Generic <span class="mathjax-formula">$2$</span>-coverings of finite groups of Lie type

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Generic 2-Coverings of Finite Groups of Lie Type (*).

D. BUBBOLONI(**) - M. S. LUCIDO(***) - TH. WEIGEL(***) Dedicated to G. Zappa for his90^thbirthday

ABSTRACT - In [10] it was shown by R.H. Dye that in a symplectic group G:Sp2`(2^f)Iso(V;h:; :i) defined over a finite field of characteristic2every element inGstabilizes a quadratic form of maximal or non-maximal Witt index inducing the bilinear form h:; :i. ThusGis the union of the two G-conjugacy classes of subgroups isomorphic toO_2`(2^f) andO_2`(2^f) embedded naturally. In this paper we classify all finite groups of Lie type (G;F) with this generic2- covering property (Thm. A). In particular, we will show that there exists also an interesting example in characteristic 3, i.e., in the finite group of Lie type G:F4(3^f) every element inG is conjuagte to an element of the subgroup B4(3^f)F4(3^f) or of the subgroup3:³D4(3^f)F4(3^f).

1. Introduction.

LetGbe a finite group and letH;K^<₄Gbe proper subgroups ofG. The pair of subgroupsfH;Kgis called a 2-coveringofG, if every element inGis conjugate to an element inHorK, i.e.,

G [

g2G

H^g[[

g2G

K^g: 1:1

IfGis a Frobenius group with complementHand kernelKthenfH;Kgis obviously a 2-covering of G. Another example of a finite group with a 2-

(*) 2000Mathematics Subject Classification. Primary 20G40; Secondary 20E32 (**) Indirizzo dell'A.: DIMAD UniversitaÁ di Firenze, Via C. Lombroso 6/17, I- 50134 Firenze, Italy; e-mail: daniela.bubboloni@dmd.unifi.it

(***) Indirizzo dell'A.: Dipartimento di Matematica ed Informatica, UniversitaÁ di Udine, Via delle Scienze 206, I-33100 Udine, Italy; e-mail: mslucido@dimi.uniud.it

(***) Indirizzo dell'A.: Dipartimento di Matematica ed Applicazioni, UniversitaÁ di Milano-Bicocca, U5-3067, Via R.Cozzi, 53, I-20125 Milano, Italy;

e-mail: thomas.weigel@unimib.it

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covering is the groupG:SL2(q),qp^f. It can be 2-covered by the pair fH;Kg, where His the normalizer of ap-Sylow subgroup, and K is the normalizer of a maximal non-split torus. A more sophisticated class of examples is the following: LetF₂^f be a finite field of characteristic 2 and order 2^f, let V be an F₂f-vector space of dimension 2` and let h:; :i:VV !F₂f be a non-degenerate symmetric bilinear form onV. In particular,G:Iso (V;h:; :i)Sp2`(2^f) is the symplectic group of degree 2`defined over the finite fieldF₂^f. LetQ:V !V be a quadratic form of maximal Witt index inducingh:; :i, and letQ⁰:V !Vbe a quadratic form of non-maximal Witt index inducing h:; :i. Then M₁:Iso (V;Q)'O_2`(2^f) and M₂:Iso (V;Q⁰)'O_2`(2^f) are maximal subgroups of G, and it was shown by R.H. Dye ([10]) that every element of G is contained in a G- conjugate ofM1orM2, i.e.,fM1;M2gis a 2-covering ofGSp2`(2^f).

The study of covering properties of finite groups was originally moti- vated by number theoretic questions. In [15], W. Jehne discovered that Kronecker equivalence of number fields is related to the existence of a certain type of covering of a finite group (cf. [17, §1C]). Therefore, one calls the proper subgroupUof the finite groupGacovering subgroup of G, if G S

a2Aut (G)U^a. The study of finite groups with a covering subgroup was initiated by R. Brandl (cf. [2]), who also conjectured that a finite group with solvable covering subgroup must itself be solvable. Finally, J. Saxl proved this conjecture in showing that a non-abelian finite simple group does not have a covering subgroup (cf. [19]).

The examples mentioned above show that in contrast to the non-existence of covering subgroups there are non-abelian finite simple groups possessing 2-coverings. Special classes of finite simple groups with this covering property have been investigated in [3] and [4]. For finite groups of Lie type there exists a more particular type of covering - a generic 2- covering - which allows to analyze the class of groups with this covering property systematically. The main purpose of this paper will be to classify all generic 2-coverings of finite groups of Lie type.

LetGbe a simple algebraic group defined over the algebraically closed fieldF. A closed subgroupHofGis calledof maximal rank, if it contains a maximal torusTofG. By afinite group of Lie type(¹) (G;F) we understand a simple algebraic groupGdefined over the algebraic closureF_pof a finite field of characteristic p together with a Frobenius morphism F:G!G.

(¹) The finite groupGF of elements fixed underF is usually called the finite group of Lie type.

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For such a group a pair of proper closedF-stable subgroupsfH;Kgwill be called ageneric2-covering of(G;F), if

(i) everyF-stable maximal torusTisGF-conjugate to a subgroup of HorK,

(ii) fH_F;K_Fgis a 2-covering ofG_F,

where _Fdenotes the subgroup of elements fixed underF. In [25, §1] it was shown that there is a canonical one-to-one correspondence between GF- conjugacy classes ofF-stable reductive subgroupsHof maximal rank ofG andW-orbits of certain cosets of Weyl subgroups ofW, whereW denotes the Weyl group ofG. We refer to theW-orbit of this coset as thetype of H.

This correspondence extends the well-known correspondence betweenGF- conjugacy classes ofF-stable maximal tori, andF-conjugacy classes in the Weyl group ofG(cf. [8, Prop. 3.3.3]). The main purpose of this article is to establish the following theorem (Thm. 4.5, § 5).

THEOREMA. Let(G;F)be a finite simple group of Lie type, and let fH;Kgbe a generic2-covering of(G;F).

(a) Assume that for one of the subgroups - say H - the connected component His not reductive. Then(G;F)is of type A1, A2, A3or²A3, H is maximal parabolic, and K is reductive. Moreover, the types of H=Ru(H)and K are as listed in Table 1.1. Every pair of F-stable subgroups fH;Kg, H parabolic, K:N_G(K), satisfying the conditions of Table 1.1 defines a generic2-covering of(G;F).

TABLE1.1. - Parabolic-type generic 2-coverings

type(G;F) type H=RuH type K

A`; `1 A1 1^W 2^W

A2 W(A1)^W 3^W

A3 W(A2)^W 2:W(A²₁)^W

2A3 2:W(A²₁)^W 2:W(A2)^W

(b) Assume that Hand Kare reductive subgroups. Then the types of(G;F), H and K are as listed in Table 1.2 modulo interchanging the role of H and K. Moreover, the characteristic of the algebraically closed fieldF must be as indicated in the5^th-column. Every pair of F-stable subgroups

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fH;Kg, H:NG(H), K:NG(K), satisfying the conditions of Table 1.2, defines a generic2-covering of(G;F).

