MODULAR REPRESENTATIONS OF FINITE REDUCTIVE GROUPS

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1 MODULAR REPRESENTATIONS OF FINITE REDUCTIVE GROUPS Marc Cabanes Institut de Mathématiques de Jussieu Université Paris Diderot Bâtiment Sophie Germain Paris Cedex 13, France We report on the main results about linear representations of finite reductive groups or finite groups of Lie type. Following the historical order, we comment on representations in the defining characteristic, ordinary characters and representations in non-defining characteristic. Contents 1 Representations 3 2 Group Algebras 4 3 Coxeter Groups 5 4 Iwahori-Hecke Algebras 8 5 BN-Pairs and Simple Groups 10 6 Reductive Groups 13 7 Finite Reductive Groups 17 8 Rational G-Modules and Weights 19 9 Defining Characteristic: The Simple Modules Steinberg s Tensor Product and Restriction Theorems Weyl Modules and Lusztig s Conjecture More Modules Harish Chandra Philosophy Deligne-Lusztig Theory Unipotent Characters and Lusztig Series l-blocks Some Derived Equivalences Decomposition Numbers Some Harish Chandra Series 41 1

2 2 M. Cabanes References 43 INTRODUCTION This survey deals with linear representations of a class of finite groups strongly related to linear algebra itself, finite reductive groups. Finite reductive groups also provide almost all finite simple groups and their essential central extensions. They are therefore omnipresent, explicitly or not, in finite group theory. Finite reductive groups are finite analogues of reductive algebraic groups whose structure is in turn close to that of Lie groups. This also explains the terminology groups of Lie type which is often used. Let us mention also the term Chevalley groups, which pays tribute to Chevalley s construction of those finite groups from integral forms of semi-simple complex Lie algebras [Ch55]. The later approach is also described in [Cart1], though the prevailing approach is now to see them as fixed point subgroups G F under a Frobenius endomorphism F : G G of some algebraic group G, an approach due to Steinberg and allowing to take advantage of the geometry of algebraic groups. The pace of this survey is intended as quite slow, giving details necessary to understand most definitions. This should suit beginners or more experienced readers from other branches of representation or group theory. We tried to comment on some examples, mainly in type A (general linear groups, finite or not), and we strongly recommend Bonnafé s book [Bn] where much of our matter, and more, is thoroughly explored for SL 2. In Part I, we describe roughly the main features of representation theory, that is mainly the study of module categories for rings that are mostly finite groups algebras or close analogues. This also leads to the study of associated categories, most famously the derived and the homotopic categories. We did not comment on the already spectacularly successful methods of categorification, referring instead to Rouquier s series of talks. Part II comments on the constructions and structure of those groups both from the elementary point of view of split BN-pairs and of algebraic groups. Part III deals with linear representations most naturally associated with finite reductive groups since performed over the field F p defining the ambient algebraic group. The classical theorems of Chevalley and Steinberg are supplemented by more recent contributions by Lusztig, though it seems that some questions remain as mysterious as they were 25 years ago.

3 Modular Representations of Finite Reductive Groups 3 Part IV, by far the longest, comments on both the theory by Deligne- Lusztig leading to a very precise description of ordinary characters of finite groups of Lie type and more recent investigations on representations in characteristics different from p. The first is presented in many references (see [Sri], [DiMi], [Cart2]) and we quickly recall the main results. The second part was largely initiated by the papers of Fong-Srinivasan ([FoSr82], [FoSr89]) relating ordinary characters of finite classical groups G F with blocks of F l (l p). Many contributions followed by Dipper (decomposition numbers [Dip85a], [Dip85b]), and Geck-Hiss-Malle (l-modular Harish Chandra theory [GeHiMa94], [GeHiMa96]). Broué in [Br90] gave several conjectures, the one on Jordan decomposition of characters being later proved by Bonnafé-Rouquier (see [BnRo03]). We report on all those subjects, which leads us to many theorems on l-blocks, decomposition numbers and modular Harish Chandra series. I. MODULAR REPRESENTATIONS We refer to the first chapter of [Be] or the short book [Sch] for most of the information we need. 1. Representations The framework is the one of representations of non commutative rings A that are mainly finite dimensional algebras over a commutative field K. The finite dimensional representations can be seen as objects of the category A-mod of finitely generated A-modules. It may also be that A is an O-free algebra of finite rank over a local ring O, in which case we require that the objects of A-mod to be O-free of finite rank. We do not recall here the terminology of simple, indecomposable, or projective A-modules. The Jacobson radical of A is denoted by J(A), the group of invertible elements of A is denoted by A. Simple modules are equivalently called irreducible representations, and Irr(A) denotes their set of isomorphism types. When A is over a field K, we denote by K 0 (A) the Grothendieck ring of A, which may be seen as the commutative group ZIrr(A) endowed with the multiplication induced by tensor product of modules over K. The multiplicity of a simple A-module S as a composition factor in the Jordan-Hölder series of an A-module M is denoted by [M : S]. We also use the notion of blocks as minimal indecomposable two-sided ideal direct summands of A. Note that this also makes sense when A is an O-free algebra of finite rank over a local ring O.

