In Search of Bounds on the Dimension of Ext Between Irreducible Modules for Finite Groups of Lie Type

In Search of Bounds on the Dimension of Ext between Irreducible Modules for Finite Groups of Lie Type

Veronica Shalotenko Sterling, Virginia

B.S., Cooper Union for the Advancement of Science and Art, 2012

A Dissertation presented to the Graduate Faculty of the University of Virginia in Candidacy for the Degree of Doctor of Philosophy

Department of Mathematics

University of Virginia May, 2018 Contents

0 Preface 5

I Background 9

1 The Ext Functor and Group Cohomology 9 1.1 The Ext Functor ...... 9 1.2 Ext and Extensions ...... 10 1.3 Some Properties of the Ext Functor ...... 11 1.3.1 The Long Exact Sequence in Ext ...... 11 1.3.2 The Eckmann-Shapiro Lemma ...... 11 1.3.3 Ext1 between Irreducible Modules ...... 13 1.4 Group Cohomology ...... 15 1.4.1 Group Cohomology via the Standard Resolution ...... 15 1.4.2 Group Cohomology via Cochains ...... 16 1.4.3 Group Cohomology in Low Degree ...... 17 1.4.4 The inﬂation homomorphism on group cohomology...... 17 1.5 The Restriction and Corestriction Homomorphisms on Group Cohomology . 18 1.5.1 The Restriction Homomorphism ...... 18 1.5.2 The Corestriction Homomorphism ...... 18 1.6 Composition of Restriction and Corestriction ...... 19

2 Groups with a BN-pair 21 2.1 The Axioms of a Group with a BN-pair ...... 21 2.2 Parabolic Subgroups ...... 22 2.3 Finite Groups with a Split BN-Pair ...... 23

3 Finite groups of Lie type 24 3.1 Steinberg endomorphisms ...... 24 3.2 The action of F on the character group of T ...... 26 3.3 The classiﬁcation of the ﬁnite groups of Lie type ...... 28 3.4 The Root System and Root Subgroups of the Finite Group of Lie Type GF . 30 3.5 Construction of a BN-pair for the Finite Group of Lie Type GF ...... 31 3.6 The Levi Decomposition of a Parabolic Subgroup of a Finite Group of Lie Type G with a Split BN-Pair ...... 32 3.7 An Order Formula for a Finite Group of Lie Type Arising from a Simple Linear Algebraic Group ...... 33

4 Harish-Chandra Series 34 4.1 Harish-Chandra Induction and Restriction ...... 34 4.2 Some Properties of Harish-Chandra Induction and Restriction ...... 35 4.3 Cuspidal Modules and Harish-Chandra Series...... 35 4.4 The Principal Series Representations ...... 38

1 5 The Steinberg Module 39 5.1 Steinberg’s Original Results on the “Steinberg Module” ...... 39 5.1.1 A Basis for the Steinberg Module ...... 39 5.1.2 An Irreducibility Criterion for the Steinberg Module ...... 40 5.2 A New Proof of the Irreducibility Criterion for the Steinberg Module . . . . 43 5.2.1 Introduction ...... 43 5.2.2 The Proof of the Irreducibility Criterion ...... 43

6 Hecke Algebras 46 6.1 The Hecke algebra associated to a Coxeter system ...... 46 6.2 Parabolic Subalgebras of He ...... 46 6.3 Specializing the Generic Hecke Algebra ...... 47 6.4 The “Double Coset” Hecke Algebra ...... 48 6.4.1 Deﬁnition of a Hecke Algebra given in [12] ...... 48 6.4.2 The case of a group with a BN-pair ...... 49 6.5 Hecke Algebra Associated with a Harish Chandra Series ...... 53

7 Highest Weight Categories and Quasi-hereditary Algebras 57 7.1 Preliminaries ...... 57 7.2 Homological Properties of the Standard and Costandard Objects ...... 58 7.3 The Ringel Dual ...... 59 7.4 Global Dimension of a Quasi-hereditary Algebra ...... 59 7.5 Heredity Ideals ...... 61

8 The q-Schur algebra 63

9 The Indexing of the Irreducible kGLn(q)-modules 65 9.1 The First Indexing of the Irreducible kGLn(q)-Modules (CPS) ...... 65 9.1.1 A Key Theorem ...... 65 9.1.2 A decomposition of OGLn(q) into sums of blocks ...... 66 9.1.3 An OG-module which satisﬁes the hypotheses of the key theorem . . 66 9.1.4 A Morita Equivalence for GLn(q)...... 68 9.1.5 An Indexing of the Irreducible kGLn(q)-modules ...... 69 9.2 The Second Indexing of the Irreducible kGLn(q)-Modules ...... 70 9.3 The Third Indexing of the Irreducible kGLn(q)-Modules ...... 71 9.4 Indexing of the Irreducible kGLn(q)-Modules belonging to the Unipotent Prin- cipal Series ...... 72

II Generalizing the Results of Guralnick and Tiep to Find Bounds on the Dimension of Ext1 between Irreducible Modules 73

10 A Summary of the Results of Guralnick and Tiep [35] 73 10.1 Introduction ...... 73 10.2 Irreducible Modules V with V B 6=0...... 75 10.3 Irreducible Modules V with V B =0...... 76

2 11 A Bound on the Dimension of Ext1 between Irreducibles in the Unipotent Principal Series 80 11.1 Introduction ...... 80 11.2 A Preliminary Result ...... 80 11.3 Case I: r - |B| ...... 81 11.4 Case II: r | |B| ...... 82

12 A New Proof of Guralnick and Tiep’s Bound on the Dimension of H1(G, V ) in the case that V B = 0 85 12.1 Introduction ...... 85 1 12.2 A ﬁrst result on the dimension of H (G, V ) when V/∈ Irrk(G|B)...... 85 12.3 Groups of Rank 1 ...... 86 12.4 Groups of Higher Rank ...... 88

13 A Bound on the Dimension of Ext1 Between Principal Series Irreducible Modules in Cross Characteristic 91 13.1 Introduction ...... 91 1 13.2 A Bound on the Dimension of ExtkG(Y,V ) when Y and V belong to distinct Principal Series ...... 92 1 13.3 A Bound on the Dimension of ExtkG(Y,V ) when Y and V belong to the same Principal Series ...... 94 13.3.1 Case I: r - |B| ...... 95 13.3.2 Case II: r | |B| ...... 95

14 A Bound on the Dimension of Ext1 between a Unipotent Principal Series Representation and an Irreducible Outside the Unipotent Principal Series 98 14.1 Introduction ...... 98 14.2 Some Preliminaries from [29, Section 4]: The Steinberg Module and Harish- Chandra Series ...... 98 14.2.1 Introduction ...... 98 14.2.2 The BN-Pair of G ...... 99 14.2.3 An r-modular system...... 99 14.2.4 An OG-lattice which yields the Steinberg module ...... 100 14.2.5 The Gelfand-Graev module ...... 100 14.2.6 Harish-Chandra series arising from the regular character σ ...... 101 1 14.3 A Bound on dim ExtkG(Y,V )...... 103

III Explicit Computations of Bounds on the Dimension of Ext1 Between Irreducible Modules for GLn(q) in Cross Characteristic106

15 Introduction 106

16 The Harish-Chandra Vertex of the Module D2(1, λ) 107

17 Some Examples of Bounds on the Dimension of Ext1 109

3 IV Generalizing the Results of Cline, Parshall, and Scott [9] to Calculate Ext Groups between Irreducible kGLn(q)-Modules 112

18 Self-Extensions of Irreducible Modules for GLn(q) in Certain Cross Char- acteristics 112 18.1 Introduction ...... 112 1 18.2 ExtkG(D(1, λ),D(1, λ)) in the case that λ ` n is l-restricted...... 113

19 Calculations of Higher Ext Groups Between Certain Modules for GLn(q) in Cross Characteristic 116 19.1 Introduction ...... 116 19.2 A Generalization of [9, Theorem 12.4] ...... 116

V Conclusion 124

VI Appendices 126

20 Cohomology of Cyclic r-Groups 126

21 The Steinberg Module in Deﬁning Characteristic 128

22 The Character of the Steinberg Module in Characteristic 0 130 22.1 Introduction ...... 130 22.2 A generalized character of G ...... 130

22.3 The character of StC ...... 132

4 0 Preface

In the second half of the 20th century, one of the biggest open problems in mathematics was the classification of the finite simple groups. Research which would eventually serve as the basis for this classification emerged in the 1950s, but there was no organized effort to classify the finite simple groups until the 1970s. A major breakthrough toward the classification came in 1963 with the Feit-Thompson Theorem, which showed that every finite group of odd order is solvable. In 1972, Daniel Gorenstein gave a series of lectures at the University of Chicago in which he outlined a program for the classification of the finite simple groups [27]. The mathematical community responded to Gorenstein’s plan with great enthusiasm. In the years after Gorenstein presented his program, over 100 mathematicians across the world worked to implement his ideas. In 1981, Gorenstein announced that the proof of the classification theorem was complete. However, upon review, several gaps were discovered in the proof, and work on the classification theorem continued. The problems with the proof of the classification theorem were highly non-trivial; it took Aschbacher and Smith seven years (and two books) to resolve the last gap [27]. Ashbacher and Smith finished their work in 2004, and the consensus now is that the proof of the classification theorem is complete. (Goren- stein died in 1992, twelve years before the proof of the classification theorem was completed.)

The current proof of the classification theorem is immense - it consists of over 10,000 pages across about 500 journal articles [27]. Due to the scope of this work, few mathematicians understand the full proof of the classification theorem. In the 1980s, Gorenstein initiated an effort to address this issue and began working together with Lyons and Solomon to revise and shorten the proof of the classification theorem. After Gorenstein’s death, Lyons and Solomon continued working on this “revisionism” project. The American Mathematical Society is publishing their work in a multi-volume series [32]; the first six volumes of this series were published between the years of 1994 and 2005 (the first three are introductory), and more are expected in the future [27].

According to the classification theorem, there are 18 infinite families of finite simple groups as well as 26 sporadic groups. In this dissertation, we will be concerned with one particular class of finite simple groups - the finite groups of Lie type (which are discussed in Chapter 3). Since most of the finite simple groups are either finite groups of Lie type or closely related to finite groups of Lie type [36, Theorem 1.1], the representation theory of finite groups of Lie type is of particular interest.

Let q be a power of a prime p, and let G(q) be a finite group of Lie type. Let k be an algebraically closed field of characteristic r. There are three distinct cases to consider in the representation theory of kG(q): (1) r = 0 (the characteristic 0 case), (2) r = p (the defining characteristic case), and

5 (3) r > 0, r 6= p (the non-defining characteristic, or cross-characteristic, case). In each case, one area of interest is the parameterization of the irreducible kG(q)-modules. This task is almost complete in the characteristic 0 and in the defining characteristic cases [36]. However, only partial results exist for the alternating groups and the finite groups of Lie type [36, 4.2.2]. In the defining and non-defining characteristic cases, another area of interest is the homological algebra of kG(q) (when char(k) = 0, kG(q) is semisimple by Maschke’s theorem and there are no non-zero cohomology groups in positive degree). One particular problem in both the defining and non-defining characteristic cases is to find bounds on the dimension of the cohomology groups Hi(G(q),V )(i ≥ 1), where V is any irreducible kG(q)- module.

In 2009, working in the deﬁning characteristic, Cline, Parshall, and Scott [10] showed that the dimension of H1(G(q),V ) (as a k-vector space) is bounded in the following sense.

Theorem 0.0.1. ([10]) Suppose that char(k) = p > 0, let q be a power of p, and let G(q) be a ﬁnite group of Lie type over Fq. Then, there is a constant C which depends only on the rank of G(q) such that dim H1(G(q),V ) < C for all irreducible kG(q)-modules V .

After Cline, Parshall, and Scott discovered their bound on the dimension of H1(G(q),V ) in defining characteristic, Guralnick and Tiep worked to find a bound on the dimension of H1(G(q),V ) in non-defining characteristic. In 2011, Guralnick and Tiep published the following result.

Theorem 0.0.2. ([35]) Suppose that q is a power of p, G(q) is a ﬁnite group of Lie type over Fq, and char(k) = r > 0, with r 6= p. Let W be the Weyl group of G(q), and let e be the Lie rank of G(q). If V is an irreducible kG-module, then ( 1 if V B = 0 dim H1(G(q),V ) ≤ |W | + e if V B 6= 0

(where B is a Borel subgroup of G(q)).

Guralnick and Tiep’s bounds on the dimension of H1(G(q),V ) (where V is an irreducible kG(q)-module and char(k) = r6 | q) inspired much of the original research presented in this dissertation. To simplify notation, let G(q) = G. Our task was to generalize Guralnick i and Tiep’s results in order to ﬁnd bounds on the dimension of ExtkG(Y,V ), i ≥ 1, when char(k) = r > 0, r6 | q, and Y and V are irreducible kG-modules.1

In Part II of this thesis, we combine the techniques of Guralnick and Tiep [35] with tech- 1 niques of modular Harish-Chandra theory to ﬁnd bounds on the dimension of ExtkG(Y,V ) 1Bendel, Nakano, Parshall, Pillen, Scott, and Stewart [2] showed that the dimensions of the Ext groups between irreducible kG-modules in the deﬁning characteristic are bounded in all homological degrees (with the bounds depending on the homological degree and the root system Φ of G).

6 in cross characteristic in the case that G has a split BN-pair.2 Let B be a Borel subgroup of G containing a maximal torus T . We consider the following cases.

1. Y,V ∈ Irrk(G|B) = Irrk(G|(T, k)) (where Irrk(G|B) is the unipotent principal Harish- Chandra series, deﬁned in Section 4.4)

If Y and V are both unipotent principal series representations, we show that 1 dim ExtkG(Y,V ) ≤ |W | + (dim Y )e (Theorem 11.4.3). (In fact, by an argument 1 analogous to that given in Corollary 13.3.4, it follows that dim ExtkG(Y,V ) ≤ |W | + min(dim Y, dim V)e). Here, the key idea is to work with the r-Sylow subgroup Tr of the abelian group T . Since Tr is an abelian r-group, it is possible to compute the exact 1 1 dimension of ExtkTr (k, k). This, in turn, yields a bound on dim ExtkTr (k, V ) for any 1 kTr-module V (Lemma 11.4.2). Finally, to ﬁnd a bound on dim ExtkG(Y,V ), we use a 1 long exact sequence in Ext to reduce the calculation of ExtkG(Y,V ) to a calculation of an Ext group over kTr. 0 0 2. Y ∈ Irrk(G|(T,X)), V ∈ Irrk(G|(T,X ) (where X and X are any irreducible kT -modules)

We assume that Y and V are principal series representations (though not necessarily belonging to the unipotent principal series). When Y and V belong to distinct prin- 1 cipal Harish-Chandra series, we show that dim ExtkG(Y,V ) = 0 (Corollary 13.2.3). And, when Y and V belong to the same principal series Irrk(G|(T,X)), we show that 1 2 dim ExtkG(Y,V ) ≤ |W | + min(dim Y, dim V)e (Corollary 13.3.4). In the case that Y and V belong to distinct Harish-Chandra series, we generalize [35, Theorem 2.2] to show 1 that dim ExtkG(Y,V ) is equal to the number of times V occurs as a composition factor G of a certain submodule of RT (X) (Theorem 13.2.1). We then use properties of Harish- Chandra induction and restriction to show that V cannot occur as a composition factor G of RT (X) (Proposition 13.2.2), and the desired result follows. In the case that Y and V belong to the same principal series Irrk(G|(T,X)), we obtain the desired bound on 1 dim ExtkG(Y,V ) by reducing to the subgroup Tr of T .

3. Y ∈ Irrk(G|B), V 6∈ Irrk(G|B)

We assume that Y is an irreducible kG-module belonging to the unipotent principal series, and that V is an irreducible kG-module outside of the unipotent principal series. We G assume additionally that the pair (G, k) is such that every composition factor of k|B belongs to a Harish-Chandra series arising from a regular character of the unipotent radical U of B (in other words, the pair (G, k) has property (P), as defined in Chapter 14). In this case, we show that there is a subset J ⊆ S (where S is a set of fundamental reflections in W ) such 1 that dim ExtkG(Y,V ) ≤ [W : WJ ] (Theorem 14.3.3). To prove that this bound holds, we use 1 the fact that dim ExtkG(Y,V ) is equal to the number of times V occurs as a composition G factor of a certain submodule of k|B. In this case, though, the basic properties of Harish- 2“Split” BN-pairs are defined in Section 2.3. The assumption that G is split is needed in order to be able to use the results of modular Harish-Chandra theory given in [31, Section 4].

7 G Chandra induction and restriction are not sufficient to find a bound on [k|B : V ]. Therefore, we introduce the Gelfand-Graev module for G and, using properties of the Gelfand-Graev G 1 module, find a bound on [k|B : V ], which yields a bound on dim ExtkG(Y,V ).

In Part III of this thesis, we compute some specific examples of bounds on the dimension 1 of Ext in the case that G = GLn(q) is the general linear group over the finite field Fq (as above, we assume char(k) = r > 0 with r6 | q). There are three widely used parameterizations of the irreducible kGLn(q)-modules (we describe these parameterizations in Chapter 9). In our examples, we work with the parameterization given by Dipper and Du in [19, 4.2.11, (6)]. Using the results of their papers [18] and [19], Dipper and Du determined how the irreducible kGLn(q)-modules are partitioned into Harish-Chandra series. In [19, Section 4], Dipper and Du describe an algorithm which can be used to find the Harish-Chandra series containing an irreducible kGLn(q)-module indexed as in [19, 4.2.11, (6)]. Since our bounds on the dimension of Ext1 between irreducible kG-modules depend on the Harish-Chandra series containing these irreducibles, Dipper and Du’s indexing is an ideal choice for the computations of Part III.

Finally, in Part IV of this thesis, we turn to calculations of higher Ext groups. In this case, our main reference is [9], in which Cline, Parshall, and Scott connect Hi calculations for i 3 G = GLn(q) in cross characteristic to Ext calculations over a q-Schur algebra. Following [9], we assume that r 6 | q(q − 1). The case of i = 1 is handled in [9, Theorem 10.1], where CPS compute H1(G, L) for an irreducible kG-module L. A key component of the proof of [9, Theorem 10.1] is the fact that k (the trivial irreducible kG-module) is in the head of the G permutation module k|B. But, any irreducible kG-module in the unipotent principal series G Irrk(G|B) is in the head of k|B. This observation allows us to generalize [9, Theorem 10.1] in 1 Chapter 18. We give our generalization in Theorem 18.2.1, where we compute ExtkG(Y,V ) between an irreducible kG-module Y belonging to the unipotent principal series and an irreducible kG-module V belonging to any Harish-Chandra series.

Cline, Parshall, and Scott’s calculation of the higher cohomology groups Hi(G, L) (where L is an irreducible kG-module) is given in [9, Theorem 12.4]. The proof of [9, Theorem 12.4] is signiﬁcantly more challenging than the proof of [9, Theorem 10.1] and requires more background machinery. The key component of the proof of [9, Theorem 12.4] is the existence of a resolution of the trivial kG-module k which possesses certain desirable properties [9, Lemma 12.3]. We give our generalization of [9, Theorem 12.4] in Chapter 19. Using techniques of Parshall and Scott [44], we construct a resolution analogous to that of CPS in Lemma 19.2.5. And, in Theorem 19.2.6, we use this resolution to relate certain higher Ext groups for GLn(q) in cross-characteristic to Ext groups over a q-Schur algebra.

3Exti calculations over a q-Schur algebra can be translated to Exti calculations in the category of inte- ˜ i grable modules for the quantum enveloping algebra Uq(gln). So, the connection between Ext calculations i for GLn(q) and Ext calculations over a q-Schur algebra opens the possibility of using the theory of quantum groups to learn more about the structure of Ext groups for GLn(q).

8 Part I Background

1 The Ext Functor and Group Cohomology

1.1 The Ext Functor

In this section, we deﬁne the Ext functor, using [49, 2.5] as our main reference. Let R be a ring (with unit), and let R − mod be the category of left R-modules. For any R-module M, the functor HomR(M, −) is left exact. Since R − mod has enough injectives, we can i construct the right derived functors R HomR(M, −)(i ≥ 0). Let V ∈ R − mod, and let 0 → V → I0 → I1 → I2 → · · · be an injective resolution of V by left R-modules. Consider the complex HomR(M,I) obtained by applying the covariant left exact functor HomR(M, −) to the injective resolution of V given above:

0 1 2 0 → HomR(M,I ) → HomR(M,I ) → HomR(M,I ) → · · · .

i i For i ≥ 0, R HomR(M, −)(V ) = H (HomR(M,I)) is the ith cohomology of this complex 0 ∼ i i (where H (HomR(M,I)) = HomR(M,V ) and H (HomR(M,I)) = Ker(HomR(M,I ) → i+1 i−1 i HomR(M,I ))/Im(HomR(M,I ) → HomR(M,I )) for i ≥ 1).

i i i For i ≥ 0, set ExtR(M, −) = R HomR(M, −). The functors ExtR(M, −)(i ≥ 0) are i called the Ext functors. For any V ∈ R − mod and i ≥ 0, ExtR(M,V ) is called an Ext i group. (The deﬁnition of ExtR(M,V ) is independent of the choice of injective resolution of V .)

Remark 1.1.1. If G is a ﬁnite group and k is a ﬁeld, we can compute Ext groups over the group algebra R = kG. If M and V are any kG-modules, then HomkG(M,V ) has the struc- i ture of a k-vector space. Hence, the Ext groups ExtkG(M,V )(i ≥ 0) are also k-vector spaces.

i It is also possible to compute ExtR(M,V )(i ≥ 0) using a projective resolution of M. Let · · · → P 2 → P 1 → P 0 → M → 0 be a projective resolution of M in R − mod. Applying the contravariant left exact functor HomR(−,V ) to this projective resolution of M, we obtain the complex

0 1 2 0 → HomR(P ,V ) → HomR(P ,V ) → HomR(P ,V ) → · · · . The ith cohomology of this complex gives the Ext group Exti(M,V ) for i ≥ 0. (This deﬁni- i tion of ExtR(M,V ) is independent of the choice of projective resolution of M.)

9 Proposition 1.1.2. ([49, 2.5.1]) The following statements are equivalent. (1) V is an injective R-module.

(2) HomR(−,V ) is an exact functor. 1 (3) ExtR(M,V ) = 0 for all M ∈ R − mod. i (4) If i ≥ 1, ExtR(M,V ) = 0 for all M ∈ R − mod.

Proposition 1.1.3. ([49, 2.5.1]) The following statements are equivalent. (1) M is a projective R-module.

(2) HomR(M, −) is an exact functor. 1 (3) ExtR(M,V ) = 0 for all V ∈ R − mod. i (4) If i ≥ 1, ExtR(M,V ) = 0 for all V ∈ R − mod.

1.2 Ext and Extensions

If M,V ∈ R − mod, an extension of M by V is a short exact sequence of the form 0 → V → X → M → 0 for some X ∈ R − mod. Two extensions 0 → V → X → M → 0 and 0 → V → X0 → M → 0 are said to be equivalent if there is a commutative diagram 0 V X M 0

= =∼ = 0 V X0 M 0. An extension is called split if it is equivalent to 0 → V → V ⊕ M → M → 0 (where V → V ⊕ M is the natural inclusion map and V ⊕ M → M is the natural projection map).

1 Proposition 1.2.1. ([49, 3.4.1]) If ExtR(M,V ) = 0, then every extension of M by V is split.

1 Proposition 1.2.2. ([49, 3.4.3]) There is a 1-1 correspondence between ExtR(M,V ) and equivalence classes of extensions of M by V (in which the split extension corresponds to 1 0 ∈ ExtR(M,V )). The higher Ext groups can be analogously interpreted in terms of extensions. Given an integer n > 0, an exact sequence in R − mod of the form

0 → V → Xn−1 → · · · → X0 → M → 0

is called an extension of degree n of M by V . If E and E0 are two degree n extensions of M by V , we say that E is congruent to E0 and write E ` E0 if there is a chain map E → E0 such that the maps on M and V are the identity maps [7, pg. 14]. The congruence relation ` is not an equivalence relation on the set of degree n extensions (` need not be symmetric). But, ` can be used to deﬁne an equivalence relation ∼ on the set of degree n extensions of

10 M by V . We will say that E is equivalent to E0 (and write E ∼ E0) if there exists a sequence F0,F1,...,F2t (t ∈ N) of degree n extensions of M by V such that

0 E a F0 ` F1 a F2 ` · · · a F2t ` E .

n Proposition 1.2.3. ([7, pg. 14]) There is a 1-1 correspondence between ExtR(M,V ) (n > 0) and equivalence classes of degree n extensions of M by V (under the equivalence relation ∼ described above).

1.3 Some Properties of the Ext Functor

Ext functors have been extensively studied, and there is a wide range of literature covering the properties and applications of Ext. In this section, we will describe several fundamental properties of the Ext functors which will be used in the original proofs of this dissertation.

1.3.1 The Long Exact Sequence in Ext

The long exact sequence in Ext was an indispensable tool in our search for bounds on the dimension of Ext groups.

Theorem 1.3.1. ([7, 1.5.6]) Suppose that 0 → M1 → M → M2 → 0 is a short exact sequence of R-modules. Given any R-module V , there is a long exact sequence

1 1 0 → HomR(M2,V ) → HomR(M,V ) → HomR(M1,V ) → ExtR(M2,V ) → ExtR(M,V ) →

1 n−1 n−1 n n ExtR(M1,V ) → · · · → ExtR (M,V ) → ExtR (M1,V ) → ExtR(M2,V ) → ExtR(M,V ) → · · · .

Theorem 1.3.2. ([7, 1.5.7]) Suppose that 0 → V1 → V → V2 → 0 is a short exact sequence of R-modules. Given any R-module M, there is a long exact sequence

1 1 0 → HomR(M,V1) → HomR(M,V ) → HomR(M,V2) → ExtR(M,V1) → ExtR(M,V ) →

1 n−1 n−1 n n ExtR(M,V2) → · · · → ExtR (M,V ) → ExtR (M,V2) → ExtR(M,V1) → ExtR(M,V ) → · · · .

1.3.2 The Eckmann-Shapiro Lemma

Suppose that A is a subring of a ring R. Given an R-module M, we can restrict the action R of R on M to an action of A on M and define the restricted A-module M ↓A. Given an R A-module N, we can define an induced R-module N ↑A= R ⊗A N. And, viewing R as an A − R-bimodule by left and right multiplication, we can define a co-induced R-module R N ⇑A= HomA(R,N).

Proposition 1.3.3. ([3, Lemma 2.8.2]) Suppose that A is a subring of R, M is an R-module, and N is an A-module. Then,

11 R ∼ R (1) HomA(N,M ↓A) = HomR(N ↑A,M), and R ∼ R (2) HomA(M ↓A,N) = HomR(M,N ⇑A).

R R Therefore, the restriction functor ↓A has left adjoint ↑ and right adjoint ⇑ .

Remark 1.3.4. Suppose that k is a field, G is a finite group, and R = kG is the group algebra of G. If H is a subgroup of G, then kH is a subalgebra of kG. Given a kG-module M and kG G kG G kG G a kH-module N, we write M ↓kH = M ↓H , N ↑kH = N ↑H , and N ⇑kH = N ⇑H . In the case G ∼ G that G is a finite group, induction and coinduction coincide on A−mod, so N ↑H = N ⇑H for any subgroup H ≤ G and kH-module N [3, 3.3]. Hence, Proposition 1.3.3 may be restated as follows.

Proposition 1.3.5. Suppose that H ≤ G and k is a ﬁeld. Then, G ∼ G (1) HomkH (N,M ↓H ) = HomkG(N ↑H ,M), and G ∼ G (2) HomkH (M ↓H ,N) = HomkG(M,N ↑H ).

The isomorphism of Proposition 1.3.5 (1) is known as Frobenius reciprocity.

We now return to the more general setting in which R an arbitrary ring with unit and A a subring of R.

Proposition 1.3.6. (Eckmann-Shapiro Lemma) Suppose that A is a subring of R, and that R is projective as an A-module. And, suppose that M is an R-module and N is an A-module. Then, for any n ≥ 0, n R ∼ n R (1) ExtA(N,M ↓A) = ExtR(N ↑A,M), and n R ∼ n R (2) ExtA(M ↓A,N) = ExtR(M,N ⇑A).

Proof. We will prove the statement of (1) (the statement of (2) can be proved using similar arguments). We follow Benson’s proof in [3, 2.8.4]. Suppose that · · · → P2 → P1 → P0 → N → 0 is a projective resolution of N as an A-module. Since R is projective as an A-module, R the induction functor ↑A is exact. Therefore, R R R R · · · → P2 ↑A→ P1 ↑A→ P0 ↑A→ N ↑A→ 0 R is an exact sequence of R-modules with Pi ↑A projective for i ≥ 0.

n R For n ≥ 0, ExtR(N ↑A,M) is the nth cohomology of the complex R R R 0 → HomR(P0 ↑A,M) → HomR(P1 ↑A,M) → HomR(P2 ↑A,M) → · · · . n R Now, ExtA(N,M ↓A) is the nth cohomology of the complex R R R 0 → HomA(P0,M ↓A) → HomA(P1,M ↓A) → HomA(P2,M ↓A) → · · · .

12 R ∼ R But, for any i ≥ 0, HomA(Pi,M ↓A) = HomR(Pi ↑A,M) by Proposition 1.3.3 (1), and the desired result follows.

Remark 1.3.7. Suppose that G is a ﬁnite group, H ≤ G, and k is a ﬁeld. In this case, kG is projective (in fact, free) as a kH-module and we have the following version of the Eckmann-Shapiro Lemma.

Proposition 1.3.8. If M is a kG-module, and N is a kH-module, then for any n ≥ 0, n G ∼ n G (1) ExtkH (N,M ↓H ) = ExtkG(N ↑H ,M), and n G ∼ n G (2) ExtkH (M ↓H ,N) = ExtkG(M,N ↑H ).

For n = 0, the statement of Proposition 1.3.8 (1) is simply the statement of Frobe- nius reciprocity. Thus, the Eckmann-Shapiro Lemma may be viewed as a generalization of Frobenius reciprocity to higher Ext groups.

1.3.3 Ext1 between Irreducible Modules

Assume that R is an Artinian ring. The purpose of this section is to give an interpretation 1 0 0 of dim ExtR(S,S ) when S and S are irreducible R-modules. We will follow [3, Section 2.4].

We deﬁne several important submodules and quotient modules of an R-module M. The socle of M (denoted soc(M)) is the direct sum of all of the irreducible submodules of M. The radical of M (denoted Rad(M)) is the intersection of the maximal submodules of M. And, the head of M is the quotient module head(M) = M/Rad(M). M is completely reducible if and only if Rad(M) = 0. In particular, since Rad(head(M)) = 0, head(M) is completely reducible.

Since R is Artinian, any ﬁnitely generated R-module M has a “minimal” projective resolution. (A projective resoluion · · · → P2 → P1 → P0 → M → 0 is minimal if each Pi is a projective cover of its image in Pi−1.)

Proposition 1.3.9. Let S and S0 be irreducible R-modules, and let P be a projective inde- 1 0 ∼ 2 0 composable module with head(P ) = S. Then, ExtR(S,S ) = HomR(Rad(P )/Rad (P ),S ).

∂3 ∂2 ∂1 ∂0 Proof. Let ··· → P2 → P1 → P0 → S → 0 be a minimal projective resolution of S. By minimality, P0 = P is the projective cover of S, and for all i ≥ 1, Pi is the projective cover 0 ∂3 ∂2 ∂1 of Im ∂i. We apply the functor HomR(−,S ) to the complex ··· → P2 → P1 → P0 to obtain the complex

0 1 2 0 δ 0 δ 0 δ 0 → HomR(P0,S ) → HomR(P1,S ) → HomR(P2,S ) → · · · . (1.1)

13 1 0 1 0 By deﬁnition, ExtR(S,S ) = Ker δ /Im δ .

1 0 0 We claim that ExtR(S,S ) = HomR(P1,S ). In fact, we will show that all of the i 1 0 maps δ in the complex (1.1) are zero maps (from which it will follow that ExtR(S,S ) = 1 0 i 0 0 Ker δ = HomR(P1,S )). For i ≥ 0, the map δ : HomR(Pi,S ) → HomR(Pi+1,S ) is 0 given by precomposition with ∂i+1 : Pi+1 → Pi. So, given f ∈ HomR(Pi,S ), the element i 0 i δ (f) ∈ HomR(Pi+1,S ) is deﬁned by δ (f) = f ◦∂i+1. Note that Im ∂i+1 = Ker ∂i ⊆ Rad(Pi), where the inclusion Ker ∂i ⊆ Rad(Pi) holds since Pi is the projective cover of Im ∂i. 0 0 i Let 0 6= f ∈ HomR(Pi,S ). Since S is simple, f is surjectve. Thus, δ (f)(Pi+1) = 0 i (f ◦ ∂i+1)(Pi+1) = f(Im ∂i+1) ⊆ f(Rad(Pi)) ⊆ RadS = 0. So, δ = 0, and the claim is proved.

Now, the module P1 in the projective resolution of S is the projective cover of Im ∂1 = Ker ∂0 = Rad(P ) (where we have used the fact that P0 = P ). To ﬁnd the projective cover of Rad(P ), we write the completely reducible module Rad(P )/Rad2(P ) as a direct sum of sim- 2 ⊕ni ple modules. Suppose Rad(P )/Rad (P ) = ⊕ Si , where each Si is a simple R-module. For i any i, let Qi be the projective indecomposable R-module with head Si. Then, the projective ⊕ni ⊕ni cover of Rad(P ) is ⊕ Qi , which means that P1 = ⊕ Qi in the minimal projective res- i i 1 0 ⊕ni 0 olution of S. Thus, by the claim of the previous paragraph, ExtR(S,S ) = HomR(⊕ Qi ,S ). i

0 ⊕ni 0 ∼ ⊕ni ⊕ni 0 ∼ Finally, since S is simple, we have HomR(⊕ Qi ,S ) = HomR(⊕ Qi /Rad(⊕ Qi ),S ) = i i i ⊕ni 0 2 0 HomR(⊕ Si ,S ) = HomR(Rad(P )/Rad (P ),S ). i

Remark 1.3.10. It is possible to generalize Proposition 1.3.9 to higher Ext. Let M be an ar- ∂3 ∂2 ∂1 bitrary ﬁnitely generated R-module, and let S be a simple R-module. Let ··· → P2 → P1 → ∂0 n ∼ P0 → M → 0 be a minimal projective resolution of M. Set Ω (M) = Ker(∂n−1) = Im(∂n).

n ∼ n We claim that ExtR(M,S) = HomR(Ω (M),S) for all n ≥ 1. To see why this is the case, apply HomR(−,S) to the minimal projective resolution of M to obtain the complex

0 1 2 0 δ 0 δ 0 δ 0 → HomR(P0,S ) → HomR(P1,S ) → HomR(P2,S ) → · · · . An argument analogous to that given in the proof of Proposition 1.3.9 shows that δi = 0 n ∼ for all i ≥ 0. Thus, for n ≥ 1, ExtR(M,S) = HomR(Pn,S). Now, since Pn is the projective cover of Im(∂n), Ker(∂n) is contained in Rad(Pn). And, since Rad(S) = 0, ∼ ∼ ∼ n HomR(Pn,S) = HomR(Pn/Ker(∂n),S) = HomR(Im(∂n),S) = HomR(Ω (M),S).

Suppose now that R is an Artinian k-algebra. If S is an irreducible R-module and M is any ﬁnitely generated R-module, dim HomR(M,S) is equal to the number of times that S appears in head(M). So, we have the following corollary to Proposition 1.3.9.

14 0 1 0 Corollary 1.3.11. If R is a k-algebra and S, S are irreducible R-modules, then dim ExtR(S,S ) is the number of times that S0 occurs in the head of Rad(P ).

1 0 2 0 Proof. By Proposition 1.3.9, dim ExtR(S,S ) = dim HomR(Rad(P )/Rad (P ),S ), which is the number of times S0 appears in head(Rad(P )/Rad2(P )) = Rad(P )/Rad2(P ) = head(Rad(P )).

In [35, Theorem 2.2], Guralnick and Tiep give a result analogous to that of Corollary 1.3.11, proving that the dimension of H1(G, V ) (where G is a finite group of Lie type, k is an algebraically closed field of characteristic different from the defining characteristic of G, and V is an irreducible kG-module with V B = 0) is equal to the multiplicity of V in the G head of a certain submodule of k|B. In order to generalize Guralnick and Tiep’s bounds on the dimension of Ext, it was necessary to generalize the result of [35, Theorem 2.2]. The fact that Corollary 1.3.11 has a homological proof suggested that [35, Theorem 2.2] should have a similar homological proof. This hypothesis turned out to be correct; in Proposition 12.2.1, we give a new homological proof of [35, Theorem 2.2], which can be generalized to extend Guralnick and Tiep’s results.

1.4 Group Cohomology

1.4.1 Group Cohomology via the Standard Resolution

In this section, we will define group cohomology (following [3, Section 3.4]). Let G be a finite group, and let R be a fixed commutative ring. We view R as an RG-module with trivial G-action. For any n ≥ 0, let Fn be the free R-module on (n+1)-tuples of elements of G with the action of G given by g(g0, . . . , gn) = (gg0, . . . , ggn) when g ∈ G and (g0, . . . , gn) ∈ Fn. There is a long exact sequence

∂2 ∂1 aug · · · → F2 → F1 → F0 → R → 0,

with the boundary map ∂n (n ≥ 1) deﬁned by

n X i ∂n(g0, . . . gn) = (−1) (g0,..., gbi, . . . gn) i=0

for any (g0, . . . , gn) ∈ Fn (the notation gi indicates that gi has been omitted). The map b P P aug : F0 → R is the augmentation map, with aug( agg) = ag for all ag ∈ R. This g∈G g∈G resolution of R by RG-modules is called the standard (or bar) resolution.

Given an RG-module M, we can apply the contravariant Hom functor HomRG(−,M) to the standard resolution to obtain the complex

δ0 δ1 δ2 0 → HomRG(F0,M) → HomRG(F1,M) → HomRG(F2,M) → · · · ,

15 n n where δ : HomRG(Fn,M) → HomRG(Fn+1,M) is given by δ (f) = f ◦ ∂n+1 for any f ∈ HomRG(Fn,M).

Deﬁnition 1.4.1. For n = 0, let H0(G, M) = Ker δ0. And, for n ≥ 1, let Hn(G, M) = Ker δn/Im δn−1. We say that Hn(G, M)(n ≥ 0) is the nth cohomology group of G with coeﬃcients in M.

From the deﬁnition, it is clear that

n ∼ n H (G, M) = ExtRG(R,M) for any n ≥ 0. In particular, this means that the properties of Ext discussed in the previous sections carry over to group cohomology. For instance, any short exact sequence of RG- modules yields a long exact sequence in group cohomology.

1.4.2 Group Cohomology via Cochains

It is possible to deﬁne group cohomology without reference to the standard resolution of R by free RG-modules. Given an RG-module M, let C0(G, M) = M, and let Cn(G, M) (n ≥ 1) be the collection of maps Gn → M (where Gn is the direct product of n copies of G). Then, Cn(G, M) is an abelian group for all n ≥ 0. (C0(G, M) = M is naturally an abelian group; and for n ≥ 1, Cn(G, M) is given the structure of an abelian group via pointwise addition of functions.) The elements of Cn(G, M) are called n-cochains.

n ∼ For any n ≥ 0, there is an isomorphism of abelian groups C (G, M) = HomRG(Fn,M). With this identiﬁcation, the the nth coboundary homomorphism δn : Cn(G, M) → Cn+1(G, M) is given by n δ (f)(g1, . . . , gn+1) = n X i n+1 gf(g2, . . . , gn+1) + (−1) f(g1, . . . , gi−1, gigi+1, gi+2, . . . , gn+1) + (−1) f(g1, . . . , gn) i=1 n n+1 for any f ∈ C (G, M) and (g1, . . . , gn+1) ∈ G .

For any n ≥ 0, the elements of Ker δn are called n-cocycles. For any n ≥ 1, the elements of Im δn−1 are called n-coboundaries. As above, we can deﬁne H0(G, M) = Ker δ0 and Hn(G, M) = Ker δn/Im δn−1 for n ≥ 1.

The following result is a basic consequence of the deﬁnition of H0(G, M).

Proposition 1.4.2. If M is any RG-module, H0(G, M) ∼= M G.

16 1.4.3 Group Cohomology in Low Degree

The low degree cohomology groups H1(G, M) and H2(G, M) can be interpreted in terms of extensions. If G is a group and M is an RG-module, then an extension of G by M is a short exact sequence 1 → M → Gˆ → G → 1 (where M is viewed as a group under addition, written multiplicatively). The extension is said to be split if there is a copy of G in Gˆ such that M and G are complementary subgroups in Gˆ (that is, if Gˆ = MG and M ∩ G = {1}).

Proposition 1.4.3. ([3, 3.7.2]) Let G be a group, let M be an RG-module, and let Gˆ be a group in which G and M are complementary subgroups (so that there is a split extension 1 → M → Gˆ → G → 1). The ﬁrst cohomology group H1(G, M) parameterizes the conjugacy classes of complements to M in Gˆ.

Proposition 1.4.4. ([3, 3.7.3]) Let G be a group, and let M be an RG-module. The second cohomology group H2(G, M) parameterizes the isomorphism classes of extensions of G by M.

1.4.4 The inﬂation homomorphism on group cohomology.

Suppose that N is a normal subgroup of G and M is an RG-module. Then, the set M N of fixed points of M under the action of N has the structure of an R(G/N)-module. Now, N let π : G G/N be the natural projection map, and let ι : M ,→ M be the inclusion of M N into M. The homomorphism π : G → G/N induces a map φn : Gn → (G/N)n for all n ≥ 1. Therefore, for any n ≥ 1, there is an induced homomorphism Cn(G/N, M N ) → Cn(G, M), given by f 7→ ι ◦ f ◦ φn for any f ∈ Cn(G/N, M N ). And, when n = 0, the inclusion map ι gives a homomorphism C0(G/N, M N ) = M N → C0(G, M) = M. The maps Cn(G/N, M N ) → Cn(G, M) defined above induce homomorphisms on cohomology inf : Hn(G/N, M N ) → Hn(G, M) for all n ≥ 0. The induced homomorphism inf on cohomology is called the inflation homomorphism.

The inflation homomorphism fits into a famous five-term exact sequence (called the inflation-restriction exact sequence).

Proposition 1.4.5. If N is a normal subgroup of G and M is an RG-module, then there is a ﬁve-term exact sequence

0 → H1(G/N, M N ) →inf H1(G, M) → H1(N,M)G/N → H2(G/N, M N ) →inf H2(G, M).

It is possible to prove the existence of this five-term exact sequence using the standard resolutions for G and G/N along with the definition of group cohomology. However, this five-term exact sequence also follows as a consequence of a Lyndon-Hochschild-Serre spectral sequence.

17 1.5 The Restriction and Corestriction Homomorphisms on Group Cohomology

The material covered in this section can be found in [49, 6.7]. Let G be a ﬁnite group, and let R be a commutative ring.

1.5.1 The Restriction Homomorphism

Let M be an RG-module, and let H be a subgroup of G. By restriction, we can view M as an RH-module. We will deﬁne a restriction homomorphism on cohomology.

n ∼ n n ∼ n ∂3 ∂2 For n ≥ 0, H (G, M) = ExtRG(R,M) and H (H,M) = ExtRH (R,M). Let ··· → P2 → ∂1 ∂0 P1 → P0 → R → 0 be a projective resolution of R as an RG-module. Since the restriction functor from the category of RG-modules to the category of RH-mod is exact, this is also a projective resolution of R as an RH-module. Applying HomRG(−,M) to this resolution, we obtain the complex

δ0 δ1 δ2 0 → HomRG(P0,M) → HomRG(P1,M) → HomRG(P2,M) → · · · .

Similarly, we can apply HomRH (−,M) to obtain the complex

δ0 δ1 δ2 0 → HomRH (P0,M) → HomRH (P1,M) → HomRH (P2,M) → · · · . In both complexes, the coboundary map δn is given by precomposition with the morphism ∂n+1 : Pn+1 → Pn.

For all n ≥ 0, we have inclusions HomRG(Pn,M) ,→ HomRH (Pn,M), and the following commutative square.