TABLE1.2. - Reductive-type generic 2-coverings

type(G;F) type H type K conditions

A`; `1 A1 1^W 2^W p2

A2 W(A1)^W 3^W p3

2A`; `2 ²A2 2:W(A1)^W 6^W p3

C`; `2 C2 W(A²₁)^W 2:W(A²₁)^W p2

C3 W(C2A1)^W 3:W(A³₁)^W p3 C` W(D~`)^W 2:W(D~`)^W p2

F4 W(B4)^W 3:W(D4)^W p3

G2 W(A2)^W 2:W(A2)^W p2

Here the connected component of an algebraic groupXis denoted by X. In Table 1.2 we used the convention that a rootsystem with a~consists of short roots. Moreover, for a subrootsystemCof the rootsystemFofG andn2N,n:W(C) will stand for a cosetn_C:W(C),n_C 2N_W(W(C)), with n_C:W(C)2N_W(W(C))=W(C) of order n. While this notations is not in- ambiguous, it will suffice for our needs, and from the context it will always be clear which particular coset is meant to be considered.

It should be remarked that Theorem A is independent of the isogeny class ofG. The proof of Theorem A is organized in several steps. Using the results in [25, § 1] one concludes that a pair of reductive subgroupsfH;Kg satisfying property (i) corresponds to a certain type of covering of the Weyl group (coset) which will be called anadmissible2-coset covering(cf. Thm.

2.2). All such coverings can be classified (cf. Thm. 3.4, Table 3.1).

In [20] closed subgroups of a simple algebraic group of positive dimension containing a regular unipotent element were classified. For- tunately, there are few examples of these groups which occur also in Table 3.1. Apart from 3 exceptions this exclusion principle will lead precisely to the examples listed in Table 1.1 and 1.2 (cf. Thm. 4.5).

In order to obtain a full classification of all generic 2-coverings, we have to show that a pair of subgroupsfH;Kgwith the properties described in

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Table 1.1 and 1.2 is indeed a generic 2-covering. This will be done in section 5, 6, 7 by a case-by-case analysis. Unfortunately, for some examples we could not find a geometric argument, and hence in those cases the only way in proving the 2-covering property is by brute force (cf. § 6, § 7).

In a subsequent paper [5] we will determine all 2-coverings of classical groups of Lie type using the classification of finite simple group. It turns out that there are also 2-coverings of classical groups of Lie type which are non-generic.

Acknowledgement:The authors would like to thank G. M. Seitz and A.

Previtali for two very illuminating discussions.

2. F-stable reductive subgroups of maximal rank

2.1 ±Weyl groups and admissible cosets

By W(F)Iso (E;(:; :)) we denote the Weyl group of the finite crystallographic rootsystemFEin the euclidean space (E;(:; :)). LetDF be a basis ofF, and letg2 Aut (W(F)) be a graph automorphism ofW(F) leavingDinvariant. In particular,

We : hgi:W(F)Gl_R(E):

2:1

The cosetg:W(F) carries a natural rightW(F)-action given by (g:w)^x:g:(g(x ¹)wx); x;w2W;

2:2

where g(x)g ¹xg. We call the coset g:W(F)We an admissible W(F)- coset.

Let (G;F) be a finite group of Lie type, i.e., G is a simple algebraic group defined over F_p, the algebraic closure the finite field F_p, and F:G!Gis a Frobenius morphism. LetBGbe anF-stable Borel subgroup of the algebraic groupGcontaining theF-stable maximal torusT.

Then W:NG(T)=T is the Weyl group of a finite irreducible crystallographic rootsystem F, i.e., WW(F). Let DF be a basis ofF corresponding to the Borel subgroupB. The Frobenius morphismF:G!Gis acting on F leaving Dinvariant, and thus corresponds to a graph automorphism g2 Aut (W). On the contrary, every pair (g;F), where Fis a finite irreducible crystallographic rootsystem and g is a graph automorphism of W(F), defines a finite group of Lie type (G;F0g) for every standard Frobenius morphismF0(cf. [8, § 1.1.18]).

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We use the same symbols which are used in the classification of finite groups of Lie type for the definition of the type of an admissible coset g:W(F), i.e., we say thatg:W(F) is of type²A`, ifFis a rootsystem of type A_`and ifgis the graph automorphism of order 2, etc.

2.2 ±Subrootsystems and saturated subrootsystems

Every subrootsystemCFof the finite crystallographic root system Fis again crystallographic. Moreover, there exists an easy algorithm described in [6, p. 8] to determine all maximal subrootsystems ofF.

For two rootsa;b2Fone defines theroot conethroughaandbby C(a;b): fk₁:ak₂:bjk₁;k₂2Ng \F:

2:3

A subrootsystem C is called saturated, if for all a, b2C, C(a;b)C.

Obviously, if in the finite irreducible crystallographic root system F all roots are of the same length, every subrootsystem CF is saturated (cf. [12, § 9.4]). But for example, the subrootsystem A~^`₁ consisting of all short roots inF(B_`) is not saturated. It is an easy exercise to show that a subrootsystemCis saturated, if and only if for every pair of rootsa;b2C, Ccontains also thea-string throughb.

Let LL(F) be the simple finite-dimensional C-Lie algebra with rootsystemFand Cartan decomposition

LCa

a2F

L_a: 2:4

ThenL(C):C `

a2CL_a is a Lie-subalgebra, if and only if CF is saturated (cf. [7, § 4.2], [12, Prop. 8.4(e)]).

LetGbe the simple simply-connected algebraic group defined over C with rootsystem F, and let TG be a maximal torus. From Che- valley's commutator formula (cf. [7, Thm. 5.2.2]) one concludes that H: hT;Uaja2Ciis a reductive subgroup with rootsystem of typeC, if and only if CFis saturated.

LetGbe the simple simply-connected algebraic group defined overF_p with rootsystem F, and letTG be a maximal torus. We call the sub- rootsystemCFp-saturated, ifH: hT;U_aja2Ciis a reductive subgroup with rootsystem of typeC. One has the following:

PROPOSITION2.1. LetFbe a finite irreducible crystallographic rootsystem and letCFbe a subrootsystem.

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(a) IfCFlongconsists only of long roots,Cis saturated.

(b) IfCis saturated, it is p-saturated for all primes p.

(c) AssumeFhas two root lengths r_l, rsand that p6r_l=rs. Then, if Cis p-saturated, it is also saturated.

(d) If F has two root lengths r_l, r_s and pr_l=r_s, every subrootsystem is p-saturated.

PROOF. Leta;b2Cbe linearly independent roots, and letG Fbe the subrootsystem generated byC(a;b). IfG is of typeA₂orA₁A₁,G coincides with the rootsystem spanned byaandb. The same is true ifG is of type B₂ orG₂and aandb are long roots (cf. [7, p. 46, fig. 1]). This yields (a). (b) is a direct consequence of Chevalley's commutator formula. Assume that C is p-saturated. The rootsystem G coincides with the rootsystem spanned by aandb unlessG is of type B₂ orG₂ anda and b are short roots. Ifpr_l=r_s, the constantsC_i;j;a;b; (cf. [7, p. 62, p.