4 4 M. Cabanes Our modules are left modules but we need also the notion of bimodule. It is defined as follows. We have two rings A, B, and T is an A-module on the left, and a B-module on the right, with both actions commuting, that is a(tb) = (at)b for any a A, t T, b B. This of course coincides with the notion of an A B opp -module where B opp is the opposite ring of B. An object T of A B opp -mod defines a functor B-mod A-mod by M T B M. The notion of a Morita equivalence is a typical example of such a functor (see [Be] Sect. 2.2). In connection with representations of finite groups, one may be led to consider the category C b (A) of bounded complexes... M i i M i+1... of A-modules ( i+1 i = 0 for all i Z, and only finitely many i s are such that M i {0}) and the related homotopic category Ho b (A). 2. Group Algebras We are mostly interested in the cases where A is a group algebra RG of a (multiplicative) finite group G over a commutative ring R. Recall that this is the R-free R-module of all sums g G r gg (for (r g ) g G any family of elements of R indexed by G) with R-bilinear multiplication extending the one of G. For us, an R-linear representation of G is an R-free RG-module M of finite rank or equivalently a group morphism G GL n (R). When H G is a subgroup, restriction Res G H has an adjoint on both sides Ind G H which is the functor RH-mod RG-mod associated with RG seen as a bimodule with translation actions of G on the left and H on the right, i.e. Ind G H(N) = RG RH N whenever N is an RH-module. When R = K is a field where G inverts and with primitive G -th roots of 1, then KG is a split semi-simple algebra = i Mat d i (K). If moreover the characteristic of K is 0, then the isomorphism type of a KG-module M is given by its trace character χ M : G K sending g to the trace of its action on M. Irreducible characters are the ones of simple KG-modules, and one denotes by Irr(G) the corresponding set of functions on G. This has values in Z[ω G ] where ω G is a primitive G -th root of 1, it is therefore independent of K. Note that Irr(KG) Irr(G). For most of character theory, see [Isa]. When R is a field k of characteristic a prime divisor of G with a primitive G p -th root of 1, the algebra kg/j(kg) is split semi-simple. Then Irr(kG) is often denoted as IBr(G) and identifies with the central functions on G p (p-regular elements of G) obtained by Brauer s method of lifting p -roots of 1 in k into an extension of Z p (see [Sch] Sect. 3.1).

5 Modular Representations of Finite Reductive Groups 5 When p is a prime number and G is a finite group, a p-modular system (O, K, k) is a triple where O is a complete discrete valuation ring containing the G -th roots of 1, free of finite rank over Z p, with K denoting the field of fractions of O (a finite extension of Q p ) and k = O/J(O) is its (finite) residue field. The decomposition of OG into blocks OG = i B i (sometimes called the p-blocks of G) gives the decomposition of kg into blocks kg = i B i k and induces a partition of both Irr(G), by Irr(G) = Irr(KG) = i Irr(B i K), and IBr(G) by IBr(G) = Irr(kG) = i Irr(B i k). The principal block of G in characteristic p is defined as the one not in the kernel of the one-dimensional trivial representation of G (where each element of G acts by 1). II. THE GROUPS 3. Coxeter Groups 3.A. Definitions References for what follows are the corresponding chapters of [Bou], [GePf]. Definition 3.1. A Coxeter graph on a set S is a non-oriented graph without loops, with nodes the elements of S, and valued edges carrying a number m st {3,..., } (of course m st = m ts ) for any s, t S. One omits m st when m st = 3 and one completes the matrix (m st ) s,t S by putting m st = 2 whenever s t and there is no edge between s and t. The associated Coxeter group is the group generated by the elements of S subject to the relations: (quadratic) s 2 = 1 for all s S. (braid) sts = tst... (with m st terms on each side) for all s t S with m st. One denotes by l S : W N the length function with regard to the generating set S. Note that the relation for s t and m st = 2 specifies that s and t commute. So the connected components of the Coxeter graph gives a partition of S into pairwise commuting sets.

6 6 M. Cabanes Example 3.2. (a) The graphs or m for m 4 give rise to the dihedral groups of order 2m (6 in the first case, infinite in the case when m = ). (b) Another important example is the one of the graph of so-called type A n as follows (n 1 nodes) whose associated group is isomorphic with the symmetric group S n+1 by the map sending the i-th generator s i in the above list (from left to right) to the transposition (i, i + 1) (Moore s presentation of the symmetric group). Several other basic properties are as follows. Fact 3.3. S injects in W and m st is the order of the product st in W. Fact 3.4. If W is finite, it has a single element w S W of maximal l S. 3.B. Parabolic subgroups, finite Coxeter groups One calls parabolic subgroups of W the subgroups generated by a subset of S. If I S, one denotes W I := <I>. The following is a consequence of the so-called exchange condition governing the way to obtain minimal decompositions (see [Bou] IV.1.5). Fact 3.5. The map I W I is a bijection between subsets of S and parabolic subgroups of W. It satisfies W I W J = W I J for all I, J S. Moreover S W I = I and the restriction of l S to W I is l I. An abstract group W with a subset S of involutions is called a Coxeter group if W is isomorphic through the canonical map with the group presented by S subject to the relations of Definition 3.1 for m st being defined as the order of the product st (which requires that S generates W ). A Coxeter group is said to be irreducible if, and only if, it comes from a connected Coxeter graph. Theorem 3.6. An irreducible Coxeter graph gives rise to a finite group W if and only if it is among the following (the index n recalls the number of nodes):

7 Modular Representations of Finite Reductive Groups 7 A n D n BC n 4 E n for n = 6, 7, 8 F 4 4 G 2 6 I 2 (m) m for m = 5 or 7 H 3 5 or H 4 5 On the other hand a typical graph producing an infinite group is as follows. Ã n. (n + 1 nodes). 3.C. Reflection representations Coxeter groups have a reflection representation. It is defined as follows. Let us start from the set S and integers m st for s t S. One then defines a real vector space V = s S Re s whose basis is in bijection with S by s e s. One defines on it a symmetric bilinear form sending (e s, e s ) to 1 and (e s, e t ) to cos(π/m st ) when s t. Denote by Isom(V ) GL R (V ) the corresponding orthogonal group. Then it is easy to see that the map sending s to the reflection through the vector e s is a group morphism W Isom(V ). A more remarkable fact is that it is injective. Another fact is that W is finite if and only if the above bilinear symmetric form is definite positive (see [Bou] Sect. V.4.8). The latter is a key fact in the proof of the above Theorem 3.6. An important generalization of the finite case above is when V is a finite dimensional complex vector space and W GL C (V ) is a finite subgroup generated by elements r W such that the image of r Id V is a line ( pseudo-reflections, remembering that r has finite order hence is semi-simple). Such a W is called a finite reflection group, a good survey is provided by [GeMa]. A defining property is that the action of W on S(V ), the ring of polynomials on V is such that the invariant subring S(V ) W is isomorphic to a polynomial ring (Chevalley, see [Bou] V.5.3).