δn HomRG(Pn,M) HomRG(Pn+1,M)

δn HomRH (Pn,M) HomRH (Pn+1,M)

Therefore, the inclusions HomRG(Pn,M) ,→ HomRH (Pn,M) give rise to a cochain map HomRG(P∗,M) → HomRH (P∗,M) and, consequently, induce a map on cohomology, called the restriction homomorphism. We denote this homomorphism by Res : Hn(G, M) → Hn(H,M).

1.5.2 The Corestriction Homomorphism

Let M be an RG-module, and let H be a subgroup of G of index m = [G : H] (since G is finite, the index m of H in G is necessarily finite). We will define the corestriction homo-

18 morphism Cor : Hn(H,M) → Hn(G, M)(n ≥ 0).

Let g1, . . . , gm be a system of representatives of the left cosets of H in G. Let n ≥ 0, and deﬁne a map Cor : HomRH (Pn,M) → HomRG(Pn,M) by

m X −1 Cor(f)(x) = gif(gi x) i=1 for f ∈ HomRH (Pn,M) and x ∈ Pn. For any f ∈ HomRH (Pn,M), h ∈ H, x ∈ Pn, and 1 ≤ i ≤ m, we have

−1 −1 −1 −1 −1 −1 gihf((gih) x) = gihf(h gi x) = gihh f(gi x) = gif(gi x).

Thus, Cor is independent of the choice of coset representatives of H in G. We claim that Cor(f): Pn → A is a G-module homomorphism for any f ∈ HomRH (Pn,M). Let g ∈ G and x ∈ Pn. Since G acts transitively on the set of left cosets of H in G, there is some permutation σ of the set {1, . . . , m} and elements hi ∈ H for all 1 ≤ i ≤ m such that m m m −1 P −1 P −1 −1 P −1 −1 g gi = gσ(i)hi. So, Cor(f)(gx) = gif(gi gx) = gif(hi gσ(i)x) = gihi f(gσ(i)x) = i=1 i=1 i=1 m m P −1 P −1 ggσ(i)f(gσ(i)x) = g gσ(i)f(gσ(i)x) = gCor(f)(x). i=1 i=1

To show that the homomorphisms Cor : HomRH (Pn,A) → HomRG(Pn,A)(n ≥ 0) induce a map on cohomology, we must check that the following square commutes:

δn HomRH (Pn,A) HomRH (Pn+1,A)

Cor Cor δn HomRG(Pn,A) HomRG(Pn+1,A).

Since ∂n+1 : Pn+1 → Pn is an RG-module homomorphism, given f ∈ HomRH (Pn,M) m n P −1 and x ∈ Pn+1, we have (Cor ◦ δ )(f)(x) = Cor(f ◦ ∂n+1)(x) = gif(∂n+1(gi x)) = i=1 m P −1 n n gif(gi ∂n+1(x)) = Cor(f)(∂n+1(x)) = δ (Cor(f))(x) = (δ ◦Cor)(f)(x). Thus, the square i=1 above commutes, which means that Cor : HomRH (P∗,M) → HomRG(P∗,A) is a cochain map. Therefore, Cor induces a homomorphism on cohomology, called the corestriction homomorphism and denoted by Cor : Hn(H,M) → Hn(G, M)(n ≥ 0).

1.6 Composition of Restriction and Corestriction

As above, let H be a subgroup of G of ﬁnite index m, and let g1, . . . , gm be a system of representatives for the left cosets of H in G.

19 Proposition 1.6.1. For any RG-module M, the composition Cor ◦ Res : Hn(G, M) → Hn(G, M) is given by multiplication by m = [G : H]. That is, for all n ≥ 0, given a cohomology class c ∈ Hn(G, M), Cor(Res(c)) = mc (where mc = c + ··· + c, m times in Hn(G, M)).

Proof. The homomorphism Cor ◦ Res : Hn(G, M) → Hn(G, M)(n ≥ 0) on cohomology is induced by the composite cochain map

Res Cor HomRG(P∗,M) → HomRH (P∗,M) → HomRG(P∗,M).

Res Cor Given any n ≥ 0, we explicitly compute the composition HomRG(Pn,M) → HomRH (Pn,M) → HomRG(Pn,M). Let f ∈ HomRG(Pn,M). Then, Res(f) = f, viewed as an RH-map by restriction. But, since f is an RG-module homomorphism, we have

m m X −1 X −1 Cor(Res(f))(x) = gif(gi x) = gigi f(x) = mf(x) i=1 i=1

for any x ∈ Pn. Thus, (Cor ◦ Res)(f) = mf for all f ∈ HomRG(Pn,M). It follows that the induced map on cohomology is given by c 7→ mc for a cohomology class c ∈ Hn(G, M).

Suppose now that R = k is a ﬁeld of characteristic r > 0.

Corollary 1.6.2. Let G be a ﬁnite group, and let H be a subgroup of G of index [G : H] = m. Suppose that k is a ﬁeld of characteristic r 6= 0, and M is a kG-module. Then, if r 6 | m, Res : Hn(G, M) → Hn(H,M) is an injective homomorphism.

Proof. If K = Ker(Res), then we have 0 = Cor(Res(K)) = mK. And, since rHn(G, M) = 0, we also have rK = 0. Now, since (r, m) = 1, we can write ar + bm = 1 for some a, b ∈ Z. Therefore, K = (ar + bm)K = arK + bmK = 0 + 0 = 0, and it follows that Res : Hn(G, M) → Hn(H,M) is an injective homomorphism.

Similarly, given RG-modules M and M 0 and a subgroup H of G, it is possible to deﬁne n 0 n 0 a restriction map Res : ExtRG(M,M ) → ExtRH (M,M ) for n ≥ 0.

Proposition 1.6.3. ([3, 3.6.18]) Let G be a ﬁnite group, and let H be a subgroup of G of index [G : H] = m. Suppose that k is a ﬁeld of characteristic r 6= 0, and let M and 0 n 0 n 0 M be kG-modules. Then, if r 6 | m, Res : ExtkG(M,M ) → ExtkH (M,M ) is an injective homomorphism.

We also mention that if G is a ﬁnite group such that r6 | |G| (where r = char(k)), then n 0 the group algebra kG is semisimple by Maschke’s Theorem and ExtkG(M,M ) = 0 for all n > 0 and for all kG-modules M and M 0.

20 2 Groups with a BN-pair

In this section, we will summarize certain basic features of groups with a BN-pair. Our main reference is Carter’s “Finite Groups of Lie Type” [8].

2.1 The Axioms of a Group with a BN-pair

Let G be a group and let B and N be two subgroups of G. The subgroups B, N form a BN-pair if the following axioms hold:

(i) G = hB,Ni. (ii) T = B ∩ N is a normal subgroup of N.

2 (iii) N/T = W is generated by a set of elements si, i ∈ I with si = 1.

(iv) Let π : N W be the natural quotient map. If ni ∈ N has image si ∈ W under π, then niBni 6= B.

(v) For each n ∈ N and each ni (i ∈ I), niBn ⊆ BninB ∪ BnB.

The group W = N/T is called the Weyl group of the BN-pair.

Given an element w ∈ W , let nw denote a preimage of w in N. We will write BwB := BnwB (this deﬁnition is independent of the choice of preimage nw of w by [8, Proposition 2.1.2]). The following result is an important consequence of the axioms of a group with a BN-pair.

Proposition 2.1.1. ([8, Proposition 2.1.2]) There exists a bijection B \ G/B ↔ W between the set of double cosets of B in G and elements of W , with the double coset BnB corresponding to the element π(n) ∈ W .

As a consequence, we have the Bruhat decomposition of G:

G = ∪ BwB, w∈W where the union is disjoint.

By [8, Proposition 2.1.7], the Weyl group W corresponding to the BN-pair of G is a Coxeter group with generators si, i ∈ I. We can deﬁne a length function l on W , where l(w) is the minimal length of an expression of w in the generators si, i ∈ I for any w ∈ W . Some basic properties of the length function l are listed in the next proposition.

21 Proposition 2.1.2. [8, Proposition 2.1.3] Let w ∈ W and let i ∈ I. If n, ni ∈ N are such that w = π(n) and si = π(ni), then

(i) l(siw) = l(w) ± 1,

(ii) if l(siw) = l(w) + 1 then niBn ⊆ BninB, and

(iii) if l(siw) = l(w) − 1 then niBn 6⊆ BninB.

By [8, 2.2], it is possible to associate a root system Φ to the Weyl group W of a BN-pair of G. In the case that G is a ﬁnite group of Lie type, Φ is a root system in the sense of [40, III] (see Chapter 3 for further information concerning root systems associated to ﬁnite groups of Lie type).

2.2 Parabolic Subgroups

Let W = hsiii∈I be the Weyl group of a BN-pair of G. Given a subset J ⊆ I, let WJ = hsiii∈J be the subgroup of W generated by the si, i ∈ J. Then, WJ is the parabolic subgroup of W corresponding to the subset J ⊆ I. The parabolic WJ is a Coxeter group with respect to the generators si, i ∈ J. There are many well-known results concerning the cosets (and double cosets) of the parabolic subgroups in W . Let J ⊆ I, and let

J J J −1 W = {w ∈ W |l(siw) > l(w) for all i ∈ J} and W = ( W ) .

J Proposition 2.2.1. ([14, Proposition 4.16 (5)]) Multiplication W × WJ → W deﬁnes a J bijection of sets. For any u ∈ W and v ∈ WJ , l(uv) = l(u) + l(v). So, u is the unique element in the coset uWJ of minimal length.

By [14, Proposition 4.16 (5)], the elements of W J are the shortest left coset representa- J tives of WJ in W . Similarly, the elements of W are the shortest right coset representatives of WJ in W .

Given subsets J, K ⊆ I, let J W K = J W ∩ W K . By [14, 4.3.2], the set J W K gives a full set of shortest (WJ ,WK )-double coset representatives in W .

The parabolic subgroups of the Weyl group W corresponding to the BN-pair of G may be used to deﬁne parabolic subgroups of G. Given a subset J ⊆ I, let WJ be the parabolic subgroup of W corresponding to J. Let NJ be the subgroup of N satisfying NJ /T = WJ . Then, PJ := BNJ B is a subgroup of G. Any subgroup of G conjugate to PJ for some J ⊆ I is called a parabolic subgroup of G. We note that when J = ∅, P∅ = B. and, when J = I, PI = G.

Proposition 2.2.2. [14, Theorem A.41]

22 J K (1) For any J, K ⊆ I, the subset W is a full set of (PJ ,PK )-double coset representatives in G. So, we have a disjoint union

G = ∪ PJ wPK . w∈J W K

J K (2) For any w ∈ W , PJ wPK = BWJ wWK B.

2.3 Finite Groups with a Split BN-Pair

Let p be a prime number, and suppose that G is a ﬁnite group such that (i) G has subgroups B and N, which form a BN-pair. (ii) B = UT , where U is a normal p-subgroup of B and T is an abelian subgroup of order prime to p. (iii) ∩ nBn−1 = T . n∈N In this case, we will say that G has a split BN-pair of characteristic p. Suppose that G has a split BN-pair. Let

U − = U w0 ,

− si Xi = U ∩ (U ) for i ∈ I,

si Ui = U ∩ U , and

w0w Uw = U ∩ U for any w ∈ W.

w0 −1 −1 Here, U = w0 Uw0 = n0 Un0, where n0 is a preimage of w0 in N (this deﬁnition is in- − si si w0w dependent of the choice of preimage n0 of w0). The subgroups (U ) , U , and U are deﬁned similarly.

We will now deﬁne the root subgroups of a (ﬁnite) group G with a split BN-pair.

−1 Proposition 2.3.1. [8, Proposition 2.5.15] The set of subgroups nXin (n ∈ N, i ∈ I) of G is in bijective correspondence with the set Φ of roots. If n is a preimage of w, then the −1 subgroup nXin corresponds to the root w(αi).

−1 The subgroups nXin (n ∈ N) are called the root subgroups of G. (As above, we will −1 −1 write nXn = wXw if w is the image of n in W .) If w(αi) = α ∈ Φ, we will denote the −1 root subgroup wXiw by Xα. Q Proposition 2.3.2. [8, Corollary 2.1.17] U = Xα (any order). α∈Φ+

23 3 Finite groups of Lie type

Our main reference is [42, III]. (All of the proof strategies of this chapter are also due to Malle and Testerman.) In this chapter, we will discuss both the split and non-split finite groups of Lie type; but, since the original research presented in this dissertation deals only with split finite groups of Lie type, the reader may choose to skip the discussions of finite groups of Lie type arising from twisted and very twisted Steinberg endomorphisms. All algebraic groups considered in this section are assumed to be linear.

3.1 Steinberg endomorphisms

Let G be a connected reductive algebraic group over an algebraically closed ﬁeld K of characteristic p, and let GLn = GLn(K) for n ≥ 1.

f Let q = p for some integer f ≥ 1. Then, the group homomorphism Fq : GLn → GLn q given by (aij) 7→ (aij) is called the standard Frobenius morphism of GLn with respect to the finite field Fq. The standard Frobenius morphism Fq is an automorphism of GLn as an ab- stract group and an endomorphism of GLn as an algebraic group, but not an automorphism Fq of GLn as an algebraic group. Note that the fixed point group of Fq is (GLn) = GLn(Fq).

The notion of a Frobenius morphism can be extended to an algebraic group G using the fact that G can be regarded as a closed subgroup of GLn for some n ≥ 1.

Deﬁnition 3.1.1. ([42, Deﬁnition 21.3]) An algebraic group endomorphism F : G → G is called a Steinberg endomorphism if there exists an integer m ≥ 1 such that F m : G → G is a Frobenius morphism.

Steinberg proved that endomorphisms of simple linear algebraic groups fall into precisely two categories, one of which consists of what are now referred to as the Steinberg endomorphisms.

Theorem 3.1.2. ([42, Theorem 21.5], due to Steinberg) Let G be a simple linear algebraic group with an endomorphism σ. Then, precisely one of the following holds:

(1) σ is an automorphism of algebraic groups, or (2) the group Gσ = {g ∈ G|σ(g) = g} is ﬁnite. An endomorphism σ : G → G falls into category (2) if and only if it is a Steinberg endomorphism.

24 Remark 3.1.3. If G is semisimple, it is still true that an endomorphism σ : G → G is a Steinberg endomorphism if and only if Gσ is ﬁnite.

The following fundamental theorem is the basis for many of the results we will state without proof in this chapter.

Theorem 3.1.4. (Lang-Steinberg, [42, 21.2]) Let G be a connected linear algebraic group over an algebraically closed ﬁeld k of characteristic p > 0, with a Steinberg endomorphism F : G → G. Then, the morphism L : G → G, g 7→ F (g)g−1 is surjective.

Deﬁnition 3.1.5. Let G be a connected reductive linear algebraic group and let F : G → G be a Steinberg endomorphism. Then, GF is called a ﬁnite group of Lie type.

To classify the finite groups of Lie type, one must classify the semisimple algebraic groups G and the Steinberg endomorphisms of these semisimple algebraic groups. The semisimple algebraic groups were classified by Chevalley. According to Chevalley’s classification theorem, a semisimple algebraic group is determined uniquely by its root datum (X(T ), Φ,Y (T ), Φ∨), where T ≤ G is a maximal torus, X(T ) is the character group of T , Φ is the root system with respect to T , Y (T ) is the cocharacter group, and Φ∨ is the set of coroots. (The root datum is discussed in detail in [42, Chapter 9].)

We now discuss the task of classifying the Steinberg endomorphisms of the semisimple algebraic groups.

Deﬁnition 3.1.6. ([42, Deﬁnition 9.14]) Let G be a semisimple algebraic group with root ∨ ∨ ∼ ∨ datum (X, Φ,Y, Φ ), and let Ω = Hom(ZΦ , Z). Since X = Hom(Y, Z) and Φ ⊆ Y , restriction gives an injective homomorphism X → Ω. Therefore, X may be viewed as a subset of Ω, and we have ZΦ ⊆ X ⊆ Ω. Now, let Λ(G) = Ω/X be the fundamental group of G. If X = Ω (which means that the fundamental group is trivial), G is of simply connected type. If X = ZΦ, G is of adjoint type.

Deﬁnition 3.1.7. ([42, 9.2]) Suppose φ : G → H is a surjective homomorphism of algebraic groups. Then, φ is called an isogeny if Ker(φ) is ﬁnite.

In particular, a Steinberg endomorphism F : G → G is an isogeny.

Theorem 3.1.8. ([42, Proposition 9.15]) Let G be semisimple, and let Φ be the root system of G. Then, there exits a simply connected group Gsc and an adjoint group Gad, each with π1 π2 root system Φ, and isogenies Gsc → G and G → Gad.

π1 Proposition 3.1.9. ([42, Proposition 9.18]) Let G, Gsc, and Gsc → G be as in Theorem 3.1.8. Then, every isogeny σ : G → G can be lifted to an isogeny σsc : Gsc → G such that π1 ◦ σsc = σ ◦ π1.

25 By Proposition 3.1.9, every Steinberg endomorphism of a semisimple algebraic group G is induced by a Steinberg endomorphism of a simply connected group. Therefore, to classify the Steinberg endomorphisms of the semisimple groups, it suffices to classify the Steinberg endomorphisms of the simply connected groups. In fact, it suffices to classify the Steinberg endomorphisms of the simple simply connected groups. Before discussing this classification, we give some necessary background theory.

3.2 The action of F on the character group of T

We will denote the root subgroup of G corresponding to α ∈ Φ by Uα. For any α ∈ Φ, let uα : Ga → Uα be a ﬁxed T -isomorphism. (With this notation, the element of Uα corresponding to a scalar c ∈ K will be denoted by uα(c).)

Lemma 3.2.1. ([42, Lemma 11.10]) Let G be a connected reductive algebraic group in characteristic p > 0, with a ﬁxed maximal torus T contained in a Borel subgroup B. Let Φ, Φ+, and Π be the corresponding root system, set of positive roots, and base of simple roots, respectively. And, let σ : G → G be an endomorphism stabilizing T and B. Then, there exists a permutation ρ of Φ+ stabilizing Π such that for all α ∈ Φ+, the following conditions hold:

(a) There exists a positive integer qα, equal to a power of char(k) > 0 such that σ(ρ(α)) := ρ(α) ◦ σ|T = qαα.

× qα (b) There exists aα ∈ k such that σ(uα(c)) = uρ(α)(aαc ) for all c ∈ k.

+ Proof. Since σ stabilizes B and T , σ permutes the root subgroups Uα, α ∈ Φ . Therefore, + + we can deﬁne a permutation ρ of Φ by ρ(Uα) = Uρ(α). Since the base of Φ is unique, ρ + must preserve Π. Now, given α ∈ Φ and c ∈ K, σ(uα(c)) ∈ Uρ(α). Therefore, there exists an endomorphism να of the additive group Ga of K such that σ(uα(c)) = uρ(α)(να(c)) for all c ∈ K. We claim that ν is a monomial. Note that for any t ∈ T and c ∈ K,

−1 tuα(c)t = uα(α(t)c).

Applying σ to the left-hand side of the equation above, we have

−1 −1 −1 σ(tuα(c)t ) = σ(t)σ(uα(c))σ(t) = σ(t)uρ(α)(να(c))σ(t) = uρ(α)(ρ(α)(σ(t))ν(c)).

Applying σ to the right-hand side, we have σ(uα(α(t)c)) = uρ(α)(ν(α(t)c)). Therefore, uρ(α)(ρ(α)(σ(t))ν(c)) = uρ(α)(ν(α(t)c)), which means that

ρ(α)(σ(t))ν(c) = ν(α(t)c) (3.1)

26 for all c ∈ K and t ∈ T .

Now, since char K = p > 0, every algebraic group endomorphism of the additive group n P pi Ga of K is of the form c 7→ h(c), where h = aiT (ai ∈ K) is a polynomial in K[T ]. In i=0 n P pi particular, ν(c) = aic for some elements ai ∈ K and for all c ∈ K. Therefore, (3.1) can i=0 be expressed as n n X pi X pi pi aiρ(α)(σ(t))c = aiα(t) c . i=0 i=0 Since the expression above holds for all c ∈ K, we have

n n X pi X pi pi aiρ(α)(σ(t))T = aiα(t) T , i=0 i=0

pi in K[T ], which means that aiρ(α)(σ(t)) = aiα(t) for 0 ≤ i ≤ n. In particular, for all i such pi that ai 6= 0, we have ρ(α)(σ(t)) = α(t) . But, this is possible for only one i. Therefore, pi ν = aiT for some i ≥ 0.

i Setting qα = p and aα = ai, we have proved part (b) of the lemma. To prove part (a), we substitute c = 1 in (3.1), getting ρ(α)(σ(t))ν(1) = ν(α(t)). Therefore, σ(ρ(α))(t) = ν(α(t)) a α(t)qα α qα (ρ(α) ◦ σ)(t) = ρ(α)(σ(t)) = = = α(t) = (qαα)(t) for all t ∈ T . ν(1) aα

For the remainder of this section, we ﬁx an F -stable Borel subgroup B and an F -stable maximal torus T of G, with T ≤ B (such a pair (T,B) exists by [42, Corollary 21.12]). Let X = X(T ) be the character group of T , and Φ ⊂ X the root system of G with respect to T . Let Φ+ be the set of positive roots corresponding to the choice of B. Suppose that the root system Φ has base Π.

We deﬁne an action of F on X as follows. Given χ ∈ X, and t ∈ T , we let F (χ)(t) := χ(F (t)).

Lemma 3.2.2. ([42, Lemma 22.1]) Let T be an F -stable maximal torus of G. Then there δ exists δ ∈ N and a power r of p = char K such that F |X(T ) = r1X(T ) on X(T ) (where 1X(T ) is the identity on X(T )).

Combining Lemmas 3.2.1 and 3.2.2, we have the following important result.

Proposition 3.2.3. ([42, Proposition 22.2]) Let G be a connected reductive algebraic group with a Steinberg endomorphism F : G → G, and let T be an F -stable maximal torus of G

27 contained in an F -stable Borel subgroup B. Let X = X(T ), and let Φ be the root system of G with respect to T and Φ+ be the system of positive roots corresponding to B. Then,

+ + (a) There is a permutation ρ of Φ and, for each α ∈ Φ , a positive integer qα > 1 such × that qα is a power of p = char K, and an element aα ∈ K such that F (ρ(α)) = qαα qα and F (uα(c) = uρ(α)(aαc ) for all c ∈ K. δ δ (b) There exists an integer δ ≥ 1 such that F |X = q 1X and F = qφ on XR for some δ positive fractional power q of p (with q a power of p) and some φ ∈ Aut(XR) of order δ which induces ρ−1 on Φ+.

The automorphism φ of XR defined in part (b) of the proposition above is key to the classification of the finite groups of Lie type, as we will see in the following section.

3.3 The classiﬁcation of the ﬁnite groups of Lie type

As previously discussed, the task of classifying the ﬁnite groups of Lie type may be reduced to classifying the Steinberg endomorphisms of simple simply connected algebraic groups. Determining the Steinberg endomorphisms of a simple simply connected algebraic group is, in turn, closely related to the classiﬁcation of the Dynkin diagram automorphisms and Cox- eter diagram automorphisms of the base Π of the corresponding root system Φ. (We refer the reader to [40] for a full discussion of root systems; Dynkin diagrams and Coxeter graphs are covered in [40, 11.2].)

A Dynkin diagram automorphism is a permutation of the nodes of the Dynkin diagram which leaves the diagram invariant. A Coxeter diagram automorphism is a permutation of the nodes of the corresponding Coxeter diagram which leaves the diagram invariant. Every Dynkin diagram automorphism is also a Coxeter diagram automorphism. But, since the Coxeter diagram does not record root length, a root system which has both long and short simple roots may admit Coxeter diagram automorphisms which are not Dynkin diagram automorphisms.

Example 3.3.1. We consider the root system of type B2. The Dynkin and Coxeter diagrams in type B2 are shown below. Dynkin diagram: 1 2

Coxeter diagram: 1 2

Since a Dynkin diagram automorphism must preserve root length, there are no non-trivial Dynkin diagram automorphisms in type B2. However, there is one non-trivial Coxeter diagram automorphism, corresponding to ﬂipping the diagram.

28 The only irreducible root systems having non-trivial Dynkin diagram automorphisms are in types An (n ≥ 2), Dn (n ≥ 4), and E6. The automorphism group in type An (n ≥ 2), Dn (n ≥ 5), and E6 is isomorphic to Z/2Z. The automorphism group in type D4 is isomorphic to the symmetric group on 3 letters. The only irreducible root systems which admit a non-trivial Coxeter diagram automorphism are B2, G2, and F4. In each case, there is one non-trivial Coxeter diagram automorphism, corresponding to ﬂipping the diagram.

We are now ready to give a classiﬁcation of the Steinberg endomorphisms of the simple simply connected groups. (Note that the following classiﬁcation is due to Steinberg.)

Theorem 3.3.2. ([42, Theore 22.5]) Let G be a simple simply connected algebraic group (deﬁned in characteristic p > 0) with root system Φ and base Π. Let Ω be the set consisting of all pairs (q, ρ) such that ρ is a Dynkin diagram automorphism of Π and q is an integral power of p or the root system of G is of type B2, G2, respectively F4, ρ is the non-trivial Cox- 2 2f+1 2f+1 2f+1 eter diagram automorphism of Π, and q = 2 , 3 , respectively 2 for some f ∈ N0. Then, there is a one-to-one correspondence between Steinberg endomorphisms F : G → G and pairs (q, ρ) ∈ Ω.

If F : G → G is a Steinberg endomorphism, then F = qφ on XR for some integral power q of p and for some φ ∈ Aut(XR). The automorphism φ of XR induces a Coxeter diagram automorphism of Π. If we deﬁne ρ to be the inverse of this automorphism, then the pair (q, ρ) corresponds to F in the classiﬁcation theorem above.

By Theorem 3.3.2, a ﬁnite group of Lie type is determined by its root system Φ, a power q of p, and the Coxeter diagram automorphism ρ of Π such that ρ−1 is induced by

the automorphism φ ∈ Aut(XR) with F = qφ on XR. We observe that a diagram automorphism of Π is determined by its order: the root systems of type An (n ≥ 2), Dn (n ≥ 5), and E6 admit exactly one non-trivial Dynkin diagram automorphism of order 2, D4 admits non-trivial Dynkin diagram automorphisms of orders 2 and 3, and B2, G2, and F4 admit a non-trivial Coxeter diagram automorphism of order 2. In particular, the diagram automorphism ρ of Theorem 3.3.2 is determined by the order δ of φ, which is either 1, 2, or 3.

Therefore, a finite group of Lie type is determined by Φ, a power q of p, and the order δ of φ. A classification of the finite groups of Lie type according to Φ, q, and δ is given in F Table 22.1 of [42], which is reproduced in Table 1 below. In Table 1, Gsc denotes the finite group obtained by taking the F -fixed points of a simply connected group Gsc with the given F root system, and Gad denotes the finite group obtained by taking the F -fixed points of an adjoint group Gad with the given root system.

The Steinberg endomorphisms F corresponding to the trivial diagram automorphisms (shown in the top section of Table 1) are called Fq-split, which means that F acts as q1X(T on the character group X(T ) of an F -stable maximal torus T . The Steinberg endomorphisms

29 Table 1: Finite groups of Lie type (Table 22.1 in [MT]) F F Φ δ Gsc Gad other isogeny types An−1 1 SLn(q) PGLn(q) Zd/e.PSLn(q).Ze, where e | d = (n, q − 1) Bn 1 Spin2n+1(q) SO2n+1(q) Cn 1 Sp2n(q) PCSp2n(q) + o + + + Dn 1 Spin2n(q) (PCO2n) (q) SO2n(q), HSpin2n(q) G2 1 G2(q) F4 1 F4(q) E6 1 (E6)sc(q)(E6)ad(q) E7 1 (E7)sc(q)(E7)ad(q) E8 1 E8(q) An−1 2 SUn(q) PGUn(q) Zd/e.PSUn(q).Ze, e | d = (n, q + 1) − o − − Dn 2 Spin2n(q) (PCO2n) (q) SO2n(q) 3 D4 3 D4(q) 2 2 E6 2 ( E6)sc(q)( E6)ad(q) 2 2 2 2 2f+1 B2 2 B2(q ) = Suz(q ), q = 2 2 2 2 2f+1 G2 2 G2(q ), q = 3 2 2 2 2 2f+1 F4 2 F4(q ) = Suz(q ), q = 2

corresponding to the non-trivial Dynkin diagram automorphisms in types An−1, Dn, and E6 (shown in the middle section of Table 1) are called twisted. A twisted Steinberg endomorphism is non-split; it is the product of an Fq-split endomorphism with an algebraic group automorphism of G. Finally, the Steinberg endomorphisms corresponding to the non-trivial Coxeter diagram automorphisms in types B2, G2, and F4 (shown in the bottom section of Table 1) are called very twisted.

2 2 The groups B2(q ) corresponding to the very twisted Steinberg endomorphisms in type 2 2 2 2 B2 are called Suzuki groups, and the groups G2(q ) and F4(q ) corresponding to the very twisted Steinberg endomorphisms in types G2 and F4 are called Ree groups.

3.4 The Root System and Root Subgroups of the Finite Group of Lie Type GF

In this section, we follow [42, Section 23.1]. Let G be a semisimple group with Stein- berg endomorphism F : G → G. Let T be an F -stable maximal torus of G, and let N = NG(T ) be the normalizer of T in G. Then, the F -ﬁxed point subset of W is given by F F ∼ F F W = (N/T ) = NGF (T )/T . (Note that NGF (T ) 6= NGF (T ) in general.)

Let B be an F -stable Borel subgroup of G containing the F -stable maximal torus T . Let Φ be the corresponding root system and Φ+ be the corresponding system of positive roots. As discussed in the previous section, the Steinberg endomorphism F induces a permutation

30 + φ ρ of Φ . Suppose F |X = qφ for some φ ∈ Aut(X ), where φ has (finite) order δ. Let X R R R 1 δ−1 denote the fixed space of φ on X , and define a projection π : X → Xφ by χ 7→ Pφi(χ). R R R δ i=0

The homomorphism π : X → Xφ can be used to deﬁne an equivalence relation ∼ on R R the root system Φ of G, where α ∼ β for α, β ∈ Φ if and only if π(α) and π(β) are positive multiplies of each other.

Given an equivalence class ω for the equivalence relation ∼, let αω be the maximal-length vector in the set {π(α) | α ∈ ω}. Let Ω denote the set of equivalence classes ω for ∼. (By + F [42, Lemma 23.4], Ω = {w(ΦI ) | I ⊆ S is an orbit under F , w ∈ W }.) Then, the set

ΦF = {αω | ω ∈ Ω}

F F is the root system of G . The root system ΦF of G has base

ΠF = {αω | ω ⊆ Π}.

We can now deﬁne the root subgroups of GF . For any equivalence class ω ∈ Ω, let

Uω = hUα|α ∈ ωi. Q By [42, Proposition 23.7], Uω (ω ∈ Ω) is F -stable and decomposes as Uω = Uα, with α∈ω uniqueness of expression for a ﬁxed order of the elements α in ω. Given any ω ∈ Ω, the F F F subgroup Uω of the Uω is a root subgroup of G . The orders of the root subgroups of G F |ω| are powers of q. By [42, Corollary 23.9], if G is simple and ω ∈ Ω, then |Uω | = q .

3.5 Construction of a BN-pair for the Finite Group of Lie Type GF

Let G be a connected reductive linear algebraic group with a Steinberg endomorphism F : G → G. Let T be an F -stable maximal torus of G, and let B be an F -stable Borel subgroup of G containing T . Let Φ and Φ+ be the associated root system and set of positive roots, respectively, and let W = NG(T )/T be the Weyl group. Note that since T is F -stable, so is NG(T ), which means that F acts on W .

The algebraic group G has Bruhat decomposition G = S BwB, where the union is w∈W − ∼ − disjoint. For every w ∈ W , we have BwB = Uw wB = Uw × {nw} × B, where nw is a − Q representative of the coset w ∈ W and Uw = Uα. α∈Φ+, w.α∈Φ− The ﬁnite group GF admits an analogous decomposition.

31 Theorem 3.5.1. ([42, Theorem 24.1]) Let G be a connected reductive algebraic group with Steinberg endomorphism F : G → G, T ≤ B an F -stable maximal torus in an F -stable Borel subgroup of G, and W the Weyl group of G with respect to T . Then,

F Y − F F G = (Uw ) wB , w∈W F

with uniqueness of expression.

The Bruhat decomposition of GF is key to proving the following theorem on the existence of a BN-pair in GF .

Theorem 3.5.2. ([42, Theorem 24.10]) Let G be connected reductive with Steinberg endomorphism F : G → G, T ≤ B an F -stable maximal torus in an F -stable Borel subgroup of F F F G, and N = NG(T ) the normalizer of T in G. Then, B , N is a BN-pair in G with Weyl group W F .

3.6 The Levi Decomposition of a Parabolic Subgroup of a Finite Group of Lie Type G with a Split BN-Pair

Suppose G is a finite group of Lie type with a split BN-pair (in characteristic p). Let W be the Weyl group associated to the BN-pair of G (with generators si, i ∈ I). When G is a finite group of Lie type, W is necessarily finite. Let Φ be the associated root system, and let Π = {α1, α2, . . . , αl} be the set of simple roots in Φ. Given a subset J ⊆ I, let ΠJ = {αi|i ∈ J} and ΦJ = WJ (ΠJ ) (where WJ is the parabolic subgroup of W corresponding to J). And, define LJ = hT,Xα|α ∈ ΦJ i. The subgroup LJ of G is called a Levi subgroup.

Let BJ be the subgroup of LJ deﬁned by BJ = Uw0J T (where w0J is the longest element of WJ ), and let NJ be the subgroup of LJ with NJ /T = WJ .

Proposition 3.6.1. [8, Proposition 2.6.3] The group LJ has a split BN-pair (of characteristic p) corresponding to the subgroups BJ and NJ .

For a subset J ⊆ I, let PJ be the parabolic subgroup of G corresponding to J. Deﬁne w0J UJ = U ∩ U . Then, UJ = Op(PJ ) is the largest normal p-subgroup of PJ (informally, we will refer to UJ as the unipotent radical of PJ ). We have the decomposition,

PJ = UJ LJ = LJ UJ , where UJ ∩ LJ = {1}. The decomposition of PJ given above is called a Levi decomposition of PJ and LJ is called a Levi complement in PJ .

32 3.7 An Order Formula for a Finite Group of Lie Type Arising from a Simple Linear Algebraic Group

The order formulas for the finite groups of Lie type are well-known and can be found in many books and papers dealing with the structure of finite groups of Lie type. The account given here is based on [42, Section 24]. Let q be a power of a prime p. Let G be a simple linear algebraic group with Steinberg endomorphism F : G → G, and let T ≤ B be an F -stable maximal torus in an F -stable Borel subgroup. Let Φ be the root system associated to B, F F and let W be the Weyl group. Then, B and NG(T ) form a BN-pair in the finite group of Lie type GF , with Weyl group W F .

The order of the finite group of Lie type GF is given by |GF | = |BF | P ql(w), where w∈W F P ql(w) is the Poincarépolynomial of W F evaluated at q, which gives the index [GF : BF ] w∈W F of BF in GF . The Poincarépolynomial of W F has a factorization of the form P xl(w) = w∈W F n di P l(w) Q x − εi F x = , where n is the rank of W , d1, . . . , dn are certain positive integers, w∈W F i=1 x − εi and the εi are roots of unity (a more detailed description of the di and εi can be found in n qdi − ε [42]). So, [GF : BF ] = P ql(w) = Q i . The order of the Borel subgroup BF is w∈W F i=1 q − εi |BF | = |U F ||T F | (where U F is the unipotent radical of BF ), and the order of U F is |U F | = n |Φ+| F F Q q . The order of T may be expressed in terms of the εi; in particular, |T | = (q − εi). i=1 n F |Φ+| Q Therefore, we have |B | = q (q − εi) and i=1

n n n di + q − ε + F F X l(w) |Φ |Y Y i |Φ |Y di |G | = |B | q = q (q − εi) = q (q − εi). q − εi w∈W F i=1 i=1 i=1

In the case that G is an arbitrary connected reductive algebraic group with a Steinberg endomorphism F : G → G, the order of GF is given by [42, Exercise 30.9]. Let T ≤ B be an F -stable maximal torus of G contained in an F -stable Borel subgroup. Suppose that G has root system Φ and positive root system Φ+. Then,

F Y F X |G | = qα |T | qw, α∈Φ+ w∈W F

Q F where qw = qα for all w ∈ W (qα is deﬁned as in [42, Lemma 11.10]). α∈Φ+,w(α)∈Φ−

33 4 Harish-Chandra Series

Our summary of Harish-Chandra theory is based on Section 4.2 of Geck and Jacon’s “Rep- resentations of Hecke Algebras at Roots of Unity” [31]. Let G be a finite group of Lie type, defined in characteristic p > 0 (so, G is the fixed point subgroup of a connected reductive ¯ algebraic group G over Fp under an endomorphism F : G → G such that some power of F is a Frobenius morphism). Assume, additionally, that G has subgroups B, N which form a split BN-pair of characteristic p, and let T = B ∩ N. Let k be an algebraically closed field of characteristic r > 0, r 6= p.

4.1 Harish-Chandra Induction and Restriction

+ Given a subset J ⊆ S, let PJ = hT,Uα | α ∈ Φ ∪ ΦJ i be the standard parabolic subgroup of G corresponding to J. Let UPJ = Op(PJ ) (the subgroup UPJ will be informally referred to as the unipotent radical of PJ ), and let LJ denote the Levi subgroup of PJ . Using n n the notation of [31, 4.2.1], let PG = { PJ | J ⊆ S, n ∈ N} and LG = { LJ | J ⊆ S, n ∈ N}.

Let P ∈ PG and let L ∈ LG be a Levi complement in P , so that P = UP o L. Let kL − mod denote the category of (finite dimensional) left kL-modules, and let kG − mod denote the category of (finite dimensional) left kG-modules. Then, there is a Harish-Chandra induction functor G RL⊆P : kL − mod → kG − mod, (4.1) G G ˜ ˜ defined by RL⊆P (X) = IndP (X) for all X ∈ kL − mod, where X denotes the inflation of X from L to P via the surjective homomorphism P → L with kernel UP .

There is also a Harish-Chandra restriction functor

∗ G RL⊆P : kG − mod → kL − mod. (4.2)

∗ G UP Given a kG-module Y , RL⊆P (Y ) = Y (which has the structure of a kL-module since UP is a normal subgroup of P ).

A key feature of Harish-Chandra induction is the following independence property, which was proved by Howlett and Lehrer [39] and Dipper and Du [18]. Let L, M ∈ LG, and suppose that L is a Levi complement of P ∈ PG and M is a Levi complement of Q ∈ PG. Let X ∈ kL − mod and X0 ∈ kM − mod. If M = nL and X0 ∼= nX for some n ∈ N, then G ∼ G 0 RL⊆P (X) = RM⊆Q(X ). As a particular application of this independence property, we have G that the Harish-Chandra induction functor RL⊆P is independent of the choice of parabolic ∗ G subgroup P ∈ PG containing L. Similarly, the Harish-Chandra restriction functor RL⊆P is independent of the parabolic subgroup P containing L. Therefore, we will omit the parabolic G ∗ G subgroup P and write RL and RL for the Harish-Chandra induction and restriction functors.

34 4.2 Some Properties of Harish-Chandra Induction and Restriction

G ∗ G Adjointness. Given a Levi subgroup L ∈ LG, the functors RL and RL are exact. The G ∗ G functors RL and RL are each other’s two-sided adjoints. So, for any Y ∈ kG − mod and X ∈ kL − mod, we have

G ∼ ∗ G HomkG(RL (X),Y ) = HomkL(X, RL (Y )),

and G ∼ ∗ G HomkG(Y,RL (X)) = HomkL( RL (Y ),X).

Transitivity. Harish-Chandra induction and restriction are transitive. Suppose that L, M ∈ LG are such that L ⊆ M. If X ∈ kL − mod, then

G ∼ G M RL (X) = RM (RL (X)).

If Y ∈ kG − mod, then ∗ G ∼ ∗ M ∗ G RL (Y ) = RL ( RM (Y )).

Mackey decomposition. As in the case of ordinary induction and restriction, we have a Mackey decomposition formula for Harish-Chandra induction and restriction. Suppose L, M ∈ LG are Levi complements of the parabolic subgroups P,Q ∈ PG, respectively. Let X be a kL-module, and let D(Q, P ) denote a full set of (Q, P )-double coset representatives in G. Given an element n ∈ D(Q, P ), we can deﬁne a k(nL)-module structure on X by setting nln−1.x = l.x for any l ∈ L and x ∈ X. The resulting k(nL)-module will be denoted by nX. The Mackey formula provides the following direct sum decomposition of the kM-module ∗ G G RM (RL (X)): ∗ G G ∼ M ∗ nL n RM (RL (X)) = ⊕ RnL∩M ( RnL∩M ( X)). (4.3) n∈D(Q,P )

Harish-Chandra induction and the linear dual. Let L ∈ LG be the Levi complement ∗ of a parabolic subgroup P ∈ PG, and suppose that X is a left kL-module. Let X be the G ∼ G ∗ k-linear dual of X, viewed as a right kL-module. We claim that RL (X) = RL (X ). By [3, pg. 60], ordinary induction commutes with taking duals in the case of ﬁnite groups. G ∗ ˜ G ∗ ∼ ˜ ∗ G So, we have (RL (X)) = (X|P ) = (X) |P . But, since the unipotent radical UP of P ˜ ˜ ∗ ˜ ∗ ∼ ∗ acts trivially on X, UP acts trivially on X , and it follows that (X) = Xf . Therefore, ˜ ∗ G ∼ ∗ G G ∗ (X) |P = Xf |P = RL (X ).

4.3 Cuspidal Modules and Harish-Chandra Series.

∗ G A kG-module Y is called cuspidal if RL (Y ) = 0 for all L ∈ LG such that L ( G. (This ∗ L deﬁnition extends to kL-modules for any L ∈ LG; a kL-module X is cuspidal if RL0 (X) = 0

35 0 0 for all L ∈ LG such that L ( L.)

Let Irrk(G) denote a full set of non-isomorphic irreducible kG-modules. (We will assume that all of the modules in Irrk(G) are ﬁnite-dimensional over k.) Given a pair (L, X) with L ∈ LG and X an irreducible cuspidal kL-module, let Irrk(G|(L, X)) be the subset of ∗ G Irrk(G) consisting of all Y ∈ Irrk(G) such that L ∈ LG is minimal with RL (Y ) 6= 0 and ∗ G X is a composition factor of RL (Y ). The set Irrk(G|(L, X)) is the Harish-Chandra series corresponding to the pair (L, X).

0 Following [31], let LG = {(L, X)|L ∈ LG and X ∈ Irrk(L) is cuspidal}. There is an 0 0 0 equivalence relation ∼N on LG given by (L, X) ∼N (L ,X ) if and only if there exists some n ∈ N such that L0 = nL and X0 ∼= nX. We will show that the Harish-Chandra series 0 Irrk(G|(L, X)), (L, X) ∈ LG/ ∼N partition Irrk(G). We begin with two technical lemmas.

Lemma 4.3.1. ([31, Lemma 4.2.3]) Let Y be an irreducible kG-module, and let L ∈ LG be ∗ G ∗ G minimal with RL (Y ) 6= 0. Then, every composition factor of RL (Y ) is cuspidal.

∗ G Proof. Let X be a composition factor of the kL-module RL (Y ). Let M be a Levi subgroup ∗ L ∗ L ∗ G of G with M ⊆ L and suppose that RM (X) 6= 0. Then, RM ( RL (Y )) 6= 0 since X is a ∗ G composition factor of RL (Y ). By transitivity of Harish-Chandra induction, we have

∗ G ∼ ∗ L ∗ G RM (Y ) = RM ( RL (Y )) 6= 0.

So, by minimality of L, we must have M = L, which means that X is a cuspidal kL-module.

Lemma 4.3.2. ([31, Lemma 4.2.4]) Let Y be an irreducible kG-module. Let L, M ∈ LG, X ∈ Irrk(L), and Z ∈ Irrk(M). And, assume that the following conditions are satisﬁed: ∗ G ∗ G (1) L is minimal such that RL (Y ) 6= 0 and X is a composition factor of RL (Y ). G (2) Z is cuspidal and Y is in the head or socle of RM (Z). Then, there is some n ∈ N such that L = nM and X ∼= nZ.