77]) vanish whenever iajb is a long root. This yields (d). However, if p6rl=rs, they do not vanish. Thus, ifCisp-saturated,C(a;b)Cmust

hold which implies (c). p

2.3 ±g-Subrootsystems

Letgbe a graph automorphism of the finite irreducible crystallographic rootsystemFand letCFbe a subrootsystem ofF. ThenCis called ag- subrootsystem, if there exists an elementw2W(F) such that^gC^wC.

LetGa simple algebraic group with rootsystem of typeFdefined over Fp, and letF:G!Gbe a Frobenius morphism acting onFthroughg. Let TGbe a maximal torus and letCFbe a subrootsystem ofF. In [25, Prop. 2] it was shown that G contains an F-stable reductive group H hT;U_a ja2Ci^g,g2G, if and only ifCisp-saturated and ifCis ag- subrootsystem. Additionally, for all twisted types the g-subrootsystems were determined.

2.4 ±Admissible2-coset coverings for Weyl groups

Let F be a finite irreducible crystallographic rootsystem, let W W(F) be its Weyl group, and letg:W!Wbe as in § 2.1. LetGFbe a finite group of Lie type, whereGis a simple algebraic group defined over Fpwith rootsystemFand assume that the Frobenius morphismF:G!G

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is acting on F through g. Let B be an F-stable Borel subgroup of G containg the maximal torusTand letUa,a2F, denote the root groups for TinBandB .

For a p-saturated g-subrootsystem CF and n_C2N_We(W(C))\

\g:W(F), the W-orbit (n_C:W(C))^W of the coset n_C:W(C) corresponds to a G_F-conjugacy class of finite reductive groups H_F, where H: hT;Uaja2Ci^g is F-stable, F(g)g ¹2N_G(T), p(F(g)g ¹)nC, p:N_G(T)!W the canonical projection. This follows from [25, Prop. 4]

bearing in mind that left translation with g, g: :W!g:W, is a bi- jection of right W-sets, where W carries the right W-action given by w^xg(x ¹)wx, x;w2W. In order to analyze the problem further we make the following definition:

DEFINITION2.1. Letg:W be an admissibleW-coset of the finite Weyl groupWW(F). LetC,JFbe properg-subrootsystems ofF, and let n_C,n_J2g:Wbe elements with the following properties:

(i) nC2N_We(W(C)),nJ 2N_We(W(J)), (ii)

(g:W)^W(n_C:W(C))^W[(n_J:W(J))^W: 2:5

Then we call (n_C:W(C);n_J:W(J)) anadmissible2-coset coveringofg:W.

From the definition and [25, § 1] one concludes the following:

THEOREM2.2. Let(G;F)be a finite group of Lie type with rootsystem Fwhose F-action is given byg2 Aut (F). Put W:W(F). Let H and K be F-stable reductive groups of maximal rank with G_F-conjugacy class corresponding to the W-conjugacy class (n_C:W(C))^W and (n_J:W(J))^W, respectively, nC, nJ2g:W (cf.[25, Prop. 4]).

(a) (n_C:W(C);n_J:W(J))is an admissible2-coset covering ofg:W, if and only if every F-stable maximal torus T is G_F-conjugate to a torus in H or K.

(b) If (nC:W(C);nJ:W(J)) is an admissible 2-coset covering of g:W, every semisimple element s2G_F is G_F-conjugate to an element in H_F or K_F.

PROOF. For (a) see [25, Prop. 4, Thm. 5]. Since every semisimple ele- ments2GF is contained in a maximalF-stable torus, (b) is a direct con-

sequence of (a). p

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3. Admissible2-coset coverings for Weyl groups 3.1 ±The Coxeter class

Let F be a finite irreducible crystallographic rootsystem and let W W(F) denote its Weyl group. There exists a very particularW-conjugacy classw^W, the class ofCoxeter elements(cf. [7, § 10.4]). Elements of this class have the following well-known property:

PROPOSITION3.1. LetFbe a finite irreducible crystallographic rootsystem and let WW(F)denote its Weyl group. Let w be a Coxeter element of W and letCFbe a subrootsystem ofF, such that w2W(C). ThenCF.

PROOF. We denote byEC:ECthe complexification ofEand denote by (:; :)_Cthe standardW-invariant hermitian form induced by (:; :). Let CFbe a subrootsystem such thatw2W(C).

The Coxeter element w2W is an element of orderk: jFj=`, where

`rk (F) is the rank ofF. It is also regular, i.e., there exists a primitivek^th- root of unityj2C such that the eigenspaceVj(w) ofwonECfor the eigenvalue j is one-dimensional, and all eigenvectors v2Vj(w)n f0g are regular, i.e., they are not invariant under any non-trivial element ofW(cf.

[23, Thm. 4.2]).

LetC: S^r

i1C_ibe the decomposition ofCin irreducibility components, and let Vi:span_C(Ci),V0:(V1 Vr)^?, where ?denotes the orthogonal complement with respect to the hermitian form (:; :)_C. The subspaces V_i, i0; ::;r; are W(C)-invariant subspaces, and hence in particular w-invariant. Sincew does not have the eigenvalue 1 (cf. [7, Prop.

10.5.6]),V₀0 andCmust be of rank`rk (F).

Assumer2. AsVj(w) coincides with thej-Fitting component ofwon EC, it must be contained in one of theV_i,i2 f1; ::;rg. HenceW(C_j),j6i, fixes V_j(w), a contradiction. Hence C must be an irreducible subrootsystem of maximal rank.

The elementw2W(C) is also regular inW(C), and hence its ordero(w) is a regular number of C. In particular, o(w) jCj=`(cf. [23, § 5]). This

yieldsCF. p

3.2 ±Conjugacy classes in W(A_`)

The Weyl group WW(A`) of type A` is isomorphic to S`1. The conjugacy classes ofW can be parametrizised by the set of partitions of

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`1. The mapping is given by assigning each elementw2Wits cycle type of the action on the set f1; ::; `1g. The Coxeter elements are the elements of cycle type (`1). We will make use of the following property:

PROPOSITION 3.2. (a) Let US_n, n2, be a subgroup containing elementsr,s,tof cycle type(2;1; ::;1),(n)and(1;n 1). Then US_n.

(b)Let USn, n4, be a subgroup containing elementsr,s,tof cycle type(2;1; ::;1),(1;n 1)and(2;n 2). Then USn.

PROOF. (a) The subgroup U is 2-transitive on f1; ::;ng, and thus in particular primitive. Thus the claim follows from Jordan's theorem (cf. [14, Satz 4.5(b)]).

