartI,thenitegroupsweconsiderarebuiltfrom(afneconnected) reductive F-groups G, where F is an algebraically closed eld ofnon-zero characteristic (we refer to the books [Borel], [Hum81], [Springer]).When G, as a variety, is dened over a nite subeld of F andF: G G isthe associated Frobenius endomorphism (think of applying a Frobenius eldautomorphism to the matrix entries in GLn(F)), the nite group GFof xedpoints is specically what we call a nite reductive group.Parts III to V of the book give proofs for the main theorems on modularaspectsofcharactertheoryofnitereductivegroups, i.e. thetypeoftheo-remsthat startedthesubject, historicallyspeaking. Just ascharactersareahandy computational tool for approaching representations of nite groups overcommutative rings, these theorems should be considered as hints of what thecategories OGFmodshouldlooklike(Oisacompletediscretevaluationring), either absolutely, or relative to OLFmod or OWmod categories forxiv PrefacecertainF-stable Levi subgroups L or Weyl groupsW(see Chapter 23). Theresults in Parts III to V are thus less complete than the ones in Part II.The version we prove of FongSrinivasan theorems (Theorem 22.9) is ourgeneralization[CaEn94], introducingandusingpolynomial ordersfortori,ande-generalizedHarish-Chandratheory[BrMaMi93]. Thisallowsustocheck Brou es abelian defect conjecture when e = 1 (Theorem 23.12). Asfor decomposition numbers, we prove the version of GruberHiss [GruHi97],giving the relation between decomposition numbers for the unipotent blocks ofGFand the decomposition numbers of q-Schur algebras (see Theorem 20.1).The framework is an extended linear case which comprises (nite) generallinear groups, and classical groups with the condition that bothand the orderof q mod.are odd.Chapter16givesafull proofofatheoremofLusztig[Lu88]about therestriction of characters from GFto [G, G]F. This checking consists mainly ina quite involved combinatorial analysis of conjugacy classes in spin groups.The general philosophy of the book is that proofs use only results that havepreviously appeared in book form.Insteadofgivingconstantreferencestothesamesetofbooksincertainplaces, especially in Part II, we have provided this information in three appen-dices at the end of the book. The rst gathers the basic knowledge of derivedcategories and derived functors. The second does the same for the part of al-gebraic geometry relevant to this book. The third is about etale cohomology.Subsections and results within the appendices are referenced using A1, A2, andA3 (i.e. A2.12 etc.).Historical notes, indicating authors of theorems and giving references forfurther reading, are gathered at the end of each chapter.We thank C edric Bonnaf e and Rapha el Rouquier for having provided earlypreprints of their work, along with valuable suggestions and references.To conclude, we should say that there are surely many books to be writ-ten on neighboring subjects. For instance, we have not included AsaiShojisdeterminationoftheRGLfunctor,whichisacrucialstepinthedenitionofgenericblocks[BrMaMi93]; seealso[Cr95]. Charactersheaves, KazhdanLusztig cells, or intersection cohomology are also fundamental tools for severalaspects of representations of nite reductive groups.TerminologyMost of our terminology belongs to the folklore of algebra, especially the grouptheoretic branch of it, and is outlined below.The cardinality of a nite set S is denoted by [S[. When G is a group andH is a subgroup, the index of H in G, i.e. the cardinality of G/H, when nite,is denoted by [G :H[.The unit of groups is generally denoted by 1.Group actions and modules are on the left unless otherwise stated.If Gacts on the set S, we denote by SGthe subset of xed points {s S [gs = s for all g G].The subgroup of a groupGgenerated by a subset Sis denoted by.In a group G, the action by conjugation is sometimes denoted exponentially,that ishg = hgh1and gh= h1gh. The center of Gis denoted by Z(G). IfSis a subset of G, we denote its centralizer by CG(S) := {g G [ gs = sgfor all s S]. We denote its normalizer by NG(S) := {g G [ gSg1= S].The notation HG means that His a normal subgroup of G. The notationG = K> H means that G is a semi-direct product of its subgroups K and H,with Hacting on K.Forg, h G, we denote their commutator by [g, h] = ghg1h1. IfH,H/ are subgroups of G, one denotes by [H, H/] the subgroup of Ggeneratedby the commutators [h, h/] for h H, h/ H/.Ifis a set of primes, we denote by/its complementary set in the setof all primes. If n 1 is an integer, we denote by nthe biggest divisor of nwhich is a product of powers of elements of . If G is a group, we denote byGthe set of elements of nite order nsatisfying n = n. We call them the-elements of G. A-group is any group of nite order n such that n = n.The /-elements of G are sometimes called the -regular elements of G. Anyelement of nite order g G is written uniquely as g = gg/ = gg/whereg Gand g/ G/ . We then call gthe -part of g.xvxvi List of terminologyIf n 1 is an integer, we denote by n(x) Z[x] the nth cyclotomic poly-nomial, dened recursively byxn 1 = dd(x) where the product is overdivisors d 1 of n.Let A be a (unital) ring. We denote by J(A) its Jacobson radical. If M is anA-module, we denote the head ofMby hd(M) =M/J(A).M. We denote bysoc(M) the sum of the simple submodules of M (this notion is considered onlywhen this sum is non-empty, which is ensured with Artin rings, for instancenite-dimensional algebras over a eld).If n 1 is an integer, we denote by Matn(A) the ring of nn matrices withcoefcients in A (generally for a commutative A). We denote the transpositionof matrices by X .tXfor X Matn(A).We denote byA

the group of invertible elements ofA, sometimes calledunits. We denote byAoppthe opposite ring.We denote byAMod (resp. Amod) the category of A-modules (resp.of nitely generatedA-modules). Note that we use the sign for objects incategories, so M Amod means that M is a nitely generatedA-module.WhenM Amod, we denote by GLA(M) the group of automorphismsof M. For a eld F and an integer n 1, we abbreviate GLn(F) = GLF(Fn).If A, Bare two rings, anA-B-bimoduleMis anABopp-module, thatisthedatumofstructuresofleft A-moduleandright B-moduleonMsuchthata(mb) = (am)bforalla A, b Bandm M. Recallthat M B theninducesafunctor BMod AMod. WhenA =B, wejustsayA-bimodule.ThecategoryofnitelygeneratedA-B-bimodulesisdenotedbyAmodB.If C is a commutative ring and G is a group, we denote by CG (sometimesC[G]) the associated group ring, or group algebra, consisting of nite linearcombinations

gG cgg of elements of G with coefcients in C endowed withthe C-bilinear multiplication extending the lawof G. The trivial module for thisring is C with the elements of G acting by IdC. This CG-module is sometimesdenoted by 1.The commutative ring Cis sometimes omitted from the notation. For in-stance, if HisasubgroupofGtherestrictionofaCG-moduleMtothesubalgebra CH is denoted by ResGHM. In the same situation, CG is consideredasaCG-CH-bimodule,sowehavetheinductionfunctorIndGHdenedbytensor product CG CH .Let O be a complete local principal ideal domain (i.e. a complete discretevaluation ring) with eld of fractions Kand residue eld k = O/J(O). LetAbe an O-algebra which is O-free of nite rank over O. Then O is said to bea splitting system forA ifA OKandA O k/J(A O k) are products ofmatrix algebras over the elds K and k, respectively. Note that this implies thatList of terminology xviiA OKis semi-simple. For group algebras OG(Ga nite group) and theirblocks, this is ensured by the fact that O has characteristic zero.If G is a nite group andis a prime, a triple (O, K, k) is called an -modularsplitting system for G if O is a complete discrete valuation ring containing the[G[th roots of 1, free of nite rank over Z

, Kdenoting its eld of fractions (anite extension of Q

) and k its residue eld (with [k[ nite, a power of ). ThenO is a splitting system for OG, i.e. KG (resp. kG/J(kG)) is split semi-simpleoverK(resp.k); see [NaTs89] 3.6. Note that if (O, K, k) is an-modularsplitting system for G, it is one for all its subgroups.Let Gbe a nite group. We denote by Irr(G) the set of irreducible char-acters of G, i.e. trace maps G C corresponding with simple CG-modules.Generalized characters are Z-linear combinations of elements of Irr(G). Theyare considered as elements of CF(G, C), the space of central functions G Cof which Irr(G) is a base. Since nite-dimensional CG-modules may be re-alized over Q[] for a [G[th root of 1, classically Irr(G) is identied withthe trace maps of simpleKG-modules for any eldKof characteristic zerocontaining a [G[th root of 1. They form a basis of CF(G, K) (central functionsG K).Classicallyweconsider onCF(G, K) thescalar product f,f /)G:=[G[1

gGf (g) f /(g1) for which Irr(G) is orthonormal.PART IRepresenting nite BN-pairsThis rst part is an elementary introduction to the remainder of the book.Instead of nite reductive groups G := GFdened as the xed points undera Frobenius endomorphism F: G Gin an algebraic group, we consider nitegroups G endowed with a split BN-pair. This is dened by the presence in G ofsubgroups B, Nsatisfying certain properties (see Chapter 2 for precise deni-tions). Part I gathers many of the results that can be proved about representationsof Gwithin this axiomatic framework. Though some results are quite recent,this should not mislead the reader into the idea that nite reductive groups canbe studied without reference to reductive groups and algebraic varieties.However, many important ideas are evoked in this part. We shall comment onHarish-Chandra induction, cuspidality, Hecke algebras, the Steinberg module,the duality functor, and derived categories.The six chapters are almost self-contained. We assume only basic knowledgeofmoduletheory(see, forinstance, therstchapterof[Ben91a]). Wealsorecall some elementary results on BN-pairs (see, for instance, [Asch86] 43,[Bour68] IV).Agroup with a split BN-pair of characteristic p is assumed to have parabolicsubgroups decomposed asP = UP >L, the so-called Levi decomposition,where L is also a nite group with a split BN-pair. A leading r ole is played bythe G-L-bimodulesRG.e(UP)where R = Z[ p1] and e(UP) = |UP|1