In the next proposition, we give an alternate characterization of the Harish-Chandra series Irrk(G|(L, X)) (where L ∈ LG and X ∈ Irrk(L) is cuspidal).

0 Proposition 4.3.3. ([31, 4.2.6]) Let (L, X) ∈ LG. If Y is an irreducible kG-module, then the following statements are equivalent.

(a) Y ∈ Irrk(G|(L, X)), G (b) Y is a composition factor of head(RL (X)), and G (c) Y is a composition factor of soc(RL (X)).

36 ∗ G Proof. Suppose that Y ∈ Irrk(G|(L, X)). Then, RL (Y ) 6= 0, which means there is some ir- 0 ∗ G reducible kL-module X in the socle of RL (Y ). By adjointness of Harish-Chandra induction and restriction, we have

0 ∗ G ∼ G 0 0 6= HomkL(X , RL (Y )) = HomkG(RL (X ),Y ),

G 0 0 which means that Y is in the head of the kG-module RL (X ). Now, by Lemma 4.3.1, X is a cuspidal irreducible kL-module. So, by Lemma 4.3.2, there exists an element n ∈ N n ∼ n 0 G 0 ∼ G with L = L and X = X . Therefore, RL (X ) = RL (X) by the independence property G of Harish-Chandra induction, and it follows that Y is in the head of RL (X). A similar ar- G gument shows that every irreducible kG-module Y ∈ Irrk(G|(L, X)) is in the socle of RL (X).

G G Suppose now that Y is in the socle or in the head of RL (X). If Y ⊆ soc(RL (X)), then adjointness of Harish-Chandra induction and restriction yields

G ∼ ∗ G 0 6= HomkG(Y,RL (X)) = HomkL( RL (Y ),X).

G If Y ⊆ head(RL (X)), then adjointness yields G ∼ ∗ G 0 6= HomkG(RL (X),Y ) = HomkL(X, RL (Y )).

∗ G ∗ G In either case, RL (Y ) 6= 0 and X is a composition factor of RL (Y ). So, to show Y ∈ ∗ G Irrk(G|(L, X)), it remains to check that L is minimal with RL (Y ) 6= 0. Suppose M ⊆ L ∗ G (M ∈ LG) is minimal with RM (Y ) 6= 0, and let Z ∈ Irrk(M) be a composition factor of ∗ G RM (Y ). Then, Z is cuspidal by Lemma 4.3.1. So, by Lemma 4.3.2 (with the roles of L and M and the roles of X and Z reversed), M = nL for some n ∈ N. But, since M ⊆ L, we have M = L, as needed.

Theorem 4.3.4. ([31, 4.2.6]) The set Irrk(G) is partitioned by the distinct Harish-Chandra series; there is a disjoint union a Irrk(G) = Irrk(G|(L, X)). 0 (L,X)∈LG/∼N

∗ G Proof. Let Y ∈ Irrk(G), and let L be minimal with RL (Y ) 6= 0. If X is any composition ∗ G factor of RL (Y ), then X is cuspidal by Lemma 4.3.1, which means that Y ∈ Irrk(G|(L, X)). Thus, every Y ∈ Irrk(G) belongs to a Harish-Chandra series. Suppose now that Y belongs 0 0 0 0 0 to Irrk(G|(L, X)) and Irrk(G|(L ,X )) for some pairs (L, X) and (L ,X ) in LG. Since ∗ G Y ∈ Irrk(G|(L, X)), L is minimal with RL (Y ) 6= 0 and X is a composition factor of ∗ G 0 0 G 0 0 RL (Y ). And, since Y ∈ Irrk(G|(L ,X )), Y is in the head of RL0 (X ). So, taking Z = X and M = L0 in Lemma 4.3.2, we have L = nL0 and X ∼= nX0 for some n ∈ N. Therefore, 0 0 (L, X) ∼N (L ,X ).

We note that the original proof of the fact that the distinct Harish-Chandra series partition Irrk(G) is due to Hiss [37].

37 4.4 The Principal Series Representations

Since every kT -module is cuspidal, there is a Harish-Chandra series of the form Irrk(G|(T,X)) for every irreducible kT -module X. The irreducible representations of G belonging to a Harish-Chandra series of the form Irrk(G|(T,X)) are called principal series representations. (Since T is abelian, an irreducible kT -module X must be one-dimensional.)

The principal Harish-Chandra series corresponding to the pair (T, k) (where k is viewed as the trivial irreducible kT -module) is called the unipotent principal series and is denoted by 4 G ∼ G Irrk(G|B). Since RT (k) = k|B, Irrk(G|B) consists of the irreducible kG-modules which can G be found in both the head and socle of the permutation module k|B. If V ∈ Irrk(G), then the G G G multiplicity of V in the head of k|B is [k|B : V ] = dim HomkG(k|B,V ) = dim HomkB(k, V ) = B G ∼ dim V (the second equality follows since HomkG(k|B,V ) = HomkB(k, V ) by Frobenius G B reciprocity). Similarly, [soc(k|B): V ] = dim V . Thus, an irreducible kG-module V belongs B to the unipotent principal series Irrk(G|B) if and only if V 6= 0.

4The unipotent principal series representations are particularly important to the representation theory of finite groups of Lie type. According to results of Bonnaféand Rouquier [4], [5], the study of general blocks for finite groups of Lie type reduces to the study of unipotent blocks.

38 5 The Steinberg Module

The Steinberg module plays a key role in Geck’s results in [29, Section 4], which are needed to prove the bound on the dimension of Ext1 in Chapter 14 of this dissertation. While the bound of Chapter 14 is the most obvious application of the Steinberg module in this dissertation, the Steinberg module had a profound influence on all of the original research presented here. First and foremost, Geck’s treatment of the Steinberg module in [29] inspired the idea to apply modular Harish-Chandra theory to the task of finding bounds on the dimension of Ext between irreducible modules for finite groups of Lie type in cross characteristic. It is also important to mention that the Steinberg module is in the background of many of the results of Cline, Parshall, and Scott [9] and of Dipper and James [21], which are used in the original proofs of Part IV of this dissertation.

5.1 Steinberg’s Original Results on the “Steinberg Module”

5.1.1 A Basis for the Steinberg Module

Robert Steinberg originally introduced the Steinberg module in 1957 [47]; however, he did not refer to it as the “Steinberg module” in his own papers. In this section, we will give an overview of Steinberg’s results in [47]. (All subsequent literature involving the Steinberg module is based on Steinberg’s work in [47].)

Let G be a split finite group of Lie type, defined in characteristic p. Let B be a Borel subgroup of G, and let U denote the unipotent radical of B (so, U is a normal p-subgroup of B and B = U o T , where T is a maximal torus of G contained in B). Let W = NG(T )/T be the Weyl group of G with respect to T , and let S be the set of fundamental reflections generating W . Let Φ be the corresponding root system, and let Φ+ denote the set of positive roots. 5 P l(w) Given a field k (of any characteristic) , we define an element e ∈ kG by e = (−1) nwb, w∈W P where b = b and nw ∈ NG(T ) is a representative of the coset w ∈ W = NG(T )/T . Then, b∈B the left ideal Stk := kGe of kG is the Steinberg module over k (this is Steinberg’s original definition in [47], though Steinberg works with right rather than left kG-modules).

Given a root α ∈ Φ, let Uα denote the corresponding root subgroup of G. Given a closed + subset Γ ⊆ Φ , let UΓ be the subgroup of G generated by the root subgroups Uα, α ∈ Γ. Q Then, UΓ = Uα. (Γ is closed in the sense that for α, β ∈ Γ such that α + β is a root, α∈Γ + − α + β ∈ Γ.) Now, if w ∈ W , then the set Φw = {α ∈ Φ | w(α) ∈ Φ } is a closed subset + of Φ . We will denote the subgroup UΦw of G by Uw. The complement of Φw is the closed 0 + + 0 subset Φw = {α ∈ Φ | w(α) ∈ Φ }; we will denote the corresponding subgroup UΦw by Uw.

5The case with k = C is discussed further in the Appendix (Chapter 22).

39 The following technical lemma concerning the element e ∈ kG is crucial to many of the properties of the Steinberg module.

Lemma 5.1.1. ([47, Lemma 1]) Fix a simple reﬂection s ∈ S, and let α be the simple root such that s = sα. Then,

1. nse = −e.

2. If uα ∈ Uα with uα 6= 1, then there exist some elements u˜α, u¯α ∈ Uα, and t ∈ T such that −1 nsuαns =u ¯αtnsu˜α.

3. With uα and u¯α as in (2), (nsuα − u¯α + 1)e = 0.

Steinberg uses [47, Lemma 1] to ﬁnd a k-basis for Stk, which is given in the next theorem.

Theorem 5.1.2. ([47, Theorem 1]) The set Ω = {ue | u ∈ U} is a k-vector space basis for Stk. Therefore, dimk Stk = |U|.

P l(w0) Corollary 5.1.3. ([47]) Let x = γuue ∈ Stk (with γu ∈ k for all u). Then, γu is (−1) u∈U P times the coeﬃcient of unw0 b in x. And, γu is the coeﬃcient of b in x. If x ∈ G and u∈U P xe = γuue, then γu = 1, −1, or 0, and at most |W | of the γu are non-zero. u∈U

5.1.2 An Irreducibility Criterion for the Steinberg Module

In the remainder of his paper, Steinberg proves the following irreducibility criterion for the Steinberg module Stk:

Stk is irreducible if and only if char(k)6 | [G : B].

P P l(w) Lemma 5.1.4. ([47, Lemma 2]) Let m = [G : B], and, let u = u. Then, ( (−1) nw)ue = u∈U w∈W me.

In the next two theorems, we will establish the irreducibility criterion for the Steinberg module (following [47]). Due to the importance of these theorems, we will present Steinberg’s proofs of these results (providing extra details as necessary).

Theorem 5.1.5. ([47, Theorem 2])

1. Stk restricted to U is isomorphic to the left regular representation of U.

2. If char(k) is relatively prime to m = [G : B], then Stk is irreducible.

40 Proof. 1. The left regular representation of U is kU, viewed as a left kU-module via left multiplication. Since {ue | u ∈ U} is a k-basis for Stk, there is a k-vector space isomorphism Stk → kU, ue 7→ u. One can check that this map is a U-module homomorphism ∼ and therefore gives a kU-isomorphism Stk = kU.

2. We will show that any non-zero element of Stk generates Stk as a left kG-module when m is relatively prime to char(k). Let 0 6= α ∈ Stk. To show that α generates Stk, it suffices to show that e ∈ kGα. Since {ue | u ∈ U} is a k-basis for Stk and each ue is k-linear combination of elements of the form xb (x ∈ G, w ∈ W ), the same is true for α. So, if α 6= 0, there must be some x ∈ G such that the coefficient γ of xb in α is not −1 −1 P equal to zero. Then, the coefficient of b in x α is γ 6= 0. And, writing x α = γuue u∈U P (γu ∈ k for all u ∈ U), we have γ = γu by Corollary 5.1.3. u∈U

Now, since γ and m are both invertible in k, we can carry out the following computation.

−1 −1 X l(w) −1 −1 −1 X l(w) X γ m ( (−1) w)ux α = γ m ( (−1) w)u γuue w∈W w∈W u∈U −1 −1 X l(w) X = γ m ( (−1) w) γu(uu)e w∈W u∈U −1 −1 X l(w) X = γ m ( (−1) w) γuue w∈W u∈U X = γ−1m−1( (−1)l(w)w)γue w∈W X = m−1( (−1)l(w)w)ue w∈W = m−1me = e.

(The equality ( P (−1)l(w)w)ue = me follows by Lemma 5.1.4.) w∈W We have shown that e ∈ kGα, completing the proof.

Finally, we show that the converse of part 2 of the previous theorem holds. So, in particular, Stk is irreducible if and only if m = [G : B] is relatively prime to char(k).

Theorem 5.1.6. ([47, Theorem 3]) If char(k) divides m, then Stk is reducible.

Proof. We will show that Stk has a non-zero proper submodule. Consider the element P P l(w) ue ∈ Stk. We claim that ue 6= 0. We have ue = (−1) unwb. To show that ue 6= 0, it u∈Uw∈W suﬃces to show that the coeﬃcient of nw0 b in ue is non-zero. But, suppose that unwb = nw0 b

41 for some u ∈ U, w ∈ W . Then, unwB = nw0 B, and by the Bruhat decomposition we must have w = w0. But, unwB = nw0 B implies that u = 1. Therefore, the coeﬃcient of nw0 b in l(w0) ue is (−1) 6= 0, which means that ue 6= 0. Thus, kGue is a non-zero kG-submodule of Stk.

To show that Stk is not irreducible, it remains to show that kGue is a proper submodule P l(w) of Stk. We claim that e 6∈ kGue. First, we will check that ue( (−1) nw) = 0. By w∈W P l(w) Lemma 5.1.4, we have ( (−1) nw)ue = me. But, since char(k) | m, m = 0 in k, so that w∈W P l(w) P l(w) ( (−1) nw)ue = 0. Now, by deﬁnition of u and e, we have 0 = ( (−1) nw)ue = w∈W w∈W P l(w) l(w0) ˜ (−1) (−1) nwunw0 b. Note that two elements nwunw0 b and nw˜un˜ w˜0 b in this sum w,w0∈W, u∈U,b∈B −1 ˜ −1 are equal if and only if their inverses (nwunw0 b) and (nw˜un˜ w˜0 b) are equal. So, setting τ = Pt, we have t∈T

X l(w) l(w0) −1 0 = (−1) (−1) (nwunw0 b) w,w0∈W, u∈U,b∈B X l(w) l(w0) −1 −1 −1 −1 = (−1) (−1) b nw0 u nw w,w0∈W, u∈U,b∈B X l(w0) −1 X l(w) −1 = ( (−1) bnw0 u)( (−1) nw ) w0∈W w∈W X l(w0) −1 X l(w) −1 = ( (−1) uτnw0 u)( (−1) nw ) w0∈W w∈W X l(w0) −1 X l(w) −1 = ( (−1) unw0 τu)( (−1) nw ) w0∈W w∈W X l(w0) −1 X l(w) −1 = ( (−1) unw0 b)( (−1) nw ) w0∈W w∈W X l(w0) X l(w) = ( (−1) un(w0)−1 b)( (−1) nw−1 ) w0∈W w∈W X l(w) = ue( (−1) nw). w∈W

−1 (The second-to-last equality follows since nw is equal to nw−1 modulo T and T ⊆ B. The last equality follows since l(w) = l(w−1) for any w ∈ W .)

P l(w) Since ue( (−1) nw) = 0, every element of kGue is annihilated by right multiplica- w∈W P l(w) P l(w) tion by (−1) nw. So, to show that e 6∈ kGue, it suﬃces to show that e( (−1) nw) 6= w∈W w∈W 0. One of the properties satisﬁed by the unipotent radical U of B is that there is an element 0 P l(w) u ∈ U such that u 6∈ Uw for all w 6= 1. We claim that u appears in e( (−1) nw) = w∈W

42 P l(w0) l(w0) (−1) (−1) nw0 bnw with non-zero coefficient. The element u appears in one of w0,w∈W 0 −1 the sums nw0 bnw (w , w ∈ W ) if and only if u ∈ nw0 Bnw or, equivalently, nw0 u ∈ Bnw. If −1 this is the case, nw0 u is contained in both BnwB and Bn(w0)−1 B and, by the disjointness 0 −1 −1 of the Bruhat decomposition, we must have w = (w ) . Therefore, u ∈ nw Bnw, which −1 0 means that nwunw ∈ U, so that u ∈ Uw. By assumption on u, we must have w = 1, and it 0 P l(w) also follows that w = 1. So, u appears only in the summand b of e( (−1) nw). Since w∈W P l(w) P l(w) the coefficient of b in e( (−1) nw) is 1, the coefficient of u in e( (−1) nw) is also w∈W w∈W 1.

5.2 A New Proof of the Irreducibility Criterion for the Steinberg Module

5.2.1 Introduction

Let G be a finite group of Lie type, defined in characteristic p and having a split BN-pair, where B is a Borel subgroup of G containing a maximal torus T and N = NG(T ) is the normalizer of T in G. Let U be the unipotent radical of B (U is a normal p-Sylow subgroup of B and B = U oT ). Let W = NG(T )/T be the Weyl group of G with respect to T , and let S be the set of fundamental reflections generating W . And, let Stk := kGe be the Steinberg module over k.

In [47], Steinberg proves the following irreducibility criterion for Stk:

Stk is irreducible if and only if char(k)6 | [G : B]. Steinberg’s original proof of the irreducibility criterion relies on a series of computations based on various properties of a group with a BN-pair (outlined in the previous section). In this section, we will provide a new (and more conceptual) proof of Steinberg’s irreducibility criterion using the Hecke algebra.

5.2.2 The Proof of the Irreducibility Criterion

If char(k) = p (where p is the defining characteristic of G), then p 6 | [G : B] since U is a p-Sylow subgroup of G, and it is well known that the Steinberg module is irreducible.6 So, in what follows, we will assume that char(k) 6= p. Additionally, we will assume that k is sufficiently large (meaning that it is a splitting field for G and all of its subgroups).

G G ∼ Let k|B be the permutation module on the cosets of B in G. Since k|B = kGb, the G G Steinberg module Stk is a submodule of k|B. Let k|B = M1 ⊕ · · · ⊕ Mn be a decomposition of

6 We give a proof of the irreducibility of Stk in deﬁning characteristic in the Appendix (Section 21).

43 G k|B as a direct sum of indecomposable kG-modules. Each indecomposable direct summand G Mi of k|B has a simple socle and a unique simple quotient (this was shown by Green in [34]). Now, by [29, Remark 2.4], there is a unique index i (1 ≤ i ≤ n) such that Stk ⊆ Mi. The G indecomposable summand Mi of k|B containing the Steinberg module is called the Steinberg G component of k|B and will be denoted by MSt.

G The endomorphism algebra of k|B is the opposite of a Hecke algebra; speciﬁcally, we have G op ∼ EndkG(k|B) = Hk, where Hk is the Hecke algebra with k-basis {Tw|w ∈ W }, satisfying the relations ( Tsw if l(sw) > l(w) TsTw = cs cs q Tsw + (q − 1)Tw if l(sw) < l(w)

cs 7 for s ∈ S and w ∈ W (with the positive integers cs given by |Us| = q for all s ∈ S).

There is a Hom functor

G Fk : kG − mod → Hk − mod,V 7→ HomkG(k|B,V ).

By [29, Theorem 3.1], the Steinberg module Stk is mapped by Fk to the sign representation SGN of Hk. The trivial kG-module k is mapped by Fk to the index representation IND of Hk [29, Example 3.3].

Let Irrk(G) be a complete set of non-isomorphic irreducible kG-modules, and let Irrk(G|B) denote the set of unipotent principal series representations of G. An irreducible kG-module G G Y is in Irrk(G|B) if and only if Y is in head(k|B), which is true if and only if Y is in soc(k|B) [31, Theorem 4.2.6]. By [31, Theorem 4.2.9], the functor Fk induces a bijection

1−1 Irrk(G|B) → Irr(Hk)

from the unipotent principal series onto the set of irreducible Hk-modules.

Lemma 5.2.1. The irreducible socle of each indecomposable module Mi in the decomposition G k|B = M1 ⊕ · · · ⊕ Mn is equal to its irreducible head.

Proof. Fix an index i (1 ≤ i ≤ n), and let V = soc(Mi) and Y = head(Mi). As remarked in the discussion above, both V and Y are irreducible. Furthermore, since Mi is a direct G summand of k|B, both V and Y belong to the unipotent principal series Irrk(G|B). By [31, Corollary 4.1.10 (c)], the Hk-module Fk(Mi) is the projective indecomposable with head Fk(Y ). On the other hand, since the functor Fk is left exact, we have Fk(V ) ⊆ Fk(Mi). Since Fk establishes a bijection Irrk(G|B) → Irr(Hk), Fk(V ) is an irreducible Hk-module contained in the socle of the projective indecomposable module Fk(Mi). But, with the assumption that char(k) 6= p, we have that Hk is a symmetric algebra. Thus, the socle of Fk(Mi) is equal to its head, and it follows that Fk(V ) = Fk(Y ). Since Fk : Irrk(G|B) → Irr(Hk) is a bijection,

7See Chapter 6 for further discussion of Hecke algebras.

44 we conclude that V = Y .

Theorem 5.2.2. Stk is irreducible if and only if char(k)6 | [G : B]

Proof. As remarked above, we may assume that char(k) 6= p (if char(k) = p, then p6 | [G : B] and Stk is irreducible). Suppose ﬁrst that char(k) 6 | [G : B]. Then, by [31, Lemma 4.3.2], G G the permutation module k|B is completely reducible. Thus, every composition factor of k|B G is contained in the unipotent principal series Irrk(G|B). Since Stk ⊆ k|B, the same is true for all of the composition factors of Stk. But, by [29, Proposition 3.2], Stk has a unique G composition factor belonging to k|B. Therefore, Stk must be irreducible.

Conversely, suppose that Stk is an irreducible module. We claim that in this case, G MSt = Stk (where MSt is the indecomposable Steinberg component of k|B). Assume, for contradiction, that Stk is properly contained in MSt. By Lemma 5.2.1, MSt has Stk as both its head and socle. So, if MSt 6= Stk, then Stk occurs at least twice as a composition factor G G of MSt, which means that [k|B : Stk] ≥ 2 (where [k|B : Stk] denotes the multiplicity of the G Steinberg module as a composition factor of k|B). Now, since U ≤ B, there exists a surjective G G homomorphism k|U k|B. By Frobenius reciprocity, the multiplicity of Stk as a composition G factor of k|U is equal to the multiplicity of k as a composition factor of the restriction Stk ↓U of Stk to U. By [47, Theorem 1], the set {ue | u ∈ U} forms a k-basis for Stk. Thus, the restriction of Stk to U is the regular representation kU of U, and [Stk ↓U : k] = dimk(k) = 1, G G G which means that [k|U : Stk] = 1. Since k|B is a homomorphic image of k|U , we must have G G [k|B : Stk] ≤ [k|U : Stk] = 1, a contradiction.

By [31, Corollary 4.1.10 (c)], Fk(MSt) is the projective indecomposable Hk-module with head Fk(Stk) = SGN. Thus, if Stk is irreducible, we have by the previous paragraph that Fk(MSt) = Fk(Stk) = SGN, which means that SGN is a projective Hk-module. Φ Φ But, IND = SGN , where SGN is the twist of the sign representation of Hk by the k- l(w) −1 automorphism Φ : Hk → Hk, which is given by Tw 7→ (−1) qwTw−1 for any w ∈ W cs cs (where qw = q 1 ··· q m if w = s1 ··· sm is any reduced expression for w). The “twisting Φ functor” Hk − mod → Hk − mod, M 7→ M is exact. So, since SGN is projective, the index representation IND of Hk is projective as well.

G Now, since k ∈ Irr(G|B), there is an indecomposable component Mj of k|B having k as its irreducible head and socle. By [31, Corollary 4.1.10(c)], Fk(Mj) is the projective cover of Fk(k) = IND. Since IND is projective, we must have Fk(Mj) = IND. But, by [31, Corollary 4.1.10 (b)], the multiplicity [Mj : k] of k as a composition factor of Mj is equal to dimk EndkG(Mj). By [31, Lemma 4.1.2], dimk EndkG(Mj) = dimk EndHk (Fk(Mj)) = ∼ dimk EndHk (IND) = 1. Thus, [Mj : k] = 1 and, since Mj is indecomposable with head(Mj) = ∼ k and soc(Mj) = k, we must have Mj = k. So, k is a direct summand of the permutation G module k|B, which can occur only if char(k)6 | [G : B].

45 6 Hecke Algebras

6.1 The Hecke algebra associated to a Coxeter system

Let Z = Z[t, t−1] be the ring of Laurent polynomials over Z in an indeterminate t. Let (W, S) be a ﬁnite Coxeter system, and let He = He(W, Z) denote the associated Hecke algebra over Z. He is free over Z, with basis {τw}w∈W , satisfying the relations

( τsw if sw > w τsτw = tτsw + (t − 1)τw otherwise for s ∈ S, w ∈ W [9].

Let l denote the length function of W with respect to S. By [44, (2.0.3)], there is a l(w) −1 Z-involution Φ : He → He with Φ(τw) = (−t) τw−1 for all w ∈ W . Given a right He-module Mf, let MfΦ denote the He-module obtained by twisting the action of He on Mf by Φ. The assignment Mf 7→ MfΦ deﬁnes an exact functor Φ : mod − He → mod − He (where mod − He denotes the category of right He-modules).

There is also a Z-anti-involution ι : He → He given by ι(τw) = τw−1 for any w ∈ W [44, (2.0.5)]. Given a right He-module Mf, let Mfι denote the left He-module obtained by twisting the action of He on Mf by ι. The anti-automorphism ι is used primarily as a means to convert the left action of He on the dual of a right He-module to a right action. Let ∗ (−) = HomZ (−, Z) denote the linear dual on the category mod − He of right He-modules. Then, given a right He-module Mf ∈ mod − He, the dual Mf∗ of Mf is a left He-module. Thus, twisting the action of He on Mf∗ by ι yields a right He-module. For any Mf ∈ mod − He, we D ∗ι will write Mf He := Mf .

D The functor D : mod − H → mod − H, M → M He is a contravariant duality functor. He e e f f D2 ∼ And, given any Z-free right He-module Mf, we have Mf He = Mf.

6.2 Parabolic Subalgebras of He

Given a subset λ ⊆ S, let Wλ = hsis∈λ be the parabolic subgroup of W corresponding to λ. For every subset λ ⊆ S, we can deﬁne a parabolic subalgebra of He by setting Heλ = hτsis∈λ.

l(w) Let IND : He → Z be the linear character on He deﬁned by IND(τw) = t for all w ∈ W . l(w) And, let SGN : He → Z be the linear character on He deﬁned by SGN(τw) = (−1) for all w ∈ W . The behavior of the H-modules IND and SGN under the functors Φ and D is e He given in [44, (2.0.6)]:

46 INDΦ ∼= SGN, SGNΦ ∼= IND and D D IND He ∼= IND, SGN He ∼= SGN. For any λ ⊆ S, let IND = IND| and SGN = SGN| . λ Heλ λ Heλ

P P λ(w) Given λ ⊆ S, deﬁne xλ = τw ∈ He and yλ = (−t) τw ∈ He. For any w ∈ Wλ, w∈Wλ w∈Wλ the action of a basis element τw on the elements xλ and yλ of He is given in the following lemma.

Lemma 6.2.1. ([25, Lemma 1.1]) Let w ∈ Wλ. Then

(1) τwxλ = INDλ(τw)xλ, and

(2) τwyλ = SGNλ(τw)yλ.

∼ He ∼ He By Lemma 6.2.1, it follows that Hxe λ = Ind INDλ and Hye λ = Ind SGNλ (where Heλ Heλ INDλ and SGNλ are the left Heλ-modules deﬁned by the linear characters INDλ and SGNλ, respectively). Similarly, if INDλ and SGNλ are the right Heλ-modules deﬁned by the linear ∼ He ∼ He characters INDλ and SGNλ, respectively, then xλHe = Ind INDλ and yλHe = Ind SGNλ. Heλ Heλ

6.3 Specializing the Generic Hecke Algebra

0 0 0 0 0 If Z is a commutative Z-algebra, we write He = HeZ0 = He ⊗Z Z . The Z -algebra He has Z-basis consisting of elements τw ⊗ 1, w ∈ W , which satisfy relations analogous to the deﬁning relations of He. By abuse of notation, we will denote τw ⊗ 1 (w ∈ W ) by τw. The Z-automorphism Φ and the Z-anti-automorphism ι of He extend to the specialized Hecke algebra He 0.

In the following lemma, we list several important results concerning the permutation 0 0 modules He xλ and He yλ.

Lemma 6.3.1. ([25, Lemma 1.1]) Let λ, µ ⊆ S. Then,

0 ∗ ∼ 0 0 ∗ ∼ 0 ∗ 0 0 1. (xλHe ) = He xλ and (yλHe ) = He yλ (where () = HomZ0 (−, Z ) is the Z -linear dual). 2. (x H0)Φ ∼ y H0 and Hom (y H0, y H0) ∼ Hom (x H0, x H0). λ e = λ e He 0 λ e µ e = He λ e µ e 0 ∼ 0 0 ∼ 0 3. If the subgroups Wλ and Wµ are W -conjugate, then xµHe = xλHe and yµHe = yλHe .

47 6.4 The “Double Coset” Hecke Algebra

6.4.1 Deﬁnition of a Hecke Algebra given in [12]

In [12, 11.22], Curtis and Reiner give the following definition of a Hecke algebra. Let G be a finite group, and let B be any subgroup of G. Now, consider the group algebra KG, where K is a subfield of the complex numbers which is a splitting field for G and all of its subgroups. For a subset X ⊆ G, define an element X of KG by X = P x. Let H(G, B) x∈X 1 be the subset of KG consisting of all K-linear combinations of elements BgB, g ∈ G. |B| H(G, B) is called a Hecke algebra (we show in Proposition 6.4.1 below that H(G, B) is a K-algebra).

For finite groups of Lie type, there is a useful modification of the definition of a Hecke algebra. Let Z = Z[q, q−1] be the ring of Laurent polynomials over Z in the indeterminate q. Let (W, S) be a finite Coxeter system, and let {cs} be a fixed system of index parameters for this Coxeter system, i.e. a system of integers cs such that cs = ct whenever s and t are W -conjugate. Given w ∈ W , let w = s1 ··· sm (s1, . . . , sm ∈ S) be a reduced expression for cs cs W . Then, define qw = q 1 ··· q m (one can check that this definition is independent of the choice of reduced expression for w). Then, the generic Hecke algebra He over Z is the algebra with Z-basis {τw}w∈W , and relations

( τsw if l(sw) > l(w); τsτw = qsτsw + (qs − 1)τw if l(sw) < l(w) for all s ∈ S, w ∈ W . Additionally, if Z0 is any commutative Z-algebra, we can tensor He 0 0 with Z to obtain a Hecke algebra HeZ0 over Z . Then, HeZ0 has generating set {τw ⊗ 1}w∈W , subject to relations analogous to those given above.

The “double coset” Hecke algebra H(G, B) can be defined without reference to a Coxeter system. Therefore, at first glance, the “double coset” definition of a Hecke algebra seems unrelated to the “Coxeter system” definition. However, it can be shown that the “double coset” definition is, in fact, a generalization of the “Coxeter system” definition. We will demonstrate that this is the case using the example of a group with a BN-pair (which is naturally equipped with a Coxeter system). Specifically, Let G be a group with a BN-pair, and let W = N/(B ∩ N) be the Weyl group of G. Let S = {s1, . . . , sn} be a generating set for W as a Coxeter group. We will show that H(G, B) admits a presentation by generators and relations analogous to the one given above.

48 6.4.2 The case of a group with a BN-pair

The following material is based on [12, 11D] and [13, 65A, 67A]. Let G be a ﬁnite group with a BN-pair, and let W denote the Weyl group of G.

If nw ∈ N is a representative of a coset w ∈ W , we write wB for the coset nwB (since B ∩ N ⊂ B, one can check that the deﬁnition of wB is independent of the choice of representative nw of w). We can decompose G as a disjoint union of (B,B)-double cosets G = S BwB (Bruhat decomposition). In particular, every (B,B)-double coset in G is w∈W of the form BwB for some w ∈ W . Therefore, H(G, B) (as deﬁned in Curtis and Reiner) 1 consists of K-linear combinations of elements BwB of KG. In fact, by the Bruhat de- |B| 1 composition, the elements BwB , w ∈ W , form a K-basis of H(G, B). For convenience, |B| 1 we will denote BwB by a . |B| w Proposition 6.4.1. H(G, B) is a K-algebra.

1 Proof. First, we observe that the element a = B ∈ KG is idempotent. Therefore, 1 |B| a1KGa1, is an algebra, with identity element a1. We claim that H(G, B) is equal to a1KGa1. Note that a1KGa1 is spanned by elements of the form a1xa1, x ∈ G, and we have 1 1 1 X a xa = Bx B = b xb . (6.1) 1 1 |B| |B| |B|2 1 2 b1,b2∈B

−1 −1 Now, if b1xb2 = x for some b1, b2 ∈ B, then b1 = xb2 x . Therefore, the number of times P −1 that x appears in the sum b1xb2 is |B ∩ xBx |. Similarly, we can show that any b1,b2∈B −1 P element b1xb2 ∈ BxB appears |B ∩ xBx | times in the sum b1xb2. b1,b2∈B

Thus, (6.1) is equal to

1 X 1 |B ∩ xBx−1| b xb = |B ∩ xBx−1| BxB. |B|2 1 2 |B|2 b1xb2∈BxB

Now, by the Bruhat decomposition, there exists some wx ∈ W such that BxB = BwxB. 1 1 1 So, |B ∩ xBx−1| BxB = |B ∩ xBx−1| Bw B = |B ∩ xBx−1| a . Consequently, |B|2 |B|2 x |B| wx 1 we have a xa = |B ∩ xBx−1| a . This calculation shows both that H(G, B) ⊆ a KGa 1 1 |B| wx 1 1 and that a1KGa1 ⊆ H(G, B). Thus, H(G, B) = a1KGa1, as claimed.

49 In the next lemma, we establish a useful formula for the multiplication of basis elements aw of H(G, B). We will use without proof the fact that the group algebra KG can be iden- tiﬁed with the set of K-valued functions on G, with multiplication given by the convolution P product. Explicitly, an element αxx ∈ KG can be identiﬁed with the function f : G → K x∈G given by f(x) = αx. If f and g are two K-valued functions on G, then their convolution product is given by (f · g)(x) = P f(xy)g(y−1) for x ∈ G. y∈G

P 1 Lemma 6.4.2. For u, v ∈ W , we have auav = µu,v,waw, where µu,v,w = |BuB ∩ w∈W |B| w(BvB)−1|.

Proof. Since the elements aw, w ∈ W , form a K-basis for H(G, B), we can write X auav = µu,v,waw, w∈W

0 where µu,v,w is some structure constant in K for u, v, w ∈ W . Fix an element w ∈ W . Identi- P fying KG with the set of K-valued functions on G, we have (auav)(nw0 ) = ( µu,v,waw)(nw0 ). w∈W P P 0 By definition ( µu,v,waw)(nw0 ) is the coefficient of nw0 in µu,v,waw. Since nw0 ∈ Bw B w∈W w∈W and the double cosets BwB (w ∈ W ) are disjoint, nw0 occurs only in aw0 , with a coefficient 1 of . So, |B| X 1 (a a )(n 0 ) = ( µ a )(n 0 ) = µ 0 . (6.2) u v w u,v,w w w |B| u,v,w w∈W P −1 On the other hand, by definition of the convolution product, (auav)(nw0 ) = au(y)av(y nw0 ). y∈G 1 Since a = BuB, an element y ∈ G occurs with non-zero coefficient in a if and only if u |B| u −1 y ∈ BuB. That is, au(y) 6= 0 if and only if y ∈ BuB. Similarly, av(y nw0 ) 6= 0 if and only if −1 −1 0 −1 −1 y nw0 ∈ BvB, or, equivalently, y ∈ nw0 (BvB) = w (BvB) . Thus, au(y)av(y nw0 ) 6= 0 if and only if y ∈ BuB ∩ w0(BvB)−1. Therefore,

X −1 (auav)(nw0 ) = au(y)av(y nw0 ). (6.3) y∈BuB∩w0(BvB)−1

1 P −1 Comparing the equations (6.2) and (6.3), we have µu,v,w0 = au(y)av(y nw0 ), |B| y∈BuB∩w0(BvB)−1 P −1 1 −1 so that µu,v,w0 = |B| au(y)av(y nw0 ). And, since au(y) = and av(y nw0 ) = y∈BuB∩w0(BvB)−1 |B| 1 0 −1 P 1 1 1 −1 for y ∈ BuB∩w (BvB) , µu,v,w0 = |B| = |BuB∩w(BvB) |. |B| y∈BuB∩w0(BvB)−1 |B| |B| |B|

50 Next, we will use Lemma 6.4.2, along with several properties of groups with a BN- pair, to compute asiaw (si ∈ S, w ∈ W ) when l(siw) = l(w) + 1 and l(siw) = l(w) − 1. We state without proof the properties of G which will be needed to carry out these computations.

Proposition 6.4.3. Let G be a group with a BN-pair. Let W be the Weyl group of G associated with the BN-pair, and let S = {s1, . . . , sn} be the set of distinguished generators of W . The following statements hold.

(a) ([13, 65.2]) For si ∈ S and w ∈ W , siBw ⊆ BwB ∪ BsiwB.

(b) ([13, 65.2]) If l(siw) > l(w), then siBw ⊆ BsiwB.

We may extend the result of Proposition 6.4.3(a) as follows. Let w, w0 ∈ W . Let 0 0 w = sj(1) ··· sj(r) be a reduced expression for w . Then, by Proposition 6.4.3(a),

0 wBw = (wBsj(1)) ··· sj(r) ⊆ (BwB ∪ Bsj(1)wB)sj(2) ··· sj(r).

Repeated application of Proposition 6.4.3(a) shows that

0 wBw ⊆ ∪Bwsj(k1) ··· sj(kp)B,

where the union is taken over all subsequences of 1 ≤ k1 < k2 < ··· < kp ≤ r of 1, . . . , r (1 ≤ p ≤ r). (This result appears in the proof of [12, 65.8].)

Deﬁnition 6.4.4. Let G be a group with a BN-pair. Let W be the Weyl group of G, and let S = {s1, . . . , sn} be the set of distinguished generators of W . Then, for 1 ≤ i ≤ n, the index parameter of si is ind(si) = qi = [B : B ∩ siBsi].

In fact, we may deﬁne an index parameter [B : B ∩ wBw−1] for any w ∈ W . This −1 deﬁnition gives rise to an “index” map ind : H(G, B) → C, given by aw 7→ [B : B ∩ wBw ]. It can be shown that ind is an algebra homomorphism [12, Exercise 11.19].

Lemma 6.4.5. For any s ∈ S, a2 = q a + (q − 1)a . i si i 1 i si

Proof. By Proposition 6.4.3 (a), (BsiB)(BsiB) = BsiBsiB ⊆ B(B ∪ BsiB)B = B ∪ BsiB. 1 1 Now, a2 = Bs B Bs B = P xy. In particular, every element xy appearing in si 2 i i 2 |B| |B| x,y∈BsiB a2 is contained in (Bs B)(Bs B) ⊆ B ∪ Bs B. Therefore, if we write a2 = P µ a as si i i i si si,si,w w w∈W in Lemma 6.4.2, the only non-zero structure coeﬃcients are µsi,si,1 and µsi,si,si , and it follows that a2 = µ a + µ a . si si,si,1 1 si,si,si si

1 |Bs B| |Bs Bs | Now, by Lemma 6.4.2, µ = |Bs B ∩ Bs B|. And, i = i i = si,si,1 |B| i i |B| |B| |BsiBsi/B| = |B/B ∩ siBsi| = [B : B ∩ siBsi] = qi (where the third equality follows

51 by the Second Isomorphism Theorem). Thus, a2 = q a + µ a . si i 1 si,si,si si

To compute µsi,si,si , we use the homomorphism ind : H(G, B) → C. Applying ind to both sides of a2 = q a + µ a , we have q2 = q (1) + q µ . So, µ = q − 1, and si i 1 si,si,si si i i i si,si,si si,si,si i a2 = q a + (q − 1)a , as needed. si i 1 i si

Theorem 6.4.6. Let w ∈ W and si ∈ S. Then, ( asiw if l(siw) > l(w) asi aw = qiasiw + (qi − 1)aw if l(siw) < l(w).

Proof. First, we consider the case of l(siw) > l(w). By Lemma 6.4.2, we have as aw = P i µsi,w,uau. Now, by Proposition 6.4.3(b), siBw ⊆ BsiwB. So, µsi,w,u = 0 when u 6= siw, u∈W

which means that asi aw = µsi,w,siwasiw. So, to prove the theorem in the case of l(siw) > l(w), it suﬃces to show that µsi,w,siw = 1.

But, by Lemma 6.4.2, we have 1 1 1 µ = |Bs B ∩ s w(BwB)−1| = |Bs B ∩ s wBw−1B| = |s Bs B ∩ wBw−1B|. si,w,siw |B| i i |B| i i |B| i i

−1 By Proposition 6.4.3 (a), siBsi ⊆ B ∪ BsiB. So, since B ⊆ wBw B, we have

−1 −1 −1 siBsiB ∩ wBw B ⊆ (B ∪ BsiB) ∩ wBw B = B ∪ (BsiB ∩ wBw B).

−1 Thus, the claim that µsi,w,siw = 1 will follow if we we can show that BsiB∩wBw B = ∅ (for, −1 −1 in that case, we will have siBsiB∩wBw B ⊆ B, which means that siBsiB∩wBw B = B, 1 |B| so that µ = |Bs B ∩ s w(BwB)−1| = = 1). si,w,siw |B| i i |B|

−1 −1 We show that BsiB∩wBw B = ∅ by contradiction. So, assume that BsiB∩wBw B 6= −1 ∅. Let w = sj(1) ··· sj(r) be a reduced expression for w. Then, w = sj(r) ··· sj(1) is a reduced expression for w−1. By the extended version of Proposition 6.4.3 (a), we have wBw−1 ⊆

∪Bwsj(k1) ··· sj(kp)B, where the union is taken over all subsequences of r ≥ k1 > k2 > ··· > −1 kp ≤ 1 of 1, . . . , r (1 ≤ p ≤ r). Since BsiB ∩ wBw B 6= ∅, BsiB ∩ (∪Bwsj(k1) ··· sj(kp)B) 6= ∅. By the disjointness of the double cosets appearing in the Bruhat decomposition of G,

we must have BsiB = Bwsj(k1) ··· sj(kp)B for some subsequence k1, . . . , kp of 1, . . . , r. Thus, −1 −1 si = wsj(k1) ··· sj(kp), so that w si = sj(k1) ··· sj(kp). And, l(siw) = l(w si) ≤ p ≤ r = l(w), which contradicts the assumption that l(siw) > l(w).

0 0 0 Now, we consider the case of l(siw) < l(w). Let w = siw. Then, l(siw ) = l(w) > l(w ). 2 So, applying the ﬁrst case, we have a a 0 = a . Using Lemma 6.4.5, a a = a a 0 = si w w si w si w 0 0 0 (qia1 + (qi − 1)asi )aw = qiaw + (qi − 1)asi aw = qiasiw + (qi − 1)aw.

52 Corollary 6.4.7. H(G, B) is the K-algebra with generators aw, w ∈ W , subject to the relations ( asiw if l(siw) > l(w) asi aw = qiasiw + (qi − 1)aw if l(siw) < l(w). for si ∈ S, w ∈ W .

6.5 Hecke Algebra Associated with a Harish Chandra Series

Let G be a finite group of Lie type defined in characteristic p > 0 (so, G is the fixed point sub- ¯ group of a connected reductive algebraic group G over Fp under a Steinberg endomorphism), and assume that the BN-pair of G is split. Let U be the unipotent radical of B (i.e., the largest normal p-subgroup of B), so that B = UT (where T = B∩N). Let (W, S) be the Cox- eter system corresponding to the BN-pair structure on G. Let k be an algebraically closed n field of characteristic r > 0 with r 6= p. As in Section 4, let PG = { PJ | J ⊆ S, n ∈ N} n be the collection of parabolic subgroups in G, and let LG = { LJ | J ⊆ S, n ∈ N} be the collection of Levi subgroups in G.

Following [31], we will work with left kG-modules in this section. Let L ∈ LG and let X G be a cuspidal irreducible (left) kL-module. Let RL be the Harish-Chandra induction functor from the category of left kL-modules to the category of left kG-modules. We deﬁne G op H (L, X) := EndkG(RL (X)) , G op G where EndkG(RL (X)) denotes the opposite of the endomorphism algebra EndkG(RL (X)). The algebra H (L, X) is called the Hecke algebra associated with the pair (L, X).

