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# Representation Theory MATH 8852

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lrreducible Modular Representations of Finite and Algebraic Groups Notes by Christopher Drupieski and Terrell Hodge based on lectures by Leonard Scott June 8 2007 Abstract In these notes we outline some aspects of the modular representation theories of nite groups of Lie type in de ning and cross characteristics with particular interest paid to how these theories relate to the modular representation theory of algebraic groups and the characteristic 0 representation theory of Lie algebras and quantum groups We begin by summarizing some classical results on the representation theory of complex semisimple Lie algebras and Lie groups and then compare the classical theory to the representation theory of algebraic groups discussing some of the issues encountered in moving to elds of positive characteristic and discussing some of the progress that has been in resolving these issues We then discuss how the study of maximal subgroups leads to the study of linear representations in cross characteristic and conclude with a discussion of how the theory of quantum enveloping algebras quantum groups helps us to understand this situation 1 Characteristic Zero Lie Theory In order to establish our notation and to provide a reference to which we will later refer we begin by summarizing some well known results on the structure and representation theory of complex semisimple Lie algebras and Lie groups Readers who are already familiar with this material may skip ahead to Section 2 For further reference consult 6 16 11 Structure of Complex Semisimple Lie Algebras Let g be a semisimple Lie algebra over C Fix a Cartan subalgebra b of g Readers not familiar with these notions may want to look ahead to Example 111 It is a fact that b is diagonalizable for any nite dimensional representation of g Specializing to the adjoint This document is based on two lectures delivered by Leonard Scott on April 25 2007 and May 2 2007 in the University of Virginia Algebra Seminar and in Professor Scottls course MATH 852 Representation Theory Some additional material has been added especially from 16 and 37 for the rst part of the notes and from 8 20 and other references listed herein for the second part of the notes representation g decomposes as a direct sum of weight spaces for the adjoint action of b on g g b 69 LI 904 OLEQ Here g0 denotes the weight space of weight 04 E W Note that go We call the nonzero weights Q C if the roots of g The R span E RQ of the roots in if is an Z dimensional Z dimc if euclidean space in which Q is a root system and on which we have a non degenerate symmetric bilinear form denoted lt which is invariant under the Weyl group W of Q Fix a base H 041 04 of the root system Q We refer to the elements of H as the simple roots Then every element of Q can be written as an integral linear combination of the simple roots with all coef cients of like sign and Q Q4r Ll Q where Q4r is the collection of positive roots ie those roots which can be written as a non negative integral combination of the 04 and where Q iQJr is the collection of negative roots lf 0 E Q and 04 2f1004 de ne the height of 04 by hta 2L1 0 We have a partial order 3 on A in fact on all of if de ned by M S A if A 7 M is a non negative integral combination of positive roots So Q4r 04 E Q 04 gt 0 04 E Q hta gt 0 and Q a Qalt0a Qhtalt0 Notethatl ll7lla Qhta1 To each root 0 E Q we have its associated coroot 04V 204lt04 04 The set of all coroots QV 04V 04 E Q forms a root system in E called the dual or coroot system of Q Evidently lt0404V 2 for all 04 E Q The set l lV 041V 04X is a base for the coroot system QV Let 0 denote the Z gtlt Z Cartan matrix C 07 lt0404 The matrix C is sym metrizable that is there exists an Z gtlt Z diagonal matrix D diagd1 d with entries in Z4r such that DC is symmetric For 04 E Q de ne the re ection 50 E GLE by saw z 7 lt04V04 for all z E E The Weyl group W of Q is the nite re ection group generated by the 50 In fact W is a nite Coxeter group generated by the simple re ections s 50 04 E H If A E E is a vector such that ltA 04V 6 Z for all 04 E Q we call A an integral weight Let w E E denote the unique vector such that ltw047V 6 Kronecker delta for all simple roots 04 6 H We call in the fundamental dominant integral weight corresponding to the simple root 0 The free abelian group A generated by the fundamental dominant weights W1 w is called the weight lattice As a set A A E E ltA04V E Z V 04 E Q the set of all integral weights Note that 04 E A for all 04 E Q We call the subgroup A of A generated by the roots 0 E Q the root lattice It is a subgroup of nite index in A If M E A and ltM04V 2 0 for all 04 E Q we call M a dominant integral weight and we denote the collection of all dominant weights by A We have M E AJr if and only if M is equal to a non negative integral combination of the fundamental dominant weights Associated to the choice H of a base for Q we have the triangular decomposition g n 69 b 69 If where b is a Cartan subalgebra of g and n resp n is the subalgebra generated by all positive resp negative root spaces We often denote the subalgebra bEBt by 1 resp b n 69 Denote the universal enveloping algebras of g If f etc by Ug HOV Ub etc By the PBW Basis Theorem the natural multiplication maps de ne isomorphisms of vector spaces Ug E HOV c 16 c 2101 E 2101 CUb Example 111 Let g 5nCC Let 1 denote the subalgebra of trace zero diagaonal matrices Then 1 is a Cartan subalgebra in g For 1 S 239 S 71 let 61 E if be de ned by 6idiaga1an 11 Then ltIgt Ei76jii7 j For 1 S 239 S 7171 let 041 el 7 6111 Then H a1an1 is a base for I7 and with respect to this base we have lt13 r eiiejz ltj For 1 Signilwe have m 616i Foriy jwe have gerej CEM7 where E E g denotes the matrix having 1 in the thy position and zeros elsewhere Then n is the subalgebra of strictly upper triangular matrcies7 and 1 is the subalgebra of trace zero upper triangular matrices 12 Irreducible g modules Fix a base H C I7 and hence a corresponding triangular decomposition g n 69 6 Let V be a g module7 and suppose that there exists a weight vector 0 6 V such that 1101 0 Then we call M a maximal vector of weight A in V Suppose furthermore that V is generated as a g module by if Then all other weights M of V satisfy M 3 A7 and we call A the highest weight of V Suppose nally that V is an irreducible g module Then the line Cf C V is uniquely determined by the fact that M is a maximal vector7 and any other irreducible g module generated by a maximal vector of highest weight A is necessarily isomorphic to V As a speci c example ofthe above setup7 given A E if de ne a g module of highest weight A as follows Let CA be the one dimensional CC vector space with basis element denoted by A De ne a one dimensional representation of E11r on CA by bA 0 and hA AhA for all h E 1 Set VA Ug 8145 CA the induced module for CA from 1 to g Then VA is a g module of highest weight A7 called the Verma module or standard module of highest weight A As a vector space7 VA 2101 C CA Every g module of highest weight A is a homomorphic image of VA The Verma module VA has a unique maximal submodule7 and its irreducible head is denoted LA The necessary and suf