DOCSLIB.ORG

The Maximal Subgroups of Positive Dimension in Exceptional Algebraic Groups

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Maximal Subgroups of Positive Dimension in Exceptional Algebraic Groups The maximal subgroups of positive dimension in exceptional algebraic groups Martin W. Liebeck Gary M. Seitz Department of Mathematics Department of Mathematics Imperial College University of Oregon London SW7 2AZ Eugene, Oregon 97403 England USA 1 Contents Abstract 1. Introduction 1 2. Preliminaries 10 3. Maximal subgroups of type A1 34 4. Maximal subgroups of type A2 90 5. Maximal subgroups of type B2 150 6. Maximal subgroups of type G2 175 7. Maximal subgroups X with rank(X) ≥ 3 181 8. Proofs of Corollaries 2 and 3 196 9. Restrictions of small G-modules to maximal subgroups 198 10. The tables for Theorem 1 and Corollary 2 212 11. Appendix: E8 structure constants 218 References 225 v Abstract In this paper we complete the determination of the maximal subgroups of positive dimension in simple algebraic groups of exceptional type over algebraically closed fields. This follows work of Dynkin, who solved the problem in characteristic zero, and Seitz who did likewise over fields whose characteristic is not too small. A number of consequences are obtained. It follows from the main theo- rem that a simple algebraic group over an algebraically closed field has only finitely many conjugacy classes of maximal subgroups of positive dimension. It also follows that the maximal subgroups of sufficiently large order in finite exceptional groups of Lie type are known. Received by the editor May 22, 2002. 2000 Mathematics Classification Numbers: 20G15, 20G05, 20F30. Key words and phrases: algebraic groups, exceptional groups, maximal subgroups, finite groups of Lie type. The first author acknowledges the hospitality of the University of Oregon, and the second author the support of an NSF Grant and an EPSRC Visiting Fellowship. vi 1 Introduction Let G be a simple algebraic group of exceptional type G2,F4,E6,E7 or E8 over an algebraically closed field K of characteristic p (where we set p = ∞ if K has characteristic zero). In this paper we determine the maximal closed subgroups of positive dimension in G. Taken together with the results of [25, 30, 40] on classical groups, this provides a description of all maximal closed subgroups of positive dimension in simple algebraic groups. We obtain a variety of consequences, including a classification of maxi- mal subgroups of the associated finite groups of Lie type, apart from some subgroups of bounded order. The analysis of maximal subgroups of exceptional groups has a history stretching back to the fundamental work of Dynkin [11], who determined the maximal connected subgroups of G in the case where K has characteristic zero. The flavour of his result is that apart from parabolic subgroups and reductive subgroups of maximal rank, there are just a few further conjugacy classes of maximal connected subgroups, mostly of rather small dimension compared to dim G. In particular, G has only finitely many conjugacy classes of maximal connected subgroups. The case of positive characteristic was taken up by Seitz [31], who de- termined the maximal connected subgroups under some assumptions on p, obtaining conclusions similar to those of Dynkin. If p > 7 then all these as- sumptions are satisfied. This result was extended in [21], where all maximal closed subgroups of positive dimension in G were classified, under similar assumptions on p. In the years since [31, 21], the importance of removing the character- istic assumptions in these results has become increasingly clear, in view of applications to both finite and algebraic group theory. For example, [24, Theorem 1] shows that any finite quasisimple subgroup X(q) of G, with q a sufficiently large power of p, can be embedded in a closed subgroup of positive dimension in G; this is used to prove that maximal subgroups of finite exceptional groups Gσ (σ a Frobenius morphism) are, with a bounded number of exceptions, of the form Xσ with X a maximal closed subgroup of positive dimension in G (see [24, Corollaries 7,8]). Here we complete the solution of this problem. We determine all maximal closed subgroups of positive dimension in G in arbitrary characteristic. For the purposes of one of our applications to finite groups of Lie type, we in fact prove a slightly more general result, admitting the presence of field endomorphisms and graph automorphisms of G. Henceforth we simply refer 1 2 MARTIN