The category of left kG-modules is related to the category of left H (L, X)-modules via the Hom functor G F G : kG − mod → H (L, X) − mod,Y 7→ HomkG(R (X),Y ). RL (X) L

To simplify the notation, we will denote the Hom functor F G by Fk. RL (X)

The following theorem (which was ﬁrst proved in [30]) describes the connection between the Harish-Chandra series Irrk(G|(L, X)) and the Hecke algebra H (L, X). We have chosen to include the proof of this result because some of the techniques used here are needed in the original proofs of this dissertation.

Theorem 6.5.1. ([31, Theorem 4.2.9]) Let L ∈ L G, and let X be a cuspidal irreducible (left) kL-module. Let P ∈ PG be such that P = UP o L. The functor Fk induces a bijection 1−1 Fk : Irrk(G|(L, X)) → Irrk(H (L, X)), where Irrk(H (L, X)) denotes the set of irreducible left H (L, X)-modules, up to isomorphism.

53 Proof. We follow the proof given by Geck and Jacon [31, Theorem 4.2.9]. Let PX be the G projective cover of X (as a kL-module). Since the Harish-Chandra induction functor RL is exact, the natural surjective kL-module homomorphism PX X induces a surjective G G kG-module homomorphism RL (PX ) RL (X).

By [31, Lemma 4.1.7, Proposition 4.1.8], to prove the statement of the theorem, it suﬃces to prove that

G G G G dimk HomkG(RL (X),RL (X)) = dimkHomkG(RL (PX ),RL (X)). First, using the adjointness of Harish-Chandra induction and restriction, we have G G ∼ ∗ G G HomkG(RL (X),RL (X)) = HomkL(X, RL (RL (X))). Now, by the Mackey decomposition, ∗ G G ∼ L ∗ nL n RL (RL (X)) = ⊕ RnL∩L( RnL∩L( X)), n∈D(P,P ) where D(P,P ) denotes a full set of (P,P )-double coset representatives in G. Therefore, we have G G ∼ L ∗ nL n HomkG(RL (X),RL (X)) = ⊕ HomkL(X,RnL∩L( RnL∩L( X))). n∈D(P,P ) n n ∗ nL n n n Since X is a cuspidal k L-module for any n ∈ D(P,P ), RnL∩L( X) = 0 when L∩L ( L. n Now, for an element n ∈ D(P,P ), L∩L = L if and only if n ∈ D(P,P )∩NG(L). Therefore, G G ∼ n HomkG(RL (X),RL (X)) = ⊕ HomkL(X, X). n∈D(P,P )∩NG(L) A similar calculation shows that G G ∼ n HomkG(RL (PX ),RL (X)) = ⊕ HomkL(PX , X). n∈D(P,P )∩NG(L)

n But, since PX is a projective cover of X, and X is an irreducible kL-module when n ∈ n ∼ n D(P,P ) ∩ NG(L), HomkL(PX , X) = HomkL(X, X) and the desired result follows.

Geck and Jacon [31] also give the following semisimplicity criterion for H (L, X).

Proposition 6.5.2. ([31, Proposition 4.2.10]) Let L be a Levi subgroup in G and let X be a G cuspidal irreducible (left) kL-module. Then, RL (X) is a semisimple kG-module if and only if H (L, X) is a semisimple algebra.

Given a pair (L, X), where L ∈ LG is a Levi subgroup of G and X is a cuspidal irre- n ∼ ducible kG-module, let W (L, X) := {n ∈ (NG(L) ∩ N)L | X = X}/L. Geck and Jacon [31] refer to W (L, X) as the inertia group of X.8

8 G Dipper and Du [19] refer to W (L, X) as the ramiﬁcation group of RL (X).

54 Proposition 6.5.3. Let L ∈ LG, and let X be a cuspidal irreducible (left) kL-module. Then, dimk H (L, X) = |W (L, X)|.

We will now consider the special case with L = T and X = k. Under these assumptions, we have Irrk(G|(L, X)) = Irrk(G|(T, k)) = Irrk(G|B), which is the unipotent principal G G G Harish-Chandra series. In this case, RL (X) = RT (k) = k|B, and the associated Hecke alge- G op bra is Hk = H (T, k) = EndkG(k|B) .

G P Note that the permutation module k|B is isomorphic to kGb, where b = b. Given b∈B ˙ G op ˙ an element w ∈ W , let Tw ∈ Hk = EndkG(k|B) be the kG-module homomorphism Tw : kGb → kGb given by ˙ X Tw(xB) = yB yB∈kGb with x−1y∈BwB

for any x ∈ kG. Now for any s ∈ S, let qs = |BsB/B|, and letq ¯s = qs1k be the image of the integer qs in k. When s, t ∈ S are W -conjugate,q ¯s =q ¯t ([31, 4.3.1]). The Hecke algebra Hk ˙ has k-basis {Tw | w ∈ W }, subject to the relations

( T˙ if l(sw) > l(w), T˙ T˙ = sw s w ˙ ˙ q¯sTsw + (¯qs − 1)Tw if l(sw) < l(w)

for all s ∈ S and w ∈ W [31, 4.3.1]. (The presentation of Hk given above was originally found by Iwahori.)

We now give a modiﬁed presentation of Hk. Since the ﬁeld k is assumed to be large 1/2 1/2 1/2 enough, there exist square rootsq ¯s ∈ k for all s ∈ S, which satisfyq ¯s =q ¯t when s, t ∈ S −1/2 ˙ are W -conjugate. Given any s ∈ S, let Ts =q ¯s Ts. For an arbitrary element w ∈ W , let

Tw = Ts1 ··· Tsn when w = s1 ··· sn (s1, . . . , sn ∈ S) is a reduced expression for w. Then, the set {Tw | w ∈ W } is a k-basis for Hk, satisfying the relations

( Tsw if l(sw) > l(w), TsTw = 1/2 −1/2 Tsw + (¯qs +q ¯s )Tw if l(sw) < l(w)

for all s ∈ S and w ∈ W [31, 4.3.1]. From the presentation of Hk given above, it follows that Hk is a specialization of a generic Hecke algebra, as deﬁned in [25].

G op Remark 6.5.4. In [31], Geck and Jacon define the Hecke algebra Hk as EndkG(kB ) , where G G k|B is a left kG-module. With this definition, the Hom functor Fk = HomkG(k|B, −) is a mapping from the category kG − mod of left kG-modules to the category Hk − mod of left Hk-modules. However, it is possible to define an analogous Hom functor relating the category of right kG-modules to the category of right modules for a Hecke algebra. Viewing G G k|B is a right kG-module, one can define the Hecke algebra as Hk = EndkG(k|B) (this is the G definition used by Dipper and Du [19]). In this case, the Hom functor Fk = HomkG(k|B, −)

55 is a mapping from the category mod − kG of right kG-modules to the category mod − Hk of right Hk-modules.

56 7 Highest Weight Categories and Quasi-hereditary Al- gebras

7.1 Preliminaries

Let A be a finite dimensional algebra over an algebraically closed field k. Let (Λ, ≤) be a finite poset which indexes the distinct isomorphism classes of irreducible objects in A − mod (where A − mod is the category of finite dimensional left A-modules). For λ ∈ Λ, let L(λ) ∈ A − mod be an irreducible module in the class of λ. Let P (λ) be the projective indecomposable module (PIM) with head L(λ).

Deﬁnition 7.1.1. ([14, C.8]) A − mod, together with the poset structure ≤ on Λ, is called a highest weight category provided that the following conditions hold: (HWC1) For λ ∈ Λ, there exists an object ∆(λ) ∈ A − mod with simple head L(λ) and with the property that all composition factors L(µ) of the radical rad(∆(λ)) satisfy µ < λ. (HWC2) For λ ∈ Λ, there exists a ﬁltration P (λ) = F λ ⊃ F λ ⊃ · · · ⊃ F λ = 0 such that 0 1 tλ l λ ∼ λ λ ∼ F0/F1 = ∆(λ) and, for 0 < i < tλ, Fi /Fi+1 = ∆(µi), for some µi ∈ Λ with µi > λ.

If A−mod is a highest weight category, the modules ∆(λ), λ ∈ Λ are called the standard objects in A − mod. There is also a costandard object ∇(λ) in A − mod for every element λ ∈ Λ [14, pg. 707]. If I(λ) is the injective envelope of L(λ) in A − mod, then I(λ) has a ∇-ﬁltration with bottom section ∇(λ) and upper sections ∇(µ), µ > λ.

Example 7.1.2. Let A be a semisimple algebra over an algebraically closed field k, and let Λ be any set indexing the irreducible A-modules. We can define a partial order ≤ on Λ by setting ≤= {(λ, λ)|λ ∈ Λ}. For any λ ∈ Λ, we define ∆(λ) = ∇(λ) = P (λ) = I(λ) = L(λ) (since A is semisimple, each irreducible A-module L(λ) is projective and injective). Then, A − mod (together with the poset structure ≤ on Λ is a highest weight category.

Proposition 7.1.3. ([14, C.2]) If A − mod is a highest weight category with poset (Λ, ≤) and Aop is the opposite algebra of A, then Aop − mod is a highest weight category with the opposite poset Λop (obtained by reversing the relations in Λ).

The category Aop − mod may be identified with the category of finite dimensional right A-modules [14, pg. 707]. With this identification, the standard object of Aop − mod cor- op ∗ responding to the weight λ ∈ Λ is ∆ (λ) = ∇(λ) = Homk(∇(λ), k) (where ∇(λ) is the costandard object in A−mod corresponding to λ). And, the costandard object of Aop −mod op ∗ corresponding to the weight λ ∈ Λ is ∇ (λ) = ∆(λ) = Homk(∆(λ), k) (where ∆(λ) is the standard object in A − mod corresponding to λ).

57 Proposition 7.1.4. ([14, Corollary C.11]) If A and B are ﬁnite dimensional k-algebras such that A − mod and B − mod are equivalent categories (i.e., if A and B are Morita equivalent k-algebras), then A − mod is a highest weight category if and only if B − mod is a highest weight category.

Deﬁnition 7.1.5. A is a quasi-hereditary algebra if A-mod is a highest weight category.

Given the deﬁnition above, the result of [14, C.11] may be restated as follows.

Proposition 7.1.6. If A and B are Morita equivalent ﬁnite-dimensional k-algebras, then A is a quasi-hereditary algebra if and only if B is a quasi-hereditary algebra.

7.2 Homological Properties of the Standard and Costandard Ob- jects

Let A − mod be a highest weight category with poset Λ. We state without proof several homological properties of modules in A − mod.

1 Proposition 7.2.1. ([14, C.12]) Let λ, µ ∈ Λ. If ExtA(∆(λ),L(µ)) 6= 0, then µ > λ. And, 1 if ExtA(L(λ), ∇(µ)) 6= 0, then µ > λ.

Let A−mod(∆) be the full subcategory of A−mod consisting of all modules in A−mod which have a ∆-filtration (i.e., modules which have a filtration in which each section is isomorphic to ∆(λ) for some λ ∈ Λ). Let A − mod(∇) be the full subcategory of A − mod consisting of all modules in A − mod which have a ∇-filtration (i.e., modules which have a filtration in which each section is isomorphic to ∇(λ) for some λ ∈ Λ).

Proposition 7.2.2. ([23, A2.2]) 1 (1) A module M ∈ A − mod has a ∆-ﬁltration if and only if ExtA(M, ∇(λ)) = 0 for all λ ∈ Λ. 1 (2) A module M ∈ A−mod has a ∇-ﬁltration if and only if ExtA(∆(λ),M) = 0 for all λ ∈ Λ.

Proposition 7.2.3. ([14, C.13]) Let A − mod be a highest weight category with poset Λ. n (1) If M ∈ A − mod(∆) and N ∈ A − mod(∇), then ExtA(M,N) = 0 for all n > 0. n n (2) Let λ, µ ∈ Λ. If ExtA(∆(λ),L(µ)) 6= 0 for some n > 0, then µ > λ. So, if ExtA(∆(λ), ∆(µ)) 6= 0 for some n > 0, then µ > λ. n n (3) Let λ, µ ∈ Λ. If ExtA(L(λ), ∇(µ)) 6= 0 for some n > 0, then λ > µ. So, if ExtA(∇(λ), ∇(µ)) 6= 0 for some n > 0, then λ > µ.

58 7.3 The Ringel Dual

The results of Sections 7.3-7.5 are not used directly in the original research presented in Parts III and IV of this dissertation, so the reader may choose to skip ahead to Chapter 8.

In this section, we follow [14, C.2]. Let A − mod be a highest weight category with poset Λ. Let A − mod(∆) be the subcategory of A − mod consisting of modules which have a ∆-ﬁltration, and let A − mod(∇) be the subcategory of A − mod consisting of modules which have a ∇-ﬁltration. A module M ∈ A−mod(∆)∩A−mod(∇) is called a tilting module.

Given a weight λ ∈ Λ, there exists an indecomposable tilting module X(λ) ∈ A − mod(∆) ∩ A − mod(∇) such that [X(λ): L(λ)] = 1 and for any composition factor L(µ) of X(λ) with µ 6= λ, µ < λ. (Here, [X(λ): L(λ)] denotes the number of times that L(λ) occurs as a composition factor of X(λ)). The indecomposable tilting module X(λ)(λ ∈ Λ) is unique up to isomorphism. If M ∈ A − mod(∆) ∩ A − mod(∇) is any tilting module, then M has a decomposition as a direct sum of the indecomposable tilting modules X(λ).

Let X ∈ A − mod(∆) ∩ A − mod(∇) be a tilting module which contains at least one direct summand isomorphic to X(λ) for every weight λ ∈ Λ, and let E = EndA(X). The algebra E is called the Ringel dual of A. The category E − mod is a highest weight category with weight poset Λop.

7.4 Global Dimension of a Quasi-hereditary Algebra

Given a ﬁnite dimensional k-algebra A, let gl dim A denote the global dimension of A (i.e., the supremum of the set of projective dimensions of A-modules). And, given an A-module M, let pd(M) denote the projective dimension of M.

Lemma 7.4.1. If A is a ﬁnite dimensional k-algebra, then gl dim A ≤ 1 if and only if A is a hereditary algebra (i.e., every submodule of a projective A-module is projective).

Proof. Suppose ﬁrst that gl dim A ≤ 1. Let M be a projective A-module, and let N be a submodule of M. Then, there is a short exact sequence of A-modules 0 → N → M → M/N → 0, which means that pd(N) ≤ max(pd(M), pd(M/N) − 1). Now, since M is projective, pd(M) = 0. And, since gl dim A ≤ 1, pd(M/N) − 1 ≤ 1 − 1 = 0. Thus, pd(N) = 0, and it follows that N is projective.

Suppose now that A is a hereditary algebra. Let M be an A-module, and let φ : P M be a projective cover of M. Since Ker(φ) is a submodule of the projective A-module P and A is hereditary, Ker(φ) is a projective A-module, which means that 0 → Ker(φ) → P → M → 0 is a projective resolution of M. Thus, pd(M) ≤ 1 for all A-modules M, and it follows that

59 gl dim A ≤ 1.

Proposition 7.4.2. If A is a ﬁnite dimensional k-algebra with gl dim A ≤ 1, then A is a quasi-hereditary algebra.

Proof. We will show that A − mod is a highest weight category. Since A is Artinian, we ∼ ⊕ k1 ⊕ kn can write AA = P1 ⊕ · · · ⊕ Pn for some positive integers ki (1 ≤ i ≤ n), where the Pi (1 ≤ i ≤ n) are distinct projective indecomposable A-modules. For any i, 1 ≤ i ≤ n, let Li = head(Pi). Then, the set {L(1),...,L(n)} is a full set of non-isomorphic irreducible 0 0 A-modules. Let Λ = {1, . . . , n}, and deﬁne a partial order ≤ on Λ by i ≤ j ⇔ Pi ⊆ Pj. Given any i ∈ Λ, let ∆(i) = P (i). To show that A-mod is a highest weight category with poset Λ, it suﬃces to show that if [∆(i): L(j)] 6= 0 for some i, j ∈ Λ, then j ≤0 i. But, if L(j) is a composition factor of ∆(i) = Pi, then the projective A-module Pi has a submodule M with head L(j). Since gl dim A ≤ 1, A is a hereditary algebra, which means that M is ∼ 0 projective. Therefore, M = Pj, and it follows that Pj ⊆ Pi and j ≤ i.

In fact, Dlab and Ringel [15] showed that a stronger result holds.

Proposition 7.4.3. ([15, Theorem 2]) If A is a ﬁnite dimensional k-algebra with gl dim A ≤ 2, then A is a quasi-hereditary algebra.

(Dlab and Ringel’s result cannot be improved - when gl dim A=3, A need not be quasi- hereditary.)

Theorem 7.4.4. If A is a quasi-hereditary algebra, then gl dim A < ∞.

Proof. We follow Donkin’s proof [23, A.2.3].

Suppose that the highest weight category A − mod has ﬁnite weight poset Λ. Given λ ∈ Λ, let l(λ) be the length of the longest chain λ0 < λ1 < ··· < λl = λ in Λ. And, let l(Λ) i be the maximum of the lengths l(λ), λ ∈ Λ. We claim that ExtA(L(λ),L(µ)) = 0 for λ, µ ∈ Λ and i > l(λ) + l(µ). Following Donkin, we use induction on l(λ) + l(µ). For the base case, suppose l(λ) + l(µ) = 2. Then, l(λ) = 1 and l(µ) = 1, so L(λ) = ∆(λ) and L(µ) = ∆(µ). i i And, since µ 6> λ, ExtA(∆(λ), ∆(µ)) = ExtA(L(λ),L(µ)) = 0 for i > l(λ) + l(µ) by [14, C.13(2)].

i 0 0 0 0 0 0 For the inductive step, assume that ExtA(L(λ ),L(µ )) = 0 for all λ , µ with l(λ )+l(µ ) < l(λ)+l(µ) and i > l(λ0)+l(µ0). Since L(λ) is the head of ∆(λ), we have a short exact sequence 0 → N → ∆(λ) → L(λ) → 0. For every composition factor L(ν) of N, ν < λ, which means that l(ν) < l(λ). Suppose i > l(λ) + l(µ). The long exact sequence in Ext induced by the short exact sequence 0 → N → ∆(λ) → L(λ) → 0 yields

i−1 i i ExtA (N,L(µ)) → ExtA(L(λ),L(µ)) → ExtA(∆(λ),L(µ)). (7.1)

60 Suppose L(ν) is a composition factor of N. Then, ν < λ, so that l(ν) + l(µ) < l(λ) + l(µ) < i. And, since l(ν) ≤ l(λ) − 1, i − 1 > l(ν) + l(µ). By the inductive hypothesis, i−1 i−1 ExtA (L(ν),L(µ)) = 0 for every composition factor L(ν) of N. Therefore, ExtA (N,L(µ)) = i ∼ i i 0, and (7.1) yields ExtA(L(λ),L(µ)) = ExtA(∆(λ),L(µ)). So, to prove that ExtA(L(λ),L(µ)) = i 0, it suﬃces to prove ExtA(∆(λ),L(µ)) = 0.

Since L(µ) is the socle of ∇(µ), we have a short exact sequence 0 → L(µ) → ∇(µ) → Q → 0, with every composition factor L(ν) of Q satisfying ν < µ. This short exact sequence induces a long exact sequence in Ext, which yields the exact sequence

i−1 i i ExtA (∆(λ),Q) → ExtA(∆(λ),L(µ)) → ExtA(∆(λ), ∇(µ)). (7.2)

If L(λ0) is a composition factor of ∆(λ) and L(ν) is a composition factor of Q, then l(λ0) + l(ν) ≤ l(λ) + l(ν) < l(λ) + l(µ). And, l(λ) + l(µ) < i, so l(λ0) + l(ν) < i − 1 and i−1 0 i−1 0 ExtA (L(λ ),L(ν)) = 0 by the inductive hypothesis. Therefore, ExtA (∆(λ ),Q) = 0. Also, i i ExtA(∆(λ), ∇(µ)) = 0 for i > 0 by [23, A.2.2(ii)]. Thus, (7.2) gives ExtA(∆(λ),L(µ)) = 0.

i We have proved that ExtA(L(λ),L(µ)) = 0 for λ, µ ∈ Λ and i > l(λ) + l(µ). Therefore, we have gl dim A ≤ 2l(Λ) < ∞.

7.5 Heredity Ideals

In this section, we will discuss another characterization of a quasi-hereditary algebra.

2 Deﬁnition 7.5.1. An ideal J E A is idempotent if J = J.

2 Theorem 7.5.2. If J E A with J = J, then there exists an idempotent e s.t. J = AeA.

Deﬁnition 7.5.3. J E A is called a heredity ideal if (1) J = AeA for some idempotent e,

(2) AJ is projective, and (3) eAe is a semisimple k-algebra.

Proposition 7.5.4. Suppose J is a heredity ideal and M and N are A/J modules. Then, n ∼ n ExtA/J (M,N) = ExtA(M,N).

n Proof. We follow Parshall’s proof in [43, Lemma 1.3]. First, we construct a map ExtA/J (M,N) n → ExtA(M,N). Let M be an A/J-module. The natural quotient map A → A/J deﬁnes an A-module structure on M, and we denote the resulting A-module by i∗M. A morphism f : M → N of A/J-modules deﬁnes a morphism i∗f : i∗M → i∗N of A-modules. The functor i∗ : A/J-mod → A-mod is exact.

61 n An element of ExtA/J (M,N) can be represented by an equivalence class of n-extensions 0 → N → Q1 → · · · → Qn → M → 0 of A/J-modules. Applying i∗ to an n-extension n of A/J-modules yields an n-extension of A-modules, giving us a map ExtA/J (M,N) → n ExtA(M,N). We will show that this map is an isomorphism.

n We claim that it is enough to show that ExtA(A/J, N) = 0 for all A/J-modules N and for n all n > 0. In this case, ExtA(P,N) = 0 for all projective A/J-modules P , all A/J-modules N, and all n > 0. Now, given an A/J-module M, ﬁx a projective module P s.t. there is an epimorphism P M and consider the resulting short exact sequence of A/J-modules 0 → M 0 → P → M → 0. This short exact sequence induces the long exact sequences

n−1 n−1 0 n n · · · → ExtA/J (P,N) → ExtA/J (M ,N) → ExtA/J (M,N) → ExtA/J (P,N) = 0

n−1 n−1 0 n n · · · → ExtA (P,N) → ExtA (M ,N) → ExtA(M,N) → ExtA(P,N) = 0. So, we have the following commutative diagram.

n−1 n−1 0 n ExtA/J (P,N) ExtA/J (M ,N) ExtA/J (M,N) 0

n−1 n−1 0 n ExtA (P,N) ExtA (M ,N) ExtA(M,N) 0

n ∼ n We proceed to show that ExtA/J (M,N) = ExtA(M,N) by induction on n. We have ∼ HomA/J (M,N) = HomA(M,N) = HomA(i∗M, i∗N) by deﬁnition of the A-module structure n−1 ∼ n−1 of i∗M and i∗N. Suppose now that n − 1 ≥ 0 and ExtA/J (M,N) = ExtA (M,N). We have ∼ n−1 n−1 HomA/J (P,N) = HomA(P,N). And, for n − 1 > 0, ExtA/J (P,N) = 0 and ExtA (P,N) = 0 since P is a projective A/J-module. It follows that the ﬁrst two downward arrows of the n ∼ n commutative diagram above are isomorphisms, and we have ExtA/J (M,N) = ExtA(M,N), as needed.

n Thus, to prove the statement of the proposition, it remains to check that ExtA(A/J, N) = n 0 for all A/J-modules N and for all n > 0. First, we show that ExtA(J, N) = 0 for all n ≥ 0. In degree 0, we must check that HomA(J, N) = 0. But, suppose f ∈ HomA(J, N). Since J is heredity, J 2 = J, so f(J) = f(J 2) = Jf(J) = 0 (here, Jf(J) = 0 since f(J) is contained in n the A/J-module N). And, for n > 0, ExtA(J, N) = 0 since J is projective as an A-module. Now, consider the short exact sequence 0 → J → A → A/J → 0. For n > 0, the long exact sequence in Ext yields the exact sequence

n−1 n n 0 = ExtA (J, N) → ExtA(A/J, N) → ExtA(A, N) = 0.

n Therefore, ExtA(A/J, N) = 0.

Deﬁnition 7.5.5. A is a quasi-hereditary algebra if there is a sequence 0 = J0 ⊆ J1 ⊆ J2 ⊆ · · · ⊆ Jn = A of ideals s.t. Ji/Ji−1 is heredity in A/Ji−1 for 1 ≤ i ≤ n.

62 8 The q-Schur algebra

Let Z = Z[t, t−1] be the ring of Laurent polynomials in an indeterminate t, and let (W, S) = (Sm,S) be the Coxeter system in which Sm is the symmetric group on m letters and S = {(1, 2), (2, 3),..., (m − 1, m)} is the set of fundamental reﬂections. Let He = He(Sm, Z) be the corresponding generic Hecke algebra (deﬁned as in [9, pg. 24, (1)]).

Let V be a free Z-module of rank n > 0 and let {v1, . . . , vn} be an ordered basis of V .

Given a sequence J = (j1, . . . , jm) of integers with 1 ≤ ji ≤ n, let vJ = vj1 ⊗ · · · ⊗ vjm . The ⊗m elements vJ give a basis of V . For σ ∈ Sm, let Jσ = (jσ−1(1), . . . , jσ−1(m)). Then, for any ⊗m s ∈ S, we can deﬁne an action of the generator τs of He on V by ( tvJs if ji ≤ ji+1 vJ τs = vJs + (t − 1)vJ otherwise.

This action extends to give a right action of He on V ⊗m, and the endomorphism algebra S (n, m) := End (V ⊗m) is the t-Schur algebra of bidegree (n, m) over Z (this is the deﬁni- t He tion given in [9, (8.2)]).

There is another description of the the tensor product V ⊗m, involving partitions of m. Let Λ(n, m) denote the set of compositions of m with at most n non-zero parts, and let Λ+(n, m) denote the set of partitions of m with at most n non-zero parts. Note that by rearranging parts, every composition λ corresponds to a unique partition Λ+. To every composition λ ∈ Λ(n, m), we may associate a subset J(λ) of S (J(λ) consists of those s ∈ S which stabilize the rows of the standard tableau of shape λ described in [9, Section 8]. Then, ⊗m ∼ ∼ as a right He-module V = ⊕ xλHe. Now, xλHe = xµHe if WJ(λ) and WJ(µ) are conjugate λ∈Λ(n,m) in W , which occurs if and only if λ+ = µ+. In particular, for every λ ∈ Λ(n, m), we have ∼ xλHe = xλ+ He. Therefore,

⊗m ∼ ⊕kλ V = ⊕ xλHe (8.1) λ∈Λ+(n,m) + for some non-negative integers kλ, λ ∈ Λ (n, m). And, it follows that the algebra A = End ( ⊕ x H) is Morita equivalent to S (n, m). He λ e t λ∈Λ+(n,m)

For a commutative ring R and a homomorphism Z → R, t 7→ q, let St(n, m) ⊗Z R = S (n, m) = S (n, m). Note that S (n, m) ∼= End (V ⊗m), where H(S , R, q) is the t R q q He(Sm,R,q) R e m specialization of He to R. Moreover, if R = k is a ﬁeld and there is an algebra homomorphism Z → k, t 7→ q, the category mod − Sq(n, m) of right Sq(n, m)-modules is a highest + weight category with poset Λ (n, m) [24, Theorem 1]. The poset structure E on the set + Λ (n, m) is the dominance order, deﬁned by λ = (λ1, λ2,...) E µ = (µ1, µ2,...) if and only + if λ1 ≤ µ1, λ1+λ2 ≤ µ1+µ2,.... For every λ ∈ Λ (n, m), there exists a right St(n, m)-module

63 ∆(e λ) such that ∆(e λ)k = ∆(λ) is the standard object in mod − Sq(n, m) corresponding to + the partition λ. Similarly, for every λ ∈ Λ (n, m), there exists a right St(n, m)-module ∇e (λ) such that ∇e (λ)k = ∇(λ) is the costandard object in mod − Sq(n, m) corresponding to the + partition λ. The irreducible Sq(n, m)-modules are indexed by Λ (n, m). The irreducible + k Sq(n, m)-modules corresponding to a partition λ ∈ Λ (n, m) will be denoted by L (λ).

In this dissertation, we will work primarily with right Sq(n, m)-modules in order to make use of the Morita equivalence given in [9, Theorem 9.17]. The category Sq(n, m) − mod of + left Sq(n, m)-modules is also a highest weight category with poset Λ (n, m). We will denote + left the standard object of Sq(n, m) − mod corresponding to λ ∈ Λ (n, m) by ∆ (λ), and we + will denote the costandard object of Sq(n, m) − mod corresponding to λ ∈ Λ (n, m) by left ∇ (λ). The standard objects of Sq(n, m) − mod are related to the costandard objects of + left ∗ mod − Sq(n, m) via the linear dual; for any λ ∈ Λ (n, m), ∇(λ) = ∆ (λ) . Similarly, the costandard objects of Sq(n, m) − mod are related to the standard objects of mod − Sq(n, m) via duality: for any λ ∈ Λ+(n, m), ∆(λ) = ∇left(λ)∗ [14, pg. 707].

64 9 The Indexing of the Irreducible kGLn(q)-modules

Let q be a power of a prime p, and let G = GLn(q) be the general linear group over the ﬁnite ﬁeld Fq of q elements. Let B be the Borel subgroup of G consisting of all invertible upper triangular n × n matrices. Let U be the unipotent radical of B (that is, the subgroup of B consisting of invertible upper triangular matrices with 1’s along the main diagonal), and let T be the maximal torus in B consisting of all invertible diagonal matrices.

Let k be an algebraically closed ﬁeld of characteristic r > 0 such that r 6 | q. In this dissertation, it will be necessary to use results of Cline, Parshall, and Scott [9] together with those of Dipper [16], [17], Dipper and James [21], and Dipper and Du [19]. The authors of all of the papers referenced here work with right kG-modules. However, the indexing of the irreducible kG-modules is not consistent across these papers: in particular, the indexing used by CPS diﬀers from the two indexings used by Dipper and James and Dipper and Du. In this section, we will present the three indexings of the irreducible kG-modules which are necessary to understand the results of CPS, DJ, and DD. Since the methods used by CPS in [9] are most relevant to the proofs appearing in Part IV of this dissertation, we will describe the indexing used by CPS in [9] in greater detail than the indexings used by DJ [21] and DD [19].

9.1 The First Indexing of the Irreducible kGLn(q)-Modules (CPS)

In this section, we will describe a Morita equivalence which Cline, Parshall, and Scott construct in Section 9 of [9] and describe how this Morita equivalence can be used to obtain an indexing of the irreducible right kG-modules. (CPS work with right, and not left, modules in [9].)

9.1.1 A Key Theorem

As above, let k be an algebraically closed field of characteristic r > 0. Let (O, K, k) be an r-modular system (so, O is a discrete valuation ring, K is the quotient field of O, and k is the residue field). Assume that the quotient field K of O is large enough so that it is a splitting field for GLn(q). We begin with the following key result.

Theorem 9.1.1. ([9, Theorem 9.2]) Let R be an O-algebra which is free of finite rank over O and has the property that RK is a semisimple algebra over K. Suppose that there is an exact sequence 0 → N → P → M → 0 of right R-modules which are free of finite rank over O, that the RK -modules NK and MK have no composition factors in common, and that P is a projective R-module. Let J = AnnRM, and assume that every irreducible R/J-module lies in the head of the right R-module M. Then the functor F (−) = HomR/J (M, −) defines a Morita equivalence F : mod − R/J → mod − EndR(M)

65 of right module categories. Following [9], we will construct an R-module M which satisﬁes the assumptions of The- orem 9.1.1.

9.1.2 A decomposition of OGLn(q) into sums of blocks

Let C denote a fixed set of representatives of the conjugacy classes in GLn(q). The finite group GLn(q) may be obtained as the set of fixed points of the algebraic group GLn (over the algebraic closure of Fq) under a Frobenius morphism. Let Gss denote the set of semisimple elements in the algebraic group GLn, and let Gss = Gss ∩ GLn(q). We set Css = Gss ∩ C (so, Css consists of the representatives of the conjugacy classes in GLn(q) which are semisimple when viewed as elements of the algebraic group GLn). Let Css,r0 denote the set of elements in Css of order prime to r.

m(s) ∼ Q ai(s) Given s ∈ Gss, the centralizer of s is of the form ZG(s) = GLni(s)(q ), where i=1 m(s) P + ai(s)ni(s) = n. For any s ∈ Gss, we let n(s) = (n1(s), . . . , nm(s)(s)). Let Λ (n(s)) be the i=1 set of multipartitions of n(s). An element λ ∈ Λ+(n(s)) is of the form λ = (λ(1), . . . , λ(m(s))), (1) (m(s)) where λ ` n1(s), ..., λ ` nm(s)(s). The ordinary irreducible characters of GLn(q) (which were ﬁrst parameterized by Green) may be indexed by pairs (s, λ), where s ∈ Css and + λ ∈ Λ (n(s)) [9, pg. 29]. We will denote the irreducible character of GLn(q) corresponding to the pair (s, λ) by χs,λ.

The group algebra OGLn(q) = OG decomposes as a direct sum of two-sided ideals called blocks, with every irreducible OG-module belonging to a unique block. Given an element s ∈ Css,r0 , let Bs,G be the sum of the blocks of OG which contain a character of the form

χst,λ, where t ∈ ZGLn(q)(s) is an r-element. OG has the direct sum decomposition

OG = ⊕ Bs,G. s∈Css,r0

9.1.3 An OG-module which satisﬁes the hypotheses of the key theorem

Given an element s ∈ Css,r0 , the centralizer of s in GLn(q) is of the form

m(s) ∼ Y ai(s) ZG(s) = GLni(s)(q ), i=1

66 m(s) P where ai(s)ni(s) = n. Let Ls(q) be the subgroup of G = GLn(q) deﬁned by i=1

m(s) Y Ls(q) = GLai(s)ni(s)(q). i=1

Let Hs(q) be the subgroup of Ls(q) deﬁned by

m(s) Y ni(s) Hs(q) = GLai(s)(q) . i=1

(So, Hs(q) consists of block-diagonal matrices in Ls(q).)

Ls(q) is a Levi subgroup of a parabolic subgroup of G, and Hs(q) is a Levi subgroup of a parabolic subgroup of Ls(q). Therefore, we have Harish-Chandra induction functors

RLs(q) : mod − OH (q) → mod − OL (q), and Hs(q) s s RGLn(q) : mod − OL (q) → mod − OGL (q). Ls(q) s n

Now, to an element s ∈ Css,r0 , we can associate a KHs(q)-module CK (s), which is a tensor product of certain irreducible cuspidal modules deﬁned by Dipper and James (a more precise description of the module CK (s) can be found in [9, 9.6]). The representation CK (s) contains an OHs(q)-lattice CO(s), and we can deﬁne a right OLs(q)-module associated to the element s ∈ Css,r0 by M = RLs(q) C (s). s,Ls(q),O Hs(q) O

By [21, (2.17)], the endomorphism algebra of the OLs(q)-module Ms,Ls(q),O is a tensor product of Hecke algebras. Speciﬁcally,

m(s) ∼ ai(s) EndOLs(q)(Ms,Ls(q),O) = ⊗ H(Sni(s), O, q ). i=1 Now, given a multipartition λ ` n(s), let

m(s) ai(s) yλ = yλ(1) ⊗ · · · ⊗ yλ(m(s)) ∈ ⊗ H(Sni(s), O, q ) i=1

m(s) ∼ ai(s) (the elements are deﬁned in Section 6.2). Since EndOLs(q)(Ms,Ls(q),O) = ⊗ H(Sni(s), O, q ), i=1 m(s) ai(s) the element yλ ∈ ⊗ H(Sni(s), O, q ) acts on Ms,Ls(q),O as an OLs(q)-module endomor- i=1 p phism. In particular, yλMs,Ls(q),O is an O-submodule of Ms,Ls(q),O. Let yλMs,Ls(q),O denote the smallest O-submodule of Ms,L (q),O such that yλMs,L (q),O ⊆ Ms,L (q),O and p s s s Ms,Ls(q),O/ yλMs,Ls(q),O is O-torsion free.

67 ∼ ⊕kµ(ni(s),ni(s)) For any 1 ≤ i ≤ m(s), we have ⊕ xµH = ⊕ xµH for + µ∈Λ(ni(s),ni(s)) µ∈Λ (ni(s),ni(s)) some positive integers kµ(ni(s), ni(s)) (the integers kµ(ni(s), ni(s)) are deﬁned in (8.1)). m(s) (1) (m(s)) Q Given a multipartition λ = (λ , . . . , λ ) ` n(s), let mλ = kλ(i) (ni(s), ni(s)) and let i=1

p ⊕mλ Mcs,Ls(q),O = ⊕ yλMs,Ls(q),O . λ`n(s)

Then, Mcs,Ls(q),O is an OLs(q)-module, and by [9, Lemma 9.11],

∼ m(s) EndOLs(q)(Mcs,Ls(q),O) = ⊗ Sqai(s) (ni(s), ni(s))O. i=1

We deﬁne an OGLn(q)-module Mcs,GLn(q),O by

M = RGLn(q)M . (9.1) cs,GLn(q),O Ls(q) cs,Ls(q),O

(To simplify the notation, we will denote Mcs,GLn(q),O simply by Mcs,G,O.) The endomorphism algebra of Mcs,G,O over OGLn(q) is

∼ m(s) EndOGLn(q)(Mcs,G,O) = ⊗ Sqai(s) (ni(s), ni(s))O. i=1

9.1.4 A Morita Equivalence for GLn(q).

Cline, Parshall, and Scott [9] show that for any s ∈ Css,r0 , the OGLn(q)-module Mcs,G,O deﬁned in (9.1) satisﬁes the hypotheses of Theorem 9.1.1 [9, Lemma 9.15]. Viewing Mcs,G,O as a module for the subalgebra Bs,G of OGLn(q), let Js = AnnBs,G (Mcs,G,O). Then, applying Theorem 9.1.1 with R = Bs,G, J = Js, and M = Mcs,G,O, we see that the functor

Fs(−) = HomBs,G/Js (Mcs,G,O, −) : mod − Bs,G/Js → mod − EndBs,G (Mcs,G,O) gives a Morita equivalence.

∼ ∼ m(s) In fact, since EndBs,G (Mcs,G,O) = EndOGLn(q)(Mcs,G,O) = ⊗ Sqai(s) (ni(s), ni(s))O, the i=1 functor Fs gives a Morita equivalence

∼ m(s) mod − Bs,GLn(q)/Js(q) → mod − ⊗ Sqai(s) (ni(s), ni(s))O. (9.2) i=1 P Let J = Js. Since the group algebra OGLn(q) may be decomposed as s∈Css,r0

OGLn(q) = ⊕ Bs,G, s∈Css,r0

68 ∼ it follows that OGLn(q)/J = ⊕ Bs,G/Js. Therefore, taking direct sums over s ∈ Css,r0 in s∈Css,r0 (9.2), we have a Morita equivalence

F (−) = HomOGLn(q)/J ( ⊕ Mcs,G,O, −): s∈Css,r0

m(s) mod − OGLn(q)/J → mod − ⊕ ⊗ Sqai(s) (ni(s), ni(s))O. s∈Css,r0 i=1 By [9, Theorem 9.17], this Morita equivalence remains valid upon base change to k; so, the category of right kGLn(q)/Jk-modules is Morita equivalent to the category of right m(s) ⊕ ⊗ Sqai(s) (ni(s), ni(s))k-modules. More precisely, we have a Morita equivalence s∈Css,r0 i=1 ¯ F (−) = HomkGLn(q)/Jk ( ⊕ Mcs,G,k, −): s∈Css,r0

m(s) mod − kGLn(q)/Jk → mod − ⊕ ⊗ Sqai(s) (ni(s), ni(s))k s∈Css,r0 i=1

(where Mcs,G,k = Mcs,G,O ⊗O k).

9.1.5 An Indexing of the Irreducible kGLn(q)-modules

By [9, Theorem 9.17], the algebras kGLn(q) and kGLn(q)/Jk have the same irreducible modules. In [9] (pg. 35), CPS use the Morita equivalence

m(s) ¯ F (−) : mod − kGLn(q)/Jk → mod − ⊕ ⊗ Sqai(s) (ni(s), ni(s))k s∈Css,r0 i=1

to index the irreducible kGLn(q)-modules. Since the irreducible Sqai(s) (ni(s), ni(s))k-modules m(s) + are indexed by the set Λ (ni(s)) of partitions of ni(s), the irreducible ⊗ Sqai(s) (ni(s), ni(s))k- i=1 modules are indexed by the set of multipartitions Λ+(n(s)). Therefore, the irreducible m(s) ⊕ ⊗ Sqai(s) (ni(s), ni(s))k-modules (and, consequently, the irreducible kGLn(q)-modules) s∈Css,r0 i=1 are indexed by pairs (s, λ), where s ∈ Css,r0 and λ is a multipartition of n(s). We will denote the irreducible kGLn(q)-module corresponding to the pair (s, λ) by D(s, λ).

A special class of irreducible kG-modules is obtained by taking s = 1 above. Since

ZGLn(q)(1) = GLn(q), n(1) = (n). Therefore, the irreducible kG-modules corresponding to the element s = 1 may be labeled as D(1, λ), with λ a partition of n. By [45, Theorem 2.1], the irreducible kG-modules D(1, λ), λ ` n, are precisely the composition factors of G the permutation module k|B. (An irreducible module D(1, λ) may appear more than once G as a composition factor of k|B.) By [9, Remark 9.18(b)], the trivial module k may be parameterized as D(1, (1n)), where (1n) is the partition of n with each part equal to 1.

69 9.2 The Second Indexing of the Irreducible kGLn(q)-Modules

In this section, we will describe the indexing of the irreducible kG-modules given in [19, (4.2.3)] (where G = GLn(q)). This indexing is based on the work of Dipper and James [22] and Dipper [17].

For any semisimple element s ∈ G, let Ls denote the corresponding standard Levi subgroup of G (defined in [19, (4.1.2)]). By parts (3) and (4) of [19, 4.2.3], every Harish-Chandra series of irreducible right kG-modules is of the form Irrk(G|(Ls,Ms)), where Ms is the cuspidal irreducible right kLs-module defined in [19, Section 4.1]. Given any semisimple element G G s ∈ G, let W (Ls,Ms) be the ramification group of RLs (Ms) and let Hs = EndkG(RLs (Ms)) be the Hecke algebra (over k) associated with W (Ls,Ms). Suppose that Ls is of the form Q ×mi + Ls = GLdi (q) for some integers di and mi, and let ~ms = (m1, m2,...). Let Λ (~ms) i ~ (1) (2) (i) denote the set of multipartitions of ~ms. Also, let ls = (l , l ,...), where l is the minimal (i) positive integer such that 1 + qdi + q2di + ··· + q(l −1)di ≡ 0 (mod r).

~ + By [19, (4.2.2,(1))], for every λ ∈ Λ (~ms), there is a right Hs-module S~λ called a Specht ~ ~ (i) 9 module. By [19, (4.2.2,(2))], if λ = (λ1, λ2, ··· ) is ls-regular (meaning that λi is l -regular

for all i), then S~λ has a unique maximal submodule, with the corresponding factor module ~ ~ ~ denoted by D~λ. The set {D~λ | λ ` ~ms, λ is ls − regular} is a complete set of non-isomorphic irreducible Hs-modules.

Now, the category of right kG-modules is connected to the category of right Hs-modules via the “Hecke functors.” First, there is a functor Hs : mod − kG → mod − Hs, defined as follows. Let β : Q → Ms be a fixed minimal projective cover of Ms (as a right kLs- G module), and let P = RLs (Q). Let Jβ be the ideal of EndkG(P) consisting of the en- G G domorphisms of P with image contained in the kernel of RLs (β): P → RLs (Ms). The functor Hs is defined as Hs = HomkG(P, −)/HomkG(P, −)Jβ. (Given a right kG-module 0 V , the right Hs-module structure of Hs(V ) is described in [16, 2.2].) If V and V are right kG-modules and f : V → V 0 is a right kG-module homomorphism, then there is a map 0 HomkG(P,V ) → HomkG(P,V ) given by φ → fφ for any φ ∈ HomkG(P,V ). As discussed in 0 [16, 2.2], this map induces an Hs-module homomorphism Hs(f): Hs(V ) → Hs(V ).