8 8 M. Cabanes 4. Iwahori-Hecke Algebras A deformation of the group algebra of a Coxeter groups, Iwahori-Hecke algebras were defined in connection with representations of finite reductive groups. 4.A. Generalities All that follows can be taken from [GePf]. Definition 4.1. If R is a commutative ring, q R and W is a Coxeter group (with subset S and integers m st ), one denotes by H R,q (W ) the R- algebra with generators (A s ) s S subject to the relations: (quadratic) (A s + 1)(A s q) = 0 for any s S, (braid) A s A t = A t A s... (m st terms on each side) for any distinct s, t S. Note that the choice q = 1 gives us the group algebra RW. More generally, one has freeness over R for an R-basis indexed by W. Theorem 4.2. If w = s 1... s ls (w) is a minimal expression of w as a product of elements of S, the product A s1... A sk is 0 and depends only on w. One denotes it by A w := A s1... A sk (with A 1 := 1). Then H R,q (W ) = w W RA w. Many properties ensue, among them the relation with parabolic subgroups of W, or with scalar extension. 4.B. Semi-simplicity in characteristic zero The first assertion below is due to Gyoja-Uno ([GyUn89], see also a generalization with several parameters in [GeP f]), the second is due to Tits. Theorem 4.3. Assume R = C and W is finite. Let q C. (a) H C,q (W ) is semi-simple if, and only if, q w W ql S(w) 0. (b) (Deformation theorem). If H C,q (W ) is semi-simple, then H C,q (W ) = CW. Note that the so-called Poincaré polynomial P W (X) = w W Xl S(w) has a factorization P W (X) = Π S i=1 Xd i +1 1 X 1 where the d i are the exponents of W, that is the degrees of the generators of the invariants of the action of W on the symmetric powers of the reflection representation. So we see that complex Iwahori-Hecke algebras over W are isomorphic to the group

9 Modular Representations of Finite Reductive Groups 9 algebra CW except for finitely many values of q that are either 0 or a root of unity. Variants of Iwahori-Hecke algebras have been defined in the case of complex (finite) reflection groups (see [BrMaM93]). 4.C. The Kazhdan-Lusztig polynomials Let (W, S) be a Coxeter group, possibly with W finite. If w = s 1... s ls (w) is a minimal expression of w as a product of elements of S, then the set [1, w] W defined by all sub-products s i1... s it for 1 i 1 < < i t l S (w), is independent of the minimal expression chosen for w. One then defines the Bruhat order on W by x y if and only if x [1, y]. See [GePf] Ex. 1.7 for this. See also Sec. 6.B below for an interpretation in terms of Zariski closure in reductive groups. Let q 1/2 be an indeterminate whose powers are denoted by (q 1/2 ) m = q m/2. Let R = Z[q 1/2, q 1/2 ] the ring of Laurent polynomials in those indeterminates. Let H be the Iwahori-Hecke algebra H R,q (W ) (see Definition 4.1 above). Note that q being invertible, each generator A s (s S) is invertible with (A s ) 1 = q 1 A s + (q 1 1)A 1, so that any A w (w W ) is also invertible. Definition 4.4. Let H H, x x be the Z-linear map sending q m/2 to q m/2 and A w to (A w 1) 1. This is a ring endomorphism. Here is the existence theorem for Kazhdan-Lusztig s polynomials (see [KL79]). Theorem 4.5. There exists a unique set of polynomials P v,w Z[q] for v w with P w,w = 1, P v,w has degree (l S (w) l S (v) 1)/2 for v < w and such that C w := q l S(w)/2 v w P v,wa w satisfies for any w W. C w = C w It has long been conjectured that the above polynomials have coefficients in N. This was proved by Kazhdan-Lusztig for finite W and also affine type like Ãn above by showing the relation with cohomology of Schubert varieties BwB/B (see Sec. 6B below). The positivity in the general case was proved recently by Elias-Williamson [EW12]. The polynomials allow Kazhdan-Lusztig to define certain subsets of the group W, called cells, and associated cell representations of the Iwahori- Hecke algebra. See more generally the notion of cellular algebras [GraL].

10 10 M. Cabanes The new basis {C w} of Theorem 4.5 also allows a more explicit version of the isomorphism in Theorem 4.3 above. More general Iwahori-Hecke algebras can be given a family of parameters (q s ) s indexed by elements of S modulo W -conjugacy. See [Lus03] Ch. 14 for several conjectures about those algebras some of them having been checked since then. 5. BN-Pairs and Simple Groups BN-pairs, first axiomatized by Tits, are a basic trait common to all groups of Lie type, be them algebraic, p-adic or finite. 5.A. The axioms and basic properties Definition 5.1. A group G is said to have a BN-pair if it possesses two subgroups B and N, and a subset S N/B N such that T := B N is normal in N and one has s 2 = 1 in the group W := N/T for any s S, G = <B, N> and W = <S>, for all s S, sbs B, for all s S, w N/T, one has BsBwB = BswB BwB. The groups B, T, W are often called the Borel subgroup, a maximal torus and the Weyl group of G, respectively. Fact 5.2. The elements of S are the only non-trivial elements of W such that B BsB is a subgroup of G (this is why they can be considered implicit in the definition of the BN -pair). Fact 5.3. (W, S) is a Coxeter group. Fact 5.4. Bruhat decomposition. The double cosets BwB for w W are pairwise disjoint and are all the elements of B\G/B: G = w W BwB. Fact 5.5. The set of subgroups of G containing B is in bijection with the set of subsets of S by I P I := BW I B = B<I>B. We get in particular P I P J = P I J for any I, J S. 5.B. Split BN-pairs and Levi decompositions The group G with a BN-pair is said to be split whenever the group B is a semi-direct product B = U T. Some authors sometimes add the

