On Self-Normalising Sylow $2$-Subgroups in Type A

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by  Amanda Schaeffer Fry

A
On Self-Normalising Sylow 2-Subgroups in Type
Amanda Schaeffer Fry and Jay Taylor
Abstract. Navarro has conjectured a necessary and sufficient condition for a finite group G to
have a self-normalising Sylow 2-subgroup, which is given in terms of the ordinary irreducible
characters of G. The first-named author has reduced the proof of this conjecture to showing that
certainrelatedstatementsholdwhenGisquasisimple. Inthisarticleweshowthattheseconditions
aresatisfiedwhenG/Z(G)isPSL (q),PSU (q),orasimplegroupofLietypedefinedoverafinite
7 n n
1 fieldofcharacteristic2.
0
2

n
a
J 1. Introduction

1

] 1.1. For any integer n > 1 we will denote by Qn the nth cyclotomic field, obtained from the rationals
T
Q by adjoining aprimitive nth rootofunity. In [SF16], thefirst-named author began an investigation into
R
. the following conjecture.
h
t
a Conjecture 1.2 (Navarro). LetGbeafinitegroupandletσ Gal(Q /Q)beanautomorphismfixing2-rootsof
m ∈ |G|
unityandsquaring2-roots ofunity. Then G has aself-normalising Sylow2-subgroup ifandonlyifevery ordinary
′
[
irreducible character of G with odd degree isfixed by σ.
1
v
1.3. This statement would be an immediate consequence of the Galois-McKay conjecture, which is a
2
7 refinement of the well-known McKay conjecture due to Navarro, see [Nav04, Conjecture A]. For a finite
2
0 group G we denoteby Irr(G) the set ofordinary irreducible characters and given a prime ℓ we denoteby
0 Irrℓ (G) Irr(G) thoseirreduciblecharacters whosedegreeiscoprime to ℓ. TheGalois-McKay conjecture
. ′ ⊆
1 then posits that for any finite group G, prime ℓ, and Sylow ℓ-subgroup P 6 G, there should exist a
0
7 bijection between Irrℓ (G) and Irrℓ (NG(P)), as predicted by the McKay conjecture, which behaves nicely
1 ′ ′
: with respectto the action of certain elements of the Galois group.
v
i
X 1.4. While the McKay conjecture has been reducedto proving certain inductive statementsfor simple
r groupsin [IMN07], and even recentlyproven for ℓ = 2 in [MS15], no such reduction yetexistsfor Galois-
a
McKay. Further, a reduction for Galois-McKay seems further from fruition in the case ℓ = 2 than for
odd primes. However, a proof of Conjecture 1.2 would provide more evidence for the conjecture, and
we consider it to be a weak form of the Galois-McKay refinement for ℓ = 2. We hope that some of the
observations made in the course of proving Conjecture 1.2 will be useful in working with an eventual
reduction for Galois-McKay for ℓ = 2. We also remark that the corresponding weak form for odd ℓ has
been proven in [NTT07].
1.5. The main result of [SF16] is a reduction of Conjecture 1.2 to certain inductive statements for
simple groups, which we recall below in Section 2, and the verification of these statements for some
simple groups. The goal of this work is to extend and simplify the proofs there in order to complete
the verification for simple groups of Lie type in characteristic 2 and simple groups of type A in all
characteristics. Specifically we prove the following.
2
Theorem A. Assume G is a simple and simply connected algebraic group defined over K = F , an algebraic
p
closure of the finite field F of prime order p > 0, and let F : G G be a Frobenius endomorphism of G. If either
p
→
p = 2 or G = SL (K), then whenever the quotient GF/Z(GF) issimple, it is SN2S-Good.
n
1.6. One of our key tools used in the proof of Theorem A is Kawanaka’s generalised Gelfand–Graev
representations (GGGRs). These are a family of characters which have already shown themselves to
be remarkably useful for deducing the action of automorphisms of a finite reductive group on the set of