W. LIEBECK AND GARY M. SEITZ to these as “morphisms of G”. Let G be of adjoint type, and define Aut (G) to be the abstract group generated by inner automorphisms of G, together with graph and field mor- phisms. In the statement below, by a subgroup of maximal rank we mean a subgroup containing a maximal torus of G, and Symk denotes the sym- ¯ metric group of degree k. Also Fp denotes the algebraic closure of the prime field Fp. Recall also that a Frobenius morphism of G is an endomorphism σ whose fixed point group Gσ is finite. Here is our main result. Theorem 1 Let G1 be a group satisfying G ≤ G1 ≤ Aut (G); in the case ¯ where G1 contains a Frobenius morphism of G, assume that K = Fp. Let X be a proper closed connected subgroup of G which is maximal among proper closed connected NG1 (X)-invariant subgroups of G. Then one of the following holds: (a) X is either parabolic or reductive of maximal rank; 2 (b) G = E7, p 6= 2 and NG(X) = (2 × D4).Sym3; (c) G = E8, p 6= 2, 3, 5 and NG(X) = A1 × Sym5; (d) X is as in Table 1 below. The subgroups X in (b), (c) and (d) exist, are unique up to conjugacy in Aut(G), and are maximal among closed, connected NG(X)-invariant sub- groups of G. Table 1 G X simple X not simple G2 A1 (p ≥ 7) F4 A1 (p ≥ 13),G2 (p = 7), A1G2 (p 6= 2) E6 A2 (p 6= 2, 3),G2 (p 6= 7), A2G2 C4 (p 6= 2),F4 E7 A1 (2 classes, p ≥ 17, 19 resp.), A1A1 (p 6= 2, 3),A1G2 (p 6= 2), A2 (p ≥ 5) A1F4,G2C3 E8 A1 (3 classes, p ≥ 23, 29, 31 resp.), A1A2 (p 6= 2, 3),A1G2G2 (p 6= 2), B2 (p ≥ 5) G2F4 MAXIMAL SUBGROUPS OF EXCEPTIONAL ALGEBRAIC GROUPS 3 Theorem 1 determines the maximal subgroups M of any such group G1 such that M ∩ G is closed and has positive dimension. Below we shall give applications with G1 = G and with G1 = Ghσi, where σ is a Frobenius morphism of G. In Tables 10.1 and 10.2 at the end of the paper we present further infor- mation concerning the subgroups X in Table 1: (1) We give the precise action (as a sum of explicit indecomposable modules) of X on L(G), and also, in the cases G = F4,E6,E7, on the module V , where V is the restricted irreducible G-module of high weight λ4, λ1, λ7 respectively (of dimension 26 − δp,3, 27, 56). These actions are recorded in Tables 10.1 and 10.2, and proofs can be found in Section 9. (2) We give the values of |NG(X): X|; this is always at most 2. In all cases where X has a factor A2, NG(X) induces a graph automorphism on this factor, and the only other case where |NG(X): X| = 2 is that in which G = E8 and X = A1G2G2, where NG(X) has an element interchanging the two G2 factors. The subgroups of G of type (a) in Theorem 1 are well understood. Max- imal parabolic subgroups correspond to removing a node of the Dynkin diagram (possibly two nodes if G1 contains an element involving a graph or graph-field morphism). Subgroups which are reductive of maximal rank are easily determined. They correspond to various subsystems of the root system of G, and we give a complete list (with a proof in Section 8) of those whose normalizers are maximal in G, in Table 10.3 in Section 10. Likewise, Table 10.4 lists the maximal connected subgroups of maximal rank (again with a proof in Section 8). Theorem 1 determines the maximal subgroups M of G1 such that M ∩G is closed of positive dimension. In particular, application of Theorem 1 with G1 = G gives a complete determination of the maximal closed subgroups of positive dimension in G - they are just the subgroups NG(X) for X as in (a)-(d). We state this formally for completeness: Corollary 2 (i) The maximal closed subgroups of positive dimension in G are as follows: maximal parabolics; normalizers of reductive subgroups of 2 maximal rank, as listed in Table 10.3; the subgroups (2 × D4).Sym3 < E7 and A1 × Sym5 < E8 in Theorem 1(b,c); and subgroups NG(X) with X as in Table 1. 4 MARTIN W. LIEBECK AND GARY M. SEITZ (ii) The maximal closed connected subgroups of G are as follows: max- imal parabolics; maximal closed connected subgroups of maximal rank, as listed in Table 10.4; and all subgroups X in Table 1, omitting the subgroup A1G2G2 < E8. The subgroup A1G2G2 < E8 in Table 1 lies in a subgroup F4G2 so is not maximal connected; however its normalizer in E8 interchanges the two G2 factors, and indeed NE8 (X) is maximal in E8. On glancing at the main results of [21, 31] and comparing them with our Theorem 1, the reader will notice that the conclusions are very similar. Indeed, the only subgroups present in our Table 1 which are not already in [21, 31] are G2 < E6 for p = 2, 3, A2 < E7 for p = 5, B2 < E8 for p = 5.