cient condition for LA to be nite dimensional is for A to be a dominant integral weight Consider an arbitrary nite dimensional irreducible g module V Being a nite direct sum of weight spaces for l V must contain a weight space V with VAM 0 for all 04 E lt13 r ie7 A is maximal among all weights M of V with respect to the partial order 3 on if Then any 1 E V is necessarily a maximal vector of weight A in V By irreducibility7 V is generated as a g module by 1 hence is a homomorphic image of the Verma module VA But LA is the unique irreducible quotient of VA7 so we conclude that V g LA and A E A Thus7 the nite dimensional irreducible g modules are parametrized up to isomorphism by their highest weights7 and we have a bijection between elements of A4r and the nite dimensional irreducible g modules given by A lt gt LA For future reference7 we mention the following theorem Theorem 121 Weyl7s Complete Reducibility Theorem Let V be a nite dimensional g module Then V decomposes as a direct sum of irreducible g modules In particular7 ExtLA LM 0 for A 7 M e N 13 Character and Dimension Formulae Given a not necessarily nite dimensional g module V such that V is a direct sum of nite dimensional b weight spaces V A E if we de ne the formal character of V by ch V Z dim ma Adv Here the 6A are formal symbols satisfying eAef 6AM for all A E if By Weyl7s Complete Reducibility Theorem 1217 every nite dimensional g module decomposes as a direct sum of irreducible g submodules So the problem of determining all characters of nite dimensional g modules reduces to the problem of computing the ch LA for A E A We begin by computing the formal character of the Verma module VA The Verma module VA has a nite composition series with composition factors of the form Lw A for w 6 W7 where w A wA p 7 p and p zaeq Oz ELI wi is the Weyl weight Moreover7 Lw A occurs as a composition factor of VA only if w A 3 A7 and LA occurs as a composition factor of VA with multiplicity one Now ch VA ZwEW aw ch Lw A for some non negative integers aw with al 1 A similar equation holds for each ch Vw A with w A S A We thus obtain a system of equations describing each ch VwA w E W in terms of the characters ch LyA y E Writing the coef cients of this system of equations with respect to a suitable ordering of the set w A w E Ww A S A7 we obtain an upper triangular matrix over Z having all diagonal entries equal to one lnverting this matrix we obtain an equation of the form ch LA Z bw ch Vw A 1 wEW for some bw E Z Following 167 we can look at the action of the Weyl groupL on both sides of this equation and deduce the following famous result of Weyl Theorem 131 Weyl7s Character Formula Let A E A For w E W let lw denote the length of w as a word in the generators 51 51 of W Then ch LA Z 71W ch Vw A 2 wEW From the vector space isomorphism Vw A 2101 C CwA we have that7 for each 1 E if dim Vw AM is equal to the number of ways that w A 7 M can be written as a non negative integral sum of positive roots From this we deduce the formula 5wP Ch zy6W71lyeyp and hence the following alternate formulation of Weyl7s character formula 1Actually7 to this point A could be any element of A But when A is dominant7 so that LA is nite dimensional7 the Weyl group fixes charLA is its action we 6Mquot7 M 6 A7 w E W Theorem 132 Weyl7s Character Formula Alternate Formulation Let A E A For w E W let lw denote the length of w as a word in the generators 51 51 of W Then Zw6W1lwgwP ZweWlt71lwewJ A further consequence of Weyl7s character formula is the following formula for the di mensions of the irreducible g modules ch L 3 Theorem 133 Weyl7s dimension formula Let A E A Then 11091 lt p7 04gt dimL H Mp agt 4 So far we have concentrated on the representation theory of semisimple complex Lie algebras and have ignored the corresponding Lie groups But since one can pass from the Lie algebra to the Lie group through the process of exponentiation the representation theory of semisimple complex Lie groups exactly parallels that of the Lie algebras 2 Algebraic Groups in Positive Characteristic This section is based on 32 It is well known that the nite simple groups fall into three classes the simple groups associated to nite groups of Lie type loosely also called groups of Lie type or simple groups of Lie type the alternating groups and the 26 sporadic nite simple groups with the simple groups of Lie type taking up the bulk of the simple groups in some sense Tits has suggested that the alternating groups may be considered as groups of Lie type over the eld of one element in which case the simple groups of Lie type take up all but 26 of the known nite simple groups2 We begin our study of the irreducible representations of the nite groups of Lie type with the irreducible representations of semisimple algebraic groups over elds of positive characteristic The Lie algebra g of a semisimple algebraic group G over an algebraically closed eld of characteristic zero carries much information about the structure of G and in this case one can deduce results on the representation theory of G from results on the representation theory of the semisimple Lie algebra g But the situation becomes more complicated when one passes from algebraically closed elds of characteristic zero to elds of arbitrary characteristic While the representation theory of semisimple algebraic groups in positive characteristic largely parallels that of the complex semisimple Lie algebras elucidated above in Section lithe nite dimensional irreducible modules are still parametrized by dominant integral weightsiwe lack complete information regarding the structure of the irreducible represen tations Indeed the problem of determining the formal characters and dimensions of the irreducible modules and the progress that has been made towards this end will be our central focus in the sections that follow 2This is more than a joke as it turns out A theorem of Gordon James guarantees that the irreducible modular representations of the symmetric group of degree 7 are determined by those of irreducible modular representations of the degree 7 general linear group via Schur Weyl duality 21 Notational Conventions Let p be a prime number and let k E7 An af ne algebraic group G over k is an af ne algebraic variety G C k for some n with a compatible group structure in the sense that multiplication Ggtlt G a G and inversion G a G are morphisms of algebraic varieties Denote the coordinate algebra of G by More generally7 we call G an af ne k group scheme if G is a representable functor from the category of commutative k algebras to the category of groups For this more general notion k need only be a commutative ring7 though we will always assume that k is at least a eld Thus there exists a commutative k algebra MG such that GA Hom al kG7 A for all commutative k algebras A The fact that G is a group forces MG to be a Hopf algebra We say that G is an algebraic af ne k group scheme if MG is nitely generated over k Given a k algebra A and g 6 GA Hom algkGA7 the product fg E GA is de ned by fg M o f 8 g o A where here M A 8 A a A denotes multiplication in A and A MG a MG 8 MG is the comultiplication in When MG is an af ne algebraic k group scheme that is reduced ie7 when MG has no nonzero nilpotent elements and when we specialize to the case k E17 and