ˆ ˇ The functor Hs has two right inverses, which are denoted by H and H in [19]. The ˆ ˆ G functor H : mod − Hs → mod − kG is defined by H(E) = E ⊗Hs RLs (Ms) for any ˇ E ∈ mod − Hs. The functor H : mod − Hs → mod − kG is defined as follows. Given G G E ∈ mod − Hs, let tP (E ⊗Hs RLs (Ms)) denote the kG-submodule of E ⊗Hs RLs (Ms) which G is maximal with the property HomkG(P, tP (E ⊗Hs RLs (Ms))) = 0. With this definition, ˇ G G ˆ ˇ H(E) = (E ⊗Hs RLs (Ms))/tP (E ⊗Hs RLs (Ms)). Collectively, the functors Hs, H, and H are known as the Hecke functors. 9A partition λ of n is called l-regular if every part of λ appears less than l times.

70 ~ Given a semisimple element s ∈ G and multipartition λ ` ~ms, there exists an irreducible ~ ~ ~ right CG-module SC(s, λ) indexed by (s, λ) (see [19, (4.2.3, (1))]). Let S(s, λ) denote the ~ ~ reduction modulo r of SC(s, λ). By [21, (3.1)], S(s, λ) has a unique maximal submodule. The corresponding factor module, which we will denote by D1(s, λ), is an irreducible right kG-module.10

The modules described above behave particularly well under the Hecke functors. Given ~ ~ any semisimple element s in G and multipartition λ ` ~ms, we have Hs(S(s, λ)) = S~λ. If, in ~ ~ 1 ~ ˇ addition, λ is ls-regular, then D (s, λ) = Hs(D~λ).

Before we can give a full classiﬁcation of the irreducible kG-modules, we must deﬁne an equivalence relation on the set of semisimple elements of G. Given such a semisimple element s, we can srite s = s0y, where s0 has order prime to r and y is an r-element. As in [19], we will say that the semisimple elements s and t of G are r-equivalent (written s ∼r t) 0 0 if s = t and Ls = Lt. Let Cr denote a full set of representatives of the equivalence classes of the semisimple elements of G under ∼r. Then, by [19, (4.2.3, (4))],

1 ~ ~ {D (s, λ) | s ∈ Cr, λ ` ~ms is ls − regular}

is a complete set of non-isomorphic irreducible kG-modules.

9.3 The Third Indexing of the Irreducible kGLn(q)-Modules

In this section, we will describe the indexing of the irreducible kG-modules given in [19, (4.2.11, (3))]. This indexing is based on the work of Dipper and James [22] and Dipper [17].

By [19, Theorem 4.2.7], there exits a q-Schur algebra Ss (over k) corresponding to the Hecke algebra Hs for every semisimple s ∈ G. The category of right kG-modules is related to the category of right Ss-modules via the “q-Schur functors” Ss : mod − kG → mod − Ss, ˆ ˇ Ss : mod − Ss → mod − kG, and Ss : mod − Ss → mod − kG. Given a semisimple element ~ s in G, the irreducible right Ss-modules are indexed by the multipartitions λ of ~ms; given a ~ k multipartition λ ` ~ms, let L (λ) denote the corresponding irreducible Ss-module. For any ~ ˇ k 2 0 11 λ ` ~ms, Ss(L (λ)) is an irreducible kG-module, which we will denote by D (s, λ ).

Let Css,r0 denote a set of representatives of semisimple r-regular conjugacy classes in G. Then, 2 ~ ~ {D (s, λ) | s ∈ Css,r0 , λ ` ~ms}

10In [19], the irreducible kG-module D1(s, λ) is denoted by D(s, λ). We have chosen to use the notation D1(s, λ) instead in order to distinguish this indexing from that used by CPS in [9]. 11In [19], D2(s, λ0) is denoted by D(s, λ0).

71 is a complete set of non-isomorphic irreducible kG-modules.

9.4 Indexing of the Irreducible kGLn(q)-Modules belonging to the Unipotent Principal Series

where |q (mod r)| denotes the multiplicative order of q mod r. By [19, (4.2.3, (5))], the irreducible kG-modules belonging to Irrk(G|B) are indexed by partitions of n. In particular, 1 1 2 Irrk(G|B) = {D (1, λ) | λ ` n, λ is l − regular}. By [19, (4.2.11, (3))], D (1, λ) = D (1, λ) for any l-regular partition λ ` n.

To prove the main results of Part IV of this dissertation, we need to determine the CPS parameterization of an irreducible kG-modules D1(1, λ) when λ ` n is l-regular. To do this, we will ﬁx an l-regular partition λ ` n and apply the CPS functor F¯ to the irreducible kG- 1 1 1 G module D (1, λ). Since D (1, λ) ∈ Irrk(G|B), D (1, λ) is in the head of k|B, which means G 1 G that HomkG(k|B,D (1, λ)) 6= 0. Since k|B is a direct summand of the module Mc1,G,k (by [9, Remark 9.18 (b)]) and F¯(D1(1, λ)) is irreducible, we can compute

¯ 1 1 F (D (1, λ)) = HomkG/Jk ( ⊕ Mcs,G,k,D (1, λ)) s∈Css,r0 ∼ 1 = HomkG( ⊕ Mcs,G,k,D (1, λ)) s∈Css,r0 ∼ G 1 = HomkG(k|B,D (1, λ)).

1 2 G 1 ∼ G 2 Since λ is l-regular, D (1, λ) = D (1, λ) and HomkG(k|B,D (1, λ)) = HomkG(k|B,D (1, λ)). Tracing through the deﬁnition of the functor S1 : mod − kG → mod − Sq(n, n)k given in [17, 2 Section 6] (and, using the fact that S1(D (1, λ)) must be an irreducible Sq(n, n)k-module), G 2 ∼ 2 ˇ we ﬁnd that HomkG(k|B,D (1, λ)) = S1(D (1, λ)). But, the functor S1 is a right inverse of 2 ∼ ˇ k 0 ∼ k 0 S1, which means that S1(D (1, λ)) = S1(S1(L (λ ))) = L (λ ). Therefore, we have shown that F¯(D1(1, λ)) ∼= Lk(λ0), and it follows that D1(1, λ) = D(1, λ0) in the indexing of CPS when λ is an l-regular partition of n. In particular, Irrk(G|B) = {D(1, λ) | λ is l−restricted} in the CPS indexing.12

12A partition λ of n is called l-restricted when its dual partition λ0 is l-regular.

72 Part II Generalizing the Results of Guralnick and Tiep to Find Bounds on the Dimension of Ext1 between Irreducible Modules

We start with a remark concerning our use of the terms “Borel subgroup,” “maximal torus,” and “unipotent radical” in the next three parts of this dissertation. Most readers will be familiar with these terms from the theory of algebraic groups; here, though, we will not be using this terminology in the sense of algebraic groups. Let G be a ﬁnite group of Lie type having a split BN-pair in characteristic p, let T = B ∩ N, and let U = Op(B) be the largest normal p-subgroup of B. Following [35], we will refer to B as a Borel subgroup of G, T a maximal torus, and U a unipotent radical. B is not a Borel subgroup in the sense of algebraic groups any more than T is a maximal torus or U is a unipotent radical. However, this terminology has precedent in the literature and proves to be very convenient when discussing the cohomology of ﬁnite groups of Lie type.

10 A Summary of the Results of Guralnick and Tiep [35]

10.1 Introduction

Much of the original work of this dissertation is concerned with generalizing the results of Guralnick and Tiep [35]. This section summarizes these results. We will state many of the results of [35] without proof; however, we will provide (Guralnick and Tiep’s) proofs of the results which were particularly useful to our generalization. In many cases, we will include details omitted in the original proofs of [35]. It is the hope of the author that this chapter will help readers gain a deeper understanding of [35]. Otherwise, the reader may choose to skip this chapter and proceed to Chapter 11.

Let k be an algebraically closed field of characteristic r. Let G be a finite group of Lie type over Fq, where q is a power of a prime p with r 6= p. Let B be a Borel subgroup of G with unipotent radical Q and maximal torus T , and let W = NG(T )/T be the Weyl group of G with respect to T .13 Let e denote the twisted rank of G. (As long as q is sufficiently large,

13To be consistent with Guralnick and Tiep’s notation, we will denote the unipotent radical of B by Q in this section. In all other sections, we will denote the unipotent radical of B by U.

73 e is the rank of T as an abelian group.) We remark that in [35], Guralnick and Tiep work with left kG-modules. Hence, all kG-modules we consider in Part II of this dissertation will be assumed to be left kG-modules.

Remark 10.1.1. (An explanation of the notion of “rank”) In [33, Part I, Chapter 1, Section 11], Gorenstein, Lyons, and Solomon give the following description of the rank of a group of Lie type. Every group G(q) of Lie type (q a power of a prime p) has a distinguished presentation (called the Steinberg presentation), with generators being certain elements of the root subgroups Xα, where α runs over the roots of the root system Σ of the associated complex Lie algebra. When G(q) is an untiwsted group, the Xα are isomorphic to the additive group of the finite field Fq. When G(q) is twisted, the root subgroups Xα have more complicated structures, and the root system Σ may be non-reduced. The (Lie) rank of G(q) is defined to be the rank of the root system Σ, i.e. the dimension of the Euclidean space RΣ. The Lie rank of G(q) is also referred to as the “twisted” rank of G(q) (as is done in [35]). There is also a notion of an “untwisted” rank of a finite group G(q) of Lie type. The untwisted rank of an untwisted group of Lie type is simply its (twisted) rank. And, the untwisted Lie rank of of a twisted group of Lie type is the Lie rank of the ambient algebraic group; i.e., it is the Lie rank of the untwisted group which was twisted to form G(q).

G The permutation module L = k|B is a key ingredient in most of the results proved in [35]. We note several important features of L which will be useful to us. First, L is a self-dual module. By Frobenius reciprocity, the G-ﬁxed point subspace of L is L G ∼= ∼ ∼ HomkG(k, L ) = HomkB(k, k) = k (since G is ﬁnite, induction and coinduction from B G G to G coincide, which means that ResB is both right and left adjoint to IndB). We also claim that L has a unique submodule L 0 of codimension 1. By Frobenius reciprocity, ∼ HomG(L , k) = HomB(k, k) = k. Choose an element φ ∈ HomG(L , k) that generates 0 0 HomG(L , k) over k, and let L = Ker φ. Since φ 6= 0 and k is an irreducible G-module, L is a maximal submodule of L of codimension 1. To show that L 0 is unique, let M be any codimension 1 submodule of L , so that L /M ∼= k. If φ0 : L → k is the natural quotient 0 0 0 map, then tM = Ker(φ ). Since φ is a nonzero element of HomG(L , k), φ = aφ for some a ∈ k×. Thus. M = Ker(φ) = Ker(φ0) = L 0.

The purpose of the first part of [35] is to find a bound on dim H1(G, V ) when V is an irreducible kG-module with V B 6= 0. The case in which V is an irreducible kG-module with V B = 0 is handled in the second part of the paper [35, Sections 4-6]. Somewhat surprisingly, finding a bound on dim H1(G, V ) turns out to be a much more difficult task when V B = 0. However, the bound obtained on dim H1(G, V ) is much stronger in the case when V B = 0. Guralnick and Tiep show that dim H1(G, V ) ≤ 1 when V B = 0; in the case that V B 6= 0, |W | + e though, they show only that dim H1(G, V ) ≤ , which (as they themselves state in dim(V Q) the introduction) is “almost certainly not best possible.”

74 10.2 Irreducible Modules V with V B 6= 0

We begin with a basic (yet very important) lemma.

Lemma 10.2.1. ([35, Lemma 2.1]) Let A := Or0 (B) be the largest normal subgroup of B of order prime to r. Then, for any kB-module V , the following statements are equivalent: (i) V B 6= 0; (ii) B has trivial composition factors on V ; (iii) V A 6= 0; (iv) (V ∗)B 6= 0.

Proof. (We follow the proof given in [35].) If V B 6= 0, then V B must contain k in its socle (k is the only irreducible kB-module on which B acts trivially), so (i) implies (ii). Suppose now that (ii) holds. Since r6 | |A|, V is completely reducible as a kA-module. Therefore, any copies of k appearing as composition factors of V as a kB-module become direct summands of V as a kA-module. Thus, k = kA ⊆ V A, so that V A 6= 0 and (iii) holds. Now, suppose that (iii) holds, so that V A 6= 0. Since A is a normal subgroup of B, V A is a kB-module, which means that V A is also a B/A-module. The r-group B/A has a non-zero fixed point on V A since V A is a non-zero vector space over a field of characteristic r (this is a standard fact in representation theory - see, for example [46, Proposition 26]). And, since (V A)B/A ⊆ V B (which follows because the action of B/A on V A is defined by bA.v = b.v for bA ∈ B/A, v ∈ V A), we must have V B 6= 0, which means that (i) holds.

We have proved that statements (i)-(iii) are equivalent. We will now show that statement (iii) is equivalent to (iv), which is suﬃcient to prove the lemma. Since V is completely reducible as a kA-module and k is the only irreducible trivial kA-module, V A 6= 0 if and only if k is a direct summand of V as a kA-module. Since k∗ ∼= k, this is true if and only if k is a direct summand of V ∗, which is true if and only if (V ∗)A 6= 0. But, by the equivalence of (i)-(iii), (V ∗)A 6= 0 if and only if (V ∗)B 6= 0.

Proposition 10.2.2. ([35, Proposition 3.1]) For any irreducible kG-module V , set fB(V ) := B Q dim(V ), fQ(V ) := dim(V ), and let mL (V ) := [L : V ] be the multiplicity of V as a composition factor of L . Then, the following statements hold: 1. dim L B = dim L Q = |W | 2. XQ = XB for any submodule X of L .

3. mL (V ) ≥ fB(V ). 4. P m (V ) · f (V ) ≤ P m (V ) · f (V ) = |W |. V ∈Irrk(G) L B V ∈Irrk(G) L Q Q 5. If dim(V ) = d > 0, then mL (V ) ≤ |W |/d ≤ |W |.

75 P 6. m (k) ≤ B m (V ) ≤ |W |. L V ∈Irrk(G),V 6=0 L

Corollary 10.2.3. ([35, Corollary 3.2]) Let X be a submodule of L . Then, dim H1(G, X) ≤ |W | + e0, where e0 is the r-rank of B/Q and is at most the twisted rank e of G.

Corollary 10.2.4. ([35, Corollary 3.3]) P f (V ) · dim H1(G, V ) ≤ |W | + e. In V ∈Irrk(G) B particular, X dim H1(G, V ) ≤ |W | + e, B V ∈Irrk(G),V 6=0 and |W | + e dim H1(G, V ) ≤ fQ(V ) if V B 6= 0.

Proof. (We follow the proof given in [35].) Let X = soc(L ). An irreducible module V ∈ ∼ ∼ Irrk(G) embeds in X if and only if HomkG(V, L ) 6= 0. But, HomkG(V, L ) = HomkB(V, k) = ∗ ∼ ∗ B ∗ B B HomkB(k, V ) = (V ) . And, by Lemma 2.1, (V ) 6= 0 if and only if V 6= 0. There- fore, V embeds in X if and only if V B 6= 0. And, if V B 6= 0, the multiplicity of V as a composition factor of X = soc(L ) is [X : V ] = dim HomkG(V, L ) = dim HomkB(V, k) = ∗ ∗ B ∗ Q dim HomkB(k, V ) = dim(V ) = dim(V ) , where we have used Frobenius reciprocity to conclude that dim HomkG(V, L ) = dim HomkB(V, k) and [35, Proposition 3.1(ii)] to conclude that dim(V ∗)B = dim(V ∗)Q (note that V B 6= 0 implies that (V ∗)B 6= 0, so that V ∗ ⊆ X is a submodule of L and [35, Proposition 3.1(ii)] applies). Now, since Q is a p-group and char(k) = r 6= p, V is completely reducible as a kQ-module, with V Q equal to the direct sum of all of the copies of k contained in the direct sum decomposition of V as a kQ-module. The same is true for (V ∗)Q, so that V Q ∼= (V ∗)Q. Therefore, ∗ Q Q B dim(V ) = dim V = dim V = fB(V ), where the second equality follows by [35, Proposi- tion 3.1(ii)]. Combining this with the previous chain of equalities, we have [X : V ] = fB(V ) when V B 6= 0.

⊕f (V ) ∼ B B By the arguments in the previous paragraph, X = ⊕V ∈Irrk(G),V 6=0V . In fact, B ∼ ⊕fB (V ) since fB(V ) = 0 when V = 0, we can write X = ⊕V ∈Irrk(G)V . Therefore, we have 1 1 ⊕f (V ) 1 1 ⊕f (V ) ∼ B B ∼ B H (G, X) = ⊕V ∈Irrk(G),V 6=0H (G, V ) and H (G, X) = ⊕V ∈Irrk(G)H (G, V ) . And, since dim H1(G, X) ≤ |W | + e by [35, Corollary 3.2], the three desired inequalities hold.

10.3 Irreducible Modules V with V B = 0

Guralnick and Tiep [35] show that if V is an irreducible kG-module with V B = 0, dim H1(G, V ) ≤ 1. In this section, we outline the results that lead to this bound on dim H1(G, V ).

76 Theorem 10.3.1. ([35, Theorem 2.2]) If V is an irreducible kG-module, with V B = 0, then dim H1(G, V ) is the multiplicity of V in head(L 0).

The result of [35, Theorem 2.2] is crucial to our generalization of Guralnick and Tiep’s work. We give a less group-theoretic proof of this theorem in Chapter 12.

Corollary 10.3.2. ([35, Corollary 2.3]) Let V be an irreducible kG-module. If V is not a composition factor of L , then H1(G, V ) = 0.

B Theorem 10.3.3. ([35, Theorem 6.1]) Let V be a kG-module with V = 0. Let Pi, 1 ≤ i ≤ n, denote the minimal parabolic subgroups of G containing B. If r does not divide [Pi : B] for any i, then H1(G, V ) = 0.

Proof. (We follow the proof given in [35].) Let Or0 (B) denote the largest normal subgroup of B of order prime to r. Then, the Hochschild-Serre spectral sequence gives us the 1 O 0 (B) five-term inflation-restriction exact sequence in cohomology: 0 → H (B/Or0 (B),V r ) → 1 1 B/O 0 (B) 2 O 0 (B) 2 H (B,V ) → H (Or0 (B),V ) r → H (B/Or0 (B),V r ) → H (B,V ). Since B O 0 (B) 1 O 0 (B) V = 0, V r = 0 by [35, Lemma 2.1]. Therefore, H (B/Or0 (B),V r ) = 0 and 2 O 0 (B) H (B/Or0 (B),V r ) = 0; so, by the exactness of the inflation-restriction sequence, we 1 ∼ 1 B/O 0 (B) 1 have H (B,V ) = H (Or0 (B),V ) r . But, since r 6 | |Or0 (B)|, H (Or0 (B),V ) = 0, and thus H1(B,V ) = 0.

1 1 Now, since r does not divide [Pi : B] for any i, the restriction map H (Pi,V ) → H (B,V ) 1 is injective for all i. It follows that H (Pi,V ) = 0 for every minimal parabolic subgroup Pi of G. And, since G = hP1,...,Pni, the cohomology vanishing theorem of Alperin and Gorenstein [1] allows us to conclude that H1(G, V ) = 0.14

Remark 10.3.4. The following general result is needed to prove [35, Lemma 6.3].

Proposition 10.3.5. Let G be a ﬁnite group and let H and R be subgroups of G. Assume that G G R R is a normal subgroup, so that the product HR is a subgroup of G. Then, k|HR = (k|H ) .

G Proof. Since G is ﬁnite, the induced module k|HR = kG⊗kHRk is isomorphic to the coinduced G ∼ module HomkHR(kG, k). Similarly, k|H = HomkH (kG, k). Now, since H ≤ HR, every HR- module homomorphism kG → k is also an H-module homomorphism, and it follows that G G G G R G k|HR ,→ k|H . We now check that k|HR = (k|H ) . First, let φ ∈ k|HR = HomHR(kG, k) and let r ∈ R. Since R is a normal subgroup of G, given an element g ∈ G, there exists an element r0 ∈ R such that gr = r0g. Thus, (rφ)(g) = φ(gr) = φ(r0g) = r0φ(g) = φ(g), where G R the last equality follows since the action of G on k is trivial. Therefore, φ ∈ (k|H ) , which G G R G R 0 shows that k|HR ⊆ (k|H ) . Now, let φ ∈ (k|H ) and let hr ∈ HR. Given g ∈ G, let r ∈ R be 14In [1], Alperin and Gorenstein prove the following vanishing theorem. Suppose that a group G is generated by a collection L of subgroups such that L has a minimal element. If A is a G-module with AL = 0 and H1(L, A) = 0 for all L ∈ L, then AG = 0 and H1(G, A) = 0.

77 such that gr = r0g. Then, φ(hrg) = hφ(rg) = φ(rg) = φ(gr0) = (r0φ)(g) = φ(g) = hrφ(g). G G R G Therefore, φ ∈ k|HR, and (k|H ) ⊆ k|HR.

Lemma 10.3.6. ([35, Lemma 6.3]) Let P be a parabolic subgroup properly containing B, and let R = Ru(P ) be the unipotent radical of P . Let W0 ⊂ W be a full set of double coset R ∼ P representatives for P \G/B (the set of P − B-double cosets in G). Then, L = ⊕ k|B as w∈W0 kP -modules.

Proof. (We follow the proof given in [35].) By the Mackey decomposition, we have G G P Bw w P ResP IndB(k) = ⊕ IndP ∩Bw ResP ∩Bw (k ) = ⊕ IndP ∩Bw (k). So, as a kP -module, w∈W0 w∈W0 G P R P R L = k|B decomposes as L = ⊕ k|P ∩Bw . Taking ﬁxed points, L = ⊕ (k|P ∩Bw ) . By w∈W0 w∈W0 P R ∼ P R P Remark 10.3.4, (k|P ∩Bw ) = k|(P ∩Bw)R, so L = ⊕ k|(P ∩Bw)R. w∈W0

We claim that for any w ∈ W ,(P ∩Bw)R is a Borel subgroup of G. Note that (P ∩Bw)R w is, indeed, a subgroup of G: since R E P ,(P ∩ B )R is a subgroup of P , which menas that (P ∩Bw)R is a subgroup of G. Now, since Bw is a Borel subgroup of G, Bw is solvable, which means that P ∩ Bw ≤ Bw is solvable. And, since R is unipotent (and therefore solvable), the product (P ∩ Bw)R is a solvable subgroup of G. So, to show that (P ∩ Bw)R is a Borel subgroup, it suﬃces to check that (P ∩ Bw)R contains a maximal torus and exactly one of the root groups Uα and U−α for every root α in the root system Φ of G. Now, Bw and P are both parabolic subgroups of G, so their intersection P ∩ Bw contains a maximal torus, which must also be contained in (P ∩ Bw)R. To show that one w of U±α is contained in (P ∩ B )R for every α ∈ Φ, we look closer at the structure of the parabolic subgroup P . Assume that P = PI for some subset I of the simple roots. Then, P = PI = LI o R, where LI is the Levi subgroup of PI . LI is generated by the maximal torus T ⊂ B and the root subgroups Uα for α ∈ ΦI (where ΦI ⊆ Φ denotes the set of roots that are linear combinations of the simple roots in I). And, R is generated by the root + subgroups Uα with α ∈ Φ \ ΦI . Now, let α ∈ Φ. If one of U±α is contained in R, then w it is contained in (P ∩ B )R and we are done. So, suppose that Uα 6⊆ R and U−α 6⊆ R. w Then, both Uα and U−α are contained in LI . And, since B is a Borel subgroup of G, w one of Uα and U−α must lie in B . Therefore, one of Uα and U−α lies in the intersection w w w w LI ∩ B ⊂ PI ∩ B = P ∩ B , which means that one of Uα and U−α lies in (B ∩ P )R. Thus, (P ∩ Bw)R is a Borel subgroup of G, as claimed.

R P R P P We have now shown that L = ⊕ (k|P ∩Bw ) = ⊕ k|(P ∩Bw)R = ⊕ k|B0 , where w∈W0 w∈W0 w∈W0 B0 = (P ∩ Bw)R is a Borel subgroup of G contained in P . But, any two Borel subgroups of G are conjugate and, induction functors from conjugate subgroups are naturally isomorphic. P ∼ P Therefore, we have k|B0 = k|B and the statement of the lemma follows.

In [35], Guralnick and Tiep use the lemma above to prove the following theorem.

78 Theorem 10.3.7. ([35, Theorem 6.4]) Let P be a minimal parabolic subgroup containing B, with unipotent radical Ru(P ) = R. Assume that r divides [P : B]. Let X be a submodule of L containing L G such that XQ = L G and dim XR > 1. Then the following statements hold: R G ∼ (1) Either soc(X /X ) is an irreducible kP -module, or r = 2 and [P,P ]/R = SL2(q) with R G q odd or SU3(q) with q ≡ 1(mod 4). In the latter two cases, soc(X /X ) is either irreducible or a direct sum of two nonisomorphic irreducible kP -modules.

G 1 1 (2) If X/X is irreducible, then ExtG(X, k) = 0 and dim ExtG(X/k, k) = 1.

A key step in the proof of this theorem is a reduction to the case of rank 1 groups. We will use this technique several times in the upcoming chapters as we generalize Guralnick and Tiep’s bounds on the dimension of H1(G, V ). Guralnick and Tiep’s bound on H1(G, V ) (where V is an irreducible kG-module with V B = 0) follows as a corollary of [35, Theorem 6.4].

Corollary 10.3.8. ([35, Corollary 6.5(i)]) Let V be an irreducible kG-module with V B = 0 and H1(G, V ) 6= 0. Then, dim H1(G, V ) = 1.

79 11 A Bound on the Dimension of Ext1 between Irre- ducibles in the Unipotent Principal Series

11.1 Introduction

Let G be a finite group of Lie type, defined in characteristic p > 0 (so, G is the fixed ¯ point subgroup of a connected reductive algebraic group G over Fp under an endomorphism F : G → G such that some power of F is a Frobenius morphism). Assume, additionally, that the BN-pair of G is split. ([35] handles all finite groups of Lie type; here, we restrict to split groups in order to use the methods of modular Harish-Chandra theory presented in [31, Section 4.2].) Let (W, S) be the Coxeter system corresponding to the BN-pair structure on G.

Let k be an algebraically closed ﬁeld of characteristic r > 0, r 6= p, and let e denote the r-rank of the maximal torus T (that is, e is the maximal dimension of an elementary abelian r-subgroup of T )15. In [35], Guralnick and Tiep prove that dim H1(G, V ) ≤ |W | + e for any irreducible kG-module V with V B 6= 0 (and, in the case that G is simple and V B = 0, they prove the stronger result that dim H1(G, V ) ≤ 1). Since an irreducible kG-module B V satisﬁes V 6= 0 if and only if V belongs to the unipotent principal series Irrk(G|B), Guralnick and Tiep’s result may be restated as follows:

1 If V is an irreducible kG-module with V ∈ Irrk(G|B), then dim H (G, V ) ≤ |W | + e .

In this section, we will present a partial generalization of Guralnick and Tiep’s result. 1 0 0 Speciﬁcally, we will prove that dim ExtkG(V,V ) ≤ |W | + (dim V )e if V and V are irreducible kG-modules such that V B 6= 0 and (V 0)B 6= 0 (in the case that V = k and (V 0)B 6= 0, we recover the bound of [35]).

T To generalize the result of [35], we will work with the induced module k|Tr , where Tr is an r-Sylow subgroup of T (in [35], Guralnick and Tiep work with the permutation module G k|B directly). Hence, we must break our proof into two cases, depending on whether or not |B| is divisible by the characteristic r of k.

11.2 A Preliminary Result

Lemma 11.2.1. If V is an irreducible kG-module in the unipotent principal series Irrk(G|B), G G then [k|B : V ] ≤ |W | (where [k|B : V ] denotes the number of times V occurs as a composition G factor of k|B). 15Here, the dimension of an elementary abelian r-group refers its dimension as a vector space over the ﬁnite ﬁeld Fr of r elements.

80 Proof. The proof strategy used here is based on that of [35, Proposition 3.1]. Let Irrk(G) denote the set of all (non-isomorphic) irreducible kG-modules. Given an irreducible Z ∈ G G Irrk(G), let mZ = [k|B : Z] be the composition multiplicity of Z in k|B. We consider the G U G U-ﬁxed point subspace (k|B) of k|B. Since r6 | |U|, every kU-module is completely reducible. G U U ⊕mZ Therefore, as a kU-module, (k|B) = ⊕ (Z ) , so that Z∈Irrk(G)

G U X U dim (k|B) = mZ dim(Z ).

Z∈Irrk(G)

∗ G U Now, since V ∈ Irrk(G|B), mV 6= 0 and RT (V ) = V 6= 0. Therefore, mV occurs with G U G U non-zero coeﬃcient in the sum giving dim (k|B) , and it follows that mV ≤ dim (k|B) .

G U ∼ G G But, (k|B) = HomkU (k, k ↑B↓U ). By the Bruhat decomposition, W gives a full set of (B,B)-double coset representatives in G. Since every element of W normalizes T , we have BwB = UT wB = UwT B = UwB for all w ∈ W . Therefore, W gives a full set of (U, B)-double coset representatives in G. So, applying the Mackey decomposition, we have

G G ∼ w wB U HomkU (k, k ↑B↓U ) = ⊕ HomkU (k, k ↓wB∩U ↑wB∩U ). (11.1) w∈W

Now, wk is the k(wB)-module with underlying space k and wB-action given by wbw−1.a = w w w wB b.a = a for any b ∈ B and a ∈ k. Thus, k is the trivial k( B)-module k, and k ↓wB∩U is k, viewed as the trivial k(wB ∩ U)-module. So, using (11.1) and the fact that induction is right adjoint to restriction in the case of ﬁnite groups, we have

G G ∼ U ∼ ∼ HomkG(k|U , k|B) = ⊕ HomkU (k, k ↑wB∩U ) = ⊕ Homk(wB∩U)(k, k) = ⊕ k. w∈W w∈W w∈W

G U G Therefore, dim (k|B) = |W |, which means that [k|B : V ] = mV ≤ |W |.

11.3 Case I: r - |B|

Assume that the characteristic r of k does not divide |B|.

Theorem 11.3.1. Let V and V 0 be irreducible kG-modules in the unipotent principal series 1 0 Irrk(G|B). Then, dim ExtkG(V,V ) ≤ |W |.

Proof. We use the proof strategy of [35, Corollary 3.2]. Since r6 | |B|, the group algebra kB is semisimple and every kB-module is projective and injective. In particular, k is an injective G kB-module, which means that k|B is also injective (since induction from B to G is exact) 1 G 0 0 G and ExtkG(V, k|B) = 0. Now, since V is in Irrk(G|B), V is contained in the socle of k|B. Therefore, we have a short exact sequence of kG-modules

0 G 0 → V → k|B → Y → 0

81 ∼ G 0 1 G (where Y is a kG-module with Y = k|B/V ). Since ExtkG(V, k|B) = 0, this short exact sequence induces the exact sequence

0 G 1 0 0 → HomkG(V,V ) → HomkG(V, k|B) → HomkG(V,Y ) → ExtkG(V,V ) → 0.

1 0 G Therefore, dim ExtkG(V,V ) ≤ dim HomkG(V,Y ) = [soc(Y ): V ] ≤ [Y : V ] ≤ [k|B : V ] ≤ |W |, where the last inequality follows by Lemma 11.2.1.

11.4 Case II: r | |B|

Assume that |B| is divisible by the characteristic r of k.

Let Tr be an r-Sylow subgroup of T . Since r | |B| = |U||T | and U is a p-group, r | |T |. Hence, Tr is a non-trivial r-subgroup of T . Moreover, since T is abelian, Tr is a normal subgroup of T , which means that T/Tr is a group. So, the permutation module T T k|Tr is isomorphic to k[T/Tr], which is semisimple by Maschke’s Theorem. Thus, k|Tr is a completely reducible kT -module.

n + e − 1 Lemma 11.4.1. Let A be an abelian r-group or rank e. Then, dim Hn(A, k) = n for all n ≥ 0.

Proof. We proceed by induction on e. If e = 1, then A = Z/mZ for some m = rd (d ∈ Z+), and Hn(A, k) ∼= k for all n ≥ 0 since A is a cyclic r-group (a proof of this standard result n + 1 − 1 can be found in the Appendix, Section 20). In particular, dim Hn(A, k) = 1 = , n and the statement of the lemma holds. Suppose now that e > 1 and dim Hn(A0, k) = n + e − 2 for all abelian r-groups A0 of rank e−1 and all n ≥ 0. Let m , . . . , m be positive n 1 e di + integers such that mi = r (di ∈ Z ) for 1 ≤ i ≤ e and A = Z/m1Z×Z/m2Z×· · ·×Z/meZ. By the K¨unnethformula,

n n H (A, k) = H (Z/m1Z × Z/m2Z × · · · × Z/meZ, k) n ∼ i n−i = ⊕ H (Z/m1Z × · · · × Z/me−1Z, k) ⊗ H (Z/meZ, k) i=0 n ∼ i = ⊕ H (Z/m1Z × · · · × Z/me−1Z, k) ⊗ k i=0

i + e − 2 By the inductive hypothesis, dim Hi( /m ×· · ·× /m , k) = for 0 ≤ i ≤ n. Z 1Z Z e−1Z i n i + e − 2 n + e − 1 Therefore, dim Hn(A, k) = P = . i=0 i n (Note that the last step in the computation above is a consequence of the following sum

82 n r + k r + n + 1 formula for binomial coeﬃcients: P = .) k=0 k n

Lemma 11.4.2. Let V be a kTr-module and assume that dim V = n as a k-vector space. 1 Then, dim ExtkTr (k, V ) ≤ ne (where e is the r-rank of T ).

Proof. We proceed by induction on the dimension n of V . Throughout this proof, we will use the fact that the only irreducible module for an r-group in characteristic r is the trivial module k. When n = 1, we have V = k, so we must show that 1 0 dim ExtkTr (k, k) ≤ e. But, Tr is an abelian r-group of rank e ≤ e. So, by Lemma 11.4.1, 1 + e0 − 1 dim Ext1 (k, k) = dim H1(T , k) = = e0 ≤ e. kTr r 1

Suppose now that dim V = n > 1 and that the statement of the lemma holds for all 0 kTr-modules V of dimension n − 1. Let 0 = V0 ⊂ V1 ⊂ V2 ⊂ · · · ⊂ Vn = V be a composition series for V as a kTr-module. Note that every composition factor Vi/Vi−1 (1 ≤ i ≤ n) is isomorphic to k. Therefore, we have a short exact sequence 0 → Vn−1 → V → k → 0, which induces the following long exact Ext sequence:

1 1 1 · · · → ExtkTr (k, k) → ExtkTr (V, k) → ExtkTr (Vn−1, k) → · · · .

1 1 1 By exactness, we have dim ExtkTr (V, k) ≤ dim ExtkTr (k, k) + dim ExtkTr (Vn−1, k). Now, 1 1 dim ExtkTr (k, k) ≤ e and dim ExtkTr (Vn−1, k) ≤ (n − 1)e by the base case and the inductive 1 hypothesis, respectively. Thus, dim ExtkTr (V, k) ≤ e + (n − 1)e = ne, as needed.

Theorem 11.4.3. Let V and V 0 be irreducible kG-modules in the unipotent principal series 1 0 Irrk(G|B). Then, dim ExtkG(V,V ) ≤ |W | + (dim V )e.

Proof. Since induction is a two-sided adjoint of restriction in the case of ﬁnite groups, we T ∼ ∼ have HomkT (k, k|Tr ) = HomkTr (k, k) = k, from which it follows that k occurs exactly once T as a composition factor of the completely reducible module k|Tr . Thus, we have a short T exact sequence of the form 0 → k → k|Tr → M → 0, where M is a completely reducible kTr-module which does not contain k as a composition factor. Applying the Harish-Chandra G induction functor RT (deﬁned in (4.1)), we obtain the short exact sequence

G G T G 0 → RT (k) → RT (k|Tr ) → RT (M) → 0. (11.2)

0 0 G G Now, since V ∈ Irrk(G|B), V is in the socle of k|B = RT (k). By the short exact sequence 0 G T (11.2), V is in the socle of RT (k|Tr ), so we obtain a short exact sequence

0 G T 0 → V → RT (k|Tr ) → Y → 0 for some kG-module Y , which gives rise to the long exact sequence

0 G T 1 0 0 → HomkG(V,V ) → HomkG(V,RT (k|Tr )) → HomkG(V,Y ) → ExtkG(V,V ) → 1 G T ExtkG(V,RT (k|Tr )) → · · · .

83 1 0 1 G T By exactness, dim ExtkG(V,V ) ≤ dim HomkG(V,Y ) + dim ExtkG(V,RT (k|Tr )). So, to prove the theorem, it is enough to show dim HomkG(V,Y ) ≤ |W | and that 1 G T dim ExtkG(V,RT (k|Tr )) ≤ (dim V )e.

First, we show that dim HomkG(V,Y ) ≤ |W |. We have dim HomkG(V,Y ) = [soc(Y ): G T G T G G V ] ≤ [Y : V ] ≤ [RT (k|Tr ): V ]. By (11.2), [RT (k|Tr ): V ] = [RT (k): V ] + [RT (M): V ], G G G G so dim HomkG(V,Y ) ≤ [RT (k): V ] + [RT (M): V ]. But, [RT (k): V ] = [k|B : V ] ≤ |W | by Lemma 11.2.1. So, to show that dim HomkG(V,Y ) ≤ |W |, it suﬃces to show that G G [RT (M): V ] = 0. Suppose, for contradiction, that [RT (M): V ] 6= 0. Since W gives a full set of (B,B)-double coset representatives in G,

∗ G G ∼ T ∗ wT w RT (RT (M)) = ⊕ RwT ∩T ( RwT ∩T ( M)) w∈W by the Mackey decomposition (4.3). For w ∈ W , wM is the wT -module with underlying space w −1 M and T -action given by wtw .m = t.m for any t ∈ T , m ∈ M. Since W = NG(T )/T , w −1 T ∗ wT T = wT w = T for any w ∈ W . Therefore, the functors RwT ∩T and RwT ∩T are equal to ∗ G G ∼ the identity functor on kT − mod for all w ∈ W , and it follows that RT (RT (M)) = ⊕ M. w∈W ∗ G G Since M is a completely reducible kT -module, RT (RT (M)) is a completely reducible kT - module. Thus, ∗ G G ∼ ∗ G ⊕mZ RT (RT (M)) = ⊕ RT (Z) , Z∈Irrk(G) G G where mZ = [RT (M): Z] is the composition multiplicity of Z in RT (M) for any Z ∈ Irrk(G). ∗ G ∗ G Now, since V ∈ Irrk(G|B), RT (V ) 6= 0 and k is a composition factor of RT (V ). But, we G ∗ G ∗ G G assumed mV = [RT (M): V ] 6= 0, so RT (V ) is a submodule of RT (RT (M)) = ⊕ M. w∈W ∗ G Since k is a composition factor of RT (V ), k must also be a composition factor of M. This is a contradiction since M has no trivial composition factors; so, we must have G [RT (M): V ] = 0.

1 G T It remains to check that dim ExtkG(V,RT (k|Tr )) ≤ (dim V )e. Using the deﬁni- G 1 G T tion of the functor RT and the Eckmann-Shapiro Lemma, we have ExtkG(V,RT (k|Tr )) = Ext1 (V, (k|T )|G) ∼ Ext1 (V, k|T ). Now, since B = U T and r 6 | |U|, the r-Sylow kG gTr B = kB gTr o subgroup Tr of T is also an r-Sylow subgroup of B. So, in particular, r 6 | [B : T ], and the restriction map Ext1 (V, k|T ) → Ext1 (V, k|T ) = Ext1 (V, k|T ) is injective. It fol- kB gTr kT gTr kT Tr lows that dim Ext1 (V,RG(k|T )) = dim Ext1 (V, k|T ) ≤ dim Ext1 (V, k|T ). Apply- kG T Tr kB gTr kT Tr 1 T ∼ 1 ing the Eckmann-Shapiro Lemma once more, ExtkT (V, k|Tr ) = ExtkTr (V, k). Therefore, 1 G T 1 dim ExtkG(V,RT (k|Tr )) ≤ dim ExtkTr (V, k) ≤ (dim V )e, where the last inequality holds by Lemma 11.4.2.

84 12 A New Proof of Guralnick and Tiep’s Bound on the Dimension of H1(G, V ) in the case that V B = 0

12.1 Introduction

Let G be a finite group of Lie type, defined over the finite field Fq, where q is a power of a prime p > 0 (so, G is the fixed point subgroup of a connected reductive algebraic ¯ group G over Fp under an endomorphism F : G → G such that some power of F is a Frobenius morphism). In order to use results of modular Harish-Chandra theory, we assume additionally that the BN-pair of G is split. Let (W, S) be the Coxeter system corresponding to the BN-pair structure on G.

Let k be an algebraically closed ﬁeld of characteristic r > 0, r 6= p. In [35], Guralnick and Tiep prove that if V is an irreducible kG-module with V B = 0, then dim H1(G, V ) ≤ 1. B Since V 6= 0 if and only if V ∈ Irrk(G|B), Guralnick and Tiep’s result may be restated as follows.

Theorem 12.1.1. If V is an irreducible kG-module with V 6∈ Irrk(G|B), then dim H1(G, V ) ≤ 1.

The purpose of this section is to present a new proof of this result. While many of the proof strategies used here diﬀer from those used by Guralnick and Tiep, the idea of using induction on the rank of the Weyl group of G was inspired by a reading of [35].

12.2 A ﬁrst result on the dimension of H1(G, V ) when V/∈ Irrk(G|B).

G ∼ ∼ By Frobenius reciprocity, HomkG(k|B, k) = HomkB(k, k) = k. Therefore, there exists a G 0 G surjective homomorphism k|B → k. Let L ⊂ k|B be the kernel of this homomorphism, so 0 G 0 that the sequence 0 → L → k|B → k → 0 is exact (the notation L is consistent with [35]). [35, Theorem 2.2] establishes that the dimension of H1(G, V ) is determined by the structure 0 B of the head of L when V = 0 (or, equivalently, when V/∈ Irrk(G|(T, k)) = Irrk(G|B)). We give this crucial result in Proposition 12.2.1 below, along with an updated (less group-theoretic) proof.

Proposition 12.2.1. Suppose that V is an irreducible kG-module which does not belong to the unipotent principal series. Then, dim H1(G, V ) = [head(L 0): V ] (where [head(L 0): V ] denotes the multiplicity of V in head(L 0)).

1 Proof. First, we claim that H (B,V ) = 0. Let A = Or0 (B) be the biggest normal subgroup of B of order prime to r. Then, B/A is an r-group. (The fact that B/A is an r-group can

85 be seen as follows. Let Tr be the unique r-Sylow subgroup of the abelian group T , and let 0 ∼ 0 0 T be the subgroup of T with T = Tr × T . then A = U o T is the biggest normal subgroup ∼ B A B/A of B of order prime to r, and B/A = Tr is an r-group.) Now, since 0 = V = (V ) , we have V A = 0 (otherwise, the r-group B/A would have a non-zero ﬁxed point on the k-vector space V A).

Consider the ﬁve-term inﬂation-restriction exact sequence

0 → H1(B/A, V A) → H1(B,V ) → H1(A, V )B/A → H2(B/A, V A) → H2(B,V ).

Since V A = 0, H1(B/A, V A) = 0. Also, since r 6 | |A|, kA is semisimple by Maschke’s Theorem, from which it follows that H1(A, V ) = 0. Therefore, H1(B,V ) = 0 by the exactness of the sequence above.

To prove that dim H1(G, V ) = [head(L 0): V ], we use the short exact sequence 0 → 0 G L → k|B → k → 0, which induces the long exact sequence

G 0 1 1 G 0 → HomkG(k, V ) → HomkG(k|B,V ) → HomkG(L ,V ) → ExtkG(k, V ) → ExtkG(k|B,V ) → · · · .

B G Since V = 0, V is not in the head or socle of k|B, so it follows that HomkG(k, V ) = 0 G 1 G ∼ 1 ∼ and HomkG(k|B,V ) = 0. By the Eckmann-Shapiro Lemma, ExtkG(k|B,V ) = ExtkB(k, V ) = 1 1 ∼ 0 H (B,V ) = 0. So, by the exactness of the sequence above, ExtkG(k, V ) = HomkG(L ,V ). 1 1 0 0 Thus, dim H (G, V ) = dim ExtkG(k, V ) = dim HomkG(L ,V ) = [head(L ): V ].