11 Modular Representations of Finite Reductive Groups 11 condition that T is abelian. An important property of the same nature is the following. Assume G has a split BN-pair B = UT with finite W. Definition 5.6. For I S, denote by w I the element of maximal length of (W I, I) and U I := U w I Uw I. Also B = w S Bw S and L I := BW I B B W I B P I = BW I B. The latter is called the standard Levi subgroup associated to I. One says that P I satisfies a Levi decomposition if U I P I and P I = U I L I. 5.C. Examples, finite and p-adic Example 5.7. The case of GL n (F). Let F be a field, n 1 an integer, and G := GL n (F). Let B (resp. U, resp. T ) be the subgroup of upper triangular (resp. unipotent upper triangular, resp. diagonal) matrices in G. Let N be the subgroup of monomial matrices in G (invertible matrices with only one non-zero element on each row and on each column). Then N/T = S n (permutation matrices) and the permutations (1, 2),..., (n 1, n) used in Example 3.2.(b) above give a subset S N/T with properties of a BN-pairs. Whenever I S generates W I = Sn1 S nk, one gets for P I = A 1 0 A 2 BW I B the subgroup of matrices in the form A k with A i GL ni (F). We also have the Levi decomposition P I = U I L I where U I (resp. L I = GLn1 (F) GL nk (F)) is the subgroup of matrices in the I n1 A I n2 form (resp. 0 A with A i GL ni (F)) I nk A k Example 5.8. A p-adic example. Let O be a finite extension of some Z p, with fraction field K and residual field k = O/J(O). The group G = SL n (K) has the following BN-pair (Iwahori-Matsumoto). Reduction of matrix entries modulo J(O) gives a surjective group morphism SL n (O) SL n (k). Let B SL n (O) be the inverse image of the upper triangular subgroup, a Borel subgroup, of SL n (k). Let N be the subgroup of monomial matrices in G. Then T := B N is the subgroup of

12 12 M. Cabanes diagonal matrices with coefficients in O and determinant 1. The quotient W = N/T is an affine Coxeter group of type Ãn 1 when n 3 (see Sec. 3.B above), Ã1 = I 2 ( ) if n = 2. Note that the proper parabolic subgroups for this BN-pair (sometimes called parahoric subgroups of G = SL n (K)) are finite unions of double cosets BwB since they are associated to finite subgroups of W. 5.D. Simple groups Recall that a simple group is any group having no other normal subgroup than itself and the trivial subgroup. A perfect group is any group G generated by its commutators, i.e. G = [G, G]. We have the following important criterion of simplicity. Theorem 5.9. Let G a group with a BN-pair B, N, S and associated Weyl group (W, S). Assume the hypotheses: (W, S) is irreducible, B is solvable and G is perfect. Then G/ g G gbg 1 is simple non-abelian. The proof is remarkably easy. It suffices to show that any normal subgroup of G is either G or a subgroup of B. So let H G. Then BH is a subgroup of G containing B, therefore BH = BW I B for some I S (see Fact 5.5 above). Let us show that if s S \ I and w I, then they commute. We have sws 1 = sws sbhs sbsh BH BsBH since H G, BW I B BsBW I B B(W I sw I )B by the axioms of BN-pairs. The uniqueness of the Bruhat decomposition (see Fact 5.4) then implies that sws 1 or s W I, and therefore sws 1 W I in both cases. Then sws 1 P I P {s,w} = P {w} = B BwB by Fact 5.5. Again by Bruhat decomposition, we get now sws 1 = w. In view of the irreducibility hypothesis we must have I = S or, that is HB = G or H B. If HB = G, then G/H = B/B H, so the last assumptions of the theorem imply that this factor group is at the same time solvable and equal to its derived subgroup. So it is indeed trivial and H = G. So BN-pairs can be seen as a way to construct simple groups, finite or not. Understandably, finite BN-pairs have been classified (see [FoSe74], [HeKaSe72]) as a (small) part of the classification of finite simple groups.

13 Modular Representations of Finite Reductive Groups 13 This in turn can be seen as a (very difficult) converse of the above, see below Sec. 7.C. 6. Reductive Groups We fix F an algebraically closed field. One considers F-varieties essentially as locally closed subvarieties of affine varieties. Algebraic groups over F are affine F-varieties such that multiplication (x, y) xy and inversion x x 1 are F-algebraic morphisms. Much of their abstract theory was done by Borel (see the reference [Bo] and the textbooks [MT], [Spr]). The group GL n (F) is clearly such an algebraic group (n 0 an integer), and in fact all algebraic groups over F are closed subgroups of some GL n (F). The latter is related with the existence of linear representations of algebraic groups, that is algebraic morphisms G GL n (F) (see [MT] 5.5). An important property of algebraic groups is the notion of unipotent and semi-simple elements, along with the Jordan decomposition of elements of G, x = us = su where u, resp. s, is sent to a unipotent (resp. semi-simple) element by any linear representation of G. 6.A. Reductive groups Definition 6.1. The unipotent radical R u (G) of an algebraic group G is its largest normal connected subgroup whose elements are all unipotent. The group G is said to be reductive if and only if it is connected and R u (G) = 1. It is said to be semi-simple if in addition its center Z(G) is finite (this is also equivalent to being perfect, i.e. [G, G] = G, and connected). When F is the field of complex numbers, the above notion coincides with the one of semi-simple Lie groups. Those are classified essentially by use of Lie algebras and the classification of root systems (see [Hum1]). Some similar results can be obtained for reductive groups over algebraically closed fields F (Chevalley, 1956), leading to presentations by generators and relations (Steinberg, 1962) and identification for non exceptional Dynkin types to classical groups. We recall below some main notions and steps for this classification, mainly because they are also crucial to the description of linear representations of both reductive groups and their finite analogues. 6.B. Borel subgroups, tori and root data In what follows G is a reductive groups over F.