A k then GA is an af ne algebraic group over k in the classical sense of an af ne algebraic variety having coordinate algebra We will feel free to consider our algebraic groups both in the classical sense as af ne algebraic varieties7 and in the functorial sense as af ne algebraic k group schemes We say that an af ne algebraic group G is de ned over a sub eld kg of k provided that there exists a Hopf algebra A0 over k0 such that the natural map k k0 A0 a MG is Hopf algebra isomorphism We identify A0 with an algebra of kO valued functions on G In what follows we will generally assume G to be de ned over some nite sub eld E1 of k E17 q pT a prime power Let G be a semisimple algebraic group over k E17 Fix a maximal torus T in G7 a Borel subgroup B containing T and the opposite Borel subgroup Bf Let U4r RUB the unipotent radical of B7 and let U Denote the character group of T by X XT The torus T acts on the Lie algebra g LieG through the adjoint map Ad7 and g decomposes as a direct sum of weight spaces ga 04 E X for T The nonzero weights form a root system ltIgt in E X 8 R and the choice of Borel subgroup B4r determines a positive system lt13 r in ltIgt and hence a base H C lt13 r for ltIgt Identify the co character group Y YT HomltGrm7 T with the group HomZ XTZ via the pairing XT gtlt YT a Z where WW A o Mt for all t 6 T7 A E XT7 M E YT Note that HomGmGm Z Now identify E Y 8 R with E via the pairing lt Then for each root a E ltIgt we have the associated coroot 04V 6 Y The abstract weights in E ie7 the M E X ZR satisfying ltM7 04V 6 Z for all a E ltIgt span a lattice A containing the root lattice AT subgroup generated by all a E ltIgt as a subgroup of nite index In fact7 we have AT Q XT Q A If XT A we say that G is simply connected We say that a character A E X is a dominant weight if ltA7aV 2 0 for all a E if or equivalently if A can be written as a non negative integral combination of the fundamental dominant weights mi 6 A Denote the collection of dominant weights in X by X4r XT We call A E XTJr an r th restricted dominant weight if 0 S ltA7aV lt pT for all a E H Denote the collection of r th restricted dominant weights by XTT We may refer to the elements of X1T simply as the restricted dominant weights Example 211 Let G GLn Then le kdet 1Xlj 1 S 2397j S n where det E kXlj 1 S 2397j S n is the determinant function a polynomial in the variables X ldentifying GLnA with the set of all n gtlt n invertible matrices with entries in the given k algebra A7 we may take B4r to be the subgroup of all upper triangular matrices7 U4r to be the subgroup of all strictly upper triangular matrices7 and T to be the subgroup of all diagonal matrices Multiplication in G is given by ordinary matrix multiplication Comultiplication in lel is given by AXij 221Xik ij The group G GLn has unipotent radical equal to the collection of scalar matrices7 hence is reductive but not semisimple Example 212 Let G SLn Since SL7 is the subgroup of GLn de ned by the vanishing of the polynomial det 71 E kGLn7 SL7 is a closed subgroup closed subfunctor of GLn We have le k17 detXlj 1 S 2397j S n and we may take B7U7T to be the subgroups of upper triangular7 strictly upper triangular and diagonal matrices7 respectively The group G SL7 is semisimple lf 6 XiZlT is the 24th coordinate function on T7 then the characters 04 el 7 6111 1 S 239 S n 7 1 form a basis for XT and a base for ltIgt So SL7 is simply connected We have wi 61 6139 Example 213 Let p be a prime number7 q p7 a power of p7 and G GUnq the general unitary group Then G U E GLnqu2 UUt 1 where U E GLnqu2 denotes the matrix obtained from U by raising each entry to the q th power and Ut denotes the matrix transpose of U The special unitary group SUnq is the subgroup of GUnq of elements with determinant equal to one Evidently GUnq and SUnq are both closed subgroups of GLnqu2 For further reference on general properties of af ne algebraic groups7 consult the stan dard references 5 177 36 For further reference on algebraic group schemes and their representations7 consult 20 22 Chevalley Groups and other Finite Groups of Lie Type The Chevalley groups over k also called split semisimple groups of Lie type over k are certain concrete constructions of semisimple algebraic groups from representations of complex semisimple Lie algebras We brie y sketch their construction here for further reference consult 37 To begin7 let gc be a complex semisimple Lie algebra with Cartan subalgebra be and let V be a faithful nite dimensional gC module Let L1 A the weight lattice of gc cf Section 117 let L0 A the root lattice of gc and let LV denote the sublattice of L0 generated by all weights of be on V Then L0 C LV C L1 There exists a basis XmHZ 04 E ltIgt71 321 for gc with Xa 6 gm H E be called a Chevalley basis7 such that all of the structure constants of gc relative to the Chevalley basis are integers Let 1 denote the subalgebra of Ugc generated by all X08 ngnl 04 E I7 n E N This subalgebra is known as the Kostant Z form ongc Then there exists a lattice VZ in V invariant under 1 Now given t E k and 04 E I7 we de ne an automorphism of Vk VZ 8 k as follows Because V is nite dimensional and since XDLVA C VAM for all weights A of V7 multiplication by Xa is a locally nilpotent endormorphism of V Then the map exp tXa Vk a Vk de ned by exp tXa 1 a 220 X0991 t a is a well de ned automorphism of Vk Let G be the group of automorphisms of Vk generated by all extha t E 1604 E ltIgt We call G a Chevalley group In fact7 G is a semisimple algebraic group with LieG g gz 8 k where gZ is the lattice in gc preserving the Z form VZ The lattice LV is realized as the character group of a maximal torus T of G7 and the lattices L0 and L1 are realized respectively as the root and weight lattices of G with respect to T Moreover7 every semisimple algebraic group G over k can be constructed in this fashion by the choice of an appropriate faithful nite dimensional gC module V for some complex semisimple Lie algebra gc lf LV L1 we say that G is a universal Chevalley group The reason for this terminology is that if G is another Chevalley group over k constructed from a faithful nite dimensional gC module V 7 then there exists a surjective homomorphism p G a G with ker p Q ZG the center of G So if G is the universal Chevalley group constructed from gc then G is a covering group for all other Chevalley groups constructed from gc Since L1 A and LV 2 XT for some maximal torus T of G7 we see that G is universal if and only if it is simply connected 23 Frobenius Morphisrns Let G be an af ne algebraic group de ned over FF ldentify lel with k EFF A07 and de ne the P robenius comorphism F HG a lel by Fa f a fp This map is readily seen to be a Hopf algebra map7 and hence is the comorphism of an algebraic group morphism F FG G a G called the Frobenius morphism We call the r th power F7 of F the r th Frobenius morphism The comorphism F de ned above is called the geometric P robenius endomorphism of There is a second Frobenius endomorphism of MG7 called the arithmetic P robenius endomorphism7 de ned by a f gt gt ap f There are also more general notions of a P robenius morphism Let X be an af ne algebraic variety over k E7 with coordinate algebra A If F A a A is an algebra homomorphism such that F is injective7 F A Aq for some p th power q