(b) From the hypothesisn4, one concludes thatU is transitive, and the cycle type ofsyields thatUis 2-transitive. The claim then follows by

the same argument as used in (a). p

3.3 ±Conjugacy classes in W(B`)

LetWW(B_`) be the Weyl group associated to the rootsystem of type B_`. In this caseWis acting faithfully on the setE_`: f e_ij1i`gof positive and negative vectors of an orthonormal basis e1; ::;e` of the euclidean space (E;(:; :)). Assigning every elementw2W the cycle type of this action yields a parametrization ofW-conjugacy classes by partitions ( j ) with positive and negative parts. TheW-conjugacy class containing the unit element 1_W 2W corresponds to the -partition (1; ::;1j); the Coxeter class corresponds to the-partition (j`).

The Weyl groupW(B`) contains the elementw02W(B`) acting as idE

onE. Multiplication withw0yields an automorphismw0: :W^W!W^Wof order 2. It corresponds to the automorphism of the set of-partitions of

` fixing blocks of even length and changing the sign of blocks of odd length. The conjugacy class consisting of the element w0 corresponds to the the-partition (j1; ::;1).

The subrootsystemF_longFconsisting of long roots is of typeD`. A conjugacy class is contained in the Weyl subgroupW(F_long), if and only if the number of negative blocks is even. The conjugacy class of long reflections, i.e., reflections corresponding to long roots, corresponds to the -partition (2;1; ::;1j). The conjugacy class of short reflections, i.e., reflections corresponding to short roots, corresponds to the -partition (1; ::;1j1).

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There exists a unique W-orbit of subrootsystemsC of typeA` 1 consisting of long roots. TheW-conjugacy classes of elements being contained in aW-conjugate ofW(C) correspond to the-partitions with trivial negative part ( j).

Let E/W denote the normal subgroup generated by all reflections corresponding to short roots. The canonical projectionp:W !W=E'S_` is mapping reflections to reflections, and thus in particular Weyl subgroups to Weyl subgroups. Omitting the ``j'' in the notation describes its effect on the conjugacy classes. LetCFbe a subrootsystem such that p(W(C))W=E. ThenCis of typeA_` ₁,D_`orB_`. This is a consequence of the following more general property:

PROPOSITION3.3. LetFbe a crystallographic finite rootsystem with at most two root lengths. Lett:W(F)!S_m, m2;be a surjective morphism mapping long reflections to long reflections. ThenFcontains a subrootsystem of type A_m ₁ consisting of long roots.

PROOF. LetR(Sm) denote the set of reflections inSm, and letR_ldenote the set of long reflections inW(F). SinceR(Sm) is a singleSm-conjugacy class, the hypothesis implies that t induces a surjective map t_R:R_l!

!R(S_m). LetS⁰ fs⁰_ij1im 1g R(S_m) be a simple generation system ofSm, and letS fsij1im 1gbe a preimage ofS⁰under tR. For any pair (s_i;s_j) of long reflections the order of its product satisfies o(s_is_j)2 f1;2;3g, and either two of these numbers are coprime. This implies that the Coxeter matrices ofSandS⁰coincide. This yields the claim.

p

3.4 ±Conjugacy classes in W(D`)

Conjugacy classes in the Weyl group WW(D_`) can be best under- stood to think ofW(D`) as the Weyl subgroup ofW(B`) being generated by long reflections. Forw2W(D`) lett(w) denote the-partition associated towbeing considered as an element inW(B`). We callt(w) thetypeofw.

Obviously,W(D_`)-conjugate elements have the same type, and a-partition is the type of an element w2W(D_`), if and only if the number of negative blocks is even. If`is odd, one hasW(B_`) hw₀iW(D_`). Hence for every -partition l with an even number of negative blocks, t ¹(flg) consists of a singleW(D`)-conjugacy class. When`is even,t ¹(flg) consists of either 2 or 1W(D`)-conjugacy classes.

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Restriction of the mapping p:W(B`)!S` of section 3.3 yields a surjective mappingp⁰:W(D`)!S` with kernelker(p⁰)W(A^`₁ ¹). Note that the only proper rootsubsystemsCofFfor whichp⁰(W(C))Sàre of type A_` ₁(cf. Prop. 3.3). There is a uniqueW(D_`)-orbit of such subrootsystems in case thatìs odd, and there are two suchW(D_`)-orbits forèven.

3.5 ±The classification of admissible2-coset coverings for Weyl groups The information we have collected in the previous subsection can now be used to classify all admissible 2-coset coverings:

THEOREM3.4. LetFbe a finite irreducible crystallographic rootsystem, let WW(F)be its Weyl group and letg:W be an admissible2-coset (cf. 2.1). Assume that(n_C:W(C);n_J:W(J))is an admissible2-coset covering ofg:W. Then the types of(n_C:W(C);n_J:W(J))- or(n_J:W(J);n_C:W(C))- are as listed in Table 3.1. Moreover, every tuple(nJ:W(J);nC:W(C))given as in Table 3.1 is an admissible2-coset covering ofg:W.

TABLE3.1. - Admissible 2-coset coverings

typeg:W nC:W(C) nJ:W(J)

A`; `1 A1 1 2

A2 W(A1) 3

A3 W(A2) 2:W(A²₁)

2A`; `2 ²A2 2:W(A1) 6

2A3 2:W(A2) 2:W(A²₁)

B`; `2 B2 W(A~²₁) 2:W(A~²₁) B3 W(B2A~1) 3:W(A~³₁) B4 W(B3A~1) 2:W(B²₂)

B` W(D`) 2:W(D`)

F4 W(B4) 3:W(D4)

W(C4) 3:W(D~4)

G2 W(A2) 2:W(A2)

W(A~2) 2:W(A~2)

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The caseFof typeC` leads to the same examples as forFof typeB`

with short and long roots exchanged. Therefore, we have not listed these cases in Table 3.1 explicitly.

PROOF. Assume that (n_C:W(C);n_J:W(J)) is an admissible 2-coset covering of g:W. We denote by n_C and n_J the canonical homomorphic images ofnC andnJ in NW(W(C))=W(C) andNW(W(J))=W(J), respectively. We may assume thatg2nC:W(C). Thus in case thatg1,nJ:W(J) contains a Coxeter element and thus n_J61 (cf. Prop. 3.1). The proof proceeds by a case-by-case analysis depending on the type ofg:W.

3.5.1 Ðg:W of typeA`

For`3 an easy exercise shows that the cases listed in Table 3.1 are the only examples one has in this case. Since for `4 no proper Weyl subgroup of W is normal in W, (W(C);NW(W(J)) is a 2-covering of W.

From [3, Thm. 4.2] one concludes that this implies `5. It remains to analyze the cases`4;5 explicitly.

Assume`4, i.e.WS₅. For any proper non-trivial subrootsystemJ ofF,N_W(W(J))=W(J) is a 2-group. HenceJ ;must be trivial, andn_Jis an element of order 5. The maximal subrootsystems ofFare of typeA3and A2A1, and thusW(C) does not contain elements of cycle type (2;3) in the first case and does not contain elements of cycle type (1;4) in the latter, respectively. Hence admissible 2-coset coverings do not exist for`4.