Load more

Recommended publications
  • ON the SHELLABILITY of the ORDER COMPLEX of the SUBGROUP LATTICE of a FINITE GROUP 1. Introduction We Will Show That the Order C
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 353, Number 7, Pages 2689{2703 S 0002-9947(01)02730-1 Article electronically published on March 12, 2001 ON THE SHELLABILITY OF THE ORDER COMPLEX OF THE SUBGROUP LATTICE OF A FINITE GROUP JOHN SHARESHIAN Abstract. We show that the order complex of the subgroup lattice of a finite group G is nonpure shellable if and only if G is solvable. A by-product of the proof that nonsolvable groups do not have shellable subgroup lattices is the determination of the homotopy types of the order complexes of the subgroup lattices of many minimal simple groups. 1. Introduction We will show that the order complex of the subgroup lattice of a finite group G is (nonpure) shellable if and only if G is solvable. The proof of nonshellability in the nonsolvable case involves the determination of the homotopy type of the order complexes of the subgroup lattices of many minimal simple groups. We begin with some history and basic definitions. It is assumed that the reader is familiar with some of the rudiments of algebraic topology and finite group theory. No distinction will be made between an abstract simplicial complex ∆ and an arbitrary geometric realization of ∆. Maximal faces of a simplicial complex ∆ will be called facets of ∆. Definition 1.1. A simplicial complex ∆ is shellable if the facets of ∆ can be ordered σ1;::: ,σn so that for all 1 ≤ i<k≤ n thereexistssome1≤ j<kand x 2 σk such that σi \ σk ⊆ σj \ σk = σk nfxg. The list σ1;::: ,σn is called a shelling of ∆.
    [Show full text]
  • ON the INTERSECTION NUMBER of FINITE GROUPS Humberto Bautista Serrano University of Texas at Tyler
    University of Texas at Tyler Scholar Works at UT Tyler Math Theses Math Spring 5-14-2019 ON THE INTERSECTION NUMBER OF FINITE GROUPS Humberto Bautista Serrano University of Texas at Tyler Follow this and additional works at: https://scholarworks.uttyler.edu/math_grad Part of the Algebra Commons, and the Discrete Mathematics and Combinatorics Commons Recommended Citation Bautista Serrano, Humberto, "ON THE INTERSECTION NUMBER OF FINITE GROUPS" (2019). Math Theses. Paper 9. http://hdl.handle.net/10950/1332 This Thesis is brought to you for free and open access by the Math at Scholar Works at UT Tyler. It has been accepted for inclusion in Math Theses by an authorized administrator of Scholar Works at UT Tyler. For more information, please contact [email protected]. ON THE INTERSECTION NUMBER OF FINITE GROUPS by HUMBERTO BAUTISTA SERRANO A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science Department of Mathematics Kassie Archer, Ph.D., Committee Chair College of Arts and Sciences The University of Texas at Tyler April 2019 c Copyright by Humberto Bautista Serrano 2019 All rights reserved Acknowledgments Foremost I would like to express my gratitude to my two excellent advisors, Dr. Kassie Archer at UT Tyler and Dr. Lindsey-Kay Lauderdale at Towson University. This thesis would never have been possible without their support, encouragement, and patience. I will always be thankful to them for introducing me to research in mathematics. I would also like to thank the reviewers, Dr. Scott LaLonde and Dr. David Milan for pointing to several mistakes and omissions and enormously improving the final version of this thesis.