pf7 and if for each f E A there exists in 2 1 such that Fmf fqm7 then we call the coordesponding morphism of varieties F X a X a generalized P robenius morphism on X7 cf 13 Chapter 4 Associated to the r th P robenius morphism F7 are two subgroups of G The kernel of F7 is a normal subgroup called the r th P robenius kernel of G7 and is denoted by G In the language of af ne group schemes7 GT is an in nitesimal group scheme We have GTK HomkalgkGT 7K e7 the trivial group7 for any eld extension K of k So Frobenius kernels are always trivial when we consider them in the classical sense as af ne algebraic varieties7 but they may no longer be trivial when we consider G as an af ne group scheme and permit k algebras A that are not elds The xed points GFT of G under the r th P robenius morphism form a nite subgroup of G7 denoted Gq or Gqu where q p If we consider G as an af ne algebraic variety over k E7 then Gq consists of those points in G C k all of whose coordinates lie in Fq If G is a Chevalley group7 then we call Gq a nite Chevalley group also called a nite group of Lie type Other nite groups of Lie type are obtained from a Chevalley group G through various twistings of the Frobenius morphism or by taking the xed point subgroup of G under a generalized Frobenius morphism The table on page 6 of 18 lists the orders of the nite Chevalley groups Gq when G is a universal Chevalley group Example 231 Let G Ga the additive group7 considered as an af ne group scheme Then le X7 a polynomial ring in one indeterminate X7 and GA A the additive group of A7 for each k algebras A The P robenius morphism satis es Ft It for all t E GA7 so GFXA t e A t1 t and GA t e GA tPT 0 Example 232 Let G Gm the multiplicative group7 considered as an af ne group scheme Then le kXXquotl7 a Laurent polynomial ring in one indeterminate X7 and GA AX7 gtlt the multiplicative group of units in A7 for each k algebra A The P robenius morphism satis es Ft It for all t E GA7 so GFTA t e AX 29 t and GA t e AX t1 1 24 Representations of Algebraic Groups Let G be the universal Chevalley group constructed from gc as in Section 22 While the Lie algebra of an algebraic group over a eld of positive characteristic carries less information concerning the structure and representations of the group than it does in characteristic zero7 Chevalley showed that the high weight theory of complex Lie algebras and Lie groups does carry over to semisimple algebraic groups in the sense that the irreducible modules for a semisimple algebraic group G are still parameterized by dominant highest weights So7 loosely speaking7 we have the same number of irreducible modules in positive characteristic as in characteristic zero One approach to constructing the irreducible G module LA parametrized by a given dominant weight A E X4r is to construct LA as a certain submodule of the coordinate algebra lel which is a G module via the left regular representation To begin7 for a given A E X4r let A denote the one dimensional B module of T weight A Set VA indg A f 6 MG fgb Ab 1fgVg e G Vb e B the induced module of A from B to G Another common notation for indgA is H0A Then VA is a nite dimensional G submodule of lel of highest weight A One can show that LA socG VA is an irreducible G module of highest weight A E X Now let AA V7w0A One can show that AA is generated by a B stable line of highest weight A and that any other such G module is a homomorphic image of AA Moreover7 AA has a unique maximal submodule7 and AA radG AA E LA We call AA the Weyl module of highest weight A Evidently the nite dimensional Weyl module AA assumes a position in the representation theory of G similar to that held by the in nite dimensional Verma module VA in the representation theory of gc cf Section 12 Another common notation for AA is lA7 further emphasizing its similarity to the Verma modules of QC 9 A second approach to the construction of the irreducible G modules makes more clear the connection between the simple modules LAC for gc and the simple modules LA for G Given A 6 A42 let LAC denote the simple gC module of highest weight A Then by the construction in Section 227 there exists a lattice LAZ C LAC such that the group G acts naturally on LAZ 8 k This G module is no longer simple in general7 but it does have an irreducible head lndeed7 LAZ 8 k AA the Weyl module3 of highest weight A7 so its head is isomorphic to the simple G module LA Before we address the structure ofthe simple modules for the nite Chevalley group Gq7 we state the following theorem of Steinberg Theorem 241 Steinberg7s Tensor Product Theorem Let A E XTJr and write A 210pm with A e X1T Then LA e LA0 LA11l Lmm1 where Logw denotes the G module obtained by composing the structure map for LA with the j th Frobenius morphism In principle then7 the structures of the irreducible G modules LA are completely de termined by those LA with restricted weights A E X1T and by the Frobenius morphism F G a G Now7 each simple G module LA remains simple on restriction to the nite Chevalley group Gq7 cf 18 Section 212 Steinberg showed that in fact every irreducible Gq module can be obtained in this manner His result also holds for any nite group GF of Lie type7 F as discussed above Theorem 242 Steinberg7 1963 Let L be an irreducible module over k for the nite group Gq Then L is the restriction from G of an irreducible G module On the other hand7 distinct irreducible G modules may no longer be non isomorphic on restriction to Gq lndeed7 let A E X1T Then by the Tensor Product Theorem7 LpTA g LATl But Gq GFT is the xed point subgroup of G under the r th P robenius morphism7 so Gq doesnt see77 the twist on LA and we have LA g LpTA as Gq modules To parametrize the simple Gq modules7 we must then restrict our attention to some subset of the dominant weights Steinberg showed that the necessary dominant weights are precisely the r th restricted dominant weights A E X He also gives a precise description of the weights needed in the general nite group of Lie type case We stick to the Chevalley groups here and below for simplicity By the Tensor Product Theorem7 one may even restrict attention to the restricted weights A 6 X1 An important step in understanding the irreducible G modules LA with A E XTT7 and hence their restrictions to the nite group Gq7 would be to know their dimensions and the dimensions of their weight spaces Problem 1 Give a Weyl character formula77 cf Theorem 131 for the irreducible G modules LA7 A E XTT an r th restricted dominant weight 3This fact is quite nontrivial It is a consequence of Kempfls Vanishing Theorem7 for line bundles on GB equivalently7 a statement regarding the vanishing of higher derived functors Rilndg Ai gt 0 of certain induction functorsi See 20 for an exposition A second problem and one that only becomes interesting for nite dimensional represen tations in the case of elds of positive characteristic cf Theorem 121 is to understand the ways in which the irreducible G modules LA can t together77 More precisely we want to understand the morphisms between the irreducible G modules and hence recursively the structure of G modules admitting a composition series Problem 2 Determine Ext LA LM Ext5qLALM for AM E XT Both problems may and should be formulated for general nite groups GF of Lie type with a suitable modi