uUP u, and their R-dualse(UP).RG.Arst natural question is to ask whether this bimodule depends on P and not juston L. We also study Harish-Chandra induced modules RG.e(UP) RLM forMsimple kL-modules (k is a eld wherep = 0) such that (L, M) is minimalwith regard to this induction process. It is important to study the law of the2 Part I Representing nite BN-pairsHecke algebra EndkG(RG.e(UP) RLM) and show that it behaves a lot likea group algebra.The RG-bimodulesRG.e(UP) RL e(UP).RGallow us to build a bounded complex dening an equivalenceDb(RGmod) Db(RGmod)within the derived category of the category of nitely generated RG-modules.We see that the main ingredients in this module theory are in fact permutationmodules. In nite group theory, these are often used as a rst step towards thestudy of the full module category, or, more importantly, through the functorsthey dene. In our context of groups G := GF, we may see the G-sets G/UPas 0-dimensional versions of the DeligneLusztig varieties dened in G.1Cuspidality in finite groupsThe main functors in representation theory of nite groups are the restrictionto subgroups and its adjoint, called induction.We focus attention on a slight variant. Instead of subgroups, we considersubquotients VP of a nite group G. It is natural to consider the xed pointfunctorResG(P,V)whichassociateswithaG-moduleMthesubspaceMVofits restriction toPconsisting of xed points under the action of V. This is aP/V-module. When the coefcient ring (we denote it by) is such that theorder of Vis invertible in , the adjoint ofResG(P,V): Gmod P/Vmodis a kind of induction, sometimes called Harish-Chandra induction,IndG(P,V): P/Vmod Gmodwhich rst makes the given P/V-module into a V-trivial P-module, theninduces it from P to G.The usual Mackey formula, which computes ResGPIndGP/ , is then replaced bya formula where certain non-symmetric intersections (P, V) (P/, V/) :=((P P/)V/, (V P/)V/) occur. This leads naturally to a notion of -stable-regularsets Lofsubquotientsofagivennitegroup. Forsuchaset ofsubquotients, an L-cuspidal triple is (P, V, M), where (P, V) L and M is aV-trivial P-module such that ResP(P/,V/)M = 0 for all (P/, V/) L such thatV V/ P/ P and (P/, V/) ,= (P, V).The case of a simple M above (for= K a eld) has remarkable properties.The induced module IndG(P,V)M is very similar to a projective G-module. Theindecomposable summands of IndG(P,V)M have a unique simple quotient, and aunique simple submodule, each determining the direct summand that yields it.A key fact explaining this phenomenon is the property of the endomorphismalgebra EndKG(IndG(P,V)M) of being a self-injective algebra. This last property34 Part I Representing nite BN-pairsseems tobe intimatelyrelatedwithcuspidalityof M. These endomorphismalge-bras are what we call Hecke algebras.Self-injectivity is a property Hecke algebras share with group algebras. Toprove self-injectivity, we dene a basis of the Hecke algebra. The invertibility ofthese basis elements is related to the following quite natural question. Assume(P, V) and (P/, V/) are subquotients such that P P/covers both quotientsP/VandP//V/ and makes them isomorphic. Then, the V-trivialP-modulesand the V/-trivialP/-modules can be identied. The independence questionis as follows. Are IndG(P,V)and IndG(P/,V/)tranformed into one another by thisidentication? A positive answer is shown to be implied by the invertibility ofthe basis elements mentioned above.1.1. Subquotients and associated restrictionsLet G be a nite group. A subquotient of G is a pair (P, V) of subgroups of Gwith VP.Denition1.1. When VP and V/ P/are subgroups of G, let(P, V) (P/, V/) = ((P P/).V/), (V P/).V/).One denotes (P, V) (P/, V/) if and only if V/ V P P/.One denotes (P, V)(P/, V/) if and only if (P, V) (P/, V/) = (P/, V/)and (P/, V/) (P, V) = (P, V).Proposition1.2. Keep the above notation. (i) If (P, V)(P/, V/), thenV P/ = V/ P = V V/ and P/V =P//V/ =P P//V V/.(ii) ((P, V) (P/, V/))((P/, V/) (P, V)).(iii)(P, V)(P/, V/)if andonlyif (P, V) (P/, V/) = (P/, V/)and[ P/V[ = [ P//V/[.Proof. (i), (ii) are easy from the denitions.(iii) The only if is clear from (i). Assume now that (P, V) (P/, V/) =(P/, V/) and [ P/V[ = [ P//V/[. Then(P P/).V/ =P/andV P/ V/.ThenP//V/isaquotient of(P P/)/(V P/)byreductionmod. V/. But(P P/)/(V P/) is a subgroup ofP/V. Since [ P/V[ = [ P//V/[, all thosequotients coincide, so P P/ V/ = V P/ and (P P/).V =P. This gives(P/, V/) (P, V) = (P, V) and therefore (P, V)(P/, V/). Notation 1.3. When V P are subgroups of G, let P/Vmod be the cate-gory of nitely generated P-modules having V in their kernel (we sometimescall them (P, V)-modules).1Cuspidality in finite groups 5LetResG(P,V): G mod P/V modbe the functor dened by ResG(P,V)(M) =MV(xed points under the action ofV) as P-module.Denition 1.4. WhenGisanitegroup, isacommutativering,andV,V/aretwosubgroupsof Gwhoseordersareinvertiblein, let e(V) =[V[1

uV u G. If VV/ is a subgroup, then e(V)e(V/) = e(VV/). In par-ticular e(V) is an idempotent.Proof. Clear.Proposition 1.5. Letbe a commutative ring. Let VPand V/ P/ in Gwith [V[ and [V/[ invertible in . Let L P be a subgroup such that P = LV.Let Nbe a P/V-module identied with a L/(L V)-module. Denote e =e(V).(i)Ge is a G-L-bimodule andGe LN = IndG(P,V)Nby ge m .g m for g G, m N.(ii) If Mis aG-module, then ResG(P,V)M = eM. If moreover (P/, V/) (P, V), then ResP(P/,V/) ResG(P,V) = ResG(P/,V/).(iii) IndGP and ResG(P,V) induce exact functors preserving projectivity of mod-ules, and adjoint to each other between P/Vmod and Gmod.(iv) If N/ is a P//V/-module, we haveHomG_IndGPN, IndGP/N/_=