irreduciblecharacters,see[CS15]and[Tay16a].Oneofthereasonswhytheyaresousefulisthattheimage
of a GGGR under an automorphism of the group is again a GGGR and the resulting GGGR can be easily
described. Here we show that the same holds for certain Galois automorphisms, see Proposition 4.10.
The statement holds whenever the GGGRs are definedand may be of independentinterest.
1.7. In [SF16] it is shown that all sporadic simple groups and simple alternating groups are SN2S-
Good. Thus we are left with checking that most simple groups of Lie type defined over a field of odd
characteristic are SN2S-Good. In this situation one should be able to employ the Harish-Chandra tech-
niques used by Malle and Spa¨th in [MS15] to solve the McKay Conjecture for ℓ = 2. However, this is
ultimately quite different from our line of argument hereand will be consideredelsewhere.
1.8. Wenowoutlinethestructureofthepaper. InSection 2,wediscussthereductionofConjecture 1.2
to simple groups proved in [SF16]. In Section 3, we introduce some general notation regarding finite
reductive groups and the action of the Galois group on Lusztig series under specific conditions. In
Section 4, we continue this discussion by introducing generalized Gelfand-Graev characters and their
behavior under the action of the Galois group. Sections 5 and 6 are dedicated to proving Theorem A in
the case that p = 2. In the remaining sections, we prove TheoremA for G = SL (K).
n
2. The Reduction Statements for Simple Groups
2.1. In [SF16] it was shown that Conjecture 1.2 holds for any finite group if every finite simple group
is SN2S-Good. The notion of being SN2S-Good is comprised of two conditions. One condition is on the
simple group itself and the secondis on its quasisimple covering groups. Beforestating theseconditions,
we introduce some notation.
2.2. Notation. Let G be a finite group. We will denote by Aut(G) the automorphism group of G. If
Q 6 Aut(G) is any subgroup then we denote by GQ the semidirect product of Q acting on G. As in the
introduction,Irr(G) denotesthesetofordinaryirreduciblecharactersof G andIrrℓ (G) Irr(G) istheset
′ ⊆
ℓ ℓ ℓ
of those irreducible characters whose degree is coprime to , where is a prime. The set of all Sylow -
subgroupsof G will be denotedby Syl (G). If H 6 G is a subgroupof G and χ Irr(H) is an irreducible
ℓ
∈
character then we denote by Irr(G χ) the set of all irreducible characters ψ Irr(G) whose restriction
| ∈
ResG(ψ) to H contains χ as an irreducible constituent;we say that ψ covers χ. Moreover,for any element
H
g G we denote by gχ the irreducible character of gH = gHg 1 defined by gχ(h) = χ(g 1hg) for all
− −
∈
h gH. We will write Irrℓ (G χ) for the intersection Irr(G χ) Irrℓ (G).
∈ ′ | | ∩ ′
From this point forward σ Gal(Q /Q) will denotethe Galois automorphism fixing
∈ |G|
2-roots of unity and squaring 2-roots of unity, c.f., Conjecture 1.2.
′
3
Condition 2.3. Let G be a finite quasisimple group with centre Z 6 G and Q 6 Aut(G) a 2-group.
Assume there exists a Q-invariant Sylow 2-subgroup P/Z Syl (G/Z) such that C (Q) = 1. Then
∈ 2 NG(P)/P
for any Q-invariant and σ-fixed λ Irr(Z), we have χσ = χ for any Q-invariant χ Irr (G λ).
2
∈ ∈ ′ |
Condition 2.4. LetSbeafinitenonabeliansimplegroupand Q 6 Aut(S)a2-group. Assume P Syl (S)
∈ 2
is a Q-invariant Sylow then if every Q-invariant χ Irr (S) is fixed by σ we have C (Q) = 1.
∈ 2′ NS(P)/P
Definition 2.5. Let S be a finite non-abelian simple group. We say S is SN2S-Good if Condition 2.4 holds
for S and Condition 2.3 holds for any quasisimple group G satisfying G/Z = S.
∼
2.6. We end this section with some remarks concerning the above conditions. Firstly, if P 6 G and