cation of XT Also one may pose problems analogous to Problem 2 for Eth or higher Ext groups The emphasis on Extl here is partly motivated by the special role of H1 in the next section Analogous to the situation of Section 13 Equation 1 we can write an equation of the form ch LA Z bw ch Aw A uerp wA6X for some bw E Z relating the characters of the simple G module LA to the characters of the Weyl modules Aw A with w an element of the af ne Weyl group lVp pZltIgt gt4 W Since the characters ch AM are given by Weyl7s Character Formula ie by Equation 3 of Theorem 132 with AM substituted for LM cf 20 Section 11510 the solution of Problem 1 above amounts to the determination of the integers bw Lusztig7s conjecture below asserts that the coef cients bw are in effect given by the values at 1 of certain polynomials PZW called Kazhdan Lusztig polynomials associated with the Coxeter group WP Though Lusztig7s formula is known to be correct for p gt h where h 1 p 043 is the Coxeter number of ltIgt 040 is the longest short root of ltIgt a lower bound for p is not known Before stating Lusztig7s conjecture we need some terminology We say that a dominant weight 1 lies in the Jantzen region if p 043 S pp 7 h 2 Conjecture 241 Lusztig 1979 Let A be a weight in the Jantzen region which includes all restricted weights ifp 2 2h 7 2 h the Coxeter number of ltIgt Then ifp 2 h dim LA is given as follows Choose w in the af ne Weyl group lVp pZltIgt gt4 W such that A w A0 for some A0 unique with 7p 3 A0 pHD S 0 for all 04 E ltIgt We say that A0 is in the antidominant lowest alcove Let we denote the longest element of W Then dim Lo 271lltwgtlltygtPyw1 dim Aw0y A0 5 where the sum is taken over all y E W such that waAO is dominant and w0yA0 S wowA0 A Aw0y A0 is the Weyl module of highest weight woy A0 and PZW is a Kazhdan Lusztig polynomial associated with the Coxeter group WP In a helpful strengthening of the conjecture Kato has proposed that formula 5 always holds for A E X1T provided p 2 h thus not requiring p 2 2h 7 2 This strengthening seems to hold up empirically in the one meager case in which the result is known ie for SL5 over F5 cf 35 The result remains an open problem for SL6 and SL7 over F7 3 Maximal Subgroups This section is based on 33 Suppose G is a nite group and H S G is a maximal subgroup Historically the study of maximal subgroups or more precisely pairs G has been a principle topic in the theory of nite groups For example through the study of maximal subgroups one may hope to obtain structural information about groups in general through a recursive procedure As another example and one which is the principle motivating factor for the rest of these notes is the role maximal subgroups play in the theory of permutation representations of nite groups The group G acts on GH not only transitively but primitively and the permutation representations associated to the pairs G H for H running over all maximal subgroups of G constitute the building blocks for all permutation representations of G analogous for nonlinear representations to the role played by the irreducible representations in the linear case Finite automata theory provides one interesting modern application of permutation representations see Chapters 6 and 7 of 14 entitled Covering by permutation and reset machines and The theory of Krohn and Rhodes We remark that any permutation machine may be covered in the terminology of 14 by a cascade wreath like product of primitive permutation machines Determining maximal subgroups of an arbitrary nite group reduces to the case of solving this problem for simple or nearly simple groups by a theorem of Ashbacher and Scott which we loosely paraphrase below Theorem 31 Aschbacheriscott 1985 The determination up to conjugacy of all pairs G M G a nite group and M S G a maximal subgroup reduces modulo smaller or easier problems to the following 1 G is almost simple and M is maximal in G 2 G HV a semidirect product of a quasisimple nite group H and one of its irreducible modules V over 197 and M is a complement to V In this case the conjugacy classes in G of such maximal subgroups M correspond bijectively to elements of the cohomology group H1H V Remark 1 Recall a nite group G is almost simple if G can be sandwiched as G0 3 G S AutG0 for a nite simple group G0 and its automorphism group By the Schreier Conjecture now a theorem AutG0 is a solvable group usually it is fairly small 2 Recall a nite group G with center ZG is quasisimple if GZG is simple and if G is equal to its commutator subgroup ie G is perfect As mentioned previously the nite groups of Lie type constitute a large7 subcollection of the nite simple groups The nite groups of Lie type split into two collections those arising from the classical groups associated to root systems of types A B G D and those of exceptional type associated to root systems of types E6E7E8F4 G2 The following very roughly phrased theorem of Aschbacher7s thus determines the maximal subgroups for a very large selection of all nite groups Theorem 32 Aschbacher7 1984 Let G be a nite classical group associated to a vector space V7 and M S G a maximal subgroup Then one of the following holds 1 M belongs to a natural list subgroups of G suspected maximal subgroups7 constructed in relatively obvious ways7 or to a small list of non natural cases 2 M is the normalizer in G of a quasisimple subgroup H S GLV acting irreducibly on the vector space V Remark 1 Item 2 of Theorem 32 is sometimes called Dynkin7s principle 7 since Dynkin pio neered this idea in the Lie theoretic context a paper of Dynkin7s in the 1950s actually classi ed maximal connected closed subgroups of classical Lie groups through this idea Dynkin eventually determined all maximal connected closed subgroups of semisimple complex Lie groups An analogous program for nite groups was proposed by Scott in l34l O7Nan and Scott determined candidate maximal subgroups for the alternating groups 34 the rst general result of this type Candidate maximal subgroups for sporadic and exceptional groups have also been given7 cf references in 33 pages 374 As remarked in 337 many candidates77 have been shown to be maximal or nearly so 3 C40 Aschbacher7s theorem 32 is fundamental to the geometric approach7 to nite linear groups in computational group theory see 30 3 A signi cant problem stemming from part 2 of Theorem 32 is that7 while H and M may both be nite subgroups of Lie type7 one may arise as Gq and the other as Gq for some prime powers q pm7 q p but with p 31 p In this manner naturally arises the problem of determining modular representations V for a nite group Gq of Lie type in the cross characteristic7 or non describing7 case7 that is7 when V is a Gq representation over a eld of characteristic 19 that does not divide q For this problem7 the whole idea in the de ning characteristic7 ie7 relating representations for Gq to modules for G and its Frobenius kernels G7 as outlined in Section 247 is not applicable7 and other methods must be employed Problem 3 Describe all the irreducible modules over a eld k of characteristic p7 pl q7 of a nite group of Lie type Gq By and large7 current progress on the modular representation theory of nite groups of Lie type in the non describing case is constrained to G of type A7 eg7 G GLnq or G SLnq Dipper