    [Show full text]
  • Unitary Representations of Real Reductive Groups
    Unitary representations of real reductive groups Jeffrey D. Adams∗ Department of Mathematics University of Maryland Marc van Leeuwen Laboratoire de Math´ematiques et Applications Universit´ede Poitiers Peter E. Trapa† Department of Mathematics University of Utah David A. Vogan, Jr.‡ Department of Mathematics Massachusetts Institute of Technology December 10, 2012 Contents 1 First introduction 5 2 Nonunitary representations 7 3 Real reductive groups 16 arXiv:submit/0611028 [math.RT] 10 Dec 2012 4 Maximal tori 20 5 Coverings of tori 25 6 Langlands classification and the nonunitary dual 38 ∗The authors were supported in part by NSF grant DMS-0968275. The first author was supported in part by NSF grant DMS-0967566. †The third author was supported in part by NSF grant DMS-0968060. ‡The fourth author was supported in part by NSF grant DMS-0967272. 1 7 Second introduction: the shape of the unitary dual 45 8 Hermitian forms on (h,L(C))-modules 47 9 Interlude: realizing standard modules 62 10 Invariant forms on irreducible representations 70 11 Standard and c-invariant forms in the equal rank case 80 12 Twisting by θ 81 13 Langlands parameters for extended groups 88 14 Jantzen filtrations and Hermitian forms 105 15 Signature characters for c-invariant forms 118 16 Translation functors: first facts 130 17 Translation functors for extended groups 142 18 KLV theory 158 19 KLV theory for extended groups 168 20 KLV theory for c-invariant forms 173 21 Deformation to ν = 0 182 22 Hyperplane calculations 195 2 Index of notation (A, ν) continuous parameter; Definition 6.5.
    [Show full text]
  • (Hereditarily) Just Infinite Property in Profinite Groups
    Inverse system characterizations of the (hereditarily) just infinite property in profinite groups Colin D. Reid October 6, 2018 Abstract We give criteria on an inverse system of finite groups that ensure the limit is just infinite or hereditarily just infinite. More significantly, these criteria are ‘universal’ in that all (hereditarily) just infinite profinite groups arise as limits of the specified form. This is a corrected and revised version of [8]. 1 Introduction Notation. In this paper, all groups will be profinite groups, all homomorphisms are required to be continuous, and all subgroups are required to be closed; in particular, all references to commutator subgroups are understood to mean the closures of the corresponding abstractly defined subgroups. For an inverse system Λ= {(Gn)n>0, ρn : Gn+1 ։ Gn} of finite groups, we require all the homomorphisms ρn to be surjective. A subscript o will be used to indicate open inclusion, for instance A ≤o B means that A is an open subgroup of B. We use ‘pronilpotent’ and ‘prosoluble’ to mean a group that is the inverse limit of finite nilpotent groups or finite soluble groups respectively, and ‘G-invariant subgroup of H’ to mean a subgroup of H normalized by G. A profinite group G is just infinite if it is infinite, and every nontrivial normal subgroup of G is of finite index; it is hereditarily just infinite if in addition every arXiv:1708.08301v1 [math.GR] 28 Aug 2017 open subgroup of G is just infinite. At first sight the just infinite property is a qualitative one, like that of simplicity: either a group has nontrivial normal subgroups of infinite index, or it does not.