Assume `5, i.e., WS₆. There is one W-conjugacy class of Weyl subgroups W(C)^W containing a 5-cycle corresponding to the subrootsystems of typeA4. The groupW(C) does not contain elements of cycle type (2;2;2), (3;3) and (6). The only proper non-trivial subrootsystemsJ for whichN_W(W(J)) contains elements of cycle type (6) are of typeA³₁and A²₂. However, forJof typeA³₁andn_Jof order 3,n_J:W(J) does not contain elements of cycle type (2;2;2); forJof typeA²₂andn_Jof order 2,n_J:W(J) does not contain elements of cycle type (3;3). This implies thatnJ:W(J) must contain elements of cycle type (1;5). As it already contains a Coxeter element, one hasJ6 ;. This implies thatN_W(W(J))S₆(cf. Prop. 3.2(a)), and thusW(J)W, a contradiction. Hence admissible 2-coset coverings do not exist for`5.

3.5.2 Ðg:W of type²A_`

In this casegis an inner automorphism, andWe hgi:W h idEi:W, where id_E denotes the endomorphism acting as 1 on E. Thus if

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(nC:W(C);nJ:W(J)) is an admissible 2-coset covering ofg:W, the pair of admissible cosets (( idE)nC:W(C);( idE)nJ:W(J)) is an admissible 2- coset covering ofW. Thus the claim can be deduced from 3.5.1.

3.5.3 Ðg:W of typeB` (orC`)

By 3.5.1 and the previous discussion (cf. § 3.3), we have to consider three cases:

(1) p(W(C))W=E, (2) p(nJ:W(J))W=E,

(3) `4 and (p(W(C));p(nJ:W(J)) is an admissible 2-coset covering ofS`.

Note thatp(nJ:W(J)) is a group, if and only ifnJ:W(J)\E6 ;. Hence

p(nJ:W(J))W=Eimpliesp(W(J))W=E.

Cases (1) and (2): The only proper subrootsystemsCofFsatisfying (1) are of typeD_àndA_` ₁(cf. Prop. 3.3). A similar argument yields that (2) implies that J is either of type D_` orA_` ₁. Hence in order to prove the claim it suffices to show thatCof typeDìmpliesJof typeD`, and thatJ of typeDìmpliesCof typeD`.

Assume thatCF_longis of typeD`. ThennJ:W(J) contains a Coxeter element and elements of theW-conjugacy classes (2;1; ::;1j1) and (1j` 1).

Hencep(n_J:W(J))W=E(cf. Prop. 3.2(a)). Asp(n_J:W(J))p(W(J)), the rootsystemJ must be of type A` 1 orD`. Assume J is of typeA` 1. As w02NW(W(J)) and jNW(W(A` 1))=W(A` 1)j 2, one has nJ:W(J)

w0:W(J). However, in this casew0:W(J) does not contain short reflections corresponding to the-partition (1; ::;1j1) (cf. § 3.3). ThusCF_long impliesJF_long.

Assume thatJF_longand thatn_Jis inducing the graph automorphism of order 2. ThennJ:W(J) does not contain anyW-conjugacy class being contained in W(F_long). In particular, W(C) contains elements of type (2;1; ::;1j1;1), (1; ` 1j), (`j). Thusp(W(C))W=E(cf. Prop. 3.2(a)), andC is either of typeA_` ₁orD_`. However, sinceW(C) contains elements of type (2;1; ::;1j1;1) the first case is impossible. This yields the claim.

Case (3): The 3 examples forFof typeA`,`1, 2, 3 yield the 3 addi- tional cases for F of type B`, `2, 3, 4. A straightforward calculation shows that these are the only exceptions.

3.5.4 Ðg:W of typeD`

By § 3.5.1 one has to consider three cases:

(1) p(W(C))'W=E⁰,

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(2) p(nJ:W(J))'W=E⁰,

(3) `4,p(W(C))'S3,p(nJ:W(J))2:W(A²₁)S4.

Case (1): The hypothesis implies that W(C) is of typeA` 1 (cf. Prop.

3.3). HencenJ:W(J) must contain an element of type (1; ::;1j1;1), and thus p(nJ:W(J)) is a group. Furthermore,nJ:W(J) must contain elements of type (j1; ` 1), (j2; ` 2) and (2;1::;1j1;1), and this implies that p(n_J:W(J))W=E⁰(cf. Prop. 3.2(b)). So alsoW(J) must be of typeA_` ₁. But this yields that`must be odd and thatn_J:W(J)w₀:W(J). However, in this case neither W(C) nor nJ:W(J) contains elements of type (1; ::;1j1;1).

Case (2): The hypothesis implies thatp(W(J))W=E⁰, and thusJis of type A_` ₁, ` must be even n_J:W(J)w₀:W(J). So W(C) must contain elements of type (2;1; ::;1j), (j2; ` 2) and (1; ` 1j). This yields that p(W(C))W=E⁰(cf. Prop. 3.2(b)) and thusCmust be of typeA` 1. As in the previous case neither W(C) nor nJ:W(J) contain elements of type (1; ::;1j1;1). So an admissible 2-coset covering cannot exist in this case.

Case (3) can be excluded, since in this caseN_W(W(J)) is a 2-group, but must also contain a Coxeter element which is of order 6, a contradiction.

3.5.5 Ðg:W of type²D`

If`is odd,g:W(F)w₀:W(F)W(B_`), and with the same argument as used in § 3.5.2 one concludes that there are no admissible 2-coset coverings in this case. So`must be even. We may assume thatnCg. Note thatgis of type (1; ::;1j1) considered as an element inW(B`). Hencep(W(C)) is a group. By § 3.5.1, we have to consider 2 cases:

(1) p(W(C))'W=E⁰, (2) p(nJ:W(J))'W=E⁰,

(3) `4,p(W(C))'S₃,p(n_J:W(J))2:W(A²₁)S₄.

Case (1): The hypothesis implies thatCis of typeA_` ₁. However, for` even there does not exist ag-subrootsystem of typeA_` ₁(cf. [25, § 2, (C)]).

Case (2): The hypothesis implies thatp(W(J))'S_`, and thusJmust be of typeA` 1. The argument which was used in case (1) yields that this is impossible.

Case (3): By hypothesis, p(nJ:W(J)) is not a group. Thus nJ:W(J)\

\E ;. The hypothesis p(g:W(C))'S₃ yields that C is of type A₂ and thatgis centralizingW(C). So the elements ofg:W(J) are of type (1;1;1j1), (1;2j1), (3j1). Hence neitherg:W(C) nornJ:W(J) contains elements of type (1j1;1;1), and thus an admissible 2-coset covering cannot exist in this case.