14 14 M. Cabanes A Borel subgroup B G is any connected solvable closed subgroup maximal for inclusion. They form a single G-conjugacy class. Definition 6.2. A torus of G is any closed subgroup isomorphic (as an algebraic group) to a direct product of copies of GL 1 (F) = F. Maximal tori of G are a single G-conjugacy class. It is now customary to fix a maximal torus T in G and introduce several notions attached to this. Each Borel subgroup can be written B = R u (B) T for a (non-unique) maximal torus of G. Let X(T) = Hom(T, F ) (morphisms of algebraic groups), which is a lattice isomorphic to Z r whenever T = (F ) r. Similarly one defines Y (T) = Hom(F, T) and gets a pairing X(T) Y (T) Z = Hom(F, F ), (x, y) x, y = x y. Definition 6.3. A root subgroup is any minimal unipotent subgroup normalized by T. Each root subgroup associated to T is isomorphic to F by some map F a u(a) G and (consequently) there exists α X(T) such that t.u(a).t 1 = u(α(t)a) for all a F, t T. The set of α X(T) giving rise to such root subgroups is denoted by Φ(G, T) and called the set of roots of G with respect to T. One also defines a set Φ(G, T) Y (T) and a bijection Φ(G, T) Φ(G, T) with α α actually obtained as follows. For α Φ(G, T) and associated root subgroup U α G, there is some morphism ϕ α : SL 2 (F) G sending the unipotent upper triangular matrices into U α, the diagonal matrices ( into ) T. Then α Y (T) = Hom(F, T) is defined by α (λ) = λ 0 ϕ α 0 λ 1 for any λ F. This α is called the coroot associated with the root α. It is well-defined thanks to the fact that ϕ α above is unique up to T-conjugacy. The relation with the usual notion of (crystallographic) root system is that Φ(G, T) X(T) X(T) Z R endowed with the above pairing X(T) Y (T) Z and α α allow to define reflections s α : X(T) Z R X(T) Z R associated with any α and to prove that they preserve the set Φ(G, T). A positive subsystem Φ(G, T) + is obtained by selecting the α such that the corresponding root subgroups U α are subgroups of a given Borel subgroup B. Recall that a positive subsystem of a root system always contains a

15 Modular Representations of Finite Reductive Groups 15 unique basis of the root system (this allows to define many notions related with this positivity phenomenon, such as the notion of highest positive root). Such a basis determines the Dynkin type of the root system and the corresponding reflections are a generating set S := {s δ δ } for the group W GL(X(G, T) Z R) generated by all reflections above. The pair (W, S) is a Coxeter system (see Sec. 3 above). The (infinite) group G has a BN-pair for the Borel subgroup B and N = N G (T). The quotient group N/T identifies with the group W described above (associate with α the class of ϕ α ( Example 6.4. (a) The case of G = GL n (F) is as follows. Let T = (F ) n be the subgroup of diagonal matrices. Then X(T) = n i=1 Ze i where e i is the morphism sending that n-tuple (t 1,..., t n ) to t i. With that notation, for any i j in {1,..., n}, α := e i e j sends (t 1,..., t n ) to t i (t j ) 1 and this is an element of Φ(G, T) corresponding to the subgroup U α of matrices in u such that u I n has only zeros except possibly the element at row i and column j. The positive roots corresponding to the Borel subgroup B G of upper triangular matrices correspond to pairs (i, j) with j > i. The associated basis is {e 1 e 2,..., e n 1 e n }. The type of Coxeter system is A n 1 (see Theorem 3.6 above) corresponding with the above ordering of roots. (b) The case of G = Sp 2n (F). Let us denote by M M T the transposition of matrices. Let J n GL n (F) be the matrix whose only non-zero elements are 1 s on the second diagonal, that is the (i, j) element is δ n+1 i,j. Let Sp 2n (F) GL 2n (F) be defined by the equation ( ) ( ) 0 Jn 0 M M T Jn =. One can choose as maximal torus the J n 0 J n 0 subgroup T of diagonal matrices with diagonals of type (t 1,..., t n, t 1 n,..., t 1 1 ), and one obtains ( a Borel subgroup ) as UT where U is the subgroup of matrices of type X XJn S 0 J n (X 1 ) T where X U n (F) (unipotent upper triangular matrices of GL n (F)) and S Sym n (F) (symmetric n n matrices). J n It is not difficult to single out one-parameter unipotent subgroups normalized by T. They come in two types corresponding with the roots α sending the above diagonal matrix to t i t j for some 1 i, j n (α = e i +e j, first type) and to t i t 1 j for some i j (α = e i e j, second type). Then e 1 e 2,..., e n 1 e n, 2e n gives a basis of the root system and the ) ).

16 16 M. Cabanes corresponding Coxeter diagram is of type BC n 4 Relation with the Bruhat order. It is easy to prove the following interpretation of the Bruhat order in W (G, T) in terms of Zariski closure of double cosets, namely ByB = 1 x y BxB. Following the model of Lie algebras, one is often led to consider the Borel subgroups opposite the one already used. In the above case, noting that U = <U α α Φ(G, T) + >, one defines U := <U α α Φ(G, T) = Φ(G, T) + >. Note that U = w S Uw S where w S is the element of W of maximal length with regard to the basis above. The group B := TU is a Borel subgroup. The product Bw S Bw S = BB = UTU is called the big cell and is open, hence dense in G (use the closure property recalled above). 6.C. Classification in terms of root data One defines a notion of root datum which stands for quadruples (X, Φ, Y, Φ ) where X, Y are lattices endowed with a perfect pairing over Z, and Φ X, Φ Y are subsets endowed with a bijection α α satisfying axioms producing root systems in the same fashion as seen above in the case of reductive groups where a maximal torus has been chosen. Then the classification theorem of Chevalley ensures that the choice of an algebraically closed field F and a root datum (X, Φ, Y, Φ ) selects a reductive group G over F such that one of its maximal torus T is such that (X, Φ, Y, Φ ) = (X(T), Φ(G, T), Y (T), Φ (G, T)). Such a G is unique up to an isomorphism which itself is unique up to T-conjugacy. The group is said to be of simply-connected type whenever it is semisimple and Y = ZΦ. From the isomorphism theorem it is clear enough that, once Φ a root system and F are chosen, then there is only one group G sc which is of simply connected type. Any semi-simple G with same Φ and F is a central quotient G sc G (see [Cart2] p 25). The group is said to be of adjoint type whenever it is semi-simple and X = ZΦ. This is also equivalent to having trivial center. Any semi-simple G with same Φ and F satisfies G/Z(G) = G ad.