    [Show full text]
  • Arxiv:1509.08090V1 [Math.GR]
    THE CLASS MN OF GROUPS IN WHICH ALL MAXIMAL SUBGROUPS ARE NORMAL AGLAIA MYROPOLSKA Abstract. We investigate the class MN of groups with the property that all maximal subgroups are normal. The class MN appeared in the framework of the study of potential counter-examples to the Andrews-Curtis conjecture. In this note we give various structural properties of groups in MN and present examples of groups in MN and not in MN . 1. Introduction The class MN was introduced in [Myr13] as the class of groups with the property that all maximal subgroups are normal. The study of MN was motivated by the analysis of potential counter-examples to the Andrews-Curtis conjecture [AC65]. It was shown in [Myr13] that a finitely generated group G in the class MN satisfies the so-called “generalised Andrews- Curtis conjecture” (see [BLM05] for the precise definition) and thus cannot confirm potential counter-examples to the original conjecture. Apart from its relation to the Andrews-Curtis conjecutre, the study of the class MN can be interesting on its own. Observe that if a group G belongs to MN then all maximal subgroups of G are of finite index. The latter group property has been considered in the literature for different classes of groups. For instance in the linear setting, Margulis and Soifer [MS81] showed that all maximal subgroups of a finitely generated linear group G are of finite index if and only if G is virtually solvable. The above property also was considered for branch groups, however the results in this direction are partial and far from being as general as for linear groups.
    [Show full text]
  • Math 602 Assignment 1, Fall 2020 Part A
    Math 602 Assignment 1, Fall 2020 Part A. From Gallier{Shatz: Problems 1, 2, 6, 9. Part B. Definition Suppose a group G operates transitively on the left of a set X. (i) We say that the action (G; X) is imprimitive if there exists a non-trivial partition Q of X which is stable under G. Here \non-trivial" means that Q 6= fXg and Q 6= ffxg : x 2 Xg : Such a partition Q will be called a system of imprimitivity for (G; X). (ii) We say that the action (G; X) is primitive if it is not imprimitive. 1. (a) Suppose that (G; X; Q) is a system of imprimitivity for a transitive left action (G; X), Y 2 Q, y 2 Y . Let H = StabG(Y ), K = StabG(y). Prove the following statements. • K < H < G and H operates transitively on Y . • jXj = jY j · jQj, jQj = [G : H], jY j = [H : K]. (These statements hold even if X is infinite.) (b) Suppose that (G; X) is a transitive left action, y 2 X, K := StabG(y), and H is a subgroup of G such that K < H < G. Let Y := H · y ⊂ X, and let Q := fg · Y j g 2 Gg. (Our general notation scheme is that StabG(Y ) := fg 2 G j g · Y = Y g.) Show that (G; X; Q) is a system of imprimitivity. (c) Suppose that (G; X) is a transitive left action, x 2 X. Show that (G; X) is primitive if and only if Gx is a maximal proper subgroup of G.
    [Show full text]
  • UNITARY REPRESENTATIONS of REAL REDUCTIVE GROUPS By
    UNITARY REPRESENTATIONS OF REAL REDUCTIVE GROUPS by Jeffrey D. Adams, Marc van Leeuwen, Peter E. Trapa & David A. Vogan, Jr. Abstract. | We present an algorithm for computing the irreducible unitary repre- sentations of a real reductive group G. The Langlands classification, as formulated by Knapp and Zuckerman, exhibits any representation with an invariant Hermitian form as a deformation of a unitary representation from the Plancherel formula. The behav- ior of these deformations was in part determined in the Kazhdan-Lusztig analysis of irreducible characters; more complete information comes from the Beilinson-Bernstein proof of the Jantzen conjectures. Our algorithm traces the signature of the form through this deformation, counting changes at reducibility points. An important tool is Weyl's \unitary trick:" replacing the classical invariant Hermitian form (where Lie(G) acts by skew-adjoint operators) by a new one (where a compact form of Lie(G) acts by skew-adjoint operators). R´esum´e (Repr´esentations unitaires des groupes de Lie r´eductifs) Nous pr´esentons un algorithme pour le calcul des repr´esentations unitaires irr´eductiblesd'un groupe de Lie r´eductifr´eel G. La classification de Langlands, dans sa formulation par Knapp et Zuckerman, pr´esente toute repr´esentation hermitienne comme ´etant la d´eformation d'une repr´esentation unitaire intervenant dans la formule de Plancherel. Le comportement de ces d´eformationsest en partie d´etermin´e par l'analyse de Kazhdan-Lusztig des caract`eresirr´eductibles;une information plus compl`eteprovient de la preuve par Beilinson-Bernstein des conjectures de Jantzen. Notre algorithme trace `atravers cette d´eformationles changements de la signature de la forme qui peuvent intervenir aux points de r´eductibilit´e.