and James 9 described all of the irreducible representations over a eld k of characteristic p7 p l q7 of GLnq Dipper and James also considered SLnq7 but there some issues remain Their approach used the q Schur algebra this concept and the related theories of Hecke algebras and quantum groups ie7 quantum enveloping algebras will be discussed in the next section 4 Hecke Algebras Schur Algebras and Quantum Enveloping Algebras Throughout this section take G GLLF to be the general linear group over an alge braically closed eld F with Weyl group W 2 6 let gc gnCC For 7 2 1 and the standard generating set S 12 2 3 7quot 71r for 6 the pair W S 6 S is a Coxeter system and for any such system one can de ne the Hecke algebra HqW over the Laurent polynomial ring Z Zvv 1 in the indeterminate 1 to be the free Z module on basis symbols Tw w E W with relations TsTw Tm lsw 1lw s E Sw E W T 1TS iq 0 s E S where q 112 For a free Z module V of rank n HqW acts naturally on V by the right action determined by place permutations7 First de ned in 1989 9 the q Schur algebra Sqnr over Z can then realized as the endomorphism algebra Sqn r EnquWV T When n r we shall write Sqn for the q Schur algebra Sqnn Specializing q to 1 E F that is regarding F as a Z module via a morphism Z a F q gt gt 1 E F and taking the tensor product HqWF HqW 83 F there is an isomorphism of algebras H HqWF FW g F6 that is the Hecke algebra is a deformation7 of the group algebra of 6 Moreover in this case the q Schur algebra reduces to the Schur algebra Sn r EndpgV 7 for the vector space V E F Classical Schur Weyl reciprocity relates the representation theory of G GLnCC to that of the Hecke algebra H 2 C6 via the GLnCC6 bimodule V T V C which has a decomposition as a sum of certain tensor products L 8 S of irreducible rational modules L that are polynomial7 and homogeneous of degree 7quot for GLnCC and irreducible modules S Specht modules cf 31 for 6 For F C the representation theories of Snr and H are related by the famous Schur functors and the double centralizer property Snr EndHV T g SltV T For F of prime characteristic the double centralizer property still holds but the same decomposition of V8 into terms L 8 S does not necessarily hold cf 12 10 As we shall discuss further below in this section a generalization of this classical Schur Weyl duality to the so called quantum case7 will make it possible to connect the repre sentations of a quantum7 analog Uq for the group G GLLF to representations of the q Schur algebra Sqn and aspects of this picture will be central to gaining information about non describing representations of the nite group Gq when charF is positive and q is some power of charF First however we provide an alternate formulation of the rele vant specialized Hecke and q Schur algebras that may better hint at the connections with non describing representations and the theory of q Schur algebras Under the assumption F has positive characteristic and specializing q to be a power of charF the Schur algebra Sqn and the Hecke algebra HqW associated to the Weyl group W of G can be de ned in the following way Let Bq denote a Borel subgroup of Gq and let Pq denote a generic parabolic subgroup of Gq Then HqUV e Endkaq F T38 and S E d F W A717 H kGq G3 TPq P022301 where the direct sum is taken over all parabolic subgroups of Gq containing Bq Moreover we have 5201771 EHqultWgt EBF Tim with J ranging over the fundamental re ections in W A later formulation of Dipper James theory eg by Takeuchi for the unipotent represen tations cf 38 and Cline Parshall and Scott in general cf 7 results in a categorical equivalence that guarantees that the irreducible modules for Gq in the cross characteristic case can be recovered from knowledge of irreducible modules for qaischur algebras Sqa Ta Theorem 41 cf Theorem 917 of Let 0 be the ring of integers of a p adic number eld K 7139 a generator of the unique maximal ideal of O and k OTFO the residue eld of characteristic p Let Gq GLnq p l q Then there exists a quotient OGqJq Jq Q rad OGq such that OGqJq is Morita equivalent to a direct sum of tensor products 8 SW of qai Schur algebras with Z aZn n It follows from the theorem above that a parametrization of irreducible Gq modules in non describing characteristic will follow from the same data for q Schur algebras like wise character formulas for the q Schur algebras will imply the same for Gq modules in cross characteristic It was known by the early 90s that the q Schur algebra in at least characteristic zero though also later shown to be true in other characteristics at q equal to an 6th root of unity is a homomorphic image of a quantum enveloping algebra for gc eg 11 Set U Ugc to be the quantized enveloping algebra over Qv associated to the complex Lie algebra gc that is the quantum enveloping algebra UAER for a root datum realization ER associated to the Cartan matrix for go see below for a precise de nition of the algebras UAER and a realization R for gc For H Qv Z HAW and Sqnr Qv Z Sqnr and rank n free Z module V there is a surjective algebra morphism 6 U a EndV Z Qv which factors through a natural surjective algebra morphism U a Sqn r restricting 6 to a particular integral form U3 of U yields a surjec tive morphism 32 a EndV with image EnquWV T g Sqnr From this integral7 result one can base change to get a version for any Z algebra in place of Z by this means one obtains Schur Weyl duality at q a root of unity via specialization one step in a more dif cult program to obtain quantum Schur Weyl reciprocity in general 12 We now take a few moments to de ne the algebra UH SR for R a root datum realization of an arbitrary m gtlt m symmetrizable Cartan matrix C with symmetrizing diagonal matrix D4 Our presentation is taken from Recall a matrix C CM 6 MmZ is a Cartan 4For C the Cartan matrix of QC D will be simply an identity matrix and the de nitions below will simplify considerably but because of the applications in these notes of the quantum enveloping algebras beyond type A as well as applications to Kac Moody algebras not discussed in these notes we include the more general de nition matrix if CM 2 for 1 S 239 S m ii 239 31 j implies CM 3 0 and iii CM 0 iff 07 0 for all m A Cartan matrix is symmetrizable if there is a diagonal matrix D diagd1 dm 6 MmZ such that DC is symmetric By de nition a root datum realization R of the m gtlt m symmetrizable Cartan matrix C is the 4 tuple ER H X HVXV having the ingredients below 0 a free Z module XV of nite rank ms having an ordered basis 0 ozyn b1 bs where s is a xed positive integer the set of simple coroots H 0d Ozyvn C XV the linear dual X HomZXVZ also a free Z module of rank m 5 called the weight lattice of C or its realization the simple roots H 041 am determined via duality pairing X gtlt XV 7 Z ah gt gt ltahgt ah and ltozioz7Vgt 07 for all 27 assume also that for all 2 17 ltOzibjgt 6 01 are chosen so that the m s gtlt m matrix has rank m where A am consequently H consists of linearly independent vectors The root datum realization ER is minimal if s m 7 rankC it follows from the item above that s 2 m 7 rankC The root lattice of the realization HXHVXV of C is 6211Z042 C X7 b blR XV Z Example 42 For n gt 1 let H1 Kn denote basis elements for a free Z module XV with dual space X HomZXVZ and corresponding dual