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3.5.6 Ðg:W of type³D₄

If one identifies the Weyl groupW with the Weyl subgroupW(D4) W(F4) associated to the subrootsystem of long roots in the rootsystem of typeF₄,g:W3:W(D₄)W(F₄) is aW(D₄)-coset in the Weyl group W(F₄) of type F₄. Since W(F₄) has an admissible 2-covering (W(B₄);3:W(D₄)) (cf. § 3.5.11), one concludes that 3:W(D₄) contains Cox- eter elements of the Weyl group W(F4) which are of order 12 and which are the twisted Coxeter elements in 3:W(D4) (cf. [23, § 7]). Letw2g:Wbe such an element being contained in n_J:W(J). From [25, Prop. 7] one concludes thatJ ;must be trivial.

The properg-invariant subrootsystems of the rootsystemFare of typeA⁴₁, A³₁,A2,A1,;(cf. [25, § 2, (B)]). The cosetnC:W(C) must intersect the twoW- conjugacy classes of elements of order 3 being contained in 3:W(D4) non-tri- vially.HenceW(C) cannotbea2-groupwhichleavesonlythecaseCoftypeA₂. Inparticular,one may assume thatn_Cgand thatgiscentralizingW(C). Itis straightforward, thatn_C:W(C) is a subset of a Weyl subgroupW(A₂A~₂) W(F4) of typeA2A~2inW(F4). HencenC:W(C) has trivial intersection with theW(F4)-conjugacy class containingg:( idE), a contradiction.

3.5.7 Ðg:W of typeE6

We may assume that CF is a maximal subrootsystem of F. The maximal subrootsystems of the rootsystem of type E6 are of type D5, A5A1 andA³₂. From Table 3.2 one concludes that W(C) has trivial intersection with aW-conjugacy class which is not the Coxeter class. This implies thatJis not empty.

Since proper Weyl subgroups of WW(F) are not normal in W, N_W(W(J))6W. Hence it suffices to show thatW(F) cannot be covered by the W-conjugates of W(C), C a maximal subrootsystem of F, and a maximal subgroup M<W containing N_W(W(J)). The maximal subgroups M<W of W'O₆(2) are known (cf. [9, p. 26]), and from their permutation characters x_M corresponding to the left C[W] module Ind^W_M(C) one can determine theW-conjugacy classes they do not contain.

In Table 3.2 we have listed someW-conjugacy classes which have trivial intersection with some of the maximal subgroups, and also the decomposition of the permutation character x_M in terms of the characters oc- curing in the character table in [9]. We have used the symbolto express that the conjugacy class has non-trivial intersection with the maximal subgroup and the symbol R in case the intersection is trivial. By sgn:W ! f1gwe denote the sign character.

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Since M has to contain Coxeter elements (cf. Prop. 3.1) which correspond to the conjugacy class 12A, one concludes from Table 3.2 thatMhas to be conjugate to either M3 orM5. SinceM1 and M4 have trivial intersection with the conjugacy class 10A,Chas to be of typeA5A1. HenceM cannot be conjugate toM5, as this group has trivial intersection with the conjugacy class of type 9A. ButMcannot be of typeM₃either, since this group has trivial intersection with the conjugacy class of type 12C.

3.5.8 Ðg:W of type²E₆.

Sinceg:W( idE):W, the same argument which was used in § 3.5.2 and the non-existence of admissible 2-coset covers for WW(E₆) (cf.

§ 3.5.7) show that admissible 2-coset covers do not exist in this case.

We may assume thatCis a maximal subrootsystem ofF. ThusCis of typeE₆,A₇,D₆A₁orA₅A₂. Let :W!Sp₆(2) denote the canonical projection on the symplectic group of degree 6 over the field with 2 elements. Some information on maximal subgroups and conjugacy classes of the groupSp6(2) has been collected in Table 3.3 (cf. [9, p. 46]).

From Table 3.3 one concludes that there exists a W-conjugacy class which is not the Coxeter class having trivial intersection withW(C). Hence J is non-trivial. Since proper Weyl subgroups of W are not normal, NW(W(J)) is contained in a maximal subgroup M of W containing w0 idE. The same technique which was used in § 3.5.7 and the information listed in Table 3.3 show that eitherCis of typeE6andMisW- conjugate to hw₀i:W(A₇), or C is of type A₇ and M is W-conjugate to TABLE3.2. - Maximal subgroups and conjugacy classes ofO₆(2)

maximal subgroup x_M 9A 12A 8A 10A 12C

M1W(D5) x₁x₄x₉ R R R

M2W(A5A1) x₁x₈x₉ R R R R M33¹² :2S4 x₁sgnx₇x₁₀ R R M4NW(W(A³₂)) x₁x₈x₁₀ R R R

M5CW(2A) x₁x₉x₁₀ R R R

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hw₀i:W(E₆). Sincew₀is not contained in Weyl subgroups of typeA₇orE₆, w02nJ:W(J), and we may assume thatJis of typeA7orE6, respectively.

However, in the first case, neither W(E6) nor w0:W(A7) contain Coxeter elements which are of order 18; in the second case neither W(A7) nor w₀:W(E₆) contain elements of order 9. Hence admissible 2-coset coverings cannot exist.

Let x:W! f1g denote the sign character of the reflection group WW(E₈), and let :W!O₈(2) denote the canonical projection on the orthogonal group of maximal Witt index of degree 8 over the field with 2 elements. As in the previous case we may assume that C is a maximal subrootsystem ofF. HenceCis of typeE₇A₁,D₈,A₈,E₆A₂orA²₄(cf.

[6]), and thusW(C) isW-conjugate to a subgroup of one of the maximal subgroupsM₁,M₄,M₇,M₁₀,M₁₅ of WO₈(2) (cf. Table 3.4). We have chosen the enumeration of theW-conjugacy classes of maximal subgroups in such a way that it coincides with its appearance in the list of maximal subgroups in [9, p. 85].

For each of the maximal subgroupsM₁,M₄,M₇,M₁₀,M₁₅, there exists oneW-conjugacy class which has trivial intersection with the maximal subgroup and which is not the homomorphic image of the Coxeter class (cf. Table 3.4). Asx(w)1 for a Coxeter elementw, this implies thatJis not empty.

TABLE3.3. - Maximal subgroups and conjugacy classes ofSp6(2)

maximal subgroup 3B 7A 8A 9A 15A

M1W(E6) R R

M2W(A7) R R R

M3W(D6A1) R R R

M4U3(3):2 R R R

M52⁶:L3(2) R R

M62:[2⁶]:(S3S3) R R R

M7W(A5A2) R R R

M8S2(8) R R

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Since W does not contain proper normal Weyl subgroups, NW(W(J))6W and NW(W(J)) is contained in a maximal subgroup M<W. LetMdenote its homomorphic image inW. From the previous remark one concludes that x(M) f1g. Hence M is W-conjugate to one of the maximal subgroups M₁, M₄, M₇, M₁₀, M₁₃C_W(2A), M₁₄, M₁₅. As M₁₃ and M₁₄ do not have elements of order 5, they cannot contain the homomorphic image of a Coxeter element whose order is divisible by 15. Hence M is W-conjugate to one of the maximal subgroups listed in Table 3.4.