PgP/GHom(PgP/)_ResPg(P/,V/) (P,V)N, ResgP/(P,V) g(P/,V/)gN/_as -modules.Proof. (i) One has clearlyG PPe = Ge by g pe . gpe (g G,p P). So one may assume G =P. Then one has to check Pe LN =Nbype m .pem. Thisisclear, thereversemapbeingm . e msinceP = LV.(ii) It is clear that the elements of eM are V-xed. But an element xed byanyu Vis xed bye. SoeM = ResG(P,V)Mas subspaces of ResGPM. Thecomposition formula comes from e(V/).e = e.(iii) The rightP-moduleGe is projective (as direct summand ofG,whichis free), so Geis exact. The image of P/V is a projectiveG-module. SoGe sends projectiveP/V-modules to projectiveG-modules. Similarly, eGisaprojectiveright G-moduleandaprojectiveleft P/V-module, so ResG(P,V): Gmod P/Vmod is exact and pre-servesprojectives. Concerningadjunction, theclassical adjunctionbetween6 Part I Representing nite BN-pairsinduction and restriction gives HomG(IndGP(N), M) = HomP(N, ResGP(M))and HomG(M, IndGP(N)) = HomP(ResGP(M), N) for all P-modules N andG-modules M(see[Ben91a]3.3). When, moreover, NisV-trivial onemay replace the ResGPby xed points under Vsince eN =N, (1 e)N = 0and therefore for all P-modules N/, HomP(N/, N) = HomP(eN/, N) andHomP(N, N/) = HomP(N, eN/).(iv) Note rst that the expression Hom(PgP/) (ResPg(P/,V/) (P,V)N,ResgP/(P,V) g(P/,V/)gN/) makes sense since P gP/ is a subgroup of the rst termsin both g(P/, V/) (P, V) and (P, V) g(P/, V/). The Mackey formula andadjunction between induction and restriction give HomG(IndGPN, IndGP/N/) =PgP/GHom(PgP/)(ResPPgP/N, ResgP/PgP/gN/). Now, we may replace the sec-ondtermwithits xedpoints under V gP/(hence(V gP/).gV/) sinceN = e(V gP/)N. Similarly, we may replace the rst termwith its xed pointsunder P gV/ (hence (P gV/).V) since (1 e(P gV/))gN/ = 0. 1.2. Cuspidality and inductionWe xa commutative ring.Denition 1.6. A G-stable set Lof subquotients (P, V) is said to be -regularif and only if, for all (P, V) L, Vis of order invertible in. One says thatL is -stable if and only if L is G-stable, (G, {1]) L, and, for all (P, V),(P/, V/) L, one has (P, V) (P/, V/) L.For the remainder of the chapter, we assume that Gis a nite group,and L is a -regular,-stable set of subquotients of G.Example 1.7. (i) When VP are subgroups of the nite group G, and [V[ isinvertible in , it is easy to show that there is a minimal -regular, -stableset of subquotients L(P,V)suchthat (P, V) L(P,V). It consists of nite-intersections (with arbitrary hierarchy of parenthesis) of G-conjugates of(P, V).(ii) Let Fbe a nite eld, let G := GL2(F) be the group of invertible matrices_a cb d_(a, b, c, d F, ad bc ,= 0). Let B be the subgroup dened by c =0, let UB be dened by a = d = 1, c = 0. Then the pair (G, {1]) along withthe G-conjugates of (B, U) is -stable. This is easily checked by the equalityG =B BwB for w =_0 11 0_, and the (more obvious) fact that B = (B Bw)Uwith B Uw= {1].Thesystemjustdenedis-regularaslongasthecharacteristicof Fisinvertible in .1Cuspidality in finite groups 7(iii) In the next chapter, we introduce more generally the notion of groupswith split BN-pairs of characteristic p ( p is a prime). These groups G have a sub-group B which is a semi-direct product U.Twhere Uis a subgroup consistingof allp-elements of B. The subgroups of G containing B are called parabolicsubgroups.Theydecomposeassemi-directproductsP = V.L,whereVisthe largest normal p-subgroup ofP. The set of G-conjugates of pairs (P, V)is -stable (see Theorem 2.27(ii)) and of course-regular for any eld ofcharacteristic not equal top.The reader familiar with these groups may assume in what follows that oursystem L corresponds with this example. The notation, however, is the same.Denition1.8. AG-module Mis saidtobe L-cuspidal if andonlyif(G, {1]) L, andResG(P,V)M = 0foreach(P, V) Lsuchthat (P, V) ,=(G, {1]). This clearly implies that, if some pair (P, {1]) is in L, thenP = G.When (P, V) L andMis aP/V-module, Mis said to be L-cuspidal ifand only if it is LP/V-cuspidal for LP/V = {(P//V, V//V) [ L a (P/, V/) (P, V)] (this implies that the only pair (P/, V) (P, V) in L is (P, V)).Remark 1.9. If L is -stable, -regular, and (P, V) L is such that thereis a cuspidal P/V-module, then the condition(P/, V) (P, V) implies P/ =Pis a strong constraint. Applying it to the pairs (P, V)g (P, V), one gets that,if g G is such that Vg P V, then P = (P Pg).V.Notation 1.10. A cuspidal triple inGrelative to L and is any triple =(P, V, M) where (P, V) L and M is an L-cuspidal P/V-module.When / = (P/, V/, M/) is another cuspidal triple, one denotes / if andonly if (P, V)(P/, V/) and ResPPP/ M = ResP/PP/ M/.Let IG= IndGPM. Then IG=IGgfor all g G.When = kis a eld, let cuspk(L) be the set of all triples = (P, V, S)such that (P, V) is in Land S is a simple cuspidal k P/V-module (one for eachisomorphism type).Proposition 1.11. Assume that= k is a eld. Let M be a simple kG-module.Then(a)there exists a simple cuspidal triple/ cuspk(L) such that M soc(IG/ ),(a/)there exists a simple cuspidal triple cuspk(L) such that M hd(IG).8 Part I Representing nite BN-pairsProof. Let (P, V) L of minimal [ P/V[ be such that ResG(P,V)M ,= 0 (recall(G, {1]) L). Let S be a simple component of the head of ResG(P,V)M. Thereis a surjection ResG(P,V)M S. For every (P/, V/) (P, V), one has a surjec-tion ResG(P/,V/)M ResP(P/,V/)S since ResP(P/,V/) is exact. Then ResP(P/,V/)S = 0when [ P//V/[ < [ P/V[ by the choice of (P, V). So (P, V, S) cuspk(L). Thisgives (a). One would get (a/) with/ = (P, V, S/) by consideringS/ a simplecomponent of soc(ResG(P,V)M). Remark. Assume (P, V)(P/, V/) in L. For each V-trivial k P-moduleMthere is a unique V/-trivial k P/-module M/withthe same restrictiontoP P/as M. Thisclearlydenesanisomorphismbetweenk[P/V]modand k[P//V/]mod. ThenMis simple if and only ifM/ is so. Similarly onechecks that (P, V, M) is cuspidal if and only if (P/, V/, M/) is so. Indeed, ifM/ V//,= 0 for (P//, V//) (P/, V/) in L, then M/ PV//=MPV//,= 0 and there-fore M(PV//).V,= 0. But (P//, V//) (P, V) L and (P, V, M) cuspk(L),so (P//, V//) (P, V) = (P, V). Along with (P//, V//) (P/, V/)(P, V),this clearly implies (P//, V//) = (P/, V/).1.3. Morphisms and an invariance theoremIn the following, is a commutative ring, Gis a nite group and L is a-regular,-stable set of subquotients of G.Denition 1.12. When = (P, V, M),/ = (P/, V/, M/) are cuspidal triples(see Notation1.10), and g Gis suchthat g/, choose anisomor-phismg,,/ : ResPPgP/M = ResgP/PgP/gM/. Itcanbeseenasalinearisomor-phismg,,/ : M M/suchthat g,,/ (x.m) = xg.g,,/ (m)forall m Mand x P gP/. Assume 1,, = IdM.When g/, dene ag,,/ : IGIG/by IG= G PM andag,,/ (1 P m) = e(V)g P/ g,,/ (m). (1)Proposition 1.13. Assumeg/. LetLbe a subgroup such that P = LV,gP/ = L.gV/ (for instance L =P gP/). Denote e = e(V), e/ = e(V/).The equation (1) denes a unique G-morphismIGIG/ . Throughthe identications IG= Ge LMand IG/ =IGg/ = Gge/ LgM/(seeProposition 1.5(i)) the map ag,,/ identies with the map Lg,,/ :Ge LM Gge/ LgM/where: Ge Gge/ismultiplicationbyge/ on the right.Proof. SincethemorphismisclearlyamorphismofG-L-bimodules, itsufces to check the second statement to have that ag,,/ is well-dened andG-linear.1Cuspidality in finite groups 9Let usrecall that, byProposition1.5(i), G PM = Ge LMbyx m . xe m for x G, m M. Similarly IG/ =IGg/ = Gge/ LgM/ byx(ge/) m/ . xg m/forx Gandm/ M/.Nowthemapag,,/ wouldsendtheelementcorrespondingwithxe mtotheonecorrespondingwithx.e.ge/ g,,/ (m). This is clearly the image by Lg,,/ . Theorem1.14. Assume= k is a eld. Assume that for each , / in cuspk(L)and /, the map a1,,/ : IGIG/is an isomorphism.Then, whenever (P, V)(P/, V/) in L, the map kGe(V) kGe(V/) de-ned by x . xe(V/) is an isomorphism (and therefore [V[ = [V/[).We give a homological lemma used here in a special and quite elementarycase, but stated also for future reference.Lemma1.15. Let Abeanite-dimensional algebraoveraeld. Let X =(. . .di 1XidiXi 1di 1. . .) be a bounded complex of projective right A-modules. That is, the di are A-linear maps, di di 1 = 0 for any i , and Xi = 0except for a nite number of i s.Let Mbe a set of (left) A-modules such that any simple A-module is in somehd(M) for M M. ThenXis exact (that is Ker(di) = di 1(Xi 1) for all i ) ifand only ifX AM =_. . .di 1AMXi AMdiAMXi 1AMdi 1AM. . ._is exact for any M M.ProofofLemma1.15. We use the classical notation Hi(X) for the quotientKer(di)/di 1(Xi 1).SupposeXisnotexact,i.e. X AAisnotexact.Leti0bethemaximalelement in {i [ Hi 1(X AM) = 0 for all M inAmod].BytheprojectivityoftheXis, anyextension0 M3 M2 M1 0gives rise toanexact sequence of complexes 0 X AM3 X AM2 X AM1 0andthen, bythehomologylongexactsequence(see[Ben91a] 2.3.7), to the exact sequenceHi0(X AM3) Hi0(X AM2) Hi0(X AM1) 0= Hi01(X AM3).Suppose rst Hi0(X AM2) ,= 0, then either Hi0(X AM1) or Hi0(X AM3) ,= 0.ThisallowsustoassumethatthereisasimpleA-moduleSsuchthat Hi0(X A S) ,= 0. Now, there is an extension with M1 = S and M2 M.Then Hi0(X AM2) = 0 by the hypothesis, and the above exact sequence givesHi0(X A S) = 0, a contradiction. 10 Part I Representing nite BN-pairsProof of Theorem 1.14. We apply the above lemma forX = (. . . 0 . . . 0 X1 = kGe(V)X0= kGe(V/) 0 . . . 0 . . .)where(b) = b.e(V/) andA = kG/ for G/ =P P//V V/ (isomorphic tobothP/VandP//V/,byProposition1.2(i)).Thistellsusthat isaniso-morphism if and only if Mis an isomorphism for eachMin a set M ofkG/-modules satisfying the condition given by Lemma 1.15.The kG/-modules can be considered as restrictions toP P/ofV-trivialk P-modules(resp. V/-trivial k P/-modules). ByProposition1.11, aset Mmay be taken to be the set of induced modulesMi = ResPPP/ IndP(Pi,Vi)Nifor(Pi, Vi, Ni) cuspk(L) and (Pi, Vi) (P, V). Let = (P0, V0, N0) be such atriple. It is easily checked that (P/, V/) (P0, V0) = (P0, V0) (see also Exer-cise 3). Then Proposition 1.2(ii) implies (P0, V0)((P0, V0) (P/, V/)) =((P0 P/).V/, (V0 P/).V/). Solet N/0bethe(P0, V0) (P/, V/)-moduledened by having the same restriction to P0 P/ as N0.Denote / = ((P0, V0) (P/, V/), N/0). Now, recallingthat 1,,/ = Id,Denition 1.12 implies that a1,,/is the map dened bykGe(V0) P0P/ N0 kGe((V0 P/).V/) P0P/ N/0, (1)x n . xe((V0 P/).V/) nfor x kGe(V0), n N0 (see also Proposition 1.13).The hypothesis of our theorem tells us that map (1) is an isomorphism.Thanks to Lemma 1.15, we just have to check that M0is anisomorphism, where M0 = ResPPP/ IndP(P0,V0)N0 = IndPP/(P0P/,V0P/)N0 = k(P P/).e(V0 P/) P0P/ N0 (see Proposition 1.5(i)). With this description of M0, M0 is the mapkGe(V) PP/k(P P/)e(V0 P/) P0P/ N0 (2)kGe(V) PP/k(P P/)e(V0 P/) P0P/ N0,y y/ n . ye(V/) y/ nwhere y kGe(V), y/ k(P P/)e(V0 P/), n N0.Note that e(V)e(V0 P/) = e(V0) byDenition1.4andsince V.(V0 P/) =V0 (V.(P P/)) = V0 P = V0. Through the trivial G-(P0 P/)-bimoduleisomorphismskGe(V) PP/k(P P/)e(V0 P/) kGe(V0), y y/ . yy/,kGe(V/) PP/k(P P/)e(V0 P/) kGe(V/.V0 P/),z z/ . zz/,1Cuspidality in finite groups 11(for ykGe(V), y/k(P P/)e(V0 P/), z kGe(V/), z/k(P P/)e(V0 P/)), map (2) becomes the mapkGe(V0) P0P/ N0 kGe(V/.V0 P/) P0P/ N/0,yy/n .ye(V/)y/ nwiththesamenotationas above. But this is map(1) sincewemaytakey = ge(V) (g G),y/ = e(V0 P/), and we haveyy/ = ge(V)e(V0 P/) =ge(V0). So map (2) is an isomorphism. 1.4. Endomorphism algebras of induced cuspidal modulesWe keep the hypotheses of 1.3 above. We showthat, under certain assumptions,the elements introduced in Denition 1.12 give a basis of the endomorphismalgebra of the induced module IG.Proposition 1.16. Let = (P, V, M), / = (P/, V/, M/) be two cuspidaltriples (see Notation 1.10).(i) The set of g G such that g/ is a union of double cosets in PG/P/.(ii) Assumeis a subring of the algebraically closed eld Kand M K,M/ Kare simple. Ifg/, then the maps ag,,/ and a1,,g/ differ by anisomorphism IG/ IGg/.Proof. (i) is trivial. (ii) is clear from the denitions. We now study the modules IndG(P,V)Mfor cuspidal triples (P, V, M) (seeNotation 1.10). We are mainly interested in the case whereis a eld and Mis absolutely simple, but we may also need a slight variant where is not aeld. When C is a set of cuspidal triples, one denes the following.Condition 1.17. Either(a)is a splitting eld for the group algebra G and C cusp

(L),or(b)isaprincipalidealdomain,subringofasplittingeldKof KG,C has asingle element (P, V, M) suchthat (P, V, M K) cuspK(L),and, whenever g G satises (P, V, M K)(P, V, M K)g, then(P, V, M)(P, V, M)g.Proposition1.18. Let C beaset of cuspidal triples satisfyingtheaboveCondition 1.17.Consider the ag,,/of Denition 1.12 as endomorphisms of I:=

C IG.(i) Onemaydenealinearform f : EndG(I ) by f (HomG(IG,IG/ )) = 0 when ,= /, x(1 m) f (x)(1 m)

PgP,=PPgP Mwhen (P, V, M) C, m M and x EndG(IndGPM).12 Part I Representing nite BN-pairsThen(ii) f (ag/,,/ ag,,/ ) ,= 0 only if P/g/P a g1, and (, /) =(/, );(iii) [V/[. f (ag1,/,ag,,/ ) = [V[. f (ag,,/ ag1,/,) =[gV/ V[, whereg,,/ g1,/, = IdMfor

.Proof. (i) It sufces to check that, if (P, V, M) C, x EndG(G PM)and m M, then x(1 m) f (x)(1 m)