basis 61 6 Setting 04X H 7 Isl1 for 1 S 239 S n 7 1 and setting b1 Kn gives a new Z basis 041V04X71b1 for XV Taking oz 6 7 61 for 1 S 239 S n 7 1 yields vectors satisfying who 07 for C CM the Cartan matrix of gc gnCC with CM 203 011 71 for all 13 239 S n 71 and cm 0 otherwise Taking H a1 ozn1 and H 0d ozxil the 4 tuple R HXHVXV gives a root datum realization for C Replacing XV by the Z span X V 211 Zaiv and X by X 211 Z04 yields a minimal root datum realization of 0 this time corresponding at the Lie algebra level to 5LnCC More generally every root datum realization R of a symmetrizable Cartan matrix 0 gives rise to a complex Lie algebra ie the associated Kac Moody Lie algebra see 8 or With the notion of a root datum realization R of a symmetrizable Cartan matrix C in hand we need just a bit more notation in order to de ne the associated quantized enveloping algebra UASR For any positive5 integers nm set M n 7 7111717 11l2llll n lnlln7llln7m1l m mum 11l2lll 5Actually n any integer can be assumed for the rst and third de nitions belowi 16 where 0 1 and 3 1 Let 72k resp Md denote the outcome of replacing 2 in resp Ml with C For ER HXHVXV the root datum realization for the m gtlt m Cartan matrix C clj with symmetrizing matrix D diagd1 dm 6 MmZ the quantum enveloping algebra UAER is the Q2 algebra generated by EiF 1 S 2 S m K h E XV subject to the relations K0 2 1 and Khhi KhKhl for hh39 E XV QEA2 KhE ultmgthgtEKh for h e XV and 1 239 m QEA3 KhF v lt igthgtEKh for h e XV and 1 239 m QEA4 EF i FjE EL312621 for 1 27 m QEAS 71f ljivilvi 0 for 1 S 2 79739 S m st1iciyj QEA6 Z 451 5JlmF29F7FZ07for1sm jsm st1iciyj where Kdiaiv and 2 vdi for a E H The quantum Serre relations are those relations given by QEAS and QEA6 As already suggested by the terminology comparing the relations above with Serre7s relations cf 16 for de ning a complex Lie algebra with a root datum ER show the algebra UAER to be a quantized7 version of such a Lie algebra The same analogy also holds more generally when R is a root datum realization associated to a Cartan matrix for a Kac Moody Lie algebra We next consider an analogue of Kostant7s Z form in the quantum case De ne the divided powers El Fn by E l i 6 F Fl 1 7 1 71M l The Lusztig integral form UZSR is the Zisubalgebra of UAER generated by all divided powers EE39LFn 1 S 2 3 77172 2 1 and elements K for h E XV For K Kay1 S 2 S m and Km ij1 S j S 5 one has in fact that UZSR is generated by all E gtFlt gt1 2 mn 21 and K1z39 m5n 21 Finally for any commutative ring R and invertible element q E B there is a unique ring homomorphism eq Z a k with eq2 q The specialization UqRfR UZSR 83 R is obtained from the integral form In the rest of our discussion we will assume for the sake of simplicity that R is the root datum for a semisimple algebraic group utilizing other accompanying notation such as XTL etc as before in these notes and we write simply Uq for UqRSR Having been somewhat careful in giving de nitions up to this point we will now proceed to balance the resulting length and detail involved in that 17 endeavor by simply sketching the representation theory for the relevant algebras UqK and its applications of importance for these notes Suppose K is a eld of characteristic 0 and q is a primitive 6th root of unity Z 2 3 and gcdZ 3 1 if the root system has a component of type G2 Then without loss of generality in studying the representation theory of UqK we can restrict attention to integrable type 17 UqK modules see eg the brief exposition in 20 based largely on the fundamental papers 292 For each A E XTJr there is a simple and nite dimensional integrable Uq module Lq of type 1 generated by a vector 1 E Lq 1 74 0 such that Efnh 0 for all n gt 0 and for all 1 S 239 S m and each simple nite dimensional type 1 module for UN is isomorphic to some Lq In general the nite dimensional type 1 Uq modules are not completely reducible but they are direct sums of their weight spaces running over weights XT This characteristic zero representation theory of UqK q a primitive W1 root of unity models crudely but still signi cantly the modular representation theory of algebraic groups and hence gives applications to the de ning characteristic representations of nite groups of Lie type more on this connection at the end of this section At the same time for the type A case when R corresponds to gc the integral form 32 introduced earlier is closely related to the Lusztig integral form UZgc of Ugc cf 8 Ch 14 for more details and the connections between UqKgC UZgC 83 K and q Schur algebras SqnrK will prove instrumental to analyzing the non describing representations of nite groups of Lie type the story to which we now return It was in the early 1990s using many deep results that Kazhdan Lusztig 24 25 26 27 28 and Kashiwara Tanisaki 22 23 determined that the integrable type 1 irreducible modules for UlLK for any characteristic zero eld K with q an 5th root of unity parametrized by dominant weights XT have character formulas very similar to Lusztig7s formula for algebraic group representations as in Section 24 In fact there are quantum Weyl modules lq as well as quantum induced modules eg and the character for Lq an irre ducible UqK module associated to A E XTJr in a suf ciently restricted region is given by the very same formula as Conjecture 241 Equation 5 with the af ne Weyl group WI acting in place of WP However compared with the algebraic group case information about Lq is much more complete that is with a few limitations the character formula for an irreducible Lq is not a conjecture but a theorem see eg 20 H12 for a brief sketch and further comments on the references given above In particular putting this information on irreducible UqK modules together with the identi cation of Sqn modules as UqKgC modules obtained via specialization from the integral quantum Schur Weyl duality setting produces a parametrization of irreducible char acteristic zero Sqn modules at q an 6th root of unity along with character formulas for them What is needed now to complete our story for cross characteristic representations in type A is a relationship between the ordinary and modular representation theories for the q Schur algebras and this is provided by James7 conjecture stated before the advent of quantum enveloping algebras see 33 for a discussion and further references Conjecture 43 James Return to the hypotheses that O is the ring of integers of a p adic number eld K 7139 a generator of the unique maximal ideal of O and k OWO the residue eld of characteristic p Let Gq GLnq p q For p gt 71 Z xed q an W1 root of unity in K the irreducible representations of the q Schur algebra in characteristic zero at an 6th 18 root of unity reduce to those in k More precisely7 for Sqn0 the qischur algebra over 0 with q specialized a primitive ill root of unity7 all irreducible SqnO lattices in irreducible SqnnK modules reduce modulo 7T0 to given irreducible iS qnk modules6 For p gt 07 with the size of p depending upon a given 717 Gruber Hiss drawing from observations of Geck noted the validity of James7 Conjecture James 21 showed that the conjecture holds for n S 10 See 33 for more details Theorem 44 Gruber Hiss 15 Return to the hypotheses that O is the ring of integers of a p adic number eld K7 7139 a generator of the unique