Assume thatCis of type E₇A₁. ThenW(C) has trivial intersection with theW-conjugacy classes 9Band 15B. The first fact implies thatM cannot be W-conjugate toM15, while the latter implies thatM cannot be W-conjugate toM1,M4,M7orM10(cf. Table 3.4). The casesCof typeD8, A₈, E₆A₂ and A²₄ can be ruled out by a similar argument and the information listed in Table 3.4.

3.5.11 Ð g:W of typeF4

The maximal subrootsystems ofFare of typeB4,C4,C3A1,B3A~1, A~₃A₁, A₃A~₁ and A₂A~₂. Note that since there exists a surjective morphismb:W(F₄)!S²₃withker(b) a 2-group,WW(F₄) has precisely 3-conjugacy classes s^W_i ,i1;2;3;of elements of order 3. The graph automorphism of F4 permutes 2 of these conjugacy classes and leaves s^W₃ fixed. We may assume that s^W₁ has non-trivial intersection with Weyl subgroups of typeC₄, ands^W₂ has non-trivial intersection with Weyl subgroups of type B₄. Since no maximal Weyl subgroup contains Coxeter elements of the Weyl subgroup of typeB₄ - which have order 8 - and also elements from theW-conjugacy classs^W₃ ,Jcannot be trivial.

TABLE3.4. - Maximal subgroups and conjugacy classes ofO₈(2)

maximal subgroup 7A 9A 9B 15B

M1W(E7A1) R R

M4W(D8) R R R

M7W(A8) R R

M10W(E6A2) R R

M15N_W(W(A²₄)) R R R

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Since nJ:W(J) contains Coxeter elements, one concludes from [25, Prop. 7] thatNW(W(J))NW(W(D4)) orNW(W(J))NW(W(D~4)). Hence Jis either of typeD4orD~4.

AssumeJis of typeD₄. Since (W(B₄);3:W(D₄)) is an admissible 2-coset covering,W(C) must contain elements from theW-conjugacy classs^W₂ and also Coxeter elements from the Weyl groupW(B₄). Thus from the knowledge of maximal Weyl subgroups of W one concludes that W(C)W(B4) is a subgroup of a Weyl group of typeB4. Proposition 3.1 then implies thatW(C) is of typeB₄. Applying the graph automorphism yields the caseJof typeD~₄.

3.5.12 Ðg:W of typeG₂

The subrootsystems ofFare of typeA2,A~2,A1A~1,A1,A~1 or;. The W-conjugacy classes are 1,w0 idE,r^W_l ,r^W_s ,s^W,t^W, wherer_lis a long root,r_s a short root,sof order 3 andt of order 6. Straightforward arguments show that forCA₂orA~₂,n_J:W(J) must be as described in Table 3.1, and that there does not exist an admissible 2-coset covering forC of typeA1A~1.

3.5.13 Ðg:W of type²B2

Non-trivialg-invariant proper subrootsystems do not exist in this case (cf. [25, § 2, (D)]). HenceCJ ;, and this yields that admissible 2-coset coverings do not exist in this case.

3.5.14 Ðg:W of type²G2

Every non-trivial properg-subrootsystem is of typeA~₁A₁(cf. [25, § 2, (D)]). Note thatWe 'D(24) is isomorphic to a dihedral group of order 24. In particular,g:Wcontains twoW-conjugacy classes of elements of order 12 and oneW-conjugacy class of elements of order 4. AsN_We(W(A~1A1)) is a 2-group of order 8, an admissible 2-coset covering (n_C:W(C);n_J:W(J)) must satisfyCF ;andn_Candn_Fmust be of order 12. However, for this choice neithernC:W(C) nornJ:W(J) contains elements of order 4, and thus admissible 2-coset coverings cannot exist in this case.

3.5.15 Ðg:W of type²F₄

Every non-trivial proper g-subrootsystem is of type B²₂, A~2A2 or A~1A1. (cf. [25, § 2, (D)]). Let r_i, i1;2;3;4, be the simple reflection associated to a basis D of F ordered in the natural way. Then s:gr₁r₂2g:W is an element of order 24. We may assume thatn_J:W(J) contains an element W-conjugate to s. Since N_W(W(A~₁A₁)) and

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NW(W(B²₂)) are 2-groups and as NW(W(A~2A2))( idE):W(A~2A2) (cf. [6, Tab. 8]),Jmust be trivial.

Moreover, g:W contains elements of order 6 and 8. Hence the same argument yields thatCcannot be of typeA~₁A₁,B²₂ orA~₂A₂. But as g:W has more than two W-orbits, also C ;is impossible showing that admissible 2-coset coverings do not exist in this case. p

4. The proof of TheoremA - part I

In this section we will prove that a generic 2-coveringfH;Kgof the finite group of Lie type (G;F) must satisfy the conditions of Theorem A (cf. Thm.

4.5). In the subsequent 3 sections it will be proved that every pair of subgroup fH;Kgas described in Theorem A is indeed a generic 2-covering. We start with the following property - the easy proof is left to the reader.

PROPOSITION 4.1. Let (G;~ F) be a finite group of Lie type, and let i: ~G!G be a finite F-invariant central isogeny. LetfH;~ Kg~ be a pair of F- stable closed subgroups of maximal rank, and let H and K denote their canonical images underi. Then, iffH;Kgis a generic2-covering of(G;F), fH;~ Kg~ is a generic 2-covering of (G;~ F). In particular, one has the following:

(a) Let(G;F)be a finite group of Lie type of adjoint type. Assume that the pair of F-stable subgroupsfH;Kgof maximal rank is a generic2- covering. Then for every group of Lie type G isogeneous toG and~ H and~ K~ defined as above,fH;~ Kg~ is a generic2-covering of(G;~ F).

(b) Let(G;~ F)be a simply-connected finite group of Lie type, and let H,~ K be F-stable subgroups of maximal rank. Assume that~ fH~_F;K~_Fg is not a2-covering ofG~_F. ThenfH_F;K_Fg- with H and K as defined above - is not a2-covering of G_Ffor all groups of Lie type(G;F)isogeneous to(G;~ F).

4.1 ±Three exceptions

It seems remarkable to us that almost all the admissible 2-coset coverings listed in Table 3.1 give rise to a generic 2-covering - sometimes with a further characteristic restriction. However, there are three cases in Table 3.1 which will not correspond to a generic 2-covering:

(1) (G;F) of type C4,H reductive of type W(C3)W(C1),K reductive of type 2:W(C₂²),

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(2) (G;F) of type A3,Hreductive of type W(A2), K reductive of type 2:W(A²₁),

(3) (G;F) of type²A3,Hreductive of type 2:W(A2),Kreductive of type 2:W(A²₁).