17 Modular Representations of Finite Reductive Groups 17 Example 6.5. For the root system of type A n 1, one has G sc = SL n (F) and G ad = PGL n (F). In type C n, G sc is the symplectic group Sp 2n (F) (see Example 6.4.b above) and G ad is its quotient by ±Id 2n. In certain types, the simply connected covering is less naturally found. In type B n, G ad is SO 2n+1 (F) (special orthogonal group for the quadratic form defined by the sum of squares of coordinates in F 2n+1 ) but G sc is a spin group Spin 2n+1 (F) defined by means of the Clifford algebra. 7. Finite Reductive Groups Let p be a prime and F the algebraic closure of the field with p elements. 7.A. Definition and Lang s theorem Let f 1 and q := p f. Let Frob: GL n (F) GL n (F) be the raising of matrix entries to the q-th power. This turns the Frobenius automorphism of the field into a so-called Frobenius endomorphism of the algebraic group GL n (F). Definition 7.1. A finite reductive group is any group of fixed points G F := {g G F (g) = g} where F : G G is an algebraic endomorphism such that, for at least one k 1, F k is the restriction to G of a Frobenius endomorphism Frob: GL n (F) GL n (F) with Frob(G) = G. The study of finite reductive groups is made easier by the following important theorem due to Lang and generalized by Steinberg (see [CaE] 7.1). Theorem 7.2. In the above setting, if S is a closed connected F -stable subgroup of G, one has S = {g 1 F (g) g S}. In particular this implies the existence of F -stable Borel subgroups and F -stable maximal tori in G. Fact 7.3. Finite reductive groups G F have BN-pairs of type B F, N F where B, N = N G (T) make the BN-pair of the reductive group G and T B are F -stable. 7.B. Examples Taking the examples of 6.4 above, and with k = 1 in Definition 7.1 above, one finds the finite groups GL n (q) and Sp 2n (q).

18 18 M. Cabanes The case k = 2 can account for finite unitary groups GU n (q), the latter a subgroup of GL n (q 2 ). For this we take F = Frob σ where Frob is the raising of matrix entries to the q-th power and σ : GL n (F) GL n (F) is the automorphism of order two sending a matrix to its transpose-inverse. Cases where k 1 is needed in Definition 7.1 are often called twisted. The above case where Frob is composed with a so-called graph automorphism also occurs in type D n and E 6. In types B 2 and F 4 with p = 2, and in type G 2 with p = 3, one may build an endomorphism F : G G which behaves like a Frobenius composed with a graph automorphism on certain root subgroups (changing the root and applying the Frobenius to the parameter) and like a graph automorphism on others (changing the root but not the parameter), see [Cart1] Sect. 12.3, 12.4, [Cart2] Then F 2 is a Frobenius endomorphism. The corresponding groups G F sc are denoted by 2 B 2 (q) or 2 F 4 (q) for q a power of 2, and 2 G 2 (q) for q a power of 3. 7.C. Classification: Finite simple groups, quasi-simple groups The classification of finite simple groups is the collective work of dozens of mathematicians and was completed at the start of the 80s. Theorem 7.4. The finite simple groups are either any cyclic group of prime order any alternating group A n with n 5 a finite group with a BN-pair of type G/Z(G) where G is a finite reductive group, or the Tits group [ 2 F 4 (2), 2 F 4 (2)] one of the 26 sporadic groups The members of the third item above are called simple groups of Lie type, see [GLS] for an in-depth analysis of their properties. As is well-known, perfect groups G (i.e. such that [G, G] = G) have a unique maximal central extension Ĝ G (a surjective morphism with central kernel and perfect Ĝ), maximality being here defined by the property that any other would be covered by that one. The kernel of the above universal map is called the Schur multiplier, it is finite when G is. Finite quasi-simple groups are defined as perfect groups G such that G/Z(G) is simple. They are central quotients of the maximal central extensions of non-abelian finite simple groups. Apart from a finite number of exceptions, the situation for simple groups of Lie type parallels

19 Modular Representations of Finite Reductive Groups 19 the one of semi-simple groups G. Namely, the maximal quasi-simple groups are the finite reductive groups E = (G sc ) F for G of irreducible Weyl group (or root system). Then the simple quotient is of course E/Z(E). Note that the latter is not necessarily a finite reductive group, see the example below. Example 7.5. PSL n (q) := SL n (q)/z(sl n (q)) while SL n (q) = (G sc ) F G of type A n 1 over F = F q. for III. REPRESENTATIONS IN DEFINING CHARACTERISTIC 8. Rational G-Modules and Weights In this section, we consider a reductive group G over an algebraically closed field F and we assume chosen T B a maximal torus and Borel subgroup of G. We recall the set of roots Φ(G, T) (see Sec. 6.C above). We denote U = R u (B) and we recall U. We consider the rational representations of a reductive group G. Definition 8.1. A rational representation of G is any rational group morphism G GL F (M) where M is a finite dimensional F-vector space. Note that M is then a FG-module for the (infinite dimensional) group algebra FG, and we denote simply by g.m or gm the outcome of the action of g G on m M. Fact 8.2. One has Res T M = λ X(T) M λ where M λ = {m M t.m = λ(t)m for all t T, m M}. The latter are called the weight subspaces of M. Recall the exponential notation for fixed points: for instance M T = M 0 = {m M t.m = m for any t T}. Fact 8.3. If M {0}, then M U {0} Fact 8.4. FG.M U = FU.M U. The character of M, ch(m) keeps track of the dimensions of the weight subspaces. Definition 8.5. One defines formal symbols e λ indexed by elements of X(T) and satisfying e λ e µ = e λ+µ. The formal character of M is ch(m) = dim F (M λ )e λ. λ X(T)