1 We note that the idea to use the subgroup A = Or0 (B) to prove that H (B,V ) = 0 is due to Guralnick and Tiep [35].

12.3 Groups of Rank 1

In this section, we assume that the split group G has a Weyl group of rank 1, so that S = {s} and W = {1, s} for some simple reﬂection s. If α is the simple root corresponding to s, then the unipotent radical of B is of the form U = Uα; in particular, U is an abelian group and |U| = q.

1 The purpose of this section is to show that dim H (G, V ) ≤ 1 when V 6∈ Irrk(G|B). We note that since G is of rank 1, T is the only proper Levi subgroup contained in G. ∗ G U Therefore, an irreducible kG-module Y is cuspidal if and only if RT (Y ) = Y = 0.

Theorem 12.3.1. If G is of rank 1 and V is a non-cuspidal irreducible kG-module such 1 that V/∈ Irrk(G|B), then dim H (G, V ) ≤ 1.

1 0 G 0 Proof. By Proposition 12.2.1, dim H (G, V ) = [head(L ): V ] ≤ [k|B : V ] (where L is a G 0 G submodule of k|B such that 0 → L → k|B → k → 0 is exact), so it suﬃces to show that

G G U ∼ U ⊕ [k|B :Y ] (k|B) = ⊕ (Y ) . Y ∈Irrk(G)

It follows that G U X G U dim (k|B) = [k|B : Y ] dim(Y ). (12.1)

Y ∈Irrk(G) G U We claim that dim (k|B) = 2. Since W gives a full set of (U, B)-double coset representatives in G, it follows by Frobenius reciprocity and the Mackey decomposition that

G U ∼ G ∼ B G ∼ G G ∼ (k|B) = HomkU (k, k|B) = HomkG(k|U , k|B) = HomkU (k, k ↑B↓U ) =

w wB U ∼ w U HomkU (k, ⊕ k ↓wB∩U ↑wB∩U ) = ⊕ HomkU (k, k ↑wB∩U ). (12.2) w∈W w∈W Now, wk is k with wB-action given by wbw−1.x = bx for any b ∈ B and x ∈ k. Since B acts trivially on k, wk is the trivial one-dimensional k(wB)-module and we will denote wk by k. So, by (12.2) and Frobenius reciprocity, we have

G U ∼ U ∼ ∼ (k|B) = ⊕ HomkU (k, k ↑wB∩U ) = ⊕ Homk(wB∩U )(k, k) = ⊕ k. w∈W w∈W w∈W

G U Since G is of rank 1, |W | = 2, and it follows that dim (k|B) = 2, as claimed.

U U G G Since V 6= 0 and k 6= 0, [k|B : V ] and [k|B : k] appear with non-zero coeﬃcient in (12.1). Also, V 6= k since k ∈ Irrk(G|B) and V/∈ Irrk(G|B). So, by (12.1),

G G G U [k|B : V ] + [k|B : k] ≤ dim (k|B) = 2.

G G G But, k appears in the head of k|B, so [k|B : k] ≥ 1, which means that [k|B : V ] ≤ 1.

Theorem 12.3.2. If G is of rank 1 and V is a cuspidal irreducible kG-module, then dim H1(G, V ) ≤ 1.

1 0 G Proof. By Proposition 12.2.1, dim H (G, V ) = [head(L ): V ] ≤ [k|B : V ]. Therefore, it G U suﬃces to show that [k|B : V ] ≤ 1. Since V is cuspidal, V = 0, which means that k is not a direct summand of V ↓U (which is a completely reducible kU-module since r6 | |U|). Now, since U = Uα is abelian, every irreducible kU-module is one-dimensional. Therefore, V ↓U has some non-trivial one-dimensional kU-module ψ as a direct summand. By Frobenius ∼ G G reciprocity, we have 0 6= HomkU (ψ, V ) = HomkG(ψ|U ,V ), so V is in the head of ψ|U .

Now, since r 6 | |U|, every kU-module is projective; in particular, ψ is a projective kU- G module. Since induction from U to G is exact, ψ|U is a projective kG-module, which means G that ψ|U is a direct sum of projective indecomposable kG-modules. Let PV be the projective

87 G indecomposable kG-module with head V . Since V ⊆ head(ψ|U ), PV is a direct summand of G ψ|U . It follows that

G G G G [k|B : V ] = dim HomkG(PV , k|B) ≤ dim HomkG(ψ|U , k|B). (12.3)

G G ∼ G G Now, by Frobenius reciprocity, HomkG(ψ|U , k|B) = HomkU (ψ, k ↑B↓U ). As shown in the G G∼ U proof of Theorem 12.3.1, k ↑B↓U = ⊕ k ↑wB∩U , so we have w∈W

G G ∼ U ∼ HomkG(ψ|U , k|B) = ⊕ HomkU (ψ, k ↑wB∩U ) = ⊕ Homk(wB∩U)(ψ, k). w∈W w∈W

w Here, W = {1, sα}. When w = 1, B ∩ U = U, and we have Homk(wB∩U)(ψ, k) = 0 since w ψ|U is non-trivial. When w = sα, B ∩ U = {1}, and we have dim Homk(wB∩U)(ψ, k) = G G dim Homk(ψ, k) = dim ψ = 1. Therefore, dim HomkG(ψ|U , k|B) = 1, and it follows by (12.3) G that [k|B : V ] ≤ 1.

12.4 Groups of Higher Rank

In this section, we consider (split) ﬁnite groups of Lie type G of higher rank.

Proposition 12.4.1. Suppose that G is of rank (strictly) greater than 1. If V is a cuspidal irreducible kG-module, then H1(G, V ) = 0.

Proof. Given a subset I ⊆ S, let PI be the corresponding standard parabolic subgroup. Since rk(G) > 1, G is generated by the set P of proper standard parabolic subgroups PI of G (i.e., those PI such that I ( S). The set P contains B as a minimal element, and for any P ∈ P, V P ⊆ V B = 0 (V B = 0 since V is cuspidal and hence does not belong to Irrk(G|B)). So, by the vanishing theorem of Alperin and Gorenstein [1], it is enough to show that H1(P,V ) = 0 for every parabolic subgroup P ∈ P.

Let P ∈ P, and let UP and L be the unipotent radical and Levi subgroup of P , ∗ G UP respectively. Since P ( G, it follows that L ( G, so that RL (V ) = V = 0 for the cuspidal kG-module V .

We now consider the ﬁve-term inﬂation-restriction exact sequence

1 UP 1 1 P/UP 2 UP 2 0 → H (P/UP ,V ) → H (P,V ) → H (UP ,V ) → H (P/UP ,V ) → H (P,V ).

UP 1 UP Since V = 0, H (P/UP ,V ) = 0. Furthermore, since r6 | |UP | (UP is a p-group), kUP is 1 P/UP semisimple and H (UP ,V ) = 0. So, by the exactness of the sequence above, we have H1(P,V ) = 0.

88 Proposition 12.4.2. Suppose that V is an irreducible kG-module belonging to a Harish- Chandra series of the form Irrk(G|(T,X)), where X is a non-trivial irreducible kT -module and G has a Weyl group of any rank. Then, H1(G, V ) = 0.

Proof. Since X 6= k, the Harish-Chandra series Irrk(G|(T,X)) is not equal to the unipotent 1 0 principal series Irrk(G|B). Thus, if V ∈ Irrk(G|(T,X)), dim H (G, V ) = [head(L ): 0 G 0 V ] ≤ [L : V ] ≤ [k|B : V ] by Theorem 12.2.1 (where the last inequality follows since L G 1 G is a submodule of k|B). So, to show that H (G, V ) = 0, it suﬃces to show that [k|B : V ] = 0.

G ∗ G Since V ∈ Irr(G|(T,X)), V is in the head and socle of RT (X) and RT (V ) 6= 0. Now, since W gives a full set of (B,B)-double coset representatives in G,

∗ G G ∼ T ∗ wT w RT (RT (X)) = ⊕ RwT ∩T RwT ∩T ( X) w∈W by the Mackey decomposition (for w ∈ W , wX is the wT -module with underlying space w −1 X and T -action given by wtw .x = t.x for any t ∈ T , x ∈ X). Since W = NG(T )/T , w −1 T ∗ wT T = wT w = T for any w ∈ W . Therefore, the functors RwT ∩T and RwT ∩T are equal to the identity functor on kT − mod for all w ∈ W , and it follows that

∗ G G ∼ RT (RT (X)) = ⊕ X. w∈W

∗ G G ∗ G Since RT (RT (X)) is completely reducible, RT (V ) is a (non-zero) direct summand of ∗ G G ∗ G RT (RT (X)). Thus, RT (V ) is a direct sum of copies of X as a kT -module.

G G Assume, for contradiction, that V is a composition factor of k|B = RT (k). Applying the Mackey decomposition as in the paragraph above, we have

∗ G G ∼ T ∗ wT w ∼ RT (RT (k)) = ⊕ RwT ∩T RwT ∩T ( k) = ⊕ k. w∈W w∈W

G ∗ G So, if V is a composition factor of RT (k), the non-zero kT -module RT (V ) is a direct sum G of copies of k. This is impossible since k 6= X, so we conclude that [k|B : V ] = 0.

Theorem 12.4.3. Let G be a ﬁnite group of Lie type (of arbitrary ﬁnite rank) having a split BN-pair of characteristic p, and let V be an irreducible kG-module such that V/∈ Irrk(G|B). Then, dim H1(G, V ) ≤ 1.

Proof. We proceed by induction on the rank of the Weyl group of G. The base case (i.e., the case of rank 1 groups) has been handled in Section 12.3.

For the inductive step, we assume that rk(G) > 1 and that the statement of the theorem holds for all groups of rank strictly less than the rank of G. Suppose that V belongs to the Harish-Chandra series Irrk(G|(L, X)) for some cuspidal irreducible kL-module X. If L = G or L = T , the theorem is proved by Propositions 12.4.1 and 12.4.2, respectively. Thus, we

89 may assume that T ( L ( G.

G G Since V ⊆ soc(RL (X)), we have a short exact sequence of the form 0 → V → RL (X) → M → 0 for some kG-module M. We consider the induced long exact sequence G 1 0 → HomkG(k, V ) → HomkG(k, RL (X)) → HomkG(k, M) → ExtkG(k, V ) → 1 G ExtkG(k, RL (X)) → · · · . By exactness, we have 1 1 G dim ExtkG(k, V ) ≤ dim HomkG(k, M) + dim ExtkG(k, RL (X)). (12.4)

We claim that dim HomkG(k, M) = 0. We have dim HomkG(k, M) = [soc(M): k] ≤ G G [M : k] ≤ [RL (X): k] (where the last inequality follows since M is a quotient of RL (X)). G Therefore, it suﬃces to show that [RL (X): k] = 0. To prove that this is the case, we consider ∗ G G the kL-module RL (RL (X)). Assume that L = LI is the Levi complement of the standard I I I I parabolic subgroup P = PI (∅= 6 I ( S). Then, the subset W = W ∩W of W is a full set of (P,P )-double coset representatives in G (here, I W denotes the set of shortest right coset I representatives of WI in W and W denotes the set of shortest left coset representatives of WI in W ). So, by the Mackey decomposition, ∗ G G ∼ L ∗ wL w RL (RL (X)) = ⊕ RwL∩L RwL∩L( X). w∈I W I But, since X is a cuspidal kL-module, wX is a cuspidal k(wL)-module. Therefore, the only non-zero terms in the direct sum above correspond to those w ∈ I W I for which wL ∩ L = wL I I L ∗ wL or, equivalently, for which w ∈ NG(L). Now, for all w ∈ W ∩ NG(L), RwL∩L and RwL∩L are the identity functors on the category of left kL-modules. So, we have ∗ G G ∼ RL (RL (X)) = ⊕ X. I I w∈ W ∩NG(L) Since L 6= T , k is not a cuspidal kL-module. Therefore, k 6= X, which means that k cannot G ∗ G be a composition factor of RL (X) (otherwise, k = RL (k) would be a direct summand of ∗ G G the completely reducible module RL (RL (X)), which is impossible since k 6= X). Hence, G [RL (X): k] = 0, as needed.

1 1 G Since dim HomkG(k, M) = 0, (12.4) yields dim ExtkG(k, V ) ≤ dim ExtkG(k, RL (X)). 1 G 1 ˜ G ∼ 1 ˜ But, by the Eckmann-Shapiro Lemma, ExtkG(k, RL (X)) = ExtkG(k, X|P ) = ExtkP (k, X) (where P is the parabolic subgroup with Levi complement L). Since [P : L] = |UP | is a power of p (and, hence, not divisible by char(k) = r), the restriction homomorphism 1 ˜ 1 1 ˜ 1 ExtkP (k, X) → ExtkL(k, X) is injective. Therefore, dim ExtkP (k, X) ≤ dim ExtkL(k, X). 1 1 Now, since X is a cuspidal irreducible kL-module, dim ExtkL(k, X) = dim H (L, X) ≤ 1 by the inductive hypothesis, and it follows that 1 1 1 dim H (G, V ) = dim ExtkG(k, V ) ≤ dim ExtkL(k, X) ≤ 1.

90 13 A Bound on the Dimension of Ext1 Between Princi- pal Series Irreducible Modules in Cross Character- istic

13.1 Introduction

Let G be a ﬁnite group of Lie type, deﬁned in characteristic p > 0, and assume that the BN-pair of G is split. Let (W, S) be the Coxeter system corresponding to the BN-pair structure on G.

Let k be an algebraically closed ﬁeld of characteristic r > 0, r 6= p, and let e denote the r-rank of the maximal torus T (that is, e is the maximal dimension of an elementary abelian r-subgroup of T ). In [35], Guralnick and Tiep prove that dim H1(G, V ) ≤ |W | + e if V is an irreducible kG-module with V B 6= 0, and that dim H1(G, V ) ≤ 1 if V is an irreducible kG- module with V B = 0. An irreducible kG-module V satisﬁes the condition V B 6= 0 if and only if V belongs to the unipotent principal Harish-Chandra series Irrk(G|(T, k)) = Irrk(G|B). 1 ∼ 1 Since H (G, V ) = ExtkG(k, V ) for any kG-module V , Guralnick and Tiep’s result may be restated as follows. Given an irreducible kG-module V , ( 1 1 if V 6∈ Irrk(G|B) dim ExtkG(k, V ) ≤ |W | + e if V ∈ Irrk(G|B).

The purpose of this section is to present a partial generalization of this result. Let Y and V be two principal series representations (in the sense of [31, Example 4.2.8]), with Y belonging to the Harish-Chandra series Irrk(G|(T,X)) and V belonging to the Harish-Chandra series 0 0 Irrk(G|(T,X )) for some (necessarily one-dimensional) irreducible kT -modules X and X . We will show that

Remark 13.1.1. If Y and V both belong to the unipotent principal series Irrk(G|B) = 1 Irrk(G|(T, k)), the bound in (2) can be improved. By Theorem 11.4.3, dim ExtkG(Y,V ) ≤ 16 |W | + min(dim Y, dim V )e when Y and V belong to Irrk(G|B). In particular, if we take Y = k ∈ Irrk(G|B), we recover Guralnick and Tiep’s bound.

16 1 By Theorem 11.4.3, dim ExtkG(Y,V ) ≤ |W | + (dim V )e when Y,V ∈ Irrk(G|B). But, an argument 1 analogous to that given in Corollary 13.3.4 shows that dim ExtkG(Y,V ) ≤ |W | + min(dim Y, dim V )e when Y,V ∈ Irrk(G|B).

91 1 13.2 A Bound on the Dimension of ExtkG(Y,V ) when Y and V belong to distinct Principal Series

Throughout this section, we will assume that the irreducible kG-module Y belongs to the principal series Irrk(G|(T,X)) and that the irreducible kG-module V belongs to the 0 0 principal series Irrk(G|(T,X )), with Irrk(G|(T,X)) 6= Irrk(G|(T,X )). Our goal is to prove 1 G 0 that ExtkG(Y,V ) = 0. Since Y ∈ Irrk(G|(T,X)), Y is in the head of RT (X). Let L G G 0 ∼ denote a maximal submodule of RT (X) with RT (X)/L = Y . In the next result, we 1 generalize [35, Theorem 2.2] to show that the dimension of ExtkG(Y,Z) is determined by the structure of head(L 0) when Z is an irreducible kG-module which does not belong to the Harish-Chandra series Irrk(G|(T,X)).

Theorem 13.2.1. Let Y be an irreducible kG-module in the principal series Irrk(G|(T,X)) (where X is a one-dimensional kT -module). Then, if Z is an irreducible kG-module such 1 0 0 that Z 6∈ Irrk(G|(T,X)), dim ExtkG(Y,Z) = [head(L ): Z] (where [head(L ): Z] denotes the multiplicity of Z as a composition factor of head(L 0)).

Proof. By deﬁnition of L 0, we have a short exact sequence of kG-modules 0 → L 0 → G RT (X) → Y → 0, which induces the long exact sequence

G 0 1 1 G 0 → HomkG(Y,Z) → HomkG(RT (X),Z) → HomkG(L ,Z) → ExtkG(Y,Z) → ExtkG(RT (X),Z) → · · · .

Now, since Z does not belong to the principal series Irrk(G|(T,X)), Z 6= Y and Z is not in G G the head of RT (X). Therefore, HomkG(Y,Z) = 0 and HomkG(RT (X),Z) = 0, which means that the exact sequence above reduces to

0 1 1 G 0 → HomkG(L ,Z) → ExtkG(Y,Z) → ExtkG(RT (X),Z) → · · · .

Therefore, to prove the statement of the theorem, it is enough to show that 1 G 1 G 1 ∼ 0 ExtkG(RT (X),Z) = 0. For, if ExtkG(RT (X),Z) = 0, then ExtkG(Y,Z) = HomkG(L ,Z) 1 by the exactness of the sequence above, which means that dim ExtkG(Y,Z) = 0 0 dim HomkG(L ,Z) = [head(L ): Z].

1 G G We will now prove that ExtkG(RT (X),Z) = 0. By definition of RT (X) and by the 1 G 1 ˜ G ∼ 1 ˜ Eckmann-Shapiro Lemma, we have ExtkG(RT (X),Z) = ExtkG(X|B,Z) = ExtkB(X,Z) ˜ (where X is the inflation of X from T to B via the surjective homomorphism B T with 1 ˜ kernel U). Thus, it suffices to prove that ExtkB(X,Z) = 0.

Let A = Or0 (B) be the biggest normal subgroup of B of order prime to r = char(k). ˜ Then, B/A is an r-group. We claim that HomkA(X,Z) = 0. Since Z 6∈ Irrk(G|(T,X)), G G ˜ G ∼ ˜ Z 6⊆ head(RT (X)). Thus, 0 = HomkG(RT (X),Z) = HomkG(X|B,Z) = HomkB(X,Z) ˜ G ∼ ˜ (where the isomorphism HomkG(X|B,Z) = HomkB(X,Z) follows by Frobenius reci- ˜ procity). Now, the group B acts on the k-vector space Homk(X,Z) (given an element ˜ ˜ b ∈ B and a k-vector space homomorphism φ ∈ Homk(X,Z), b.φ ∈ Homk(X,Z)

92 is deﬁned by (b.φ)(x) = bφ(b−1x) for any x ∈ X˜), and we have an isomor- ˜ B ∼ ˜ ˜ phism (Homk(X,Z)) = HomkB(X,Z). So, since HomkB(X,Z) = 0, it follows ˜ B that (Homk(X,Z)) = 0. Now, since A is a normal subgroup of B, we have ˜ B ˜ A B/A 0 = (Homk(X,Z)) = [(Homk(X,Z)) ] . But, B/A is an r-group and char(k) = r, ˜ A so this is possible only if (Homk(X,Z)) = 0 (otherwise, the r-group B/A would ˜ A have a non-zero ﬁxed point on the k-vector space (Homk(X,Z)) ). Therefore, ˜ ∼ ˜ A HomkA(X,Z) = (Homk(X,Z)) = 0.

Now, applying the ﬁve-term inﬂation-restriction exact sequence on cohomology to the ˜ ∗ ˜ ∗ ˜ kB-module X ⊗k Z (where X is the k-linear dual of X), we have:

1 ˜ ∗ A 1 ˜ ∗ 1 ˜ ∗ B/A 0 → H (B/A, (X ⊗k Z) ) → H (B, X ⊗k Z) → H (A, X ⊗k Z) → 2 ˜ ∗ A 2 ˜ ∗ H (B/A, (X ⊗k Z) ) → H (B, X ⊗k Z).

1 ˜ ∗ Since r6 | |A|, kA is semisimple by Maschke’s Theorem, which means that H (A, X ⊗k B/A ˜ ∗ ∼ ˜ ˜ ∗ A ∼ ˜ A ∼ Z) = 0. Since X ⊗k Z = Homk(X,Z), we have (X ⊗k Z) = (Homk(X,Z)) = ˜ 1 ˜ ∗ A 1 ˜ ∗ HomkA(X,Z) = 0, so that H (B/A, (X ⊗k Z) ) = 0. We conclude that H (B, X ⊗k 1 ˜ ∗ ∼ 1 ˜ ∗ ∼ Z) = 0 by exactness of the sequence above. But, H (B, X ⊗k Z) = ExtkB(k, X ⊗k Z) = 1 ˜ 1 ˜ ExtkB(X,Z), so it follows that ExtkB(X,Z) = 0, as needed.

Proposition 13.2.2. Let Y be an irreducible kG-module in the principal series 0 Irrk(G|(T,X)), and let V be an irreducible kG-module in the principal series Irrk(G|(T,X )), 0 G with Irrk(G|(T,X)) 6= Irrk(G|(T,X )). Then, [RT (X): V ] = 0.

0 G 0 Proof. Since V ∈ Irrk(G|(T,X )), V is a composition factor of the head and socle of RT (X ) ∗ G and RT (V ) 6= 0. Now, since W gives a full set of (B,B)-double coset representatives in G, ∗ G G 0 ∼ T ∗ wT w 0 RT (RT (X )) = ⊕ RwT ∩T RwT ∩T ( X ) w∈W by the Mackey decomposition (for w ∈ W , wX0 is the wT -module with underlying space X0 w −1 0 and T -action given by wtw .x = t.x for any t ∈ T , x ∈ X ). But, since W = NG(T )/T , w −1 T ∗ wT T = wT w = T for any w ∈ W . Therefore, the functors RwT ∩T and RwT ∩T are equal to the identity functor on kT − mod for all w ∈ W , and it follows that ∗ G G 0 ∼ w 0 RT (RT (X )) = ⊕ X . w∈W

w 0 ∗ G G 0 For any w ∈ W , X is an irreducible kT -module, which means that RT (RT (X )) is com- ∗ G ∗ G ∗ G G 0 pletely reducible. Since RT is exact, RT (V ) is a non-zero kT -submodule of RT (RT (X )). Thus, there must be some subset Ω ⊆ W such that ∗ G ∼ w 0 RT (V ) = ⊕ X . (13.1) w∈Ω

G Suppose, for contradiction, that [RT (X): V ] 6= 0. Applying the Mackey decomposition as ∗ G G ∼ w in the paragraph above, we have RT (RT (X)) = ⊕ X. So, if V is a composition factor of w∈W

93 G ∗ G ∗ G G w ∗ G RT (X), then RT (V ) is a non-zero submodule of RT (RT (X)) and X ⊆ RT (V ) for some w ∈ W . By (13.1), this means that wX ∼= w0 X0 for some w0 ∈ Ω, so that X0 ∼= (w0)−1wX. But, 0 0 if X is a twist of X by an element of W , then Irrk(G|(T,X)) = Irrk(G|(T,X )), contradicting the assumption in the statement of the proposition.

Corollary 13.2.3. Let Y be an irreducible kG-module in the principal series Irrk(G|(T,X)), 0 and let V be an irreducible kG-module in the principal series Irrk(G|(T,X )), with 0 1 Irrk(G|(T,X)) 6= Irrk(G|(T,X )). Then, ExtkG(Y,V ) = 0.

0 G G 0 ∼ Proof. As above, let L be a submodule of RT (X) with RT (X)/L = Y . Then, by Theorem 1 0 0 G 13.2.1 and Proposition 13.2.2, dim ExtkG(Y,V ) = [head(L ): V ] ≤ [L : V ] ≤ [RT (X): V ] = 0.

1 13.3 A Bound on the Dimension of ExtkG(Y,V ) when Y and V belong to the same Principal Series

The results of this section generalize the results of Chapter 11. From this point on, we will assume that Y and V belong to the same principal series Irrk(G|(T,X)) (where X is any one-dimensional kT -module). Our goal is to prove 1 2 that dim ExtkG(Y,V ) ≤ |W | + min(dim Y, dim V )e (where e is the r-rank of T ). We begin with a lemma which generalizes Lemma 11.2.1 and provides a bound on the number of times an irreducible representation in the principal series Irrk(G|(T,X)) can appear as a G composition factor of the induced module RT (X).

Lemma 13.3.1. If Z is an irreducible module in the principal series Irrk(G|(T,X)), then G [RT (X): Z] ≤ |W |.

Proof. Since W gives a full set of (B,B)-double coset representatives in G and wT = T for any w ∈ W , the Mackey decomposition yields

∗ G G ∼ T ∗ wT w ∼ w RT (RT (X)) = ⊕ RwT ∩T RwT ∩T ( X) = ⊕ X. w∈W w∈W Now, wX is an irreducible one-dimensional kT -module for every w ∈ W , which means that ∗ G G ∗ G G the kT -module RT (RT (X)) is completely reducible and dim RT (RT (X)) = |W |. On the ∗ G ∗ G G other hand, since RT is exact, RT (RT (X)) has the following direct sum decomposition as a kT -module: G 0 ∗ G G ∼ ∗ G 0 ⊕[RT (X):Z ] RT (RT (X)) = ⊕ RT (Z ) . 0 Z ∈Irrk(G) Therefore, ∗ G G X G 0 ∗ G 0 dim RT (RT (X)) = [RT (X): Z ] dim RT (Z ). (13.2) 0 Z ∈Irrk(G)

94 By assumption, the irreducible kG-module Z belongs to the principal series Irrk(G|(T,X)), ∗ G G which means that RT (Z) 6= 0. Thus, [RT (X): Z] appears with non-zero coeﬃcient in G ∗ G G (13.2), and it follows that [RT (X): Z] ≤ dim RT (RT (X)) = |W |.

1 To establish the desired bound on ExtkG(Y,V ), we will work with the unique Sylow r-subgroup Tr of the abelian group T . Hence, we will have to break our proof into two cases, depending on whether or not |B| is divisible by the characteristic r of k.

13.3.1 Case I: r - |B|

Assume that the characteristic r of k does not divide |B|.

Theorem 13.3.2. Let Y and V be irreducible kG-modules in the principal series 1 Irrk(G|(T,X)). Then, dim ExtkG(Y,V ) ≤ |W |.

Proof. Since r6 | |B|, the group algebra kB is semisimple and every kB-module is projective ˜ G ˜ G and injective. In particular, X is an injective kB-module, which means that RT (X) = X|B is also injective since induction from B to G is exact. Now, since V ∈ Irrk(G|(T,X)), V is G contained in the socle of RT (X). Therefore, we have a short exact sequence of kG-modules

G 0 → V → RT (X) → M → 0

∼ G G 1 G (where M = RT (X)/V ). Since RT (X) is injective, ExtkG(Y,RT (X)) = 0, so the short exact sequence above induces the exact sequence

G 1 0 → HomkG(Y,V ) → HomkG(Y,RT (X)) → HomkG(Y,M) → ExtkG(Y,V ) → 0.

1 G Therefore, dim ExtkG(Y,V ) ≤ dim HomkG(Y,M) = [soc(M): Y ] ≤ [M : Y ] ≤ [RT (X): Y ] ≤ |W |, where the last inequality follows by Lemma 13.3.1.

13.3.2 Case II: r | |B|

Assume that |B| is divisible by the characteristic r of k. As above, let Tr be the (unique) r-Sylow subgroup of T . Since r | |B| = |U||T | and U is a p-group, r | |T |. Hence, Tr is a non-trivial (abelian) r-subgroup of T . A key step in proving that 1 2 dim ExtkG(Y,V ) ≤ |W | + min(dim Y, dim V )e will be to reduce the problem to ﬁnding a bound on the dimension of a cohomology group of Tr. We will use the results on the cohomology of Tr developed in Chapter 11.

T As in Chapter 11, we will work with the completely reducible permutation module k|Tr . Let Z be any irreducible kT -module. Then, Z is one-dimensional, which means that the restriction Z ↓Tr of Z to Tr is irreducible. Since the only irreducible module for an r-group in characteristic r is the trivial module k, it follows that Z ↓Tr = k. So, by Frobenius

95 T ∼ ∼ reciprocity, we have HomkT (k|Tr ,Z) = HomkTr (k, Z ↓Tr ) = HomkTr (k, k) = k, from which it follows that Z occurs exactly once as a composition factor of the completely reducible T T module k|Tr . Thus, k|Tr contains every irreducible kT -module as a direct summand exactly once.

Theorem 13.3.3. Let Y and V be irreducible kG-modules in the principal series 1 2 Irrk(G|(T,X)). Then, dim ExtkG(Y,V ) ≤ |W | + (dim V )e.

Proof. Since wT = T for any w ∈ W , the k(wT )-module wX is also a kT -module for each w w ∈ W . Therefore, the set Ie = { X | w ∈ W } is a subset of Irrk(T ) consisting of at most |W | distinct irreducible kT -modules. Let I be a subset of Ie consisting of one representative of each isomorphism class of irreducible kT -modules appearing in Ie. By the independence G ∼ G 0 0 property of Harish-Chandra induction, RT (X) = RT (X ) for all X ∈ I.

Since every irreducible kT -module occurs exactly once as a composition factor of the T completely reducible kT -module k|Tr , we have a direct sum decomposition

T ∼ 0 k|Tr = M ⊕ ⊕ X , X0∈I where M is a completely reducible kT -module which does not contain X0 as a composition 0 G factor for any X ∈ I. Applying the (exact) Harish-Chandra induction functor RT , we can write G T ∼ G G 0 RT (k|Tr ) = RT (M) ⊕ ⊕ RT (X ) . (13.3) X0∈I G Now, since Y ∈ Irrk(G|(T,X)), Y is in the head of RT (X), which means that Y is in the G T head of RT (k|Tr ) by (13.3). Therefore, we have a short exact sequence of kG-modules

0 G T 0 → M → RT (k|Tr ) → Y → 0

0 G T G T 0 ∼ (where M is a kG-submodule of RT (k|Tr ) such that RT (k|Tr )/M = Y ), which gives rise to the long exact sequence

G T 0 1 0 → HomkG(Y,V ) → HomkG(RT (k|Tr ),V ) → HomkG(M ,V ) → ExtkG(Y,V ) → 1 G T ExtkG(RT (k|Tr ),V ) → · · · .

1 0 1 G T By exactness, dim ExtkG(Y,V ) ≤ dim HomkG(M ,V ) + dim ExtkG(RT (k|Tr ),V ). So, 0 2 to prove the theorem, it is enough to show that dim HomkG(M ,V ) ≤ |W | and that 1 G T dim ExtkG(RT (k|Tr ),V ) ≤ (dim V )e.

0 2 0 First, we show that dim HomkG(M ,V ) ≤ |W | . We have dim HomkG(M ,V ) = 0 0 G T [head(M ): V ] ≤ [M : V ] ≤ [RT (k|Tr ): V ]. By (13.3),

G T G X G 0 [RT (k|Tr ): V ] = [RT (M): V ] + [RT (X ): V ], X0∈I

96 0 G P G 0 0 G 0 ∼ so dim HomkG(M ,V ) ≤ [RT (M): V ] + [RT (X ): V ]. But, for all X ∈ I, RT (X ) = X0∈I G G RT (X) by the independence property. Since V ∈ Irrk(G|(T,X)), [RT (X): V ] ≤ |W | by Lemma 13.3.1, which means that

X G 0 X G X 2 [RT (X ): V ] = [RT (X): V ] ≤ |W | ≤ |I| |W | ≤ |W | . X0∈I X0∈I X0∈I

0 2 G So, to show that dim HomkG(M ,V ) ≤ |W | , it suﬃces to show that [RT (M): V ] = 0. Now, since M is a completely reducible kT -module which does not contain X0 as a composition m 0 ∼ factor for any X ∈ I, M has a decomposition M = ⊕ Zi such that each Zi (1 ≤ i ≤ m) is i=0 ∼ 0 0 ∼ w an irreducible kT -module with Zi =6 X for all X ∈ I. So, for all i, 1 ≤ i ≤ m, Zi =6 X for all w ∈ W , which means that the Harish-Chandra series Irrk(G|(T,Zi)) and Irrk(G|(T,X)) G are distinct. Therefore, by Proposition 13.2.2, [RT (Zi): V ] = 0 for all i, and it follows that m m G G P G [RT (M): V ] = [ ⊕ RT (Zi): V ] = [RT (Zi): V ] = 0. i=0 i=0

1 G T It remains to check that dim ExtkG(RT (k|Tr ),V ) ≤ (dim V )e. Using the deﬁni- G 1 G T tion of the functor RT and the Eckmann-Shapiro Lemma, we have ExtkG(RT (k|Tr ),V ) = Ext1 ((k|T )|G,V ) ∼ Ext1 (k|T ,V ). Now, since B = U T and r 6 | |U| = [B : T ], kG gTr B = kB gTr o the restriction map Ext1 (k|T ,V ) → Ext1 (k|T ,V ) = Ext1 (k|T ,V ) is injective. It fol- kB gTr kT gTr kT Tr lows that dim Ext1 (RG(k|T ),V ) = dim Ext1 (k|T ,V ) ≤ dim Ext1 (k|T ,V ). Apply- kG T Tr kB gTr kT Tr 1 T ∼ 1 ing the Eckmann-Shapiro Lemma once more, ExtkT (k|Tr ,V ) = ExtkTr (k, V ). Therefore, 1 G T 1 dim ExtkG(RT (k|Tr ),V ) ≤ dim ExtkTr (k, V ) ≤ (dim V )e, where the last inequality holds by Lemma 11.4.2.

Corollary 13.3.4. Let Y and V be irreducible kG-modules in the principal series 1 2 Irrk(G|(T,X)). Then, dim ExtkG(Y,V ) ≤ |W | + min(dim Y, dim V)e.

1 2 Proof. By Theorem 13.3.3, dim ExtkG(Y,V ) ≤ |W | + (dim V )e. Since Y and V belong to the principal series Irrk(G|(T,X)), Y and V occur in both the head and the socle of G ∗ ∗ RT (X). Hence, the k-linear duals Y and V occur in both the head and the socle of G ∗ ∼ G ∗ ∗ ∗ ∗ (RT (X)) = RT (X ). Thus, Y and V both belong to the principal series Irrk(G|(T,X )), 1 1 ∗ ∗ 2 ∗ 2 and dim ExtkG(Y,V ) = dim ExtkG(V ,Y ) ≤ |W | + (dim Y )e = |W | + (dim Y )e (where 1 ∗ ∗ 2 ∗ the inequality dim ExtkG(V ,Y ) ≤ |W | + (dim Y )e follows by another application of Theorem 13.3.3).

97 14 A Bound on the Dimension of Ext1 between a Unipotent Principal Series Representation and an Irreducible Outside the Unipotent Principal Series

14.1 Introduction

Let G be a ﬁnite group of Lie type, deﬁned in characteristic p > 0, and assume that the BN- pair of G is split. Let (W, S) be the Coxeter system corresponding to the BN-pair structure on G. Let Φ be the corresponding root system, Φ+ the set of positive roots, and Π the set of + simple roots in Φ . Given a root α ∈ Φ, let Uα denote the corresponding root subgroup of G.

In this chapter, we will assume certain additional conditions on the group G in order to 1 ﬁnd a bound on dim ExtkG(Y,V ) when Y ∈ Irrk(G|B) and V 6∈ Irrk(G|B). Several key ideas behind the proof of the main theorem (Theorem 14.3.3) can be found in [29], so we begin with a summary of the relevant results of [29].

14.2 Some Preliminaries from [29, Section 4]: The Steinberg Mod- ule and Harish-Chandra Series

14.2.1 Introduction

Let G be a finite group of Lie type with a split BN-pair and associated Coxeter system (W, S). Let Φ be the corresponding root system, Φ+ the set of positive roots, and Π the set of simple roots. Let k be a field, and let Stk be the Steinberg module of G. In [29], Geck studies the composition factors of the Steinberg module. It is well known that Stk is irreducible if and only if char(k)6 | [G : B]. In the case that char(k) | [G : B] and Stk is not irreducible, Tinberg [48] proved that the socle of Stk is simple. However, many open questions remain concerning the structure of Stk when char(k) | [G : B]. For instance, the length of a composition series of Stk is not known. This question is of particular interest to Geck; in [29], he works toward finding a bound on the length of a composition series of Stk.

Geck [29] uses the Hecke algebra associated to the Coxeter system (W, S) to prove that G soc(Stk) is irreducible (Tinberg uses the endomorphism algebra EndkG(k|U ) in his original

98 proof). Geck’s proof yields new information on the structure of Stk. For example, in the case that G = GLn(q) (where q is a power of a prime p), Geck identiﬁes the partition that parameterizes the socle of Stk [29, 3.6]. In [29, Section 4], Geck studies the relationship between Stk and Harish-Chandra series of irreducible kG-modules, which allows him to determine the composition length of Stk in certain cases [29, 4.7-4.14]. While the composition length of Stk is not needed to prove any of the original results presented in this dissertation, the relationship between Stk and Harish-Chandra series is crucial to the proof of Theorem 14.3.3.

14.2.2 The BN-Pair of G

Since G is a finite group of Lie type, there exists a connected reductive algebraic group G over the algebraic closure Fp of the finite field Fp and a Steinberg endomorphism F of G such that G = GF . Let B be an F -stable Borel subgroup of G, and let T be an F -stable F F maximal torus contained in B. Then, the subgroups B = B and N = NG(T) form a BN-pair in G. If U is the unipotent radical of B, then U = UF is the unipotent radical of B.

Let [U, U] be the commutator of U (by [29, Remark 2.5], [U, U] is an F -stable closed connected normal subgroup of U), and deﬁne U ∗ = [U, U]F . (It is always true that [U, U] ⊆ U ∗; however, the reverse containment does not hold in certain small characteristics.)

14.2.3 An r-modular system.

Let k be a field of characteristic r > 0, r 6= p. We will work with an r-modular system (O, K, k), where O is a discrete valuation ring with residue field k and field of fractions K (with char(K) = 0). We will assume that the fields k and K are both large enough, meaning that they are splitting fields for G and all of its subgroups.

An OG-module M will be called a lattice if M is ﬁnitely generated and free over O. Given a lattice M, we let KM := K ⊗O M and M := k ⊗O M. It is a standard fact in modular representation theory that given a projective OG-lattice M and an OG-lattice M 0 0 0 0 (with M not necessarily projective), dim HomKG(KM,KM ) = dim HomkG(M, M ) [46, 15.4].

As recorded in [29] (pg. 14), Harish-Chandra induction and restriction are compatible with the r-modular system in the following sense. Let J ⊆ S and let LJ denote the Levi complement of the standard parabolic subgroup PJ of G. Then, if X is an OLJ -lattice, KRG (X) ∼ RG (KX) and RG (X) ∼ RG (X). LJ = LJ LJ = LJ If Y is an OG-lattice,

K∗RG (Y ) ∼ ∗RG (KY ) and ∗RG (Y ) ∼ ∗RG (Y ). LJ = LJ LJ = LJ

99 14.2.4 An OG-lattice which yields the Steinberg module

Let StO=OGe be the Steinberg module over OG (we will call StO the Steinberg lattice). ∼ Then, KStO = StK and StO = Stk. Since char(K) = 0, the Steinberg module StK is irreducible. In [29, Section 4], Geck constructs another OG-lattice which yields StK upon base change to K. In the remainder of this section, we will describe this alternate lattice.

× P Let σ : U → K be a group homomorphism, and deﬁne uσ := σ(u)u ∈ KG. Since U u∈U × is a p-subgroup of G, r6 | |U| and σ(u) ∈ O for all u ∈ U, which means uσ ∈ OG. We have

2 X 0 0 XX 0 0 X uσ = σ(u)u σ(u )u = σ(uu )uu = uσ = |U|uσ. u,u0∈U u∈Uu0∈U u∈U

1 Since |U| is a unit in O, |U| uσ is an idempotent in OG and the OG-lattice Γσ := OGuσ is projective.

By [29, 4.1], dim HomOG(Γσ, StO) = 1 and dim HomKG(KΓσ, StK ) = 1.

Proposition 14.2.1. ([29, Proposition 4.2]) For any group homomorphism σ : U → K×, 0 0 ∼ 0 there exists a unique OG-sublattice Γσ ⊆ Γσ such that K(Γσ/Γσ) = StK . If Sσ = Γσ/Γσ, the kG-module Dσ := Sσ/rad(S σ) is simple and occurs exactly once as a composition factor of Stk.

14.2.5 The Gelfand-Graev module

We will now assume that the center of G is connected. Let σ : U → K× be a ﬁxed regular ∗ character (i.e., σ is a group homomorphism such that U ⊆ Ker(σ) and σ|Uα is non-trivial for all α ∈ Π). In this case, the projective OG-module Γσ is called a Gelfand-Graev module for G (the Gelfand-Graev module is unique up to isomorphism when the center of G is connected). Since Γσ is a projective OG-lattice, the r-modular reduction Γσ of Γσ is a projective kG-module.

For any J ⊆ S, let LJ be the corresponding Levi subgroup of G. By [29, 4.3], there is a J J Gelfand-Graev module Γσ for OL . Therefore, [29, Proposition 4.2] yields an OLJ -lattice J J J 0 J J J J ∼ J Sσ = Γσ/(Γσ) such that Dσ := S σ/rad(S σ) is a simple kLJ -module and KSσ = StK J (where StK is the Steinberg module for KLJ ).

The OG-modules Γσ and Sσ behave particularly well with respect to Harish-Chandra restriction. For any J ⊆ S, we have the following isomorphisms of OLJ -modules [29, Lemma 4.4]: ∗RG (Γ ) ∼ ΓJ and ∗RG ( ) ∼ J . LJ σ = σ LJ Sσ = Sσ

100 (The isomorphism ∗RG (Γ ) ∼ ΓJ is due to Rodier.) LJ σ = σ

Since the Harish-Chandra restriction functor is compatible with the r-modular system J (O, K, k), we also have ∗RG (KΓ ) ∼ KΓJ , ∗RG (K ) ∼ K J , ∗RG (Γ ) ∼ Γ , and LJ σ = σ LJ Sσ = Sσ LJ σ = σ J ∗RG ( ) ∼ . LJ S σ = S σ

14.2.6 Harish-Chandra series arising from the regular character σ

∗ J ∗ J Let Pσ = {J ⊆ S | Dσ is a cuspidal kLJ − module}. For every J ∈ Pσ, Dσ is an irreducible cuspidal kLJ -module, which means that there is a Harish-Chandra series of the J ∗ J form Irrk(G|(LJ ,Dσ )). If J ∈ Pσ, the Harish-Chandra series Irrk(G|(LJ ,Dσ )) consists of the irreducible kG-modules in the head of RG (DJ ) (equivalently, Irr (G|(L ,DJ )) consists LJ σ k J σ of the irreducible kG-modules in the socle of RG (DJ )). Given I,J ∈ ∗, the series LJ σ Pσ I J Irrk(G|(LI ,Dσ)) and Irrk(G|(LJ ,Dσ )) contain the same irreducible kG-modules if and only if J = wIw−1 for some w ∈ W .

We include Geck’s proof of the next proposition because some of the techniques used in this proof are needed to prove Theorem 14.3.3.