PgP,=P PgP Mfor auniquescalar f (x) . Therestrictionto Pof IndGP(M) = G PMisthe direct sumof P-submodules PgP Massociatedtothe doublecosetsPgP. Then there is EndP(M) such thatx(1 m) 1 (m)

PgP,=P PgP Mfor all m M. But EndP(M) = byhypothesis,hence our claim.(ii)(iii) Theelements of EndG(I ) areinthematrixform(x,/ ),/Cwith x,/ HomG(IG, IG/ ). The linear form satises f ((x,/ ),/ ) =

f (x,).One clearly has f (ag/,,/ ag,,/ ) = 0 when (, /) ,= (/, ).Denote =(P, V, M), /=(P/, V/, M/). Let m M. Onehas ag/,/,ag,,/ (1m) =ag/,/,([V[1

uVugg,,/ (m)) =[V[1[V/[1

uV, u/V/ (ugu/g/ g/,/,g,,/ (m)). Using the direct sumdecomposition G PM =

PhPPhP M, the projection on 1 Misnon-zero only if VgV/g/ P ,= . Thus (ii).If inadditiong/ = g1, thenVgV/g1 P = V byProposition1.2(i).Thus ag1,/,ag,,/ (1 m) 1 m/

Px P,=P Px M where m/ =[V/[1[gV/ V[g1,/,g,,/ (m). So, for all m M, f (ag1,,/ ag,/,)(1 m) = [V/[1[gV/ V[(1 g1,/,g,,/ (m)). This gives us thatg1,/,g,,/is a scalar, necessarily invertible, denoted byandthereforef (ag1,,/ ag,/,) =.[V/[1[gV/ V[

. Changing (g, , /) into (g1, /, ) gives the same since, if g1,/,g,,/ = IdM, then g1,/, = (g,,/ )1and g,,/ g1,/, =IdM/ . Thus (iii). Let usrecall thefollowingnotions(see[Ben91a] 1.6, [NaTs89] 2.8,[Th evenaz] 6).Denition1.19. Let be a principal ideal domain and let A be a -free nitelygenerated -algebra.A is said to be a symmetric algebra if and only if there existsf : A a -linear map such that f (ab) =f (ba) for all a, b A, and a . (b .f (ab))induces an isomorphismA Hom

(A, ).One says that A is a Frobenius algebra (see [Ben91a] 1.6) if and only if = Kis a eld, and there exists f : A Ka K-linear map such that, for alla A {0], f (aA) ,= {0] andf (Aa) ,= {0].1Cuspidality in finite groups 13Note that A is symmetric (resp. Frobenius) if and only if the opposite algebraAoppis symmetric (resp. Frobenius). Note also that, when= K is a eld, anysymmetric algebra is Frobenius.Theorem 1.20. Let C be a set of cuspidal triples satisfying Condition 1.17.(i) Let = (P, V, M), / = (P/, V/, M/) C. Take a representative in eachdouble coset PgP/such thatg/. Then the corresponding ag,,/ form a-basis of HomG(IG, IG/ ).(ii) Incase (b) of Condition1.17(whichimplies C = {0]), the endomorphismalgebra EndG(IG0) is a symmetric algebra.(iii) In case (a) of Condition 1.17 (which implies is a eld), the -algebraEndG(

C IG) is a Frobenius algebra.(iv) In case (a) of Condition 1.17 and if L has the additional property thatany relation (P, V)(P/, V/) in Limplies [V[ = [V/[, then EndG(

C IG)is a symmetric algebra.Proof of Theorem 1.20. Consider the ag,,/ above as endomorphisms ofI =

C IG. DenoteE:= EndG(I ). Having chosen representativesg Gforeach pair , / C, denote by Tthe resulting set of triples (g, , /).Lemma 1.21. E = Tas a -module.Proof of Lemma 1.21. By Proposition 1.5(iv), one hasHomG(IG, IG/ )= PgP/GHom(PgP/)_ResPg(P/,V/) (P,V)M, ResgP/(P,V) g(P/,V/)gM/_where the summand is zero unless (P, V)g(P/, V/) by cuspidality of M andM/. By Condition 1.17 on C, the corresponding summand is isomorphic toif g/, zero otherwise. Returning to the proof of Theorem 1.20, takef as in Proposition 1.18.Let K0betheeldof factionsof . Let x =

(g,,/)Tg,,/ ag,,/ E K0 be a linear combination of the ag,,/ s with coefcients in K0. Propo-sition 1.18(ii) and (iii) yieldg,,/ =f (xag1,/,) f (ag,,/ ag1,/,)1(1)(wheref denotes also the extension of f to E K0). This implies at once thattheag,,/ s for (g, , /) T areK0-linearly independent. Then theag,,/ sfor(g, , /) T areaK0-basisof E K0byLemma1.21.But(1)aboveand Proposition 1.18(iii) showthat any x E is a combination of the(ag,,/ )(g,,/)Twith coefcients in . Thus (i) is proved.14 Part I Representing nite BN-pairsThe ag,,/ s for (g, , /) Tand the ag1,/,s for (g, , /) Tare bothbases of E by (i). The formula in (1) also implies that finduces an isomorphismbetween E and Hom(E, ), the basis dual to (ag,,/ )(g,,/)Tbeing( f (ag1,/,ag,,/ )1ag1,/,)(g,,/)T .This gives (iii). When, moreover, C has a single element, Proposition 1.18(ii)(iii) for = / (hence V = V/) gives f (aa/) =f (a/a) for all basis elements,hence for every a, a/ E. This implies our (ii). A similar result holds if in Lthe relation (P, V)(P/, V/) implies [V[ = [V/[, whence (iv) is proved. Remark 1.22. Thelinearformf givesthecoefcientonIdIG= a1,,(seeDenition 1.12) in the basis of Theorem 1.20(i).Proposition 1.23. Let Hbe a subgroup ofGand let Mbe aH-module.ThenthesubalgebraofEndG(IndGHM)consistingof f : IndGHM IndGHMsuch that f (1 M) 1 Mis isomorphic to EndH(M) by the restrictionmapf .f[1M.Let (P/, V/) (P, V) be in L, and let = (P/, V/, N) be a cuspidal triplesatisfying Condition 1.17. Then the injection above sends ag,, EndP(IP)to the element denoted the same in EndG(IG).Proof. Writing IndGHM = G HM =

HgHHG/HHg Mas a H-module, the summand MH for g H is isomorphic to M. Let E be the subalge-bra of EndG(IndGHM) of endomorphisms x suchthat x MH MH. ToshowthatEis isomorphic to EndH(M), it sufces to show that everyy EndH(M)extendstoaunique y E. Theuniquenessisensuredbythefactthat MHgenerates IndGHMas aG-module. The existence is just the functoriality ofIndGH = GH. One takes y = G Hy, dened by y(g m) = g y(m)for m M, g G. This gives our rst claim.For thesecond, let usrecall thedeningrelationforag,,: ag,,(1 P/n) = e(U/)g P/ g(n)foranyn N. It isclearthat ag,,stabilizes M =P P/ Nwheng Pandcoincides withtheelement denotedag,,inEndP(IP). 1.5. Self-injective endomorphism rings and an equivalenceof categoriesIn the following, k is a eld and A is a nite-dimensional k-algebra. One denotesbyAmod (resp. modA) the category of nitely generated left (resp. right)1Cuspidality in finite groups 15A-modules. One has the contravariant functorM .M = Homk(M, k) be-tween them.Notation 1.24. Let Y be a nite-dimensional A-module and let E:= EndA(Y).Let H be the functor from AmodtomodE denedby H(V) = HomA(Y, V),where E acts on H(V) by composition on the right.Let A modYbethefullsubcategoryof AmodwhoseobjectsaretheA-modules Vsuch that there exist l 1 and e EndA(Yl) with V = e(Yl).Theorem1.25. LetYbeanitelygeneratedA-module. Let E:= EndA(Y)andlet H = HomA(Y, ) beas above. Assumethat Eis Frobenius (seeDenition 1.19). Then(i) Hisan equivalence of additivecategories fromAmodYto mod-E.Assume moreover that all simpleA-modules are inAmodY. Then(ii) if M is inAmodYthen it is simple if and only if H(M) is simple. Thisinducesabijectionbetweenthesimpleleft A-modulesandthesimplerightE-modules.(iii) If Y/ is an indecomposable direct summand of Y, then soc(Y/), hd(Y/)are simple, and H(soc(Y/)) = soc(H(Y/)), H(hd(Y/)) = hd(H(Y/)).(iv)IfY/,Y//areindecomposabledirectsummandsofY,thensoc(Y/) =soc(Y//) (and hd(Y/) = hd(Y//)) if and only if Y/ = Y//.Over a Frobenius algebra, projective modules and injective modules coincide(see [Ben91a] 1.6.2(ii)). Considering injective hulls, we get the following.Lemma 1.26. If EisaFrobeniusalgebra,theneverynitelygeneratedE-module embeds into a free module Elfor some integer l.We shall use the following notation.Notation. If M H(V), we denote M.Y:=

mM m(Y) V.Assume that E is Frobenius.Lemma 1.27. Let Vbe inAmod.(i) H(Y) =EE (E considered as right E-module) and, if l is an integer 1,H(HomA(Yl, V)) = HomE((EE)l, H(V)).(ii) If V = e(Yl) for e EndA(Yl), then H(V).Y = V.(iii) Let l 1 and M H(Yl) = (EE)lbe a right E-submodule. Then M.Yis in AmodYand H(M.Y) =M, the latter being induced by the image by Hof the inclusion M.Y Yl.Proof of Lemma 1.27. (i) is straightforward.(ii) Writing V = e(Yl) for e EndA(Yl), H(V) clearly contains e composedwith all the coordinate maps Y Yl, hence H(V).Y = V.16 Part I Representing nite BN-pairs(iii)OnehasM.Y YlandM H(M.Y) (EE)lasright E-modules.The sum M.Y =