20 20 M. Cabanes Let us recall the pairing X(T) Y (T) Z that we denote below as,. Definition 8.6. The dominant weights are defined by X + (T) := {λ X(T) λ, α 0 for all α Φ(G, T) + }. 9. Defining Characteristic: The Simple Modules We keep the same notations as in the section above. The following is due to Chevalley. Theorem 9.1. The (isomorphism types of) simple rational representations of G are in bijection with dominant weights. One denotes by L(λ) the simple rational FG-module associated with λ X + (T). It is characterized by the property that L(λ) U = L(λ) λ and is a line. Moreover L(λ) µ {0} implies that λ µ NΦ(G, T) +. Example 9.2. The case of ( G = SL ) 2 (F). In this case we take for T the t 0 diagonal torus of matrices 0 t 1 for t F and Borel subgroup TU ( ) 1 a where U = { a F}. Then X(T) 0 1 = Z by the map associating to ( ) t 0 m Z the map 0 t 1 t m. The dominant weights correspond to N. Consider the natural action of ( G on ) F[X, Y ] = S(F 2 ) where P (X, Y ) is a c sent to P (ax + by, cx + dy ) by SL 2 (F). For p 1 m 0, let b d L (m) F[X, Y ] be the subspace of homogeneous polynomials of degree m. Then L (m) U is clearly the line FX m = L (m) m while L (m) = FU.X m (this is where the hypothesis that m p 1 is used). So we get part of the parametrization of Theorem 9.1. Note that L (m) has dimension m + 1 in that case. For the whole parametrization of simple rational representations, see Theorem 10.3 below. Note that when m is any integer 0, then FG.X m = FU.X m provides indeed a simple rational representation of G, that we should rename L(m). This is the parametrization of Theorem 9.1 by dominant weights. But we no longer have L(m) = L (m) in general and the dimension can be smaller than m + 1, (though computable, see Theorem 10.2 below). Example 9.3. L(0) = F the trivial module. In the notations of Example 6.4.a above, L(e 1 ) for GL n (F) is the natural representation, of dimension n.

21 Modular Representations of Finite Reductive Groups Steinberg s Tensor Product and Restriction Theorems We now turn to relations of simple rational modules with simple modules in the same characteristic for finite reductive groups. The theorems are by Steinberg (see [St63]) and are the main source of information on representations of finite reductive group in the defining characteristic. We keep G a reductive group over F which is assumed to be of characteristic the prime p. We keep T B G a maximal torus and Borel with corresponding definition of root system Φ(G, T) with basis. Definition If q is a power of p, we denote X + q (T) := {λ X(T) 0 λ, δ q 1 for any δ }. We now assume that G GL n (F) is a subgroup invariant under Frob the raising of matrix entries to the p-th power. If M is the underlying vector space of a rational representation of G, and i 0, one denotes by M [i] the same vector space but where the action of G is twisted by Frob i. With the new action being denoted by, we have g m = Frob i (g).m for any g G, m M. The following is Steinberg s tensor product theorem. Theorem Let λ X + (T) written as λ = i 0 pi λ i where λ i X + p (T). Then L(λ) = i 0 L(λ i ) [i]. Assume F = (Frob) f FG F -modules. where q = p f. The following gives the simple Theorem For any λ X q (T) the module L(λ) restricts into an irreducible representation of G F. Moreover, if Y (T) = ZΦ, then all simple kg F -modules are of that type. Definition When Y (T) = ZΦ, one defines the Steinberg module of G F as Res G G F L(λ) where λ X(T) is defined by λ, α = q 1 for all α. The Steinberg FG F -module is projective and has dimension U F, the order of the Sylow p-subgroup of G F.

22 22 M. Cabanes 11. Weyl Modules and Lusztig s Conjecture We return to the study of FG-modules L(λ) for λ X + (T 0 ). We assume in this section that G = G sc, that is Y (T 0 ) = ZΦ. Weyl G-modules V (λ) are a construction borrowed from the characteristic zero case. In the case of characteristic zero the formal characters of those modules are known thanks to Weyl s so-called character formula (see [Hum1] Sect. 24). So it is enough to determine the multiplicities [V (λ) : L(µ)] for any λ, µ X + (T 0 ) to have the characters of the simple modules L(λ). 11.A. Weyl modules Let g be the semi-simple Lie algebra over C with root system Φ. Any λ X + (T 0 ) can be considered as a dominant weight of g, so it gives rise to a simple g-module with highest weight λ, V (λ) C, which corresponds to L(λ) for the simply-connected algebraic group G C of root system Φ over C. Then, by a construction of Chevalley (see [Hum1] Ch. VII), it is possible to choose a lattice V (λ) Z stable under the action of a Lie subalgebra g Z such that V (λ) := F Z V (λ) Z has a compatible algebraic action of G = G sc. The formal character of V (λ) is known from the one of V (λ) C which in turn is given by Weyl s character formula (see [Hum2] Sect. 3.2). 11.B. Affine Weyl group action on X(T) and Lusztig s conjecture Let α 0 be the highest root of Φ with regard to the basis. Let ρ X(T 0 ) be the half sum of all positive roots, so that ρ, α = 1 for all α. Definition Let W a be the subgroup of GL(X(T)) generated by W (T) and the translations by elements of pφ = {pα α Φ}. This is a Coxeter (affine) group for the generators {s δ δ } {r 0 } where r 0 is the reflection in the hyperplane {x X(T) x, α0 = p}. Let W dom := {w W a w(ρ) ρ X + (T)}. Since W a is a Coxeter group for the indicated set of generating reflections, one has an associated Bruhat order, length map l : W a N, and Kazhdan-Lusztig polynomials P v,w Z[x] (v w in W a ). Lusztig s conjecture, formulated as a Problem in [Lus80], is as follows Problem Assume p > α 0, ρ and let w W dom with α 0, w(ρ) p(p h + 2) where h is the Coxeter number of Φ (see for instance [GePf] p. 29), then