∗ Proposition 14.2.2. ([29, Proposition 4.6] If J ∈ Pσ, then Stk has a unique composition J factor which belongs to the Harish-Chandra series Irrk(G|(LJ ,Dσ )). ∼ Proof. Since StK = KSσ, the kG-modules Stk and S σ have the same composition factors, with multiplicity. Therefore, it suﬃces to show that the statement of the proposition holds J for S σ. First, we show that S σ has a composition factor which belongs to Irrk(G|(LJ ,Dσ )) ∗ ∗ for every J ∈ Pσ. Let J ∈ Pσ be ﬁxed. Since Harish-Chandra induction and restriction are adjoint functors,

Hom ( ,RG (DJ )) ∼ Hom (∗RG ( ),DJ ). kG S σ LJ σ = kLJ LJ S σ σ

J Since ∗RG ( ) ∼ by [29, Lemma 4.4], we have LJ S σ = S σ

J Hom (∗RG ( ),DJ ) ∼ Hom ( ,DJ ). kLJ LJ S σ σ = kLJ S σ σ

J But, by deﬁnition of DJ , Hom ( ,DJ ) 6= 0, and it follows that Hom ( ,RG (DJ )) 6= σ kLJ S σ σ kG S σ LJ σ 0. Therefore, some composition factor of is in the socle of RG (DJ ). Since every S σ LJ σ irreducible summand of soc(RG (DJ )) is in the Harish-Chandra series Irr (G|(L ,DJ )), LJ σ k J σ S σ J has a composition factor which belongs to Irrk(G|(LJ ,Dσ )).

∗ J Next, we show that given J ∈ Pσ, only one irreducible kG-module in Irrk(G|(LJ ,Dσ )) can be a composition factor of S σ. If V is an irreducible kG-module with V ∈ Irr (G|(L ,DJ )), then V ⊆ head(RG (DJ )). Now, since there is a surjective k J σ LJ σ

101 J homomorphism Γ DJ and the Harish-Chandra induction functor RG is exact, there is a σ σ LJ J surjective homomorphism RG (Γ ) RG (DJ ). Therefore, for every V ∈ Irr (G|(L ,DJ )), LJ σ LJ σ k J σ J J V ⊆ head(RG (Γ )). Since RG (Γ ) is a projective kG-module, the projective cover P of LJ σ LJ σ V J V is a direct summand of RG (Γ ) for every V ∈ Irr (G|(L ,DJ )). LJ σ k J σ

Given the results of the previous paragraph, the total number of composition factors of J ∗ S σ belonging to the Harish-Chandra series Irrk(G|(LJ ,Dσ )) (J ∈ Pσ) is

X X [S σ : V ] = dim HomkG(PV , S σ) J J V ∈Irrk(G|(LJ ,Dσ )) V ∈Irrk(G|(LJ ,Dσ ))

= dim HomkG( ⊕ PV , S σ) J V ∈Irrk(G|(LJ ,Dσ )) J ≤ dim Hom (RG (Γ ), ). kG LJ σ S σ Now, since the Harish-Chandra induction functor RG respects the r-modular system and LJ RG (ΓJ ) is a projective OG-lattice, we have LJ σ J dim Hom (RG (Γ ), ) = dim Hom (RG (KΓJ ),K ). kG LJ σ S σ KG LJ σ Sσ Using the adjointness of Harish-Chandra induction and restriction along with [29, Lemma 4.4], we have dim Hom (RG (KΓJ ),K ) = dim Hom (KΓJ , ∗RG (K )) = KG LJ σ Sσ KLJ σ LJ Sσ J J J ∼ J J dim HomKLJ (KΓσ,KSσ ). Since KSσ = StK , it follows that dim HomKLJ (KΓσ,KSσ ) = J dim HomKLJ (KΓσ, StK ) = 1.

G Proposition 14.2.3. ([29, Proposition 4.7]) Assume that every composition factor of k|B J ∗ belongs to Irrk(G|(LJ ,Dσ )) for some J ∈ Pσ. Then, Stk is multiplicity-free and the length ∗ of a composition series of Stk is equal to the number of subsets J ∈ Pσ, up to W -conjugacy.

The result of [29, Proposition 4.7] inspired the following deﬁnition.

Deﬁnition 14.2.4. We will say that the pair (G, k) satisﬁes property (P) if every composi- G J tion factor of k|B belongs to a Harish-Chandra series of the form Irrk(G|(LJ ,Dσ )) for some ∗ J ∈ Pσ.

When the ﬁeld k is clear from context we will simply say that G has property (P). There are many examples of ﬁnite groups of Lie type G such that (G, k) has property (P). If q is a power of p, then the pair (GLn(q), k) has property (P) [29, Example 4.9]. If r is a linear prime,17 then the following pairs have property (P) [29, 4.14]:

1.( SOn(q), k), n odd and q odd, and

2.( Spn(q), k), n even and q a power of 2.

17 i−1 r is a linear prime for SOn(q) and Spn(q) if q 6≡ −1 mod r for all i ≥ 1.

102 1 14.3 A Bound on dim ExtkG(Y,V )

1 In this section, we will ﬁnd a bound on dim ExtkG(Y,V ) when the pair (G, k) has property (P) and Y and V are irreducible kG-modules such that Y ∈ Irrk(G|B) and V 6∈ Irrk(G|B). G Since Y ∈ Irrk(G|B), Y is in the head of the permutation module k|B. Therefore, there 0 G G 0 ∼ exists a kG-submodule L of k|B such that k|B/L = Y . When V 6∈ Irrk(G|B), the 1 0 dimension of ExtkG(Y,V ) is determined by the structure of the head of L .

1 0 Proposition 14.3.1. If V 6∈ Irrk(G|B), dim ExtkG(Y,V ) = [head(L ): V ] (where [head(L 0): V ] denotes the number of times that V occurs as a composition factor of head(L 0)).

Proof. This result is a special case of Theorem 13.2.1.

Corollary 14.3.2. Suppose that Y ∈ Irrk(G|B) and V 6∈ Irrk(G|B). If V is not a composi- G 1 tion factor of k|B, then ExtkG(Y,V ) = 0.

1 0 G Proof. By Proposition 14.3.1, dim ExtkG(Y,V ) = [head(L ): V ] ≤ [k|B : V ] = 0.

1 By Corollary 14.3.2, it suﬃces to ﬁnd a bound on dim ExtkG(Y,V ) in the case that G V is a composition factor of k|B. Since (G, k) has property (P), we can assume that the G irreducible kG-module V (which is a composition factor of k|B) belongs to a Harish-Chandra J ∗ series of the form Irrk(G|(LJ ,Dσ )) for some J ∈ Pσ.

Theorem 14.3.3. Suppose that the pair (G, k) has property (P), and let Y and V be irreducible kG-modules such that Y ∈ Irrk(G|B) and V 6∈ Irrk(G|B). Assume that V is a G J composition factor of k|B and that V belongs to the Harish-Chandra series Irrk(G|(LJ ,Dσ )) ∗ 1 (J ∈ Pσ). Then, dim ExtkG(Y,V ) ≤ [W : WJ ], where WJ is the parabolic subgroup of W generated by J.

1 0 G Proof. By Proposition 14.3.1, dim ExtkG(Y,V ) = [head(L ): V ] ≤ [k|B : V ], so it G J suﬃces to prove that [k|B : V ] ≤ [W : WJ ]. Since V ∈ Irrk(G|(LJ ,Dσ )), V is in the head of the kG-module RG (DJ ). By deﬁnition, DJ is in the head of the r-modular LJ σ σ J J reduction Γσ of the Gelfand-Graev module Γσ for LJ , which means that there is a surjective J kL -module homomorphism Γ DJ . Since the Harish-Chandra induction functor RG J σ σ LJ J is exact, there is a surjective kG-module homomorphism RG (Γ ) RG (DJ ). So, since LJ σ LJ σ J V ⊆ head(RG (DJ )), it follows that V ⊆ headRG (Γ ). LJ σ LJ σ

J Let PV denote the projective indecomposable kG-module with head(PV ) = V . Since Γσ J is a projective kL -module and RG is exact, RG (Γ ) is a projective kG-module. So, since J LJ LJ σ

103 J J V ⊆ headRG (Γ ), P is a direct summand of RG (Γ ) and we have LJ σ V LJ σ

J [k|G : V ] = dim Hom (P , k|G) ≤ dim Hom (RG (Γ ), k|G). B kG V B kG LJ σ B

J dim Hom (RG (Γ ), k|G) = dim Hom (RG (KΓJ ),K|G) = dim Hom (RG (KΓJ ),RG(K)). kG LJ σ B KG LJ σ B KG LJ σ T (The ﬁrst equality above holds because RG (ΓJ ) is a projective OG-lattice.) LJ σ

We will now compute dim Hom (RG (KΓJ ),RG(K)). First, since the functor ∗RG is KG LJ σ T LJ right adjoint to RG , we have LJ

Hom (RG (KΓJ ),RG(K)) ∼ Hom (KΓJ , ∗RG RG(K)). KG LJ σ T = KLJ σ LJ T

J Let PJ be the standard parabolic subgroup of G containing LJ , and let W denote the set of J shortest right coset representatives of WJ in W . Then, W gives a full set of (PJ ,B)-double coset representatives in G and it follows by the Mackey decomposition that

w J ∗ G G ∼ J LJ ∗ T w Hom (KΓ , R R (K)) Hom (KΓ , ⊕ Rw Rw ( K)) KLJ σ LJ T = KLJ σ T ∩LJ T ∩LJ w∈J W w ∼ J LJ ∗ T w ⊕ Hom (KΓ ,Rw Rw ( K)). = KLJ σ T ∩LJ T ∩LJ w∈J W

Now, wK is the K(wT )-module with underlying vector space K and wT -action given by wtw−1.x = tx for all t ∈ T and x ∈ K. So, since K is the trivial one-dimensional KT - module, wK is the trivial one-dimensional K(wT )-module. Since wT = wT w−1 = T for ∗ wT ∗ T ∗ T w all w ∈ W , Rw = R = R is the identity functor on KT -mod, K = K, and T ∩LJ T ∩LJ T LJ LJ Rw = R for all w ∈ W . Therefore, continuing the calculation above, we have T ∩LJ T

w J LJ ∗ T w ∼ J LJ ⊕ Hom (KΓ ,Rw Rw ( K)) ⊕ Hom (KΓ ,R (K)), KLJ σ T ∩LJ T ∩LJ = KLJ σ T w∈J W w∈J W

LJ ∗ LJ and since RT is right adjoint to RT ,

J LJ ∼ ∗ LJ J ⊕ HomKLJ (KΓσ,RT (K)) = ⊕ HomKT ( RT (KΓσ),K). w∈J W w∈J W

∗ LJ J ∼ ∅ Now, since T = L∅, RT (KΓσ) = KΓσ by Rodier’s result on the restriction of Gelfand- Graev modules in characteristic 0. Since T has a trivial unipotent radical, the Gelfand-Graev ∅ module KΓσ for T is equal to the group algebra KT . Therefore,

∗ LJ J ∼ ⊕ HomKT ( RT (KΓσ),K) = ⊕ HomKT (KT,K). w∈J W w∈J W

104 K appears once as a direct summand of the completely reducible KT -module KT , so dim HomKT (KT,K) = 1 and, following our chain of calculations, we have

dim Hom (RG (KΓJ ),RG(K)) = |J W |. KG LJ σ T

J Thus, [k|G : V ] ≤ dim Hom (RG (Γ ), k|G) = dim Hom (RG (KΓJ ),RG(K)) = |J W | = B kG LJ σ B KG LJ σ T [W : WJ ], as needed.

The bound of Theorem 14.3.3 is particularly strong in the case that V is a cuspidal irreducible kG-module.

Corollary 14.3.4. Suppose that the pair (G, k) has property (P), and let V be a cuspidal 1 irreducible kG-module. Then, dim ExtkG(Y,V ) ≤ 1 for any irreducible kG-module Y ∈ Irrk(G|B).

G 1 Proof. If V is not a composition factor of k|B, then ExtkG(Y,V ) = 0 by Corollary 14.3.2. G Suppose now that V is a composition factor of k|B. Since V is a cuspidal kG-module and (G, k) has property (P), the Harish-Chandra series containing V is of the form {V } = S 1 Irrk(G|(G, Dσ )). So, by Theorem 14.3.3, dim ExtkG(Y,V ) ≤ [W : W ] = 1.

105 Part III Explicit Computations of Bounds on the Dimension of Ext1 Between Irreducible Modules for GLn(q) in Cross Characteristic

15 Introduction

Let G be a finite group of Lie type defined over a finite field Fq, where q is a power of a prime p > 0 (so, G is the fixed point subgroup of a connected reductive algebraic group ¯ G over Fp under an endomorphism F : G → G such that some power of F is a Frobenius morphism). Assume that the BN-pair of G is split, and let (W, S) be the Coxeter system corresponding to the BN-pair structure on G.

Let k be an algebraically closed field of characteristic r > 0, r 6= p, and suppose that the pair (G, k) satisfies property (P) (see Definition 14.2.4). Let Y and V be irreducible kG-modules with Y belonging to the unipotent principal series Irrk(G|B) and V belonging to a Harish-Chandra series Irrk(G|(LJ ,X)) (J ⊆ S) with Irrk(G|(LJ ,X)) 6= Irrk(G|B). In Chapter 14, we showed that

 1 G dim ExtkG(Y,V ) = 0 if [k|B : V ] = 0 1 |W | G dim ExtkG(Y,V ) ≤ if [k|B : V ] 6= 0 |WJ |

G G (where [k|B : V ] denotes the multiplicity of V as a composition factor of k|B). In G particular, if V is a composition factor of k|B and V belongs to the Harish-Chandra series 1 Irrk(G|(LJ ,X)), Irrk(G|(LJ ,X)) 6= Irrk(G|B), the bound on dim ExtkG(Y,V ) depends only on the Levi subgroup LJ (for, WJ is the Weyl group of LJ ).

If the irreducible kG-module V belongs to the Harish-Chandra series Irrk(G|(LJ ,X)), we will say that LJ is a Harish-Chandra vertex of V , and that X is a Harish-Chandra source of V (this terminology is used in [18] and [19]). The Harish-Chandra vertex of V is unique up to conjugation by elements of N = NG(T ). (If L is a Levi subgroup of a parabolic subgroup of G and X is a cuspidal kL-module, then the Harish-Chandra series Irrk(G|(L, X)) and n n Irrk(G|( L, X)) are equal for any n ∈ N. Thus, if L is a Harish-Chandra vertex of an irreducible kG-module V , then nL is as well.) Given this terminology, we may restate the G last assertion of the paragraph above as follows: if V is a composition factor of k|B and V belongs to the Harish-Chandra series Irrk(G|(LJ ,X)), Irrk(G|(LJ ,X)) 6= Irrk(G|B), then

106 1 the bound on dim ExtkG(Y,V ) found in Chapter 14 depends only on the Harish-Chandra vertex of V .

1 We will explicitly demonstrate the bounds on dim ExtkG(Y,V ) given in Chapter 14 in a series of examples. In these examples, we will work with the general linear group G = GLn(q) over the finite field Fq. By [29, 4.9], the pair (GLn(q), k) satisfies property (P), and consequently the bounds of Chapter 14 apply.

We will use the parameterization of kGLn(q)-modules given in [19, (4.2.11, (6))] (see Chapter 9). In this labeling, the complete set of irreducible kGLn(q)-modules is given by 2 ~ ~ {D (s, λ) | s ∈ Css,r0 , λ ` ~ms}. In the next section, we will present an algorithm of Dipper G and Du [19] which determines the Harish-Chandra vertex of any composition factor of k|B (where G = GLn(q)). Then, we will apply Dipper and Du’s algorithm to certain irreducible 1 kGLn(q)-modules in order to obtain explicit bounds on the dimension of Ext between these irreducibles.

16 The Harish-Chandra Vertex of the Module D2(1, λ)

In [19, Section 4.3], Dipper and Du present an algorithm which may be used to determine 2 a Harish-Chandra vertex of an irreducible kGLn(q)-module D (s, λ) for any s ∈ Css,r0 and multipartition λ ` ~ms. In our examples, we will work only with the irreducible 2 kGLn(q)-modules D (1, λ), where λ ` n. Therefore, we will outline Dipper and Du’s algorithm in the special case of s = 1.

2 Let λ ` n and let D (1, λ) be an irreducible kGLn(q)-module (as indexed in [19, (4.2.3 (11))]). A Harish-Chandra vertex of D2(1, λ) may be determined as follows.

0 0 0 0 2 0 0 Let λ = λ−1 + lλ0 + lrλ1 + lr λ2 + ··· be the l − r-adic decomposition of λ , meaning 0 0 18 0 that λ−1 ` n−1 is l-restricted and λa ` na is r-restricted for a ≥ 0. Since λ ` n, the integers na ∈ Z≥0 must satisfy the relation

X a n = n−1 + lr na. a≥0

18A partition µ ` n is called l-restricted if its dual µ0 is l-regular, meaning that every part of µ0 occurs less than l times.

107 If λ0 has the l−r-adic decomposition given above, a Harish-Chandra vertex of the irreducible 2 kGLn(q)-module D (1, λ) is the Levi subgroup

×n−1 ×n0 ×n1 ×n2 L = GL1(q) × GLl(q) × GLlr(q) × GLlr2 (q) × · · ·

of GLn(q). (Any conjugate of L by an element n ∈ N gives another Harish-Chandra vertex of D2(1, λ).)

Example 16.0.1. Let char(k) = r = 2 and let G = GL2(q). Assume that 2 6 | q (since [G : B] = q + 1 in this case, we must have 2 | [G : B]). The only partitions of 2 are (2) and (1, 1) = (12). Therefore, the irreducible kG-modules which appear as composition factors G 2 2 2 2 of k|B are D (1, (2)) and D (1, (1 )). Here, D (1, (2)) = k ∈ Irrk(G|B) = Irrk(G|(T, k)). Therefore, the Harish-Chandra vertex of D2(1, (2)) is the maximal torus T .

We will apply Dipper and Du’s algorithm to determine a Harish-Chandra vertex of the irreducible kG-module D2(1, (12)). Since r = 2 and 26 | q, |q (mod r)| = |q (mod 2)| = 1. Therefore, we set l = r = 2. To ﬁnd a Harish-Chandra vertex of D2(1, (12)), we must ﬁnd the 2 − 2-adic decomposition of (2).

The 2 − 2-adic decomposition of (2) is of the form

2 (2) = λ−1 + 2λ0 + 2(2)λ1 + 2(2) λ2 + ··· ,

where λ−1 ` n−1 is l = 2-restricted and λa ` na is r = 2-restricted for a ≥ 0. Since n = 2, the integers na ∈ Z≥0, a ≥ −1, must satisfy the equation

X a 2 = n−1 + 2(2 )na. a≥0 By inspection of this equation, only two cases are possible:

1. n−1 = 2, na = 0 for a ≥ 0, or

2. n−1 = 0, n0 = 1, na = 0 for a ≥ 1.

But, since (2)0 = (1, 1) is not l = 2-regular, (2) is not 2-restricted, which eliminates the ﬁrst possibility. Therefore, the second possibility must hold - that is, we must have λ−1 = (0), λ0 = (1), and λa = (0) for a ≥ 1, which means that (2) = (0) + 2(1) = 2(1) is the 2 − 2-adic decomposition of (2). Since λ0 = (1) ` 1, n0 = 1. And, since λa = (0) for all 2 2 a 6= 0, na = 0 for a 6= 0. Thus, the Harish-Chandra vertex of D (1, (1 )) is GL2(q) = G, which means that D2(1, (12)) is cuspidal.

Remark 16.0.2. It is possible to verify that D2(1, (12)) is cuspidal in the set-up of the previous example using Geck’s results on the structure of the Steinberg module Stk [29]. By assumption, 2 | [G : B] = q + 1. Thus, the Steinberg module Stk is not irreducible and head(Stk) is an irreducible kG-module which does not belong to the unipotent principal

108 G G series. So, since Stk ⊆ k|B and all of the composition factors of k|B are isomorphic to either 2 2 2 2 2 D (1, (2)) = k or D (1, (1 )), we must have head(Stk) = D (1, (1 )) ∈/ Irrk(G|B). Now, since G = GL2(q) is of rank 1, the only proper Levi subgroup of G is T , which means that every irreducible kG-module is either a principal series representation or cuspidal. But, the G only principal series irreducibles appearing as composition factors of k|B are those belonging 2 2 to Irrk(G|B). Thus, it follows that D (1, (1 )) must be cuspidal.

17 Some Examples of Bounds on the Dimension of Ext1

Let G = GLn(q) and let k be an algebraically closed ﬁeld of characteristic r > 0, r6 | q. By G [45, Theorem 2.1] and [19, (4.2.11, (6))], the irreducible constituents of k|B are of the form D2(1, λ), λ ` n. Thus, the bounding result of Chapter 14 may be stated as follows.

Theorem 17.0.1. Let λ ` n and assume that the irreducible kG-module D2(1, λ) belongs to the unipotent principal series Irrk(G|B).

1 2 2 1. If s ∈ Css,r0 with s 6= 1 and µ ` ~ms, then ExtkG(D (1, λ),D (s, µ)) = 0. 2 2. Let µ ` n and suppose that D (1, µ) has Harish-Chandra vertex LJ 6= T (∅ 6= J ⊆ S). 1 2 2 |W | 19 Then, dim ExtkG(D (1, λ),D (1, µ)) ≤ . |WJ | Note that the ﬁrst part of Theorem 17.0.1 is a basic consequence of the fact that D2(s, µ) and D2(1, λ) belong to diﬀerent blocks when s 6= 1 (see [19, (4.2.11)]).

In the remainder of this section, we will provide several explicit examples of the bound given in Theorem 17.0.1 (2).

Example 17.0.2. G = GL3(q), char(k) = r > 0, r6 | q

In this case, W = S3, the symmetric group on the set {1, 2, 3}, and S is the set {(1, 2), (2, 3)} of fundamental reﬂections in W . There are three partitions of 3: (3), (2, 1), 3 G and (1 ). Therefore, the irreducible kG-modules which occur as composition factors of k|B 2 2 2 3 2 2 are D (1, (3)), D (1, (2, 1)), and D (1, (1 )). Since D (1, (3)) = k, D (1, (3)) ∈ Irrk(G|B) for any r (r6 | q).

Let ( r if |q (mod r)| = 1 l = |q (mod r)| if |q (mod r)| > 1.

19In the case that λ = (n) and D2(1, λ) = k, this bound can be improved. As was shown by Guralnick 1 2 1 2 2 and Tiep, dim ExtkG(k, D (1, µ)) = dim H (G, D (1, µ)) ≤ 1 when D (1, µ) ∈/ Irrk(G|B).

109 By deﬁnition, l > 1. So, since (2, 1)0 = (2, 1), (2, 1) is both an l-regular and an l-restricted partition for any r, q with r6 | q. Thus, (2, 1) is its own l−r-adic decomposition for any choice ×3 of r (r 6 | q), and Dipper and Du’s algorithm yields GL1(q) = T as the Harish-Chandra vertex of D2(1, (2, 1)). Since the only principal series representations which occur as G 2 composition factors of k|B are those in Irrk(G|B), we must have D (1, (2, 1)) ∈ Irrk(G|B) for any r, r6 | q.

2 3 G Thus, D (1, (1 )) is the only composition factor of k|B whose Harish-Chandra vertex varies with r. We will compute the Harish-Chandra vertex of D2(1, (13)) and ﬁnd the corresponding Ext bounds in several cases.

1.GL 3(q), r = 26 | q

Since r = 2 6 | q, |q (mod 2)| = 1. Thus, we set l = r = 2. The 2 − 2-adic decomposition of (3) is (3) = (1) + 2(1). So, the Harish-Chandra vertex of D2(1, (13)) is L = GL1(q) × GL2(q). Now, L is conjugate to the standard Levi subgroup LJ(2,1) = GL2(q) × GL1(q) of GL3(q). The subset J(2, 1) of S is defined as follows. Let Y (2, 1) be the Young diagram of shape (2, 1). Fill in the boxes of the first row of this Young diagram with the integers 1, 2, and fill the box of the last row with the integer 3. The result of this process is a standard tableau t(2,1) of shape (2, 1). J(2, 1) is then the subset of S consisting of those elements s ∈ S which stabilize the rows of t(2,1).

2 3 Since L is conjugate to LJ(2,1), LJ(2,1) is another Harish-Chandra vertex of D (1, (1 )). The Weyl group WJ(2,1) of LJ(2,1) consists of those w ∈ W which stabilize the rows of (2,1) t . Thus, WJ(2,1) = Sym({1, 2}) × Sym({3}) and |WJ(2,1)| = 2! · 1! = 2. So, Theorem 17.0.1 (2) yields the following bounds:

1 2 3 1 2 3 |W | 3! dim ExtkG k, D (1, (1 )) = dim H G, D (1, (1 )) ≤ = = 3, and |WJ(2,1)| 2 1 2 2 3 dim ExtkG D (1, (2, 1)),D (1, (1 )) ≤ 3.

2.GL 3(q), r = 36 | q

In this case, |q (mod 3)| is equal to 1 or 2. Repeating the same analysis as above, we obtain the following results.

If |q (mod 3)| = 1, then l = 3 and the 3 − 3-adic decomposition of (3) is (3) = 3(1). 2 3 Thus, the Harish-Chandra vertex of D (1, (1 )) is GL3(q) = G, and it follows that D2(1, (13)) is cuspidal. Since G has Weyl group W , Theorem 17.0.1 (2) yields the following bounds:

110 W dim H1G, D2(1, (13)) ≤ = 1, and W 1 2 2 3 dim ExtkG D (1, (2, 1)),D (1, (1 )) ≤ 1.

If |q (mod 3)| = 2, then l = 2 and the 2−3-adic decomposition of (3) is (3) = (1)+2(1). 2 3 So, a Harish-Chandra vertex of D (1, (1 )) is GL2(q) × GL1(q), and Theorem 17.0.1 (2) yields the bounds: dim H1G, D2(1, (13)) ≤ 3, and 1 2 2 3 dim ExtkG D (1, (2, 1)),D (1, (1 )) ≤ 3.

Remark 17.0.3. There are inﬁnitely many choices of r for which the module D2(1, (13)) belongs to the unipotent principal series Irrk(G|B). By [31, Lemma 4.3.2], the permutation G module k|B is completely reducible when r 6 | [G : B]. Thus, when r 6 | [G : B], every G composition factor of k|B is a direct summand and therefore belongs to Irrk(G|B). In 2 3 particular, D (1, (1 )) ∈ Irrk(G|B) when r6 | [G : B].

Example 17.0.4. G = GL4(q), r = 26 | q

In this case, W = S4, and l = 2. The partitions of (4) are: (4), (3, 1), (2, 2), (2, 1, 1), 4 G 2 2 2 and (1 ). Thus, the composition factors of k|B are D (1, (4)), D (1, (3, 1)), D (1, (2, 2)), D2(1, (2, 1, 1)), and D2(1, (14)). The approach of the previous example yields the following results.

2 2 Irrk(G|B) = {D (1, (4)) = k, D (1, (3, 1))}

2 0 |W | λ ` n 2 − 2-adic decomposition of λ Harish-Chandra vertex LJ of D (1, λ ) |WJ | |WJ | (4) (4) = 4(1) G 4! 4!/4! = 1 ×2 (3, 1) (3, 1) = (1, 1) + 2(1) GL2(q) × GL1(q) 2 4!/2 = 12 ×2 (2, 2) (2, 2) = 2(1, 1) GL2(q) 4 4!/4 = 6

Thus, Theorem 17.0.1 (2) yields the bounds: dim H1G, D2(1, (14)) ≤ 1, dim H1G, D2(1, (2, 1, 1)) ≤ 12, dim H1G, D2(1, (2, 2)) ≤ 6, 1 2 2 4 dim ExtkG D (1, (3, 1)),D (1, (1 )) ≤ 1, 1 2 2 dim ExtkG D (1, (3, 1)),D (1, (2, 1, 1)) ≤ 12, and 1 2 2 dim ExtkG D (1, (3, 1)),D (1, (2, 2)) ≤ 6.

111 Part IV Generalizing the Results of Cline, Parshall, and Scott [9] to Calculate Ext Groups between Irreducible kGLn(q)-Modules

18 Self-Extensions of Irreducible Modules for GLn(q) in Certain Cross Characteristics

18.1 Introduction

Let k be an algebraically closed ﬁeld of characteristic r > 0 such that r 6 | q. For the remainder of this dissertation, we will work with right kG-modules. We will show that under 1 the additional assumption r 6 | (q − 1), ExtkG(Y,V ) (where Y is an irreducible kG-module belonging to the unipotent principal series Irrk(G|B) and V is an arbitrary irreducible kG-module) is isomorphic to an Ext1 group over a q-Schur algebra. As a consequence, we will show that there are no non-split self-extensions of irreducible kG-modules belonging to the unipotent principal series. In other words, we will show that given any irreducible 1 20 kG-module Y belonging to the unipotent principal series Irrk(G|B), ExtkG(Y,Y ) = 0. The proof of this result relies on the techniques and methods of Cline, Parshall, and Scott [9].

In [9], Cline, Parshall, and Scott prove that there is an ideal Jk E kG such there is a Morita equivalence

m(s) ¯ F : mod − kG/Jk → mod − ⊕ ⊗ Sqai(s) (ni(s), ni(s))k. s∈Css,r0 i=1 ¯ Since kG and kG/Jk have the same irreducible modules [9, 9.17], the Morita equivalence F yields an indexing of the irreducible kG-module. In this indexing, the full set of irreducible

20 1 Using the methods of Guralnick and Tiep, it is only possible to show that dim ExtkG(Y,Y ) ≤ |W | = n! (where W = Sn is the Weyl group of GLn(q)).

112 kG-modules is given by {D(s, λ) | s ∈ Css,r0 , λ ` n(s)} G (see Section 9.1). The irreducible constituents of the permutation module k|B are indexed by D(1, λ), λ ` n. If ( r if |q (mod r)| = 1 l = |q (mod r)| if |q (mod r)| > 1, then the irreducible kG-modules belonging to the unipotent principal series Irrk(G|B) are indexed by D(1, λ), where λ ` n is l-restricted.

1 18.2 ExtkG(D(1, λ),D(1, λ)) in the case that λ ` n is l-restricted.

+ The q-Schur algebra Sq(n, n)k has weight poset Λ (n) = {λ|λ ` n} (the poset structure on + Λ (n) is given by the dominance order). Therefore, the irreducible Sq(n, n)k-modules are indexed by partitions λ of n. Given a partition λ ` n, let Lk(λ) denote the corresponding ¯ k irreducible Sq(n, n)k-module. By construction, F (D(1, λ)) = L (λ) for any λ ` n.

1 1 In [9], CPS establish a connection between H -calculations for G = GLn(q) and Ext calculations for the q-Schur algebra Sq(n, n)k. By [9, Theorem 10.1], under the additional assumption that char(k) = r6 | (q − 1) (so that r6 | q(q − 1)), ( Ext1 (Lk((1n)),Lk(µ)) if s = 1 H1(G, D(s, µ)) ∼= Sq(n,n)k 0 if s 6= 1

n for any s ∈ Css,r0 and any multipartition µ ` n(s). Since k = D(1, (1 )), the result of [9, Theorem 10.1] may be restated as follows: if char(k) = r6 | q(q − 1), ( Ext1 (Lk((1n)),Lk(µ)) if s = 1 Ext1 (D(1, (1n)),D(s, µ)) ∼= Sq(n,n)k kG 0 if s 6= 1 for any s ∈ Css,r0 and any multipartition µ ` n(s). The key component of the proof of [9, Theorem 10.1] is the observation that k is in G the head of the permutation module k|B. But, k is not the only irreducible kG-module G contained in head(k|B); any irreducible kG-module of the form D(1, λ), where λ is an G l-restricted partition of n, is contained in head(k|B) (this follows since λ ` n is l-restricted if and only if D(1, λ) ∈ Irrk(G|B)). Hence, the proof of [9, Theorem 10.1] holds if we replace k = D(1, (1n)) by an irreducible kG-module D(1, λ) with λ ` n l-restricted.

Theorem 18.2.1. Suppose that r 6 | q(q − 1) and that λ is an l-restricted partition of n. Then, for any s ∈ Css,r0 and any multipartition µ ` n(s),

( Ext1 (Lk(λ),Lk(µ)) if s = 1 Ext1 (D(1, λ),D(s, µ)) ∼= Sq(n,n)k kG 0 if s 6= 1

113 Proof. (The following proof is due to CPS [9, 10.1] - no modiﬁcations are necessary when k is replaced by an irreducible kG-module D(1, λ) with λ ` n l-restricted.) Let s ∈ Css,r0 , let µ ` n(s), and let D(s, µ) be the corresponding irreducible kG-module.

G Since λ ` n is l-restricted, D(1, λ) ∈ Irrk(G|B), which means that D(1, λ) ⊆ head(k|B). Therefore, there exists a short exact sequence of kG-modules

G 0 → L → k|B → D(1, λ) → 0 (18.1) G n for some submodule L of k|B. Now, since r6 | q(q − 1), we have r6 | |T | = (q − 1) and r6 | |U|. Therefore, r6 | |B| = |T ||U|, which means that every kB-module is projective. In particular, the trivial kB-module k is projective. Since induction from B to G is exact, it follows that G 1 G k|B is a projective kG-module, so that ExtkG(k|B,D(s, µ)) = 0. Therefore, the short exact sequence (18.1) induces the exact sequence

G 1 HomkG(k|B,D(s, µ)) → HomkG(L ,D(s, µ)) → ExtkG(D(1, λ),D(s, µ)) → 0. (18.2) Now, by [9, Theorem 9.17], any irreducible kG-module D(s, τ)(τ ` n(s)) is also an ir- G reducible kG/Jk-module. By [9, Remark 9.18(c)], k|B is a kG/Jk-direct summand of a projective kG/Jk-module and thus is itself a projective kG/Jk-module. It follows that (18.1) G is a short exact sequence of kG/Jk-modules in which k|B is a projective kG/Jk-module. Therefore, (18.1) also induces the exact sequence Hom (k|G,D(s, µ)) → Hom ( ,D(s, µ)) → Ext1 (D(1, λ),D(s, µ)) → 0. kG/Jk B kG/Jk L kG/Jk (18.3) The exact sequences (18.2) and (18.3) give rise to a commutative diagram

G 1 HomkG(k|B,D(s, µ)) HomkG(L ,D(s, µ)) ExtkG(D(1, λ),D(s, µ)) 0

G Hom ( ,D(s, µ)) Ext1 (D(1, λ),D(s, µ)) HomkG/Jk (k|B,D(s, µ)) kG/Jk L kG/Jk 0

which has exact rows. The two left vertical arrows of the commutative diagram above are isomorphisms, and it follows that the third vertical arrow is also an isomorphism. Therefore, Ext1 (D(1, λ),D(s, µ)) ∼ Ext1 (D(1, λ),D(s, µ)). Since F¯(D(1, λ)) = Lk(λ), it follows kG = kG/Jk by [9, Theorem 9.17] that Ext1 (D(1, λ),D(s, µ)) ∼ Ext1 (Lk(λ), F¯(D(s, µ))).21 kG/Jk = Sq(n,n)k Therefore, Ext1 (D(1, λ),D(s, µ)) ∼ Ext1 (Lk(λ), F¯(D(s, µ))). (18.4) kG = Sq(n,n)k

m(s) 21 k The irreducible Sq(n, n)k-module L (λ) has the structure of a ⊕ ⊗ Sqai(s) (ni(s), ni(s))k- s∈Css,r0 i=1 m(s) module via the natural quotient map ⊕ ⊗ Sqai(s) (ni(s), ni(s))k → Sq(n, n)k (Sq(n, n)k is the s∈Css,r0 i=1 m(s) summand of ⊕ ⊗ Sqai(s) (ni(s), ni(s))k corresponding to s = 1). So, for s 6= 1, the tensor s∈Css,r0 i=1 m(s) k 1 ∼ product ⊗ Sqai(s) (ni(s), ni(s))k acts as 0 on L (λ), and it follows that ExtkG/J(q) (D(1, λ),D(s, µ)) = i=1 k Ext1 (Lk(λ), F¯(D(s, µ))) ∼ Ext1 (Lk(λ), F¯(D(s, µ))). m(s) = Sq (n,n)k ⊕ ⊗ S (n (s),n (s)) ai(s) i i k s∈ i=1 q Css,r0

114 Now, if s 6= 1, D(1, λ) and D(s, µ) belong to diﬀerent blocks, which means that

1 ExtkG(D(1, λ),D(s, µ)) = 0.

If s = 1, then µ is a partition of n, F¯(D(s, µ)) = F¯(D(1, µ)) = Lk(µ), and the isomorphism (18.4) gives Ext1 (D(1, λ),D(s, µ)) ∼ Ext1 (Lk(λ),Lk(µ)). kG = Sq(n,n)k

To prove that there are no non-split self-extensions of irreducible kG-modules belonging to the unipotent principal series Irrk(G|B), we will require the following result on self- extensions of irreducible modules for the q-Schur algebra Sq(n, n)k.

Proposition 18.2.2. If λ is a partition of n and Lk(λ) is the corresponding irreducible S (n, n) -module, then Ext1 (Lk(λ),Lk(λ)) = 0. q k Sq(n,n)k

Proof. Let ∆(λ) denote the standard module in mod-Sq(n, n)k corresponding to the weight λ. ∆(λ) has simple top Lk(λ), which means that there is a short exact sequence

0 → rad(∆(λ)) → ∆(λ) → Lk(λ) → 0.

This short exact sequence gives rise to the long exact sequence 0 → k k k k HomSq(n,n)k (L (λ),L (λ)) → HomSq(n,n)k (∆(λ),L (λ)) → HomSq(n,n)k (rad(∆(λ)),L (λ)) → Ext1 (Lk(λ),Lk(λ)) → Ext1 (∆(λ),Lk(λ)) → · · · . Sq(n,n)k Sq(n,n)k

By deﬁnition of ∆(λ), all composition factors Lk(µ) of rad(∆(λ)) satisfy µ < λ, k which means that HomSq(n,n)k (rad(∆(λ)),L (λ)) = 0. Also, by [14, Lemma C.12], Ext1 (∆(λ),Lk(λ)) = 0. So, by exactness of the sequence above, we have Sq(n,n)k Ext1 (Lk(λ),Lk(λ)) = 0. Sq(n,n)k

Corollary 18.2.3. If λ is an l-restricted partition of n (so that D(1, λ) ∈ Irrk(G|B)), then 1 ExtkG(D(1, λ),D(1, λ)) = 0.

Proof. By Theorem 18.2.1 and Proposition 18.2.2, we have

Ext1 (D(1, λ),D(1, λ)) ∼ Ext1 (Lk(λ),Lk(λ)) = 0. kG = Sq(n,n)k

115 19 Calculations of Higher Ext Groups Between Certain Modules for GLn(q) in Cross Characteristic

19.1 Introduction

Let k be an algebraically closed ﬁeld of characteristic r > 0 such that r6 | q. In [9], Cline, Parshall, and Scott prove that there is an ideal Jk E kG such there is a Morita equivalence

m(s) ¯ F : mod − kG/Jk → mod − ⊕ ⊗ Sqai(s) (ni(s), ni(s))k. s∈Css,r0 i=1

i i CPS [9] use this Morita equivalence to relate H -calculations for GLn(q) to Ext -calculations for a q-Schur algebra. By [9, Theorem 10.1], if r6 | q(q − 1) and V is a right kG-module with Jk ⊆ AnnkG(V ), H1(G, V ) ∼ Ext1 (Lk((1n)), F¯(V )) = Sq(n,n)k k n n (here, L ((1 )) is the irreducible Sq(n, n)k-module corresponding to the partition (1 )). By m+1 [9, Theorem 12.4], if r6 | q Q (qj − 1) for some integer m ≥ 0 and V is a right kG-module j=1 with Jk ⊆ AnnkG(V ), Hi(G, V ) ∼ Exti (Lk((1n)), F¯(V )) = Sq(n,n)k for 0 ≤ i ≤ m + 1.

The result of [9, Theorem 10.1] was generalized in the previous chapter. The purpose of this chapter is to obtain a generalization of the result of [9, Theorem 12.4].

19.2 A Generalization of [9, Theorem 12.4]

Going forward, we will assume that r6 | q(q − 1). When r6 | q(q − 1), r6 | |B| (for, |B| = |T ||U|, where |T | is a power of (q − 1) and |U| is a power of p). Thus, k is a projective right G kB-module, which means that k|B is a projective right kG-module.

Let Sn be the symmetric group on n letters, and let S be the generating set of fundamental reﬂections in Sn. Let He = He(Sn,S) denote the generic Hecke algebra over −1 the Laurent polynomial ring Z = Z[t, t ] corresponding to the pair (Sn,S). Let HeO denote

116 the O-algebra obtained by base change to O, and let Hek denote the k-algebra obtained by base change to k. As in [44], we will denote Hek by H. (For consistency, we will also denote the q-Schur algebra Sq(n, n)k by Sq(n, n).)

∼ G G Since H = EndkG(kB ) and k|B is projective, the functor H1 : mod − kG → −mod − H G of [19, Section 4.1] is given by H1 = HomkG(k|B, −). Thus, we will adopt the notation of Geck and Jacon [31] and denote the functor H1 by Fk. The functor Fk has a right inverse G Gk : mod − H → mod − kG, which is given by Gk(E) = E ⊗H k|B for any right H-module G E (here, the right kG-module k|B is naturally a left module for the endomorphism algebra G ˆ G H = EndkG(k|B)). In the notation of [19], Gk = H1. Since k|B is projective, Gk is a two- sided inverse of of Fk on the full subcategory of mod − kG consisting of all V ∈ mod − kG such that

1. every non-zero submodule of V has a composition factor in Irrk(G|B), and

2. every non-zero quotient module of V has a composition factor in Irrk(G|B) [29, 4.1.14].

Lemma 19.2.1. If V is a kG-module in the image of the functor Gk : mod−H → mod−kG, then V is annihilated by Jk.

G ∼ Proof. Let E be a right H-module such that Gk(E) = E ⊗kG k|B = V . Given α ∈ kG and G G e ⊗ x ∈ E ⊗kG k|B (where e ∈ E and x ∈ k|B), we have (e ⊗ x).α = e ⊗ (x.α). But, since G G k|B is a direct summand of the right kG/Jk-module Mc1,G,k, k|B is annihilated by Jk. Thus, G e ⊗ (x.α) = e ⊗ 0 = 0. It follows that E ⊗kG k|B is annihilated by Jk, which means that the same statement holds for V .

Before we proceed, we must deﬁne a certain Sq(n, n) − H bimodule which is used in [44] to link the representation theory of H to the representation theory of Sq(n, n). As in −1 Chapter 8, let V be a free Z = Z[t, t ]-module of rank n. Given λ ` n, let kλ ∈ Z be such ⊗n ∼ kλ that V = ⊕ (xλHe) (see (8.1)). We deﬁne λ∈Λ+(n)

kλ Te = ⊕ (xλHe) . λ∈Λ+(n)

T is a right H-module by deﬁnition; and, since S (n, n) = End (V ⊗n) ∼ End (T ), there is e e t He = He e a natural left action of St(n, n) on Te. Let T = Tek be the Sq(n, n) − H bimodule obtained by reduction of Te to k.

G G G Since k|B is a right kG-module, k|B is a left module for H = EndkG(k|B), which means G ∗ that the dual (k|B) has the structure of a right H-module.

117 G ∗ Lemma 19.2.2. Suppose that r6 | q(q − 1), and let Mc1,G,k ⊗kG (k|B) be the right H-module G ∗ 22 G ∗ ∼ Φ deﬁned via the right action of H on (k|B) . There is an isomorphism Mc1,G,k⊗kG(k|B) = T of right H-modules (where T Φ denotes the right H-module obtained by twisting the action of H on T by the involution Φ of H deﬁned in [44, (2.0.3)]).

G p G ∼ 23 By [21, Lemmas 2.22, 2.15 (ii) and 2.8], HomOG(O|B, yλO|B) = yλHO. Thus,

G ∼ G ∼ kλ ∼ kλ ∼ Φ ∼ Φ HomkG(k|B, Mc1,G,k) = HomOG(O|B, Mc1,G,O)k = ⊕ (yλHO)k = ( ⊕ (yλHO) )k = Tek = T , λ`n λ`n

kλ ∼ Φ where the isomorphism ⊕ (yλHO) = Te follows by [25, Lemma 1.1 (c)]. λ`n

Before we proceed to the next result, we must compare the indexing of Specht modules for the Hecke algebra used by Du, Parshall, and Scott [25] and Parshall and Scott [44] with that used by Dipper and James [21]. In the work of DPS and PS, the Specht module in mod − HeO corresponding to a partition λ ` n is denoted by Seλ. The deﬁnition of Seλ can be found in [44, Section 2.2]. Given a partition λ ` n, Hom (y 0 H , x H ) ∼= O. HeO λ eO λ eO \ Therefore, there exist indecomposable summands Yeλ of yλ0 HeO and Yeλ of xλHeO such that Hom (Y \, Y ) ∼= O. (For λ ` n, Y \ is called a twisted Young module, and Y HeO eλ eλ eλ eλ is called a Young module.) Let ζ : Y \ → Y be a generator of Hom (Y \, Y ). Then, λ eλ eλ HeO eλ eλ ˜ the Specht module Sλ corresponding to λ is deﬁned to be the image of the homomorphism ζλ.