    [Show full text]
  • Computing the Maximal Subgroups of a Permutation Group I
    Computing the maximal subgroups of a permutation group I Bettina Eick and Alexander Hulpke Abstract. We introduce a new algorithm to compute up to conjugacy the maximal subgroups of a finite permutation group. Or method uses a “hybrid group” approach; that is, we first compute a large solvable normal subgroup of the given permutation group and then use this to split the computation in various parts. 1991 Mathematics Subject Classification: primary 20B40, 20-04, 20E28; secondary 20B15, 68Q40 1. Introduction Apart from being interesting themselves, the maximal subgroups of a group have many applications in computational group theory: They provide a set of proper sub- groups which can be used for inductive calculations; for example, to determine the character table of a group. Moreover, iterative application can be used to investigate parts of the subgroups lattice without the excessive resource requirements of com- puting the full lattice. Furthermore, algorithms to compute the Galois group of a polynomial proceed by descending from the symmetric group via a chain of iterated maximal subgroups, see [Sta73, Hul99b]. In this paper, we present a new approach towards the computation of the conju- gacy classes of maximal subgroups of a finite permutation group. For this purpose we use a “hybrid group” method. This type of approach to computations in permu- tation groups has been used recently for other purposes such as conjugacy classes [CS97, Hul], normal subgroups [Hul98, CS] or the automorphism group [Hol00]. For finite solvable groups there exists an algorithm to compute the maximal sub- groups using a special pc presentation, see [CLG, Eic97, EW].
    [Show full text]
  • Irreducible Character Restrictions to Maximal Subgroups of Low-Rank Classical Groups of Type B and C
    IRREDUCIBLE CHARACTER RESTRICTIONS TO MAXIMAL SUBGROUPS OF LOW-RANK CLASSICAL GROUPS OF TYPE B AND C KEMPTON ALBEE, MIKE BARNES, AARON PARKER, ERIC ROON, AND A.A. SCHAEFFER FRY Abstract Representations are special functions on groups that give us a way to study abstract groups using matrices, which are often easier to understand. In particular, we are often interested in irreducible representations, which can be thought of as the building blocks of all representations. Much of the information about these representations can then be understood by instead looking at the trace of the matrices, which we call the character of the representation. This paper will address restricting characters to subgroups by shrinking the domain of the original representation to just the subgroup. In particular, we will discuss the problem of determining when such restricted characters remain irreducible for certain low-rank classical groups. 1. Introduction Given a finite group G, a (complex) representation of G is a homomorphism Ψ: G ! GLn(C). By summing the diagonal entries of the images Ψ(g) for g 2 G (that is, taking the trace of the matrices), we obtain the corresponding character, χ = Tr◦Ψ of G. The degree of the representation Ψ or character χ is n = χ(1). It is well-known that any character of G can be written as a sum of so- called irreducible characters of G. In this sense, irreducible characters are of particular importance in representation theory, and we write Irr(G) to denote the set of irreducible characters of G. Given a subgroup H of G, we may view Ψ as a representation of H as well, simply by restricting the domain.