23 Modular Representations of Finite Reductive Groups 23 ch(l( w(ρ) ρ)) = v W dom v w ( 1) l(w) l(v) P v,w (1) ch(v ( v(ρ) ρ)). The above, combined with Andersen-Humphreys linkage principle and Jantzen s translation principle (see [Jn] II.6 and II.7), allows to express any ch(l(λ)) (λ X + (T 0 )) in terms of ch(v (µ)) s as long as p > h. The latter ch(v (µ)) is known in turn by Weyl s character formula. It has been proved by Andersen-Jantzen-Soergel (see [AJS94]) with additional restriction on p. Theorem Lusztig s problem has a positive answer for large p. In this work the bound on p is not explicit. An explicit bound exponential in the rank was given by Fiebig [F12]. Using Soergel s bimodules and Juteau-Mautner-Williamson s theory of parity sheaves, G. Williamson has shown that part of Lusztig s conjecture for p > h is equivalent to absence of p-torsion in certain cohomology of Schubert varieties. In [W13], Williamson gives a method to find such torsion numbers. In particular Fibonacci numbers F n and F n+1 are shown to be such numbers for G = SL 4n+7 (F), which readily excludes that Lusztig s conjecture could be proved for p f(h) with f a linear function (for SL n ). It is expected that any polynomial bound can be excluded. 12. More Modules We show here some relations between simple modules for finite reductive groups in defining characteristic and representations of modular Iwahori- Hecke algebras (Green-Tinberg-Sawada, see [CaE] Ch. 6). This leads to define certain indecomposable modules with interesting properties (see [Gre78]). We are back to a general finite reductive group G F where G is a reductive group over F of characteristic p. Then we have seen in Fact 7.3 that G = G F has a split BN-pair with subgroups B = B F, T = T F, N = N G (T) F. Recall that U = R u (B) is such that U = U F is a Sylow p- subgroup of G. Note that an immediate corollary of Bruhat decomposition is that G = n N UnU (disjoint union).

24 24 M. Cabanes 12.A. Modular Iwahori-Hecke algebra The group G acts by translation on the set G/U of left cosets with regard to U, so the vector space Y := FG/U is a FG-module. It is isomorphic with the induced module Ind G U F where F denotes here the trivial FU-module. Definition Let H F (G, U) = End FG (Y ). This is a finite dimensional algebra with basis (a n ) n N defined by a n (U) = x U/U nun 1 xnu (where we have given the image of the class U in G/U FG/U, the image of the other classes being then easily determined). Proposition (i) The simple H F (G, U)-modules are 1-dimensional. (ii) H F (G, U) is a self-injective algebra (i.e. the regular representation is injective). 12.B. Fixed point functor and simple modules We now define the following classical functor H Y from (left) FG-modules to right H F (G, U)-modules. If M is an FG-module, H Y (M) = Hom FG (Y, M) = M U. The second equality results from Y = Ind G U F and Frobenius reciprocity. Note that this second equality implies that H Y (M) {0} whenever M {0} since U is a p-group for p the characteristic of F. Using self-injectivity from Proposition 12.2 in a crucial way, one proves the following. Theorem (Green) One has a decomposition Y = a i=1y i with each Y i an indecomposable module. Each Y i had a simple head and simple socle (though not isomorphic in general). Moreover if 1 i, j a, then Y i = Yj i = j hd(y i ) = hd(y j ) soc(y i ) = soc(y j ) hd(h Y (Y i )) = hd(h Y (Y j )) soc(h Y (Y i )) = soc(h Y (Y j )). Corollary ([CE] 1.25.(i)) Let FG-mod Y be the full subcategory of the category of FG-modules whose objects are the FG-modules in the form e(y m ) for m 1 and e End FG (Y m ). Then H Y induces an equivalence between FG-mod Y and the category of right H F (G, U)-modules. 12.C. The p-blocks The following is due to Humphreys (see [CaE] 6.18).

25 Modular Representations of Finite Reductive Groups 25 Theorem If S = G F /Z(G F ) is simple non-abelian, then FS = F St B 0 (FS) a sum of the one-dimensional block corresponding to the Steinberg module and the principal block (see Sec. 2 above) of S. Remark So we see that simple groups of Lie type have only two blocks in the defining characteristic. Certain groups have even less, that is just one (principal) block. This is the case for the simple Mathieu groups M 22 and M 24 with regard to the prime p = 2. On the other hand, groups with just one p-block are never a simple group of Lie type. It can even be proved that for p an odd prime and G a finite group, the group algebra F p G is just one block if and only if G has a normal p-subgroup P G such that C G (P ) P (see [Ha85] Th. 1). IV. OTHER CHARACTERISTICS We take G = G F a finite reductive group with G over F of characteristic p. We let k be an algebraically closed field of characteristic l p. The goal is to study the category kg-mod. The case l = 0 is very developed thanks to the work of Deligne-Lusztig and subsequent work of Lusztig, Shoji and others. Much less is known in the case of a positive l but Harish Chandra philosophy can be used and provides a partition of simple modules. 13. Harish Chandra Philosophy We assume G F is equipped with its BN-pair B F, T F, N, etc... as in Fact 7.3. A Levi decomposition is any G F -conjugate of some standard Levi decomposition of a standard parabolic subgroup P I = BW I B = R u (P I ) L I (see Sec. 5.B above) where I is an F -stable subset of, the basis of Φ(G, T) associated with B. We then write P F = R u (P) F L F where P = gp I g 1, L = gl I g 1, for some g G F. 13.A. Harish Chandra induction and restriction For P F = R u (P) F L F a Levi decomposition, we define e = R u (P) F 1 x kg F, x R u(p) F an idempotent commuting with the elements of L F. One defines two functors.

26 26 M. Cabanes Definition Let R G L : kl F -mod kg F -mod N kg F e kl F N defined by and R G L : kg F -mod kl F -mod defined by M M Ru(P)F = em = ekg F kg F M. The first is clearly the inflation of L F -modules to P F -modules followed by induction from P F to G F. The two functors R G L and R G L are exact and are clearly adjoint to each other on both sides. Note that R G G = R G G is the identical functor. We have a transitivity formula R G H = RG L RL H whenever Q = R u(q)h is another F -stable Levi decomposition in G with Q P and H L (so that Q L = (R u (Q) L). H is a Levi decomposition in the reductive group L). We have a Mackey formula for the compound R G L RG L : Theorem Whenever P = R u (P )L is another F -stable Levi decomposition, one has R G L R G L = g R L L gl g 1 ad g R L L g 1 Lg where g ranges over a representative system of the double cosets P F \G F /P F. In order to simplify our notation we have omitted the parabolic subgroup used to define R G L. In fact we have invariance with regard to the choice of P, for a given L. The following is due simultaneously to Dipper- Du and Howlett-Lehrer ([DipDu93], [HowL94], see also [CaE] 3.10): Theorem Whenever P = R u (P )L is another F -stable Levi decomposition with same Levi subgroup L, and one denotes e := R u (P ) F 1 kg F one has an isomorphism by the map sending x to xe. x R u(p ) F x kge kge

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