mM mYmay be turned into a nite sum sinceMis nitedimensional, so M.Yis a sub- A-module of some nite power of Y. ThereforeM.Yis inAmodY.Let us assume H(M.Y)/M ,= 0. Then there exists a right E-module N suchthat M N H(M.Y) (EE)land N/M is simple. By Lemma 1.26, N/Minjectsintosome(EE)m.Sothereisanon-zeromap f : N EEsuchthatf (M) = 0. By the self-injectivity mentioned above ([Ben91a] 1.6.2), the mod-uleEEisinjective,so f extendsinto f : (EE)lEE.Butthen f isintheform f =H(e) where e HomA(Yl, Y) (Lemma 1.27(i)). The hypothesis onf implies e(M.Y) = 0, e(N.Y) ,= 0. ButM.Y N.Y H(M.Y).Y M.Y,so N.Y =M.Y, a contradiction. Proof of Theorem 1.25. (i) Let M be a right E-module, then M is a submoduleofsome(EE)lbyLemma1.26. ThenLemma1.27(iii)applies, soonegetsM =H(V) for V =M.Y, which is inAmodY.It remains to check thatHis faithful and full. Let V, V/ beA-modules inAmodY; one must check that Hinduces an isomorphism of vector spacesbetween HomA(V, V/) and HomE(H(V), H(V/)).Obviously H is linear. If f HomA(V, V/) is in its kernel, thenf (H(V).Y) = 0 by denition ofH, butH(V).Y = Vby Lemma 1.27(ii), sof (V) = 0 andf = 0.In order to check surjectivity, one may assume that V = e(Yl), V/ = e/(Yl)for e, e/ EndA(Yl). Then H(V) and H(V/) are submodules of (EE)l. Let g HomE(H(V), H(V/)). By injectivity of (EE)l, g extends to g HomE((EE)l,(EE)l) which is H(HomA(Yl, Yl)) by Lemma 1.27(i). Theng =H(f ) for f EndA(Yl). Wehave f (V) V/since f (V) = f (H(V).Y) = ( g.H(V)).Y =(g.H(V)).Y H(V/).Y = V/. Therefore g =H( f ), wheref : V V/ is therestriction of f .This completes the proof of (i).Assume now that all simpleA-modules are inAmodY.(ii) Take VinAmodYand assume that H(V) is simple. One may assumethat there is some l such that V Yland V =H(V).Y by Lemma 1.27(ii). LetXbe a simple submodule ofV. ThenXoccurs in hd(Y), soH(X) ,= 0. ButH(X) H(V)soH(X) =H(V)andthereforeX =H(X).Y =H(V).Y =V, so Vis simple.Conversely, assume that Vis a simpleA-module. By the hypothesis on Y,Vis a submodule of Y. Let S be a simple submodule of H(V), then 0 ,= S.Y H(V).Y = V by Lemma 1.27(ii). Then S.Y = V and S =H(S.Y) =H(V) byLemma 1.27(iii).1Cuspidality in finite groups 17The equivalence of (i) then implies (ii).(iii)LetY/beanindecomposabledirectsummandofY,thenH(Y/)isa(projective) indecomposable direct summand ofEE, so its head and socle aresimple (see [Ben91a] 1.6).Nowit isclearby(ii)abovethat H(soc(Y/))isanon-zerosemi-simplesubmodule of soc(H(Y/)), whence the rst claimed equality.By what we have just checked, soc(Y/) is simple. We may now apply this toY since EndA(Y) = EndA(Y)opp. We obtain that the indecomposable directsummands of Yhave simple heads. To check the second equality of (iii), notethat wehaveanon-zeroelement inHomA(Y/, hd(Y/))whilebothmodulesare inAmodY, so by the equivalence of (i), there is a non-zero element inHomE(H(Y/), H(hd(Y/))). The rst module has simple head while the secondis simple by (ii) and what we have just said. So we have hd(H(Y/)) =H(hd(Y/))as claimed.(iv) This follows from (iii) and the fact that this is true for indecomposabledirect summands of EE. 1.6. Structure of induced cuspidal modules and seriesWe take again a nite groupG, ka eld such that kG/J(kG) is split (i.e. aproduct of matrix algebras over k), and L a k-regular -stable set of subquo-tients of G. This allows us to consider the set cuspk(L)) of cuspidal triples (seeNotation 1.10).Theorem 1.28. For each cuspidal triple (P, V, M) where (P, V) L and Mis a simple cuspidal k P/V-module, the induced module IndGPM can be writtenas a direct sum

i Yiwhere(a) each Yiis indecomposable,(b) soc(Yi) = soc(Yj) if and only if Yi = Yj,(b/) hd(Yi) = hd(Yj) if and only if Yi = Yj.(c) If moreover L has the property that any relation (P, V)(P/, V/) im-plies [V[ = [V/[, then soc(Yi) = hd(Yi) for all i .Proof. Let Y =

IndGPS where ranges over cuspk(L)). Theorem 1.20 tellsus that EndkG(Y) is Frobenius. TheH(Yi) are the indecomposable projectiveE-modules. Any simple kG-module occurs in both hd(Y) and soc(Y) by Propo-sition 1.11. We may now apply Theorem 1.25 to the module Y.When the condition of (c) is satised, Theorem 1.20(iv) tells us that Eissymmetric. Thisimpliesthat hd(H(Yi)) = soc(H(Yi))(see[Ben91a]1.6.3),whence (b) by Theorem 1.25(iv). 18 Part I Representing nite BN-pairsNotation 1.29. When , / cuspk(L), we write G/ if and only if thereexists g G such that g/.When cuspk(L), denote by E(kG, ) the set of simple components ofhd(IG).One has E(kG, ) = E(kG,g) for all g G.Theorem 1.30. Assume that L has the property that any relation (P, V)(P/, V/) in L implies [V[ = [V/[.(i)

cuspk(L)E(kG, ) gives all simple kG-modules.(ii) If E(kG, ) E(kG, /) ,= , then G/.(iii) Assume L satises the hypotheses of Theorem 1.14. Then Gis anequivalencerelationoncuspk(L), andtheunionin(i )isapartitionofthesimple kG-modules indexed by the quotient cuspk(L)/G.(iv) If S/is asimplecompositionfactor of some IG( = (P, V, N) cuspk(L)), then S/ E(kG, /) where /=(P/, V/, N/) and(P/, V/) (P, V) = (P, V) (andtherefore [ P//V/[ [ P/V[). If moreover [ P//V/[ =[ P/V[, then G/.Proof. (i) is clear from Proposition 1.11.(ii) Since the head and socle of each IGyield the same simple kG-modulesthanks to Theorem 1.28(c) above, E(kG, ) E(kG, /) ,= implies that thereis a non-zero morphism IGIG/ . Then G/ by Theorem 1.20(i).(iii) When the hypotheses of Theorem 1.14 are satised, IG=IG/when-ever G/. Sincetheconverseis true(seeNotation1.10), thereis anequivalence. Therefore Gisanequivalencerelation. ThenwealsohaveE(kG, ) = E(kG, /) as long as G/, so the union in (i) is a partition.(iv) This wont be used. We leave it as an exercise (hint: consider a projectivecover of S/). Exercises1. Findacounterexampleinacommutativegroupshowingthat isnottransitive. Find one with minimal cardinality of G.2. Let a, b be subquotients of a nite group.(a) Showthat a(a b) = a b and a(ba) =(a b) a = (b a) a = b a.Moregenerally,whenb/ b, relateb/ (b a), b (b/ a), (b a) b/,b (a b/), and (a b) b/ with b/ a and a b/.1Cuspidality in finite groups 19(b) Showthat inducesastructureofan(associative)monoidonLa,b = {a, b, a b, b b].(c) Let Mbethemonoidgeneratedbytwogeneratorsx, y, subjecttotherelationsx2= x, y2= y, xyx = yx andyxy = xy. ShowthatZ[M] = Z Z U where U is the ring of upper triangular matrices inMat2(Z).3. Let a, a/, b, b/ be subquotients of a nite group. If a b = b, a/ a, andb b/, show that a/ b/ = b/.4. Let Lbe a set of subquotients of a nite group G. Showthat a b = b afor all a, b L, if and only if there is a subgroup H G such that, for all(P, V) L, V H P.5. Show that there are groups Gwith subgroups U, Vsuch that UVUis asubgroup but e(UVU) ,= e(U)e(V)e(U).6. Prove a Mackey formula implying Proposition 1.5(iv),ResG(P/,V/)IndGPN=