    [Show full text]
  • MAT 511 Notes on Maximal Ideals and Subgroups 9/10/13 a Subgroup
    MAT 511 Notes on maximal ideals and subgroups 9/10/13 A subgroup H of a group G is maximal if H 6= G, and, if K is a subgroup of G satisfying H ⊆ K $ G, then H = K. An ideal I of a ring R is maximal if I 6= R, and, if J is an ideal of R satisfying I ⊆ J $ R, then I = J. Zorn's Lemma Suppose P is a nonempty partially-ordered set with the property that every chain in P has an upper bound in P. Then P contains a maximal element. Here a chain C ⊆ P is a totally-ordered subset: for all x; y 2 C, x ≤ y or y ≤ x.A maximal element of P is an element x 2 P with the property x ≤ y =) x = y for all y 2 P. Partially-ordered sets can have several different maximal elements. Zorn's Lemma says, intuitively, if one cannot construct an ever-increasing sequence in P whose terms get arbitrarily large, then there must be a maximal element in P. It is equivalent to the Axiom of Choice: in essence we assume that Zorn's Lemma is true, when we adopt the ZFC axioms as the foundation of mathematics. Theorem 1. If I0 is an ideal of a ring R, and I0 6= R, then there exists a maximal ideal I of R with I0 ⊆ I. Proof. Let P be the set of proper (i.e., 6= R) ideals of R containing I0, ordered by inclusion. Then S P is nonempty, since I0 2 P.
    [Show full text]
  • A New Maximal Subgroup of E8 in Characteristic 3
    A New Maximal Subgroup of E8 in Characteristic 3 David A. Craven, David I. Stewart and Adam R. Thomas March 22, 2021 Abstract We prove the existence and uniqueness of a new maximal subgroup of the algebraic group of type E8 in characteristic 3. This has type F4, and was missing from previous lists of maximal subgroups 3 produced by Seitz and Liebeck–Seitz. We also prove a result about the finite group H = D4(2), that if H embeds in E8 (in any characteristic p) and has two composition factors on the adjoint module then p = 3 and H lies in this new maximal F4 subgroup. 1 Introduction The classification of the maximal subgroups of positive dimension of exceptional algebraic groups [13] is a cornerstone of group theory. In the course of understanding subgroups of the finite groups E8(q) in [3], the first author ran into a configuration that should not occur according to the tables in [13]. We elicit a previously undiscovered maximal subgroup of type F4 of the algebraic group E8 over an algebraically closed field of characteristic 3. This discovery corrects the tables in [13], and the original source [17] on which it depends. Theorem 1.1. Let G be a simple algebraic group of type E8 over an algebraically closed field of char- acteristic 3. Then G contains a unique conjugacy class of simple maximal subgroups of type F4. If X is in this class, then the restriction of the adjoint module L(E8) to X is isomorphic to LX(1000) ⊕ LX(0010), where the first factor is the adjoint module for X of dimension 52 and the second is a simple module of dimension 196 for X.
    [Show full text]
  • The Contragredient
    The Contragredient Jeffrey Adams and David A. Vogan Jr. December 6, 2012 1 Introduction It is surprising that the following question has not been addressed in the literature: what is the contragredient in terms of Langlands parameters? Thus suppose G is a connected, reductive algebraic group defined over a local field F , and G(F ) is its F -points. According to the local Langlands conjecture, associated to a homomorphism φ from the Weil-Deligne group of F into the L-group of G(F ) is an L-packet Π(φ), a finite set of irreducible admissible representations of G(F ). Conjecturally these sets partition the admissible dual. So suppose π is an irreducible admissible representation, and π ∈ Π(φ). Let π∗ be the contragredient of π. The question is: what is the homomor- phism φ∗ such that π∗ ∈ Π(φ∗)? We also consider the related question of describing the Hermitian dual in terms of Langlands parameters. Let ∨G be the complex dual group of G. The Chevalley involution C of ∨G satisfies C(h) = h−1, for all h in some Cartan subgroup of ∨G. The L L-group G of G(F ) is a certain semidirect product ∨G ⋊ Γ where Γ is the absolute Galois group of F (or other related groups). We can choose C so L that it extends to an involution of G, acting trivially on Γ. We refer to this L as the Chevalley involution of G. See Section 2. 2000 Mathematics Subject Classification: 22E50, 22E50, 11R39 Supported in part by National Science Foundation Grant #DMS-0967566 (first au- thor) and #DMS-0968275 (both authors) 1 We believe the contragredient should correspond to composition with the L Chevalley involution of G.
    [Show full text]