Q 6 Aut(G)areasinCondition 2.3thentheconditionthatC (Q) = 1isequivalenttoGQ/Zhaving
NG(P)/P
aself-normalisingSylow2-subgroup,see[NTT07,Lemma2.1(ii)].Secondly,assumeGisquasisimplewith
simplequotientS = G/ZandletGˆ beauniversalperfectcentralextension,orSchurcover,ofS. Itiseasily
checked that if Condition 2.3 holds for Gˆ then it holds for G. Indeed, as Gˆ is a Schur cover there exists a
surjectivehomomorphismGˆ G withcentralkernel. Thisinducesaninjectivemap Irr(G) Irr(Gˆ)and
→ →
a surjective homomorphism Aut(Gˆ) Aut(G), see [GLS98, Corollary 5.1.4(a)], and the claim follows.
→
Remark 2.7. We note that a simplified version of one side of the reduction, namely Condition 2.3, has
been proven in [NT15]. However, for the purposesof this paper, we work with our strongercondition.
3. Galois Automorphisms and Lusztig Series
From this point forward we denoteby K = F an algebraic closure of the finite field of
p
ℓ
prime order p. Moreover, denotesa prime.
3.1. The Basic Setup. We introduce here the basic setup that will be used throughout this article. In
particular, G will be a connected reductive algebraic group defined over K and F : G G will be a
→
Frobeniusendomorphismadmittingan F -rationalstructure G = GF. Moreover,wedenoteby ι : G ֒ G
q
→
aregularembedding,inthesenseof[Lus88, 7].TheFrobeniusendomorphismofGwillagainbedenoteed
§
by F and G = GF will be the resulting finite reductive group. e
We asseumeefixed pairs (G⋆,F⋆) and (G⋆,F⋆) dual to (G,F) and (G,F) respectively. As before we set
G⋆ = G⋆F⋆ and G⋆ = G⋆F⋆. We now cheoose an F-stable maximal toerus T 6 G and a dual F⋆-stable
0
maximal torus T⋆e6 Ge⋆. The group T := ι(T )Z(G) is then an F-stable maximal torus of G. Recall
0 0 0
that the regular embedding ι inducesea surjective hoemomorphism ι⋆ : G⋆ G⋆ which is defiened over
→
F . If T⋆ 6 G⋆ is a torus dual to T then ι⋆(T⋆) = T⋆ and ι⋆ is uniqueeup to composing with an inner
q 0 0 0 0
automoerphisme affected by an elemeent of T⋆. e
0
e
3.2. We will denote by (G,F) the set of all pairs (T,θ) consisting of an F-stable maximal torus
C
T 6 G and an irreducible character θ Irr(TF). Note we have an action of G on (G,F) defined by
∈ ∇
g (T,θ) = (gT,gθ); wewrite (G,F)/G for theorbitsunderthis action and [T,θ] for theorbit containing
· C
⋆ ⋆ ⋆ ⋆
(T,θ). Dually, we denoteby (G ,F ) the set of all pairs (T ,s) consisting of an F -stable maximal torus
S
T⋆ 6 G⋆ and a semisimple element s T⋆F⋆. Again we have an action of G⋆ on (G⋆,F⋆) defined by
∈ S
g (T⋆,s) = (gT⋆,gs), and we write (G⋆,F⋆)/G⋆ for the corresponding orbits and [T⋆,s] for the orbit
· S
4
⋆
containing (T ,s). By [DL76, 5.21.3], see also [DM91, 13.13], we have a bijection
Π : (G,F)/G (G⋆,F⋆)/G⋆
C → S
betweentheseorbits. Notethatthisbijectiondependsonthechoiceofagroupisomorphismı : (Q/Z)
p
′ →
K× and an injective group homomorphism  : Q/Z ֒ Q×ℓ , so we implicitly assume that such homomor-
→
phisms have been chosen.
⋆
3.3. For any semisimple element s G we denote by (G,F,s) (G,F) the set of all pairs (T,θ)
∈ C ⊆ C
such that Π([T,θ]) = [T⋆,t] and t is G⋆-conjugate to s. Now, to each pair (T,θ) (G,F), there is
∈ C
a corresponding Deligne–Lusztig character RG(θ), and we denote by (G,T,θ) the set χ Irr(G)
T E { ∈ |
χ,RG(θ) = 0 of its irreducible constituents. Note we will sometimes also write RG (s) for RG(θ)
h T iG 6 } T⋆ T
when Π([T,θ]) = [T⋆,s]. The union
(G,s) = (G,T,θ)
E [ E
(T,θ) (G,F,s)
∈C
is, by definition, a rational Lusztig series. The set of all irreducible characters is then a disjoint union
⋆
Irr(G) = (G,s), wherewe runoverall G -conjugacy classes ofsemisimple elements,see[Bon06,11.8].
SE
ℓ ℓ
If H is a finite group and x H is an element then we denote by xℓ, resp., xℓ , the -part, resp., ′-part, of
∈ ′
x = xℓxℓ = xℓ xℓ. With this