P/gPG IndP/g(P,V) (P/,V/)igResgP(P/,V/) g(P,V)gN,where igis the functor making aP/ gV-trivial P/ gP-module into agV-trivial (P/ gP)gV-module.You may use the following steps in relation to P/-P-bimodules.(a) e(V/)P/Pe(V) = P/e(V/.(V P/)) PP/e((V/ P).V)P.(b) If g G, then e(V/)P/gPe(V) = P/e(V/.(gV P/)) gPP/e((V/ gP).gV)gP gPgP.(c) Decompose e(V/)G GGe(V) = e(V/)Ge(V).7. Show Theorem 1.20(i) more directly, without using the linear formf orthe rank argument.8. If = (P, V, S) cuspk(L), show that NG(V) NG(P).9. Let L be a -stable set of subquotients (P, V) (V P G).(a) Assume aa/ in L. Showthat x . x a/ and x/ . x/a induceinverse isomorphisms between the intervals {x L [ x a] and {x/ L [ x/ a/] in L. Show that xx a (resp. x/x/ a/) for allx a (resp. x/ a/).(b) If a = (P, V)a/ = (P/, V/), dene a set LPP/ of subquotients ofP P/ in bijection with the above intervals. Show that cuspk(LPP/ )injects into cuspk(L) in two ways. Apply this to the proof of Theorem1.14.(c) See which simplication of that proof can be obtaind by assuming theexistence of a subgroup L suchthat P = LV, P/ = LV/ are semi-directproducts.20 Part I Representing nite BN-pairs10. Assume Lisa -stable-regularsetofsubquotientsofG. Assumethat, for every relation (P, V)(P/, V/) in L with [ P[ = [ P/[, the mapx . xe(V/) is an isomorphism from Ge(V) to Ge(V/). Show that, forall (P, V), (Q, W) in L, one hasGe(V)e(W) = Ge(V Q)e(W) =Ge(W)e(V Q).11. Show a converse of Theorem 1.14.12. Let A be a nite-dimensional k-algebra. Let i, j A be two idempotentssuch that i j = 1. Show that A is symmetric if and only if the followingconditions are satised:(a) i Ai and j Aj aresymmetricforforms fi, fjsuchthat fi(i aj bi ) =fj( j bi aj ) for all a, b A,(b) for all 0 ,= x i Aj , x Ai ,= 0 and for all 0 ,= y j Ai , y Aj ,= 0.Application (see Theorem 1.20): if Y = Y1 . . . Ynis a sum ofA-modules,showthatEndA(Y)isasymmetricalgebraifandonlyifeachalgebraEndA(Yi) is symmetricfor aformfisuchthat, for all xi, j HomA(Yj, Yi) and yj,i HomA(Yi, Yj), one has fi(xi, jyj,i) =fj(yj,ixi, j)and, if xi, j ,= 0, there is a yj,isuch that xi, jyj,i ,= 0.13. CheckTheorem1.25assumingthat bothEandEoppareself-injectiveinstead of Frobenius. Recall that a ring is said to be self-injective when the(projective) regular module is also injective.14. Prove Theorem 1.25 assuming that only Eoppis self-injective.Hint: only the proofs of (iii) and (iv) need some adaptation. Let Y/ bean indecomposable direct summand of Y. Show that hd(Y/) is simple usingthe following steps. Show that EndA(hd(Y/)) = EndE(H(Y/), H(hd(Y/))).Then use a decomposition of hd(Y/) as a sum of simple A-modules and itsimage by H.15. Let Y be an A-module such that EndA(Y) is Frobenius and the semi-simpleA-modules soc(Y) and hd(Y) have the same simple components (possiblywith different multiplicities). Prove a version of Theorem 1.25 where thesimpleA-modules are replaced by the ones occurring in soc(Y).NotesModular versions of Harish-Chandra induction (see, for instance, [DiMi91] 6for the characteristic zero version) were used by Dipper [Dip85a], [Dip85b].Thegeneral denitionforBN-pairsisduetoHiss[Hi93]andwasquicklyfollowed by DipperDu [DipDu93] who partly axiomatized it and gave a proofof the independence with regard to (P, U) of the Harish-Chandra induction (our1Cuspidality in finite groups 21Theorem1.14). 1.5comes from[Gre78] and[Ca90] (see alsoChapter 6below).The application to generalized Harish-Chandra theory is due to Linckelmannand GeckHiss; see [GeHi97] and [Geck01]. Exercise 14 is due to Linckelmann(see [Geck01] 2.10).For a more general approach to Harish-Chandra induction and restriction,see [Bouc96]. For more general category equivalences induced by the functorHomA(Y, ), see [Ara98] and [Aus74].2Finite BN-pairsThe aimof this chapter is to give a description of a -stable set of subquotients(see the introduction to Chapter 1) present in many nite simple groups. The ax-iomatic setup of BN-pairs (see [Asch86] 43, [Bour68] IV, [CuRe87] 65) hasbeen devised to cover the so-called Chevalley groups and check their simplicity.Such a Ghas subgroupsB, Nsuch that B NNand the quotient groupN,B Nis generated by a subset S such that the unionsPs:=B Bs B aresubgroups of G for any s S. More generally, the subgroups of G containingB (standard parabolic subgroups) are in bijection with subsets of S:I S .PI.Under certain additional hypotheses, dening a notion of a split BN-pair ofcharacteristic p ( p a prime), each PIhas a semi-direct product decomposition,called a Levi decomposition,PI = UILI.where UIis the biggest normal p-subgroup of PIand LIis a group with a splitBN-pair given by the subgroups B LI, N LIand the set I .Among other classical properties, we show that the set L of G-conjugatesof subquotients (PI. UI) (Iranging over the subsets of S) is -stable.The approach we follow uses systematically the reection representation ofthe group W =N,B N and the associated nite set of so-called roots. The set+ of roots has a very rich structure which gives us a lot of information aboutWand the structure ofGitself (root subgroups). While the axiomatic studyof BN-pairs involves many (elementary) computations on double cosets BnB,once the notion of root subgroups B is introduced, the description of subgroupsof B of the form B Bn1 Bn2 . . . for n1, n2, . . . W and of double cosetsBnB is much easier.We develop several examples.222Finite BN-pairs 23Thetheoryofcuspidal simplemodulesandtheirinducedmodulesthenapplies to nite groups with a split BN-pair. Lusztig has classied the cuspidaltriples(PI. UI. M) (seeDenition1.8) over eldsof characteristic0(see[Lu84]). Chapters19and20givetherst stepstowardsaclassicationofthe cuspidal triples over elds of non-zero characteristic (see Theorem 19.20for groups GLn(Fq)).2.1. Coxeter groups and root systemsInthepresent section, (W. S) isaCoxeter systemintheusual sense(see[Asch86] 29, [Bour68] IV, [CuRe87] 64.B, [Hum90] 5). We consider it asacting on a real vector space RLwith a basis Lin bijection with S ( . s) anda symmetric form such that . /) = cos(,[-ss/ >[). Then Wacts faith-fully on RLby a morphism which sends s to the orthogonal reection through.Dening+ := {n() [ n W. L] (the root system of W), each el-ementof+isalinearcombinationofelementsofL(simpleroots)withcoefcientseitherall 0orall 0.Thisgivesthecorrespondingpartition+ = + + (see [CuRe87] 64.18 and its proof, [Hum90] 5.3).We use subsets of Las subsets of S and denote accordingly WI the subgroupof Wgenerated by the elements of S corresponding with elements of I , +I =+ RI , for I L.We use diagrams to represent the setL of simple roots. These are graphs,where, in the examples given below, a simple (resp. double) link between twoelements means an angle of 2,3 (resp. 3,4). This also means that the productof the two corresponding reections is of order 3 (resp. 4). There is no link whenthe angle is ,2 (commuting reections).In this notion of root system, each root could be replaced by the half-lineit denes. This notion is well adapted to the classication and study of Coxetergroups.In the classical notion of (nite, crystallographic) root systems (see [Bour68]or A2.4 below), roots are indeed elements of a Z-lattice X(T) and root lengthsmay be ,= 1 (this is the notion that will be used from Chapter 7 on to describealgebraic reductive groups).Example 2.1. (i) Coxeter group of type An1 (see [Bour68] Planche I).It is easy to see that the symmetric group on n letters Sn is a Coxeter group forthe subset of generators {si:= (i. i 1) [ i = 1. . . . . n 1]. Let E = Re1 . . . Renbe the n-dimensional euclidean space where the eiare orthogonal24 Part I Representing nite BN-pairsof norm2,2. The reection representation of Sn is given by the hyperplaneorthogonal to e1 en, where Snacts onEby permutation of the eis,withLA = {i:= ei 1 ei [ i = 1. . . . . n 1], represented by the followingdiagram12 n1 (An1)+ = {ei ej [ i ,=j.1 i. j n 1] and + = {ei ej [ 1 j - i n 1].(ii) Coxeter group of type BCn(see [Bour68] Planche II, III). In Eabove,take the basis LBC = {0] LA with 0 =2e1.0=====1 n1 (BCn)The corresponding reections generate the matrix group W(BCn) of permu-tationmatriceswith 1insteadofjust1s.Denotingbys/ithereectionofvector ei, every element in the Coxeter group of type BC can be written in aunique way ass/i1 . . . s/iknfor n Sn and 1 i1- . . . - ik n.(iii) Coxeter group of type Dn(see [Bour68] Planche IV). This time, takeLD = {/0] LA with /0 = e1 e2.///,/0123 n1 (Dn)ThecorrespondinggroupW(Dn)isthesubgroupofW(BCn)ofmatriceswith an even number of 1s. This corresponds to the condition that k is evenin the decomposition above.Denition2.2. Whenn W, denote +n = + n1(+). If I L, letDI = {n W [ n(I ) +]. If I. J L, let DI J =DJ (DI)1.Proposition 2.3. (i) If L, then +n if and only if l(ns) = l(n) 1.(ii) +n is nite of cardinality l(n).2Finite BN-pairs 25(iii) If :, n W, then l(:n) = l(:) l(n) +n +:n+ :1(+) n(+) +:n = +n n1(+:) +: +n1 = .Proof. (i) and (ii) are standard ([Hum90] 5.6, [Stein68a] (22) p. 270). (iii) isleft as an exercise. More generally, if+/n denotes the set of lines correspond-ing to elements of+n, one has+/:n = +/nn1(+/:) (boolean sum) for any:. n W. Proposition2.4. If I. J L, theneverydoublecoset inWIW,WJcon-tains an element of minimal length, which is inDI J. This induces a bijectionWIW,WJ DI J. (The letter D is for distinguished representatives.)Proof. Ifn is of minimal length in WInWJ, it is of minimal length innWJandWIn, whencen() +if Jandn1() +if I , thanks toProposition 2.3(i). Example 2.5. Let Sn1 Sn be the inclusion corresponding to permutationsof n letters xing the last one. In the reection representation of Example 2.1above, this corresponds to the subsetL/ = L {en en1]. When i -j , letsi. jbe the cycle of order j i 1 equal to (i. . . . . j ). When i >j , let si. j =(sj.i)1. Checking images of the elements of L/, it is easy to see that si.n D.L/for each i = 1. . . . . n. Then sn.i DL/.. Moreover, if n Sn, it is clear thatsn.n(n)n and nsn1(n).n x n, hence are in Sn1. By Proposition 2.4 above, thisimplies that{si.n [ 1 i n] =D.L/ and {sn.i [ 1 i n] =DL/..For any n Sn, one gets n sn(n).nSn1 and n Sn1sn.n1(n).Theorem 2.6. If I. J L and n DI J, let K =I n(J) L. Then WI nWJ = WK, +I n(+J) = +K, and +I n(+J ) = +K.Lemma 2.7. Given the same hypotheses as for Theorem2.6, we havenJ +II .Proof of Lemma 2.7. Let Jbe such that n() +I . Let us write n() =

/I/ /with/ 0.Then = /I/ n1(/) Lwitheachn1(/)a positive root since n1DI. One of the / is non-zero, say 0> 0. Thenn1(0) must be proportional to , hence equal to it. That is, 0 = n() I . Proof of Theorem2.6. Theinclusions WK WI nWJand+K +I n(+J) are clear.Conversely, let x WI nWJ and let us check that x WK. We use induc-tiononthe lengthof x. If x = 1this is clear. Otherwise let y = n1xn WJ 126 Part I Representing nite BN-pairsand J be such that y() +J , or equivalently y = y/s with l(y/) = l(y) 1. Sincen DJ, onehasxn() = ny() +. Onehasn() +sincen DJ. Thenn() +x +I . Now Lemma 2.7 impliesn() I . Denot-ing / = n(), one has / I n(J). Moreover, x(/) +, so x = x/s/withl(x/) = l(x) 1. One may then apply the induction hypothesis to x/ = ny/n1.Now letbe an element of+I n(+J). Let t be the element of Wcor-responding to the reection associated with in the geometric representation.Now t WI nWJ. Then t WKby what we have just proved, and therefore+t +K. But t () = , so +Kas claimed. Thus+I n(+J) = +K.Then, makingtheunionwithitsopposite, weget +I n(+J) = +K. Theequality +I n(+J ) = +Kalso follows since n(+J ) +. WeassumeWisnite. Thentheform . ) onRLispositivedef-inite(see[CuRe87] 64.28(ii), [Bour68] V.4.8, [Hum90] 6.4). Moreover,Whas a unique element of maximal length, characterized by several equiv-alent conditions, among which is the fact that it sends L to L (see [Hum90]1.8).Notation 2.8. IfI L, one denotes by nIthe element of maximal length inWI. If L I , let :(. I ) = nI {]nI.Example 2.9. (i) In Sn (see Example 2.1(i)), the element of maximal length isdened by n0(i ) = n 1 i (it is easily checked that this element makes neg-ative all i = ei 1ei LA). It is easily checked that :(n1. LA {n1]) =sn.1 (cycle of order n, see Example 2.5).(ii) For the Coxeter group of type BCn, n0 is IdEin the geometric repre-sentation (see Example 2.1(ii)). One has :(n1. LBC {n1]) = s/n.(iii) In the geometric representation of Dn, one gets n0 = s/1 . . . s/n = IdE ifn is even, n0 = s/2. . . s/n = s/1if n is odd. One has :(n1. LD {n1]) = s/1s/n.Proposition 2.10. Let I L, L I .(i) We have +nI = +I , +:(.I ) = +I {] +I , and :(. I )(I ) I {].(ii) Let n W satisfy n(I ) L. Then l(n.:(. I )1) = l(n) l(:(. I )) ifand only if n() +. Otherwise, l(n.:(. I )1) = l(n) l(:(. I )).Proof. (i) We have nI(I ) =I , so nI(+I ) =+Iand :(. I )(I ) =nI {](I ) nI {](I {]) =I {]. But l(nI) = +I , hence +nI =+I . This also gives +:(.I ) = +{]I +Isince l(n{]I) = l(nI) l(:(. I )).(ii) Let nownbesuchthat n(I ) Land n() +. Toshowthatl(n.:(. I )1) = l(n) l(:(. I )), as a result of Proposition2.3(iii), it isenough to show that +n +:(.I ). Let +:(.I ), i.e. +{]I +Ithanksto (i) above. Let us write =