In the work of Dipper and James [20], [21] (and, in the work of Dipper and Du [19]), the Specht module corresponding to the partition λ ` n is denoted by Seλ and deﬁned as λ 0 the right ideal Se = xλτw0λ yλ HeO of HeO, where w0λ is the longest element of the parabolic subgroup Wλ of W [20, Section 4].

22 See (9.1) for the deﬁnition of Mc1,G,k. 23 G In the notation of Dipper and James, O|B = M and the lemmas of [21] apply as follows. Let λ ` n. q √ G ∼ By [21, Lemma 2.22], yλO|B = yλM = Myλ = {u ∈ M | l(yλHO)u = 0}, where l(yλHO) is the left G annihilator of yλHO in HO. By [21, Lemma 2.15 (ii)], HomOG(O|B,Myλ ) = rl(yλHO), where rl(yλHO) denotes the right annihilator of l(yλHO) in HO. And, by [21, Lemma 2.8], rl(yλHO) = yλHO. Combining q G G ∼ these results, we see that HomOG(OB , yλO|B) = yλHO.

118 Lemma 19.2.3. The DPS labeling of the Specht modules for HO is consistent with the DJ ∼ λ labeling. That is, for any λ ` n, Seλ = Se as right HeO-modules.

λ Proof. By [20, Corollary 4.2], Se = xλHeOyλ0 HeO. By parts (h) and (f) of [25, Lemma 1.1], S (which is deﬁned as the image of a generator of Hom (y 0 H , x H )) is isomorphic to eλ HeO λ eO λ eO xλHeOyλ0 HeO as well.

The PS indexing of the irreducible H-modules is also consistent with the DJ indexing. Let Sλ = Seλk denote the Specht module for H obtained by base change to k. As before, let ( r if |q (mod r)| = 1 l = |q (mod r)| if |q (mod r)| > 1.

When λ is an l-regular partition of n, Dλ := Sλ/rad(Sλ) is an irreducible H-module. A full set of irreducible H-modules is given by {Dλ | λ is l − regular}. In the work of Dipper and James, the irreducible H-module corresponding to an l-regular partition λ of n λ λ ∼ is denoted by D . But, by Lemma 19.2.3, we have D = Dλ for all l-regular partitions λ of n.

As in Section 9.2, given a partition µ of n, let S(1, µ) denote the indecomposable right kG-module deﬁned by Dipper and James with head(S(1, µ)) = D1(1, µ) (where D1(1, µ) is indexed according to [19, (4.2.3)]). When λ ` n is l-restricted, the right kG-module D(1, λ) (in the CPS indexing) is isomorphic to D1(1, λ0) (see Section 9.4). Thus, when λ ` n is l-restricted, the indecomposable right kG-module S(1, λ0) of Dipper and James has the irreducible D(1, λ) (in the CPS indexing) in its head. We can now prove the following proposition, which is key to our generalization of [9, Theorem 12.4].

Proposition 19.2.4. If l > 2, λ is an l-restricted partition of n, and r 6 | q(q − 1), then F¯(S(1, λ0)) ∼= ∆(λ) (where ∆(λ) is the standard object corresponding to λ in the category of right Sq(n, n)-modules).

Proof. By [21, (3.1)], any composition factor of S(1, λ0) is of the form D1(1, µ) for some 1 partition µ ` n. By [19, (4.2.3), (4)], D (1, µ) belongs to B1,G for any µ ` n (where B1,G is the sum of the unipotent blocks of G). Now, as shown in the proof of [9, Theorem 9.17],

all composition factors of the head of the kG/Jk-module Mcs,G,k (where s ∈ Css,r0 ) belong to Bs,G. Therefore, when s 6= 1, the head of Mcs,G,k has no irreducible constituents belonging to B1,G. In particular, when s 6= 1, the head of Mcs,G,k has no irreducible constituents of 1 the form D (1, µ), µ ` n. So, viewing Mcs,G,k as a kG-module via the natural quotient map 0 kG kG/Jk, we have HomkG(Mcs,G,k,S(1, λ )) = 0 when s 6= 1. G Now, since r6 | q(q − 1), k|B is a projective right kG-module. Thus, by [19, (4.2.3, (1))] 0 ∼ 0 ∼ 0 and [29, (4.1.4)], S(1, λ ) = Gk(Fk(S(1, λ ))) = Gk(Sλ0 ). Since S(1, λ ) is in the image of 0 Gk, S(1, λ ) is a kG/Jk-module by Lemma 19.2.1, which means that we can apply the CPS functor F¯ to S(1, λ0). Using the arguments of the ﬁrst paragraph, we can compute

119 ¯ 0 0 F (S(1, λ )) = HomkG/Jk ( ⊕ Mcs,G,k,S(1, λ )) s∈Css,r0 ∼ 0 = HomkG( ⊕ Mcs,G,k,S(1, λ )) s∈Css,r0 ∼ 0 = HomkG(Mc1,G,k,S(1, λ )),

0 where HomkG(Mc1,G,k,S(1, λ )) is viewed as a right Sq(n, n)k-module via the natural left ∼ action of Sq(n, n)k = EndkG(Mc1,G,k) on the right kG-module Mc1,G,k.

0 ∼ ∼ G As shown in the previous paragraph, S(1, λ ) = Gk(Sλ0 ) = Sλ0 ⊗H k|B (here, we have used Lemma 19.2.3 to identify the Dipper-James Specht module Sλ0 with the CPS Specht G ∼ G ∗ module Sλ0 ). We claim that there is an isomorphism Sλ0 ⊗H k|B = HomH ((k|B) ,Sλ0 ) of right G ∗ kG-modules. (The right action of kG on HomH ((k|B) ,Sλ0 ) is given by (φ.α)(f) = φ(α.f) G G ∗ G ∗ for any α ∈ k|B, φ ∈ HomH ((k|B) ,Sλ0 ), and f ∈ (k|B) .) It is known that there is a vector G G ∗ space isomorphism Ω : Sλ0 ⊗H k|B → HomH ((k|B) ,Sλ0 ), given by Ω(s ⊗ x) = (f 7→ f(x)s) G for any s ∈ Sλ0 and x ∈ k|B. So, to prove the claim, it suﬃces to show that Ω respects G G ∗ the right action of kG. But, given any α ∈ kG, s ∈ Sλ0 , x ∈ k|B, and f ∈ (k|B) , Ω((s ⊗ x).α)(f) = Ω(s ⊗ xα)(f) = f(xα)s = (α.f)(x)s = Ω(s ⊗ x)(α.f) = (Ω(s ⊗ x).α)(f). Thus, Ω is a right kG-module isomorphism and the claim follows.

G ∼ G ∗ Since Sλ0 ⊗H k|B = HomH ((k|B) ,Sλ0 ) (as right kG-modules),

0 ∼ G ∗ ∼ G ∗ HomkG(Mc1,G,k,S(1, λ )) = HomkG(Mc1,G,k, HomH ((k|B) ,Sλ0 )) = HomH (Mc1,G,k⊗kG(k|B) ,Sλ0 )

as right Sq(n, n)-modules (the third isomorphism in the chain of isomorphisms above follows by tensor-hom adjunction, which preserves the right Sq(n, n)-module structure of G ∗ HomkG(Mc1,G,k, HomH ((k|B) ,Sλ0 )).

G ∗ ∼ Φ By Lemma 19.2.2, Mc1,G,k ⊗kG (k|B) = T as right H-modules. Thus, tracing through the calculations above, we have

¯ 0 ∼ Φ F (S(1, λ )) = HomH (T ,Sλ0 ),

Φ with the right action of Sq(n, n) on HomH (T ,Sλ0 ) defined via the left action of Sq(n, n) on Φ T . Now, it follows from the proof of [25, Theorem 7.7] that the right Sq(n, n)O-module Φ left β left 0 e HomHO (Te , Seλ ) identifies with ∆e (λ) , where ∆e (λ) is the standard object correspond- left βe ing to the partition λ in the category of left Sq(n, n)O-modules and ∆e (λ) is the right left Sq(n, n)O-module obtained by converting the left action of Sq(n, n)O on ∆e (λ) to a right action via the anti-automorphism β˜ defined in [25, Lemma 2.2]. But, since

left ∗βe left DS (n,n) left (∆e (λ)) = (∆e (λ)) q O ∼= ∇e (λ)

120 left βe ∼ left ∗ (where DSq(n,n) is the duality on mod − Sq(n, n)O), we have ∆e (λ) = ∇e (λ) and

Φ left β left ∗ 0 ∼ e ∼ HomHO (Te , Seλ ) = ∆e (λ) = ∇e (λ) .

Since ∇e left(λ)∗ ∼= ∆(e λ) (where ∆(e λ) is the standard object corresponding to λ in the category of right Sq(n, n)O-modules), it follows that

Φ 0 ∼ HomHO (Te , Seλ ) = ∆(e λ). Finally, when l > 2, an argument analogous to that given in the second part of [44, Lemma Φ 0 ∼ 2.4] shows that the isomorphism HomHO (Te , Seλ ) = ∆(e λ) holds upon base change to k. Therefore, when l > 2, ¯ 0 ∼ Φ ∼ F (S(1, λ )) = HomH (T ,Sλ0 ) = ∆(λ).

In order to generalize [9, Theorem 12.4], we must construct a suitable resolution of 0 S(1, λ ) when λ is l-restricted. To construct this resolution, we will use the functors Ni : mod − He → mod − He defined in [44, Section 3.1]. As above, let S denote the set of fundamental reflections in the symmetric group Sn. Since |S| = n − 1, the functor Ni is defined for 0 ≤ i ≤ n − 1 by

Y ⊗ He He Ni = Ind ◦ Res : mod − He → mod − H.e HeJ HeJ J⊆S,|J|=i Lemma 19.2.5. Suppose that r 6 | q(q − 1) and λ is an l-restricted partition of n. Then, there exists an exact sequence of right kG-modules of the form 0 → Mn−1 → · · · → · · · M1 → 0 M0 → S(1, λ ) → 0 in which every module is annihilated by Jk and Mi is projective as both a kG and a kG/Jk-module for 0 ≤ i ≤ l − 2.

Proof. By [44, Theorem 3.4], there exists an exact sequence of right HeO-modules of the form

Φ 0 → Sfλ → N0(Sfλ) → N1(Sfλ) → · · · → Nn−1(Sfλ) → 0. By [44, Remark 3.5], this sequence remains exact upon base change to k, yielding the exact sequence Φ 0 → Sλ → N0(Sλ) → N1(Sλ) → · · · → Nn−1(Sλ) → 0

of right H-modules. Now, by [44, Remark 3.10], Ni(Sλ) is a projective right H-module for i ≤ l − 2. Applying the contravariant duality functor DH : mod − H − mod − H to the exact sequence above, we obtain the exact sequence

DH DH DH Φ DH 0 → Nn−1(Sλ) → · · · → N1(Sλ) → N0(Sλ) → (Sλ ) → 0

DH Φ DH ∼ of right H-modules, in which Ni(Sλ) is projective for i ≤ l − 2. Since (Sλ ) = Sλ0 , the exact sequence above can be re-written as

DH DH DH 0 → Nn−1(Sλ) → · · · → N1(Sλ) → N0(Sλ) → Sλ0 → 0.

121 G G Now, since r6 | q(q−1), k|B is a projective right kG-module and the functor Gk(−) = −⊗H k|B ∼ is exact. Applying Gk to the exact sequence above and using the isomorphism Gk(Sλ0 ) = S(1, λ0), we obtain the exact sequence

DH DH DH 0 0 → Gk(Nn−1(Sλ) ) → · · · → Gk(N1(Sλ) ) → Gk(N0(Sλ) ) → S(1, λ ) → 0 (19.1)

of right kG-modules.

By Lemma 19.2.1, each of the right kG-modules in the exact sequence (19.1) is annihilated by Jk, which means that (19.1) is also an exact sequence of right kG/Jk-modules. DH DH For 0 ≤ i ≤ n − 1, let Mi = Gk(Ni(Sλ) ). Since Ni(Sλ) is a projective right H-module for i ≤ l − 2 and Gk is exact, Mi is a projective right kG-module for i ≤ l − 2. So, since every projective kG-module which is annihilated by Jk is also a projective kG/Jk-module, it follows that Mi is a projective kG/Jk-module for i ≤ l − 2.

We are now ready to prove our generalization of [9, Theorem 12.4].

Theorem 19.2.6. Suppose that r6 | q(q−1), l > 2, and λ is an l-restricted partition of n. If V i 0 ∼ i ¯ is a right kG-module with Jk ⊆ AnnkG(V ), then ExtkG(S(1, λ ),V ) = ExtSq(n,n)(∆(λ), F (V )) for 0 ≤ i ≤ l − 1.

Proof. Given the results of Proposition 19.2.4 and Lemma 19.2.5, the proof of [9, Theorem 12.4] goes through with virtually no change.

0 Let 0 → Mn−1 → · · · → · · · M1 → M0 → S(1, λ ) → 0 be the exact sequence obtained in Lemma 19.2.5. This sequence is exact in both the category of right kG-modules and the category of right kG/Jk-modules. For 0 ≤ i ≤ l−2, Mi is projective for both kG and kG/Jk. Let R = kG or kG/Jk, and let C•• denote the double complex obtained by applying the functor HomR(−,V ) to a Cartan-Eilenberg resolution of the complex M•. Filtering C•• by columns Ci• leads to the spectral sequence

t i+t 0 ExtR(Mi,V ) ⇒ ExtR (S(1, λ ),V ).

t But, Mi is projective for 0 ≤ i ≤ l − 2, so ExtR(Mi,V ) = 0 for t > 0 and 0 ≤ i ≤ l − 2. Since

HomkG(−, −) and HomkG/Jk (−, −) are equivalent bifunctors on mod − kG/Jk, we have

Exti (S(1, λ0),V ) ∼ Exti (S(1, λ0),V ) for 0 ≤ i ≤ l − 1. (19.2) kG = kG/Jk

Finally, since F¯(S(1, λ0)) ∼= ∆(λ) when l > 2 (by Proposition 19.2.4) and F¯ is a Morita equivalence, we have Exti (S(1, λ0),V ) ∼ Exti (∆(λ), F¯(V )) for all i. Com- kG/Jk = Sq(n,n) i 0 ∼ bining this isomorphism with the isomorphism of (19.2), we have ExtkG(S(1, λ ),V ) = i ¯ ExtSq(n,n)(∆(λ), F (V )) for 0 ≤ i ≤ l − 1.

122 Remark 19.2.7. When λ = (1n), λ0 = (n) and S(1, λ0) = S(1, (n)) = D1(1, (n)) = n i ∼ i ∼ D(1, (1 )) = k. So, in this case, Theorem 19.2.6 yields H (G, V ) = ExtkG(k, V ) = i k n ¯ ExtSq(n,n)(L ((1 )), F (V )) for 0 ≤ i ≤ l − 1.

Theorem 19.2.6 allows us to use known Ext results in the category of modules for the q-Schur algebra to obtain new Ext results in the category of right kG-module. For example, we can use the fact that mod − Sq(n, n) is a highest weight category to prove the following corollary.

Corollary 19.2.8. Suppose that r6 | q(q − 1), l > 2, and λ is an l-restricted partition of n.

i 0 (a) If µ ` n is such that µ E λ, then ExtkG(S(1, λ ),D(1, µ)) = 0 for 1 ≤ i ≤ l − 1. i 0 0 (b) If µ ` n is l-restricted and µ E λ, then ExtkG(S(1, λ ),S(1, µ )) = 0 for 1 ≤ i ≤ l − 1.

Proof. (a) By Theorem 19.2.6,

i 0 ∼ i ¯ i k ExtkG(S(1, λ ),D(1, µ)) = ExtSq(n,n)(∆(λ), F (D(1, µ))) = ExtSq(n,n)(∆(λ),L (µ))

for 1 ≤ i ≤ l − 1. Since µ λ, Exti (∆(λ),Lk(µ)) = 0 for 1 ≤ i ≤ l − 1 by [14, E Sq(n,n) Proposition C.13 (2)].

(b) By Proposition 19.2.4, F¯(S(1, µ0)) ∼= ∆(µ). Thus, Theorem 19.2.6 yields Exti (S(1, λ0),S(1, µ0)) ∼ Exti (∆(λ), ∆(µ)) for 1 ≤ i ≤ l − 1. Since µ λ, kG = Sq(n,n) E i ExtSq(n,n)(∆(λ), ∆(µ)) = 0 for 1 ≤ i ≤ l − 1 by [14, Proposition C.13 (2)].

123 Part V Conclusion

Let q be a power of a prime number p, let G = G(q) be a finite group of Lie type with a split BN-pair, and let k be an algebraically closed field of characteristic r > 0 with r 6= p. The aim of this dissertation was to study Ext groups between irreducible kG-modules. In Part 1 II we generalized the work of Guralnick and Tiep [35] to find bounds on dim ExtkG(Y,V ) in the following cases:

(1) Y,V ∈ Irrk(G|B) (Chapter 11), 0 (2) Y ∈ Irrk(G|(T,X)), V ∈ Irrk(G|(T,X )) (Chapter 13), and

(3)( G, k) has property (P), Y ∈ Irrk(G|B), V 6∈ Irrk(G|B) (Chapter 14). In Part IV, we generalized the work of Cline, Parshall, and Scott [9] to compute Ext groups between irreducible modules for GLn(q) in cross characteristic. In Chapter 18, we 1 1 related Ext groups for GLn(q) to Ext groups over a q-Schur algebra, and in Chapter 19, we related higher Ext groups for GLn(q) to higher Ext groups over a q-Schur algebra. (Both of the generalizations were done under the assumption that char(k) = r6 | q(q−1).)

We have made progress toward a more complete understanding of Ext groups for kG in non-defining characteristic, but there are still many directions for future research. First, 1 there are several other cases to consider when computing bounds on dim ExtkG(Y,V ) (where Y , V are irreducible kG-modules.) In this dissertation, we have assumed that Y ∈ Irrk(G|(T,X)) is a principal series representation. But, what if Y belongs to a Harish-Chandra series of the form Irrk(G|(L, X)) where T ( L? A natural question to 1 ask is whether we can find bounds on dim ExtkG(Y,V ) analogous to those of Chapters 13 and 14 when Y is not a principal series representation. Another question to consider when generalizing the results of Chapter 14 is whether we can drop some of the additional assumptions on G. Specifically, is there a bound analogous to that found in Chapter 14 when the pair (G, k) does not satisfy property (P)?

Another possibility for future research is to extend the bounds on the dimension of Ext1 given in this dissertation to a wider range of groups. In Part II, we worked only with finite groups of Lie type which have a split BN-pair. (We chose to focus on the split case in order to take advantage of the results on modular Harish-Chandra theory given in [31, Section 4] and [29].) However, Guralnick and Tiep’s bounds on the dimension of H1(G, V ) (where V is an irreducible kG-module) apply both to split and non-split finite groups of Lie type [35]. This suggests that there is likely a way to extend our bounds on the dimension of Ext1 to include the non-split finite groups of Lie type.

There are also many ways in which we could generalize and extend the results of Part

124 IV of this dissertation. We have related certain Ext groups over kGLn(q) to Ext groups over a q-Schur algebra and used homological properties of the q-Schur algebra to obtain Ext vanishing results for kGLn(q) (Corollaries 18.2.3 and 19.2.8). However, we have not yet used the fact that Ext calculations over the q-Schur algebra translate to Ext calculations in a category of integrable modules over a quantum group. Perhaps, we can ﬁnd bounds on the dimension of Ext in the category of integrable modules for the quantum group, which could in turn yield bounds on the dimension of Ext over kGLn(q). In particular, it would be interesting to use the quantum group to search for lower bounds on the dimension of Ext over kGLn(q). Combining such lower bounds on the dimension of Ext with the upper bounds found in this dissertation could yield the exact dimension of certain Ext groups over kGLn(q). (Or, at the very least, we would know that the dimensions of certain Ext groups lie within a range of possible values.)

We could also try to generalize the result of Theorem 19.2.6 to even higher Ext groups. Using the machinery developed by Parshall and Scott in [44], we were able calculate Exti only for i ≤ l − 1 (where l = |q mod(r)| if |q mod(r)| > 1 and l = r if |q mod(r)| = 1). However, it may be possible to generalize Parshall and Scott’s work in [44] and construct resolutions which would allow us to compute Exti for i > l − 1. Additionally, much of Parshall and Scott’s work in [44] is done under the assumption that l > 2. Perhaps, we can ﬁnd analogous results in the case that l = 2 and thus obtain Ext calculations when char(k) = 2. Once this is done, we can work to generalize our Ext calculations to an even wider range of characteristics. As in [9], we assumed that char(k) = r 6 | q(q − 1). G The assumption that r 6 | (q − 1) is needed to ensure that the permutation module k|B is projective. However, it may be possible to ﬁnd a projective module (or a collection of G projective modules) which could be used in place of k|B, allowing us to drop the assumption that r6 | (q − 1).

Finally, it may be possible to extend the Ext calculations of Part IV of this dissertation to more general finite groups of Lie type. Since the q-Schur algebra is defined in Type A, Cline, Parshall, and Scott work only with the finite general linear group GLn(q) in [9]. Therefore, our Ext calculations in Chapters 18 and 19 (which generalize Cline, Parshall, and Scott’s work in [9]) apply only in Type A. However, using recent work of Du, Parshall, and Scott [26], it may be possible to generalize our Ext calculations to other types. In [26], Du, Parshall, and Scott construct an analog of the q-Schur algebra outside of Type A. Replacing the q-Schur algebra with the new endomorphism algebra of [26] may yield cross characteristic Ext calculations for finite groups of Lie type other than GLn(q).

125 Part VI Appendices

20 Cohomology of Cyclic r-Groups

Proposition 20.0.1. Let k be a ﬁeld of characteristic r > 0, and let G = hgi be any cyclic r-group, with |g| = rl for some l ≥ 1. Then, Hn(G, k) ∼= k for n ≥ 0 (where k is viewed as a G-module with trivial action).

Proof. (Following [28]) We begin by constructing a projective resolution for k as a kG- rl−1 rl−1 P i P l module. Let aug : kG → k, aig 7→ ai (with ai ∈ k for 0 ≤ i ≤ r − 1), be the i=0 i=0 augmentation map. Then, the augmentation ideal Ker(aug) is generated by the element g − 1 ∈ kG. Now, let T = 1 + g + g2 + ··· + grl−1 ∈ kG. We claim that T = (g − 1)rl−1. To show that this is true, we consider the polynomial ring k[t] in an indeterminate t. The polynomial trl − 1 has factorization trl − 1 = (t − 1)(1 + t + · + trl−1). On the other hand, since char k = r, trl − 1 = (t − 1)rl . Therefore, (t − 1)rl = (t − 1)(1 + t + · + trl−1), from which it follows that (t − 1)rl−1 = 1 + t + · + trl−1 by the uniqueness of factorization in k[t]. Since kG ∼= k[t]/(trl − 1), we have T = (g − 1)rl−1 in kG.

Now, we consider the following sequence of kG-modules:

(g−1) (g−1) aug ··· → kG →T kG → kG → k → 0,

(g−1) where kG → kG is the map given by multiplication by (g − 1), and kG →T kG is the map given by multiplication by T . For convenience, we will replace the module kG in the nth position of the sequence above by the isomorphic module kGxn, the free kG-module on one generator xn. With this notation, the sequence above can be written as

(g−1) T (g−1) aug ··· → kGx2 → kGx1 → kGx0 → k → 0. (20.1)

Note that every one of the modules kGxi in the sequence (20.1) is a projective (in fact, free) kG-module. So, to prove that (20.1) is a projective resolution of k, we need only show that it is exact. Exactness at kGx0 follows because Ker(aug) = (g − 1)kG. To prove exactness at all of the other terms in the sequence, it suﬃces to to show that Im(g − 1) = Ker(T ) and Im(T ) = Ker(g − 1).

First, note that T (g −1) = grl −1 = 0. So, Im(g −1) ⊆ Ker(T ) and Im(T ) ⊆ Ker(g −1). It remains to check that Ker(T ) ⊆ Im(g − 1) and Ker(g − 1) ⊆ Im(T ). To show that the

126 rl−1 P j ﬁrst containment holds, let α = ajg ∈ Ker(T ) (where aj ∈ k for all j). Then, j=0

rl−1 rl−1 rl−1 rl−1 rl−1 rl−1 rl−1 rl−1 rl−1 X i X j X j X i X X i+j X X i X 0 = T α = ( g )( ajg ) = ajg ( g ) = aj( g ) = aj( g ) = ( aj)T, i=0 j=0 j=0 i=0 j=0 i=0 j=0 i=0 j=0 where the next-to-last equality follows because grl = 1, which means that gi+j = gi+j (modrl). rl−1 rl−1 P P Since 0 = ( aj)T , we must have aj = 0, which means that α ∈ Ker(aug) = Im(g − 1). j=0 j=0

rl−1 P i It remains to prove that Ker(g − 1) ⊆ Im(T ). Suppose α = aig ∈ Ker(g − 1). Then, i=0

rl−1 X i 0 = (g − 1)α = (g − 1) aig i=0 rl−1 X i+1 i = ai(g − g ) i=0 2 3 2 rl−1 = a0(g − 1) + a1(g − g) + a2(g − g ) + ··· + arl−1(1 − g ) rl−1 = (arl−1 − a0)1 + (a0 − a1)g + ··· + (arl−2 − arl−1)g .

rl−1 Since {1, g, . . . , g } is a k-basis for kG, we must have arl−1 − a0 = a0 − a1 = ··· = arl−2 − arl−1 = 0, from which it follows that a0 = a1 = a2 = ··· = arl−1. Therefore, rl−1 P i α = a0 g = a0T ∈ Im(T ). i=0

127 21 The Steinberg Module in Deﬁning Characteristic

Let G be a finite group of Lie type, defined over a finite field Fq (q a power of a prime p) and having a split BN-pair, where B is a Borel subgroup of G containing a maximal torus T and N = NG(T ) is the normalizer of T in G. So, G is the fixed point subgroup of a ¯ connected reductive algebraic group G over Fp under an endomorphism F : G → G such that some power of F is a Frobenius morphism. We assume that the algebraic group G is simple and of simply connected type.

Let U be the unipotent radical of B (U is a normal p-subgroup of B and B = U o T ), and let W = NG(T )/T be the Weyl group of G with respect to T . Given a ﬁeld k (of P l(w) P any characteristic), we deﬁne an element e ∈ kG by e = (−1) nwb, where b = b w∈W b∈B and nw ∈ NG(T ) is a representative of the coset w ∈ W = NG(T )/T . Then, the left ideal Stk := kGe of kG is the Steinberg module over k.

¯ From now on, we will assume that k = Fp, the algebraically closed ﬁeld of characteristic equal to the deﬁning characteristic of G. We will show that Stk is an irreducible kG- module, and we will classify Stk according to its highest weight. The ideas behind the proofs in this section can be found in [41, Section 2], though the details are left to the reader in [41].

The irreducibility of Stk follows by the following result of Brauer and Nesbitt (as stated in the introduction of [41]).

Theorem 21.0.1. ([6, Theorem 1]) Let G be a group of order pab, where p is prime and (p, b) = 1. An ordinary irreducible representation of degree divisible by pa remains irreducible after reduction modulo p.

The Steinberg module is irreducible in characteristic 0, and Stk can be obtained as a reduction mod p of the ordinary Steinberg representation. By [47, Theorem 1], dimk Stk = |U|. Since U is a p-Sylow subgroup of G, |U| is the biggest power of p dividing G. Thus, Stk is irreducible by Brauer-Nesbitt.

Now, an irreducible module for the simply connected algebraic group G (over the ﬁeld k) can be classiﬁed by its highest weight; there is a bijective correspondence between the dominant weights λ and irreducible kG-modules L(λ). As shown by Steinberg, the irreducible kG-modules can be obtained by restricting certain irreducible kG-modules L(λ) P to G. Given a dominant weight λ, we can write λ = ci$i, where each $i is a fundamental dominant weight and each ci is a nonnegative integer. Steinberg proved that a complete set of nonisomorphic irreducible kG-modules is obtained by restricting those L(λ) for which ci lies in [0, q − 1] for all i [36, Theorem 2.5].

128 Since Stk is an irreducible kG-module, Stk must be the restriction of some L(λ), where P λ is a dominant weight with coordinates lying in [0, q − 1]. Let ρ = $i be the sum of the fundamental dominant weights (equivalently, ρ is the half-sum of the positive roots for the root system of G).

Theorem 21.0.2. Stk = L((q − 1)ρ).

Proof. Each irreducible kG-module L(λ) is a quotient of a Weyl module V (λ). Thus, P dimk L(λ) ≤ dimk V (λ) for each dominant weight λ. Let λ = ci$i be such that ci is in [0, q − 1] for all i. By Weyl’s dimension formula, we have

Q (λ + ρ, α) α∈Φ+ Y (λ + ρ, α) dim V (λ) = = , k Q (ρ, α) (ρ, α) α∈Φ+ α∈Φ+ where Φ is the root system for G and Φ+ is the set of positive roots.

(λ + ρ, α) We will ﬁnd a bound on each term (α ∈ Φ+) in the product, which will in (ρ, α) turn give us a bound on dimk V (λ).

2 (λ + ρ, α) (α,α) (λ + ρ, α) hλ + ρ, αi First, we observe that = 2 = . (ρ, α) (α,α) (ρ, α) hρ, αi

We will denote the simple roots giving a base for Φ by αi (with the simple ∨ root αi corresponding to the fundamental weight $i). The duals αi of the sim- ∨ ple roots αi form a base for the dual root system Φ . Therefore, given an ele- + ∨ P ∨ ment α ∈ Φ , we can write α = diαi for some nonnegative integers di. So, ∨ P ∨ P hλ + ρ, αi = (λ + ρ, α ) = di(λ + ρ, αi ) = dihλ + ρ, αii. Now, for each i, hλ + ρ, αii is P P P the coeﬃcient of $i in λ + ρ. And, since λ + ρ = ci$i + $i = (ci + 1)$i, we have hλ+ρ, αii = ci +1 for all i. By our assumption on λ, ci ≤ q −1 for all i, from which it follows P P that hλ + ρ, αi = di(ci + 1) ≤ diq (with equality occuring if and only if ci = q − 1 for all i or, equivalently, λ = (q − 1)ρ). Now, hρ, αi = (ρ, α∨) = P d (ρ, α∨) = P d . Therefore, P i i i hλ + ρ, αi ( di)q hλ + ρ, αi ≤ P = q. And, = q if and only if λ = (q − 1)ρ. hρ, αi di hρ, αi

|Φ+| By the discussion of the previous paragraph, we have dimk V (λ) ≤ q for all dominant weights λ with coordinates in [0, q − 1], with equality occurring if and only if λ = (q − 1)ρ. |Φ+| But, q = |U| = dimk Stk. Thus, Stk cannot be a quotient of V (λ)(λ with coordinates in [0, q − 1]) unless λ = (q − 1)ρ. Hence, we must have Stk = L((q − 1)ρ) (which, by dimension considerations, is also equal to V ((q − 1)ρ).

129 22 The Character of the Steinberg Module in Charac- teristic 0

22.1 Introduction

Let G be a finite group of Lie type, defined in characteristic p. Let B be a Borel subgroup of G, and let U denote the unipotent radical of B (so, U is a normal p-subgroup of B and B = U o T , where T is a maximal torus of G contained in B). Let W = NG(T )/T be the Weyl group of G with respect to T , and let S be the set of fundamental reflections generating W . Let Φ be the corresponding root system, and let Φ+ denote the set of P l(w) positive roots. Given a field k, we define an element e ∈ kG by e = (−1) nwb, where w∈W P b = b. Then, Stk := kGe is the Steinberg module over k. Steinberg showed that a k-basis b∈B for Stk is given by the set {ue | u ∈ U}, so that dimk(Stk) = |U|.

Note that the Steinberg module may be deﬁned over a ﬁeld of any characterstic. However, the structure of Stk can vary greatly depending on the characteristic of k. In particular, Steinberg showed that the module Stk is irreducible if and only if char(k) does not divide [G : B].

In this section, we will assume that k = C, the ﬁeld of complex numbers. Since we are in characteristic 0, the Steinberg module StC is irreducible. We will show that the character of St is given by the formula P (−1)|J|1G , where P denotes the standard parabolic subgroup C PJ J J⊆S of G corresponding to the subset J ⊆ S, and 1G is the trivial character on P induced to PJ J G. This character formula was originally proved by Curtis in 1965 [11]. In what follows, we will use ideas from Curtis’s paper [11] as well as from [8, Chapter 6].

22.2 A generalized character of G

Following [8, 6.2], we will show that the generalized character χ := P (−1)|J|1G is, in fact, PJ J⊆S an irreducible character of G. We will use the following standard facts in our proof.

(1) Let H and K be subgroups of G, and let R be a set of double coset representatives of G with respect to H and K. Let φ be a character of H, and let ψ be a character of K. Then, the scalar product of the induced characters φG and ψG is given by the formula G G P r r r r r (φ , ψ ) = (φ, ψ)H∩rK , where ψ is the character of K deﬁned by ψ( x) = ψ(x) r∈R for x ∈ K. (This result is due to Mackey.)

130 (2) Given a subset J ⊆ S, let WJ = hsis∈J be the parabolic subgroup of W corresponding to J. Then, the generalized character ε = P (−1)|J|1W of W is given by ε(w) = (−1)l(w) WJ J⊆S for w ∈ W . Therefore, ε is the irreducible character of W corresponding to the sign representation.

Lemma 22.2.1. Let I,J ⊆ S. And, let I W J =I W ∩ W J (where I W is a set of shortest J right coset representatives of WI in W , and W is a set of shortest left coset representatives of W in W ). Then, (1G , 1G ) = |I W J | and (1W , 1W ) = |I W J |. J PI PJ WI WJ

I J Proof. Observe that W is both a set of (PI ,PJ ) double coset representatives in G and a set of (WI ,WJ ) double coset representatives in W . By Mackey’s formula, we have

G G X w (1 , 1 ) = (1, 1) w PI PJ PI ∩ PJ w∈I W J

X 1 X w = w 1(p) 1(p) |PI ∩ PJ | w w∈I W J p∈PI ∩ PJ X 1 X = w 1 |PI ∩ PJ | w w∈I W J p∈PI ∩ PJ

X 1 w = w |PI ∩ PJ | |PI ∩ PJ | w∈I W J X = 1 w∈I W J = |I W J |.

The proof that (1W , 1W ) = |I W J | is analogous. WI WJ

Proposition 22.2.2. (χ, χ) = 1.

Proof. By Lemma 22.2.1, we have XX (χ, χ) = (−1)|I|(−1)|J|(1G , 1G ) PI PJ I⊆SJ⊆S XX = (−1)|I|(−1)|J||I W J | I⊆SJ⊆S XX = (−1)|I|(−1)|J|(1W , 1W ) WI WJ I⊆SJ⊆S = (ε, ε) = 1,

where the last equality follows since ε is an irreducible character of W .

131 Using Proposition 22.2.2, we conclude that ±χ must be an irreducible character of G.

G Proposition 22.2.3. (1B, χ) = 1.

Proof. Since B = P , Lemma 22.2.1 gives (1G, 1G ) = |W J | for any subset J ⊆ S. Therefore, ∅ B PJ we can compute X X X (1G, χ) = (−1)|J|(1G, 1G ) = (−1)|J||W J | = (−1)|J|(1W , 1W ) = (1W , ε). B B PJ 1 WJ 1 J⊆S J⊆S J⊆S

W Now, by Frobenius reciprocity, (11 , ε) = (1, ε|1)1. Since the restriction of the sign character to the trivial subgroup {1} of W is the trivial character 1, we have (1, ε|1)1 = (1, 1)1 = 1.

G Now, 1B is a a character of G (and not just a generalized character). Since the coeﬃcient G G of χ in 1B is (1B, χ) = 1, χ must be an irreducible character of G.

We end this section with another computation, which will be useful in the next section.

Proposition 22.2.4. Let I ⊆ S, with I 6= ∅. Then, (1G , χ) = 0. PI

Proof. We have (1G , χ) = P (−1)|J|(1G , 1G ) = P (−1)|J||I W J | = P (−1)|J|(1W , 1W ) = PI PI PJ WI WJ J⊆S J⊆S J⊆S (1W , ε). By Frobenius reciprocity, (1W , ε) = (1, ε ) , where ε is the sign representation WI WI WI WI WI

of WI . Now, since I 6= ∅, WI is a non-trivial subgroup of W . Therefore, εWI is an irreducible

character of WI distinct from 1, which means that (1, εWI )WI = 0.

22.3 The character of StC

In this section, we follow Curtis’s arguments in [11]. Curtis gives the following characterization of the character χ = P (−1)|J|1G [11, Theorem 2 (d)]. PJ J⊆S Lemma 22.3.1. χ is the unique irreducible character of G such that χ is a component of 1G but not a component of 1G for any J 6= ∅. B PJ

Proof. We showed already that χ possesses the desired properties. It remains to check that χ is unique with these properties. But, suppose that χ0 is another such irreducible character. In particular, (χ0, 1G) 6= 0 and (χ0, 1G ) = 0 for any J 6= ∅. Therefore, (χ0, χ) = B PJ P (−1)|J|(χ0, 1G ) = (χ0, 1G) 6= 0. Since χ and χ0 are both irreducible characters, we must PJ B J⊆S have χ = χ0.

132 We can now prove that the irreducible Steinberg module StC = CGe has character χ [11, Theorem 3]. We will use without proof the following results proved by Steinberg [47].

Lemma 22.3.2. Given s ∈ S, let ns denote a representative of s in NG(T ). And, let α be the simple root such that s = sα.

1. nse = −e.

2. For any element uα 6= 1 in the root subgroup Uα, there exists an element u¯α ∈ Uα such −1 that ns uαe = (¯uα − 1)e.

Theorem 22.3.3. The character χ is aﬀorded by the module StC.

G Proof. First, we note that the character 1B is afforded by the permutation module |G ∼ Gb, and the character 1G (J ⊆ S) is afforded by the permutation module C B = C PJ |G ∼ Gp , where p = P p ∈ P . Over the complex numbers, representations are C PJ = C J J C J p∈PJ determined by their characters. Therefore, by Lemma 22.3.1, the module which affords χ is the unique irreducible CG-module which occurs in CGb but not in CGpJ for any non-empty subset J of S. So, to show that StC has character χ, it suffices to check that StC occurs in CGb and not in CGpJ for any J 6= ∅; in fact, since all CG-modules are completely reducible, it suffices to check that StC ⊆ CGb and that StC 6⊆ CGpJ for any J 6= ∅.

The fact that StC ⊆ CGb follows directly from the deﬁnition of e. Now, let J be a non-empty subset of S and assume, for contradiction, that StC ⊆ CGpJ . Observe that 2 P P 0 P pJ = pp = pJ = |PJ |pJ . Therefore, if StC ⊆ CGpJ , then StCpJ = StC. Now, 0 p∈PJ p ∈PJ p∈PJ 1 op since pJ is an idempotent in G, we have an isomorphism End G( GpJ ) → pJ GpJ , |PJ | C C C C given by φ 7→ φ(pJ ) for any φ ∈ EndCG(CGpJ ). And, since all representations over the complex numbers are completely reducible, StCpJ is a direct summand of CGpJ . Thus, ∼ op op EndCG(StCpJ ) ⊆ EndCG(CGpJ ) = pJ CGpJ . In the algebra pJ CGpJ , EndCG(StCpJ ) may op be identiﬁed with pJ StCpJ . And, since StCpJ = StC 6= 0, EndCG(StCpJ ) 6= 0, which means that pJ StC = pJ StCpJ 6= 0.

We claim that ue ∈ pJ St . We begin by showing that bSt = huei , the C-span C P C C of the element ue ∈ StC (where u = u). First, we check that hueiC ⊆ bStC (where u∈U P b = b). Since B = U o T , any element b ∈ B may be written as b = ut, with u ∈ U, b∈B t ∈ T . Now, since T ⊆ NG(U), nw ∈ NG(T ) for all w ∈ W , and uu = u, we have P l(w) P l(w) P l(w) bue = utue = uute = ut (−1) nwb = u (−1) nwtb = u (−1) nwb = ue w∈W w∈W w∈W (where the equality tb = b follows because t ∈ B). Therefore, bue = |B|ue, so that 1 ue = bue ∈ bSt and huei ⊆ bSt . Now, we check that bSt ⊆ huei . Since U ⊆ B, |B| C C C C C b B U u u e ue e StC = StC ⊆ StC = StC. And, since u = for any u ∈ U and the set {u |u ∈ U} is a C-basis for StC, we have uStC = hueiC. Thus, bStC ⊆ hueiC, as needed. Now, since

133 p St = StPJ , bSt = StB, and B ⊆ P , we have p St ⊆ bSt = huei . But, p St 6= 0 J C C C C J J C C C J C and hueiC is one-dimensional; it follows that pJ StC = hueiC, so that ue ∈ pJ StC.

Now, for any s ∈ J, a representative ns of s ∈ W = N/T is contained in PJ , which means that nspJ = pJ . And, since ue ∈ pJ StC, we must have nsue = ue for all s ∈ J. We now fix a simple reflection s ∈ J. Let α be the simple root with s = sα. We can 0 0 Q write U = UαUα, where Uα is the root subgroup corresponding to α and Uα = Uβ. β∈Φ+\{α} 0 0 0 Since every element u ∈ U can be expressed uniquely in the form u = uαuα with uα ∈ Uα P P 0 P P 0 P 0 and uα ∈ Uα, we have u = u = uαuα = ( uα)uα = u uα, where 0 0 0 0 u∈U uα∈Uα, uα∈Uα uα∈Uα uα∈Uα uα∈Uα 0 P 0 u := uα. Now, using the statements of Lemma 173, we carry out the following 0 0 uα∈Uα computation. For the simple reflection s ∈ J,

−1 −1 X 0 −1 X 0 0 −1 0 X −1 0 ns ue = ns ( u uα)e = ns ( u uα + u )e = ns u ( uα)e + ns u e uα∈Uα uα∈Uα, uα∈Uα, uα6=1 uα6=1 −1 0 −1 X −1 0 −1 0 X −1 −1 0 = ns u nsns ( uα)e + ns u e = ns u ns( ns uαe) + ns u e uα∈Uα, uα∈Uα, uα6=1 uα6=1 −1 0 X −1 0 = ns u ns( (¯uα − 1)e) + ns u e uα∈Uα, uα6=1 −1 0 X −1 0 −1 0 = ns u ns( u¯α)e − (|Uα| − 1)ns u nse + ns u e. uα∈Uα, uα6=1

+ −1 0 Q 0 Since s permutes the elements of Φ \ α, we have ns Uαns = Us(β) = Uα. β∈Φ+\{α} −1 0 0 Therefore, ns u ns = u . Also, since the assignment Uα \{1} → Uα \{1}, uα → u¯α is P P bijective, we have u¯α = uα. Continuing the calculation above, we have uα∈Uα, uα∈Uα, uα6=1 uα6=1

134 ∗ 0 X 0 −1 0 ( ) = u ( u¯α)e − (|Uα| − 1)u e + ns u e uα∈Uα, uα6=1 0 0 −1 0 = u (u − 1)e − (|Uα| − 1)u e + ns u e 0 0 −1 0 = ue − u e − (|Uα| − 1)u e − ns u nse 0 0 = ue − |Uα|u e − u e 0 = ue − (|Uα| + 1)u e 6= ue.

0 0 0 (The last assertion follows since the elements uαe, uα ∈ Uα ⊆ U are part of the C-basis 0 P 0 {ue | u ∈ U} for StC, which means that (|Uα| + 1)u e = (|Uα| + 1)uαe 6= 0.) 0 0 uα∈Uα

−1 Our calculation has shown that ns ue 6= ue, which means nsue 6= ue, contradicting the statement that nsue = ue for all s ∈ J. Thus, we conclude that StC 6⊆ CGpJ , as needed.

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