maximal ideal of O7 and k OWO the residue eld of characteristic p Let Gq GLnq7 p q For p gtgt 717 Z xed7 q an Z th root of unity in K7 the irreducible representations of the q Schur algebra in characteristic zero at an Z th root of unity reduce to those in k For non describing characteristic representations of nite groups of Lie type beyond type A7 it is not clear that q Schur algebras and hence quantum enveloping algebras will be the right objects to use Raphael Rouquier has proposed utilizing in place of the q Schur algebras an analogous class of algebras the Cherednik algebras7 and work of Michelle Broue linking Hecke type algebras for complex re ection groups and blocks for unipotent characters arising from cuspidal7 characters in Deligne Lusztig induction7 provides one suggestive replacement to substitute for the type A connection between GLnC and H or between Ugc and H and their associated specializations at q an 5th root of unity but it is not yet clear whether these will do the trick7 Currently7 for non describing representation theory in the non type A case7 there is not even a parametrization of irreducible Gq modules7 much less character formulas Although quantum enveloping algebras may not provide the right stuff7 for the cross characteristic case non type A7 we shall close these notes with a comment on their impor tance for the de ning characteristic case in any type Building on the work of Kazhdan Lusztig and Kashiwara Tanisaki see references given above that established the validity of Lusztig7s character formula for the irreducible UlLK modules Lq q a primitive 6th root of unity in the characteristic 0 eld K7 Andersen7 Jantzen and Soergel 1 proved the Lusztig Conjecture 241 for irreducible modular representations of algebraic groups7 in that for each xed root system8 there is some number such that if p is a prime greater than the number7 the Lusztig conjecture for any group G over k with char k p A key fact is that simple G modules L for A E XTL7 can be obtained via reduction modulo p from simple modules Lq for UlLK The Kazhdan Lusztig polynomials PW7 whose values appear as the needed coef cients in Lusztig7s Conjecture 2417 can themselves be characterized as coef cients aris ing from base changing from a standard basis me E m to a Kazhdan Lusztig basis7 in an appropriately de ned Hecke algebra the lwahori Hecke algebra associated to the af ne Weyl group W1 see 19 Chapter 7 for details 6Actually7 there are some hypotheses on 4 see eigi7 33 Conjecture 212 for details Also7 James7 Conjecture was stated for de ning as well as cross characteristic representations 7We hope before the start of the AIM workshop7 or at least by its end7 to add further comments on this front to these notes 8Assume that the rank of the root system is gt 3 Disclaimers These notes are only scratching the surface of the topic and then only lightly Many key themes have at present been omitted entirely or almost entirely including but not limited to the cohomology of nite groups of Lie type in both describing and non de ning char acteristics and its connections with the Lusztig conjecture via Kazhdan Lusztig theory the deep geometric underpinnings essential to current progress on Lusztig7s conjectures eg perverse sheaves and intersection cohomology the theory of quivers and Ringel Hall alge bras as related to quantum groups more on the the role of symmetric group representations and their analogues in the theory of Hecke algebras the theory of highest weight categories quasi hereditary algebras and strati ed algebras Alperin7s Conjecture and Broue 7s Conjec ture calculations in small primes support varieties and more We hope to see these topics added to these notes in the future The authors welcome suggestions and corrections please e mail Terrell Hodge at ter rellhodgewmichedu Any omission of important topics or references are the result of constraints on time and the effort to limit the survey to some signi cant background ma terial to serve as a common base for the June 2007 AIM Workshop Representations and Cohomology of Finite Groups of Lie Type Computations and Consequences and lack of knowledge on the authors7 parts not intentional slight The authors thank Brian Parshall and Len Scott for reviewing drafts of these notes but claim remaining inaccuracies as their own References n ersen an zen an oerge epresen a ions 0 quan um 1 A d H H J t J C d S l W 1994 R t t f t groups at a p th root of unity and of semisimple groups in characteristic p Asterisque B H H Andersen P Polo K Wen 1991 Representations of quantum algebras Invent Math 104 1 59 E Aschbacher M 1984 On the maximal subgroups of the nite classical groups Invent Math 76 469 514 4 Aschbacher M and Scott L 1985 Maximal subgroups of nite groups J Algebra 92 44 80 5 Borel A 1991 Linear Algebraic Groups 2nd ed Graduate Texts in Mathematics 126 Springer Verlag New York Berlin 6 Carter RW 2005 Lie Algebras 0f Finite and A ne Type Cambridge Studies in Advanced Mathematics 96 Cambridge University Press New York 7 Cline E Parshall B and Scott L 1999 Generic and q rational representation theory RIMS 35 31790 E E llOl llll 12l ll3l M l H E ll6l ll7l ll8l ll9l l D 2 PH 22l 23l 24l Deng 13 Du J Parshall 13 and Wang J P January 2 2007 Finite Dimensional Algebras and Quantum Groups draft to be published in 20072008 by the AMS Dipper R and James G 1989 The q Schur algebra Proc London Math Soc 59 23750 Donkin S 1992 Invariants of several matrices Invent Math 110 389 401 Du J 1995 A note on quantized Weyl reciprocity at roots of unity Alg Golloq 2 363 372 Du J Parshall 13 and Scott L 1998 Quantum Weyl reciprocity and tilting modules Gomm Math Physics 195 321 352 Geck M 2003 An Introduction to Algebraic Geometry and Algebraic Groups Oxford Graduate Texts in Mathematics 10 Oxford University Press Oxford New York Ginzburg Abraham 1968 Algebraic Theory of Automata Academic Press Gruber J and Hiss G 1997 Decomposition numbers of nite classical groups for linear primes J Reine Angew Math 485 55 91 Humphreys JR 1978 Introduction to Lie Algebras and Representation Theory Grad uate Texts in Mathematics 9 Springer Verlag New York Berlin Humphreys JR 1975 Linear Algebraic Groups Graduate Texts in Mathematics 21 Springer Verlag New York Berlin Humphreys JR 2006 Modular Representatiosn of Finite Groups of Lie Type London Mathematical Society Lecture Notes 326 Cambridge University Press New York Humphreys JR 1990 Reflection Groups and Gometer Groups Cambridge Studies in Advanced Mathematics 29 Cambridge University Press New York Jantzen JC 2003 Representations of Algebraic Groups 2nd ed Mathematical Sur veys and Monographs 107 American Mathematical Society James G 1990 The decomposition matrices of GLnq for n S 10 Proc London Math Soc 60 225 265 Kashiwara M and Tanisaki T 1995 Kazhdan Lusztig conjecture for af ne Lie alge bras with negative level Duke Math J 77 21 62 Kashiwara M and Tanisaki T 1996 Kazhdan Lusztig conjecture for af ne Lie alge bras with negative level 11 Non integral case Duke Math J 84 771 813 Kazhdan D and Lusztig G 1980 Schubert varieties and Poincare duality appearing in Osserman R and Weinstein A eds Geometry of the Laplace Operator Proc Honolulu 1979 Proc Symp Pure Math 36 Amer Math Soc Providence RI

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