/I/ /with> 0 and/ 0 for2Finite BN-pairs 27/ I . Thenn() = n()

/I/ n(/). Ifwehadn() +, sincen() +andn(I ) L, the non-zero coefcients inn() would be for el-ementsof n(I ). Son() +n(I ), orequivalently +I. But L I , acontradiction.It remains to show that, if n(I ) L and n() +, then l(n.:(. I )1) =l(n) l(:(. I )). We apply the implication we have just proved with n.:(. I )instead ofn, so we just have to check thatn.:(. I )() +. We have seenthat +:(.I ), so:(. I )() +I. Then its image bynis in+sincensends both Iand into + by hypothesis. Theorem2.11. Assume as above that W is nite. Let n W andL1 Lbesuchthat n(L1) L. Thenthereexist L1. . . . . LksubsetsofL, and a sequence1 L L1, . . . , k L Lk, such that, for all 1 j k 1, :(j. Lj)(Lj) = Lj 1and n = :(k. Lk) . . . :(1. L1) withl(n) =l(:(k. Lk)) l(:(1. L1)).If moreover L L1 has a single element 1, then n = 1 or :(1. L1).Proof. The rst point is by induction on the length ofn. Everything is clearwhenn = 1. Otherwise, there is Lsuchthat n(1) +. Then1 , L1andProposition 2.10(ii) allows us to writen = n1:(1. L1) with lengths adding.LettingL2 = :(1. L1)(L1), one may clearly apply the induction hypothesisto L2 and n1.We also prove the second point by induction on the length. Ifn = 1, wearedone. Otherwise, onehasn = n1:(1. L1)withlengthsaddingandn1satisfying the same conditions as n for L2 = :(1. L1)(L1) = n0(L1). Theinduction implies n/ = 1 or :(n0(1). L2). But the latter is n0:(1. L1)n0 =:(1. L1)1. We get our claim. 2.2. BN-pairsWe now dene BN-pairs.Denition2.12. ABN-pair(orTitssystem)consistsofthedataofagroupG, two subgroupsB, Nand a subset Sof the quotient N,B Nsuch that,denoting T:=B Nand W:=N,T:(TS1) T N(Wis therefore a quotient group), Wis generated by S ands S, s2= 1.(TS2) s S, n W, s Bn BnB BsnB.(TS3) B Ngenerates G.(TS4) s S, s Bs ,=B.28 Part I Representing nite BN-pairsRemark 2.13. The notation Bn is unambiguous since n is a class mod. TandT B. Similarly, ifXis a subgroup ofBnormalized by T, the notationXnmakes sense (and is widely used in what follows).The hypothesis (TS4) implies that all the elements of S are of order 2.(TS2) implies(TS2/) s S, n W, nBs BnB Bns B.For the next two theorems, we refer to [Bour68] IV, [Cart85] 2,[CuRe87] 65.Theorem2.14. (Bruhat decomposition) If G. B. N. S is a Tits system, the sub-sets (BnB)nWare distinct and form a partition of G.Denition 2.15. If G. B. N. S is a Tits system, and I S is a subset, let WI =-I >, andNIbe the subgroup ofNcontaining Tsuch thatNI,T = WI. LetPI:=BNI B = nWIBnB.Theorem 2.16. Let (G. B. N. S) be a group with a BN-pair.(i) Wis a Coxeter group with regard to S.(ii) The PIdened above are subgroups of G (parabolic subgroups) andN PI =NI. If Pisasubgroupof GcontainingB, wehaveP =PJforJ:= {s S[s P].(iii) If PIis a parabolic subgroup, (PI. B. NI. I ) is a BN-pair.(iv) If I ,Jare subsets of S, then PIG,PJ = WIW,WJ.Example 2.17. (see [Cart72b] 11.3, 14.5, [DiMi91] 15) Let F be a eld, letn 1 be an integer.(i) Let GLn(F) be the group of invertible elements in the ring Matn(F) ofn n matrices with coefcients in F. Let U (resp. T, resp. W) be the subgroupof upper triangular unipotent (resp. invertible diagonal, resp. permutation) ma-trices.Let B = UT(uppertriangularmatricesinGLn(F)), N = T WandSbe the set of elements of Wcorresponding to the transpositions (i. i 1) fori = 1. . . . . n 1.ThenB N = Tand(B. N. S)makesaBN-pairforGLn(F)(seeExer-cise 1). The associated Coxeter system (W. S) corresponds to Example 2.1(i),i.e. type An1.Aslight adaptation of the above allows us to showa similar result for SLn(F),the group of matrices of determinant 1.(ii) Assume now that F has an automorphism . of order 2. This ex-tends as g . g for g GLn(F). Letn0 GLn(F) be the permutation matrixcorresponding to i . n 1 i for i = 1. . . . . n. Denote by: GLn(F) GLn(F)the group automorphism dened by (g) = n0.tg1.n0.2Finite BN-pairs 29Let GUn(F) be the groupof xedpoints inGLn(F), i.e. g GLn(F) satisfyingg.n0.tg = n0. Let B, T, Wbe the subgroups of GUn(F) consisting of xedpointsunderinthecorrespondingsubgroupsofGLn(F).Letm := [n2]bethebiggestinteger n2.ThenitiseasilycheckedthatWisisomorphicto(Z,2Z)m>Sm and that it is generated by the set S0 of permutation matricescorresponding to the following elements of Sn: (i. i 1)(n 1 i. n i ) for2i - n, plus an element equal to (m. m 1) when n = 2m is even, and equalto (m. m 2) when n = 2m 1 is odd.This makes a Coxeter system of type BCm.From the fact that B, N, S of (i) above make a BN-pair, one may prove thatB, N, and S0 are a BN-pair for GUn(F) (see Exercise 2).When F is nite and . is non-trivial, [F[ is a square q2and = qfor all F. Then, it is also easily checked that the above group GUn(F) isisomorphic to the group of matrices satisfying a.ta = Idn (a more classical def-inition of unitary groups). For that, it sufces to nd an element g0 GLn(F)suchthat g0.tg0 = n0. Thisreducestodimension2whereonetakesg0 =_1 ( )1( )1_for ,= inFsatisfyingq1= q1= 1.Then a . ag0is the isomorphism sought.(iii) Let F be a eld of characteristic ,= 2, m 2 an integer. Recalln0 SL2m(F), the permutation matrix associated with the permutation i . 2m i 1.LetSO2m(F)denotethesubgroupofSL2m(F)consistingofmatricessat-isfyingtg.n0.g = n0. Thisisthespecial orthogonal groupassociatedwiththe symmetric bilinear form on F2mof maximal Witt index (hence the inSO). Let B/(resp.T/) be its subgroups of upper triangular, resp. diagonal,matrices. Let S/ be the set of permutation matrices corresponding to the follow-ing elements of S2m : (i. i 1)(2m i. 2m i 1) for i = 1. . . . . m 1, and(m 1. m 1)(m. m 2). Clearly S/ SO2m(F) and it generates the central-izer W/ of n0 in the group of permutation matrices in SL2m(F). Along with S/,this makes a Coxeter system of type Dm (note that the embedding of W/ in Wabove corresponds with the embedding of type Dmin type BCmsuggested inExample 2.1).Using a method similar to (ii) above (see Exercise 2), one may check thatB/, T/. W/, and S/ make a BN-pair of type Dm for SO2m(F).2.3. Root subgroupsWe keep the same notation as in 2.2. Gis a group with a BN-pair and niteW. We show how to associate certain subgroups of G with the roots of W.Assume B Bn0= T.30 Part I Representing nite BN-pairsDenition 2.18. If n W, L, let Bn =B Bn0n, B =Bs.Theorem2.19. Let GbeagroupwithaBN-pair andnite W. AssumeB Bn0= T. Let n W, s S, L.(i) If l(ns) = l(n) 1, then B BnsB Bs.(ii) B =Bs(B Bs) = (B Bs)Bs.(iii)nB depends only on n(). We writenB =Bn().(iv) There is a sequence1, . . . ,Ngiving all the elements of+with norepetition, such that B =B1 . . . BN. The corresponding decomposition of theelements of B is unique up to elements of T.(v) If . / L are such that n0() = /, then P P/n0=B BsB.Proof. (i), (ii), (iii), (iv) are classic (see [Cart85] 2.5, [CuRe87] 69.2). Theycan be deduced in a fairly elementary way from the axioms of the BN-pair (see[Asch86] Exercise 10, p. 227).(v) Let us show rst(v/) B (P/ )n0=B.Using (ii), one has P/ =B Bs/ B =B B/ s/ B. Therefore P/ =s/ P/ = s/ B s/ B/ s/ B = s/ B B/ B and (P/ )n0= sBn0 BBn0bythe denition of / from and (iii). Now B sBn0= (Bn0 sn0B)n0 = by Theorem 2.14 (Bruhat decomposition). So B (P/ )n0=B BBn0=Bsince B Bn0= T. This is (v/).Let us writeP =B BsB, again by (ii). We have s, B (P/ )n0, so(v/) implies that P (P/ )n0=B BsB as claimed. Denition 2.20. The BN-pair (G. B. N. S) is said to be split of characteristicpif and only if Gis nite, B Bn0= T, and there is a semi-direct productdecompositionB = UTwhere UBis ap-group and Tis a commutativegroupof orderprimeto p. TheBN-pairissaidtobestronglysplitwhenmoreover, for all I S, UI:= U UnIis normal in U.When +, let X be the set ofp-elements of B (see Denition 2.18).Notethat B =X.T (semi-direct product). Inthefollowing, (G. B =UT. N. S) is a split BN-pair of characteristicp.Theorem2.21. (i) UisaSylowp-subgroupof GandGhasnonormalp-subgroup ,= {1].(ii)nX depends only on n(), so we can writenX =Xn(). It is not equalto {1].(iii) There is a sequence 1, . . . , Ngiving all the elements of + with norepetition, such that U =X1 . . . XNwith uniqueness of the decompositions(i.e. [U[ = H+[X[).2Finite BN-pairs 31Proof. (i) SinceUisaSylowp-subgroupof B, it sufcestocheckthatNG(U) =B to have that U is a Sylow p-subgroup of G. We have NG(U) B,so NG(U) =PI for some I S. Then, if s I , we have Us= U and thereforeBs=B. This contradicts (TS4).There is no normalp-subgroup ,= {1] in G since such a subgroup would bein U Un0B Bn0= T, a group of order prime top.