These cases will now be analyzed by a case-by-case analysis. If the characteristic of the field is different from 2, neitherH_FnorK_Fwill contain regular unipotent elements (cf. [20]). Therefore, we may restrict our con- sideration to the case when the field of definition is of characteristic 2.

Case (1):Let (G;F) be the simply-connected finite group of Lie type of typeC4 defined over the algebraically closed fieldF2 of characteristic 2, and letHandKbeF-stable reductive subgroups of maximal rank of type W(A₁C₃)^W and 2:W(C²₂)^W, respectively. We putK :N_G(K). In particular,

G_F'Sp₈(q); H_F 'Sp₂(q)Sp₆(q); K_F'2:Sp₄(q²) 4:1

for some 2-powerq2^f. One has:

PROPOSITION4.2. The pair of subgroupsfHF;KFgis not a2-covering of G_F.

PROOF. Assume that fH_F;K_Fg is a 2-covering of (G;F). Let x:x_sx_u2G_Fbe an element wherex_sis a semisimple element generating a Coxeter torus inSp4(q) and wherexu is a regular unipotent element in Sp4(q). Hencex2Sp4(q)Sp4(q), andSp4(q)Sp4(q) Sp8(q) is sta- bilizing the 4-dimensional summands V4 and V₄⁰ of a orthogonal decomposition of the natural moduleV V₈ofSp₈(q). From the action ofxonV, it is obvious thatxcannot be contained in anyG_F-conjugate ofH_F. Hence it must be contained in aG_F-conjugate ofK_F, and thus we may assume that x2KF. Sincexu is acting as a Jordan block of size 4 onV₄⁰,xucannot be contained in K_F. Moreover, one concludes that xs 2CK_F(xu)'Sp4(q) is acting diagonally onV. But this yields a contradiction, since we assumed thatx_sis leaving a 4-dimensional subspace ofVfixed. p Case (2): Let G be the reductive group GL₄ defined over the algebraically closed fieldF2 of characteristic 2, and letHandK beF-stable reductive subgroups of maximal rank of typeW(A2)^W and 2:W(A²₁)^W, respectively. We put alsoK:N_G(K). In particular,

GF'GL4(q); HF 'GL3(q)GL1(q); KF'2:GL2(q²) 4:2

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for some 2-power q2^f. LetV4 denote the natural Fq[GL4(q)]-module, and let V :V2V₂⁰ be a direct decomposition in subspaces of Fq-dimension 2. Then using the element x:xsxu, where xs is a semisimple element acting as a Singer cycle onV₂and fixingV₂⁰, and wherex_uis a non- trivial unipotent element fixingV₂ and acting non-trivial onV₂⁰, the same argument which was used in the proof of Proposition 4.2 shows that fHF;KFg is not a 2-covering of GF. Moreover, since in this case GL4(q)SL4(q)(q 1) this yields that fSL4(q)\HF;SL4(q)\KFg is not a 2-covering ofSL₄(q). Hence one has:

PROPOSITION4.3. Let(G;F)be a simply-connected group of Lie type of type A3 defined over the algebraically closed field F2 of characteristic2.

Let H be an F-stable reductive subgroup of maximal rank of type W(A2)^W, and let K be an F-stable reductive subgroup of maximal rank of type 2:W(A²₁)^W. Then for K:N_G(K),fH;Kg is not a generic 2-covering of (G;F).

Case (3): Let G be the reductive group GL4 defined over the algebraically closed field F2 of characteristic 2, and let F be a Frobenius morphism which is the composition of a standard Frobenius morphism with a graph automorphism. Let H and K be F-stable reductive subgroups of typeW(A₂)^W and 2:W(A²₁)^W, respectively. There are obviously twoW-conjugacy classes of type 2:W(A²₁)^W. The one we are considering is the one corresponding to theGF-conjugacy class of Levi complements of the maximal parabolic subgroup of type A1A1. Then for K:N_G(K) one has:

GF 'GU4(q); HF 'GU3(q)GU1(q); KF '2:GL2(q²)^] 4:3

for some 2-powerq2^f, whereGL₂(q²)^]denotes the group of 2-by-2 matrices X over F_q2 whose norm of the determinant is equal to 1, i.e., N_F2

q=Fq(det(X))1.

Let V4 be the natural F_q²[GU4(q)]-module equipped with the non-degenerate hermitian form (:; :). LetV₄V₂V₂⁰ be an orthogonal decomposition into non-degenerate 2-dimensional F_q2-subspacesV₂ and V₂⁰. By choosing an elementxx_sx_u, wherex_uis acting as a non-trivial unipotent element onV₂⁰and fixingV2pointwise, and wherexsis acting as an element of order q 1 onV2 and fixingV₂⁰ pointwise, the same argument as pre- sented for Case (1) shows that fH_F;K_Fg is not a 2-covering of G_F

GU₄(q). Thus using Proposition 4.1 and applying the same technique

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which was used to reduce to the simply-connected case (cf. Case (2)) one obtains the following:

PROPOSITION4.4. Let(G;F)be a simply-connected group of Lie type of type²A₃ defined over the algebraically closed fieldF₂ of characteristic2.

Let H be an F-stable reductive subgroup of maximal rank of type 2:W(A2)^W, and let Kbe an F-stable reductive subgroup of maximal rank of type 2:W(A²₁)^W corresponding to the Levi complement of the F-stable parabolic subgroup of type A²₁. Then for K:N_G(K), fH;Kg is not a generic2-covering of(G;F).

4.2 ±Generic2-coverings

From Theorem 2.2 and 3.4 and the main result of [20] one concludes the following:

THEOREM4.5. LetfH;Kgbe a generic2-covering of the finite group of Lie type(G;F). Then H and K are of maximal rank.

(a) Assume that for one subgroup say H, Ru(H)61. Then(G;F)is of type A1, A2, A3 or²A3, H is maximal parabolic and K is reductive.

Moreover, the types of the Levi complement H=R_u(H)of H and of Kis as listed in Table 1.1.

(b) Assume that Hand Kare reductive. Then the types of(G;F), Hand K and the characteristic ofF is as listed in Table 1.2.

PROOF. By definition, one of the subgroups - sayH- must contain a maximal torus, and thus it is of maximal rank. Then eitherK is also of maximal rank, orHmust contain anF-stable maximal torus of eachGF- conjugacy class. Assume the latter case holds. IfH is not reductive, by enlarging the groupH we may assume thatH is anF-stable parabolic subgroup of G (cf. [13, Thm. 30.4]). In particular, an F-stable Levi complement L must contain an F-stable maximal torus of each GF- conjugacy class. Thus in either case G must contain an F-stable reductive subgroupRcontaining anF-stable maximal torus of eachG_F- conjugacy class. From Proposition 3.1 and [25, Prop. 4] one concludes that this implies that (G;F) must be of twisted type. Moreover, the arguments used in § 3.5 show that (G;F) cannot be of type ²A`,²D`, ` odd, ²E6 either. For none of the remaining types there exist an admissible 2-coset covering for the Weyl group coset (cf. Thm. 3.4). Hence