(ii) isaconsequenceof Theorem2.19(iii). Thegroup Xisnon-trivialsinceX = {1]wouldimplyB = T, andthereforeB =B BsbyTheo-rem 2.19(ii), contradicting (TS4).(iii) is a consequence of Theorem 2.19(iv). Lemma2.22. Let 1, . . ., mbemdistinctpositiverootsand, foreveryi ,let xi Xi {1]. Letn W, then x1x2. . . xm Unif and only ifn(i) is apositive root for every i .Proof. The if is clear sincen(X) =Xn() U when n() is positive (The-orem 2.21(iii)).We prove the converse by induction on the length ofn. Ifn = 1, this isclear. Ifn = swith L, one must check only thatis none of theis.Suppose on the contrary that = i0. Then in the product x1x2. . . xm, all thetermsontheleftofxi0arein Us(bytheif above),andthesameistruefor the ones on the right. So xi0 U Us, while xi0 X = U Un0s. ButBs Bn0s= Ts= T, so xi0 = 1, contradicting the hypothesis.Foranarbitrarynoflength 1, writen = n/swithl(n) = l(n/) 1.We have U Un U Usas a result of Theorem 2.19(i) so, by the case justtreated, s(i) is positive for all i . Nowdening /i = s(i), x/i = (xi)gfor g Na representative of s, we have x/1 . . . x/m Un/andthe inductionhypothesis givesour claim. Theorem 2.23. Let (G. B = UT. N. S) be a split BN-pair of characteristic p(see Denition 2.20).If A is a subset of Wcontaining 1, denote +A = _aA a1(+) and UA =_aA Ua.(i) Let 1. 2. . . . . Nbe alist of the positive roots suchthat U =X1 . . . XN(seeTheorem2.21(iii)). Let Abeasubset of Wcontaining1.Then, denoting +A = {i1. i2. . . . . im] with 1 i1- . . . - im N, one hasUA =Xi1. . . Xim = -X ; +A>and+A = { + [X UA].32 Part I Representing nite BN-pairs(ii)If A, A/, A//aresubsetsof Wsuchthat1 A A/ A//and+A +A/ +A// , then UA = (UA UA/ ).(UA UA// ) UA/ .UA// .Proof. (i) Repeated application of Lemma 2.22.(ii) One has clearly, for arbitrary subsets ofWcontaining 1, UA UA/ =UAA/ and+A +A/ = +AA/ . By(i), onehas [UA[ = H+A[X[. There-fore(UA UA/ ).(UA UA// )hascardinality [UAA/ [.[UAA// [.[UAA/A// [1=H+AA/ +AA//[X[. We have +AA/ +AA// = +A (+A/ +A// ) = +Aby hypothesis. So [(UA UA/ ).(UA UA// )[ = [UA[. This implies (UA UA/ ).(UA UA// ) = UAsincetheinclusion(UA UA/ ).(UA UA// ) UAisclear. This gives (ii). Remark. The sets +A of Theorem 2.23 coincide with the intersections of +with convex cones (see Exercise 3).WhentheBN-pairisstronglysplit,i.e.U UnI Uforall I ,therootsubgroups X satisfy a commutator formula (see Exercise 5).2.4. Levi decompositionsWe now assume thatGhas a strongly split BN-pair of characteristicp(seeDenition 2.20).Denition 2.24. Let I L. Let NI be the inverse image of WI in N, and recallthat UI = U UnI.Let LI = -BnI. NI>;thisiscalledtheLevisubgroupassociated with I .Denote L = {(PI. UI)g[I L . g G].Proposition 2.25. LIhas a strongly split BN-pair of characteristicpgivenby (BnI. NI. I ). One has a semi-direct product decomposition PI = UI>LI,and UIis the largest normalp-subgroup of PI. So L above is a set of subquo-tients.Proof. Let uscheckrst that LIhasasplit BN-pair. Theaxioms(TS1)and(TS3)areclear.(TS4)andBnI (BnI)nI= TbothfollowusingTheo-rem 2.23(i). One has BnI = (U BnI)T, a semi-direct product.It remains to check (TS2). Let s S correspond with I and let n WI.UsingTheorem2.23(ii), onehasU = UXandtherefore BnI = T(BnI U) = T(BnI U)X. One has s(+I ) = +I , so Theorem 2.23(i) im-plies that snormalizes BnI U. If n1() is positive, it is anelementof+I= +nI, so sBnIn T(BnI U)snBnI BnIsnBnI. Ifn1() isnegative, onemayapplytheprecedingcasetosn, soit sufcestoshow2Finite BN-pairs 33sBnIs BnI BnIsBnI. Now using the same decomposition of BnIas be-fore, one has sBnIs sBsBnI. Theorem 2.19(v) told us that B BsBis a group, so sBs B BsB. Thus we have our claim.Wemust showthat theBN-pair of LIisstronglysplit. Wehaveseenthat BnI =XIT, asemi-direct product whereXI = U BnI = U Ux0nI.So, given J I , wemust check XI (XI)nJXI. Wehave XI = U Un0nIandXI (XI)nJ= U Un0nI UnJ Un0nInJ.KnowingthatU UnJ UbythestronglysplitconditionsatisedinG,itsufcestocheckthat U Un0nI UnJ Un0nInJ= U Un0nI UnJ. By Theorem 2.23(i),this may be checked at the level of the corresponding subsets of +.Thisfollowsfrom+ nJ(+) = + +J, + nIn0(+) = +Iandthe fact that +I +Jis made negative by nInJ(all this follows fromProposition 2.10(i)).The strongly split condition gives UI U. But UIis clearly normalized byNI(use Theorem 2.23 (i)). Then UIPI. Now LIhas no non-trivial normalp-subgroup, so UI LI = {1]. To check that UILI =PIit sufces to checkB = UI BnI. This is clear by Theorem 2.23(ii). Denition 2.26. When I L, let WIbe the subgroup {n W [ nI =I ].Theorem2.27. Let (G. B = UT. N. S) be a strongly split BN-pair of charac-teristicp (see Denition 2.20). Let I ,J L, g G.(i) If d DI J, thend(PJ. UJ) (PI. UI) = (PK. UK) for K =I d J.(ii) L(seeDenition2.24)isk-regularand -stableforalleldskofcharacteristic ,=p.(iii) (PI. UI)g(PJ. UJ) if and only if g PId PJwhere d Wsatisesd J =I . This induces a bijection between PI{g G [ (PI. UI)g(PJ. UJ)],PJand {d W [ d J =I ]. When (PI. UI)g(PJ. UJ), one has PI =UI>L andgPJ =gUJ>L for an L which is a PI-conjugate of LI.(iv) {g G [ (PI. UI)(PI. UI)g] =PING(LI)PI =PIWIPI(see De-nition 2.26).(v) Any relation (P. V)(P/. V/) in L implies [ P[ = [ P/[ and [V[ = [V/[.Proof. (i) Let (P. V) =((dPJ PI)UI. (dUJ PI)UI) =d(PJ. UJ) (PI. UI).Letusshowrstthat P PK.WehaveT PI dPJ P.If +I ,then X PI and d1() + since d DI, therefore X P. But UI P,therefore (Theorem 2.23(ii)) B P. It now sufces to checkWK P. ButWK WI dWJ PI dPJ P.Since P PI, there exists a subset K/ such that K K/ Iand UI.(PI dPJ) =PK/ .34 Part I Representing nite BN-pairsLet us show that X Vfor all + +K. If + +I, then X UI V. If +I +K, X PI. It remains to show that Xd UJ, whichin turn comes from d1() , +J. If d(+J), then +I d(+J) = +K(Theorem 2.6). This contradicts the hypothesis. Therefore UK V.Now, since V is a normal p-subgroup in P =PK/ , one has V UK/ , whenceUK UK/ and therefore UK X = {1] forall +K/ (Theorem 2.23(i)).Then + +K + +K/, i.e. K/ K. We already had the reverse inclusion,whence the equality.(ii) WehaveUIPIand [UI[ is apower of p. It remains tocheckthat(PI. UI)g(PJ. UJ)hisin L. Thisreducesto(i)recallingthat G =PI DI J PJ a gh1.(iii) is clear by (i) and (ii). We takeL =x(LI), where x PIis such thatg xd PJand d Wsatises d J =I .(iv) We clearly have{g G [ (PI. UI)(PI. UI)g] PING(LI)PI PIWIPI. Now, if (PI. UI)(PI. UI)g, let us writeg PId PIfor d DI Isatisfying(PI. UI)(PI. UI)d. By(i), thismeansdI =I , i.e. d WIasstated.(v) follows from (iii). Example2.28. Let usgivesomeexamplesof subgroups WI(seeDeni-tion 2.26). We use the notation of Example 2.1.(i) For type An1 (n 2), L = {1. . . . . n1], let I = {1. . . . . i] (i 1). . ,, .I . ,, .I/It is easy to see that any elementn WImust correspond to a permutationwhich increases on the set {1. . . . . i 1] but is also such that {1. . . . . i 1] ispreserved. So WIcoincides with permutations xing all elements of this set,i.e. WI= -si 2. . . . . sn1> = WI/where I/ = {i 2. . . . . n1].(ii) For type BCn, LBC={0. 1. . . . . n1], let I ={0. 1. . . . . i 1] (i 1).AnelementinWImustpermutethebasiselementsejinafashionsim-ilar tothecaseabove, withtrivial signs one1. . . . . ei. Soweget WI=-s/i 1. si 1. . . . . sn1>, which is a Coxeter group of type BCnirepresentedin the space generated by LI= {ei 1. i 1. . . . . n1].I, .. ,=== __ei 1i 1(iii) For type Dn, LD={/0. 1. . . . . n1] (n 4), let I ={/0] {1. . . . . i 1] (i 1).2Finite BN-pairs 35If I = {/0], onends WI= -s1>-s/3s/4. s3. . . . . sn1>, whichis oftype A1 Dn2withsimple roots {1] {e4 e3. 3. 4. . . . . n1]. Wheni 2, onends WI= -s/1s/i 1. si 1. . . . . sn1>isomorphicwiththeCox-eter group of type BCnithrough its action on the space generated byLI={ei 1. i 1. . . . . n1].I, .. ,__ __ei 1i 12.5. Other properties of split BN-pairsInthefollowing, GisanitegroupwithastronglysplitBN-pair(G. B =UT. N. S) (see Denition 2.20). We state (and prove) the following results forfuture reference.Proposition 2.29. Let I L. Then the following hold.(i) NG(UI) =PIand UIis the largest normalp-subgroup of PI.(ii) If g G is such thatgUI U, then g PIandgUI = UI. If moreovergUI = UJfor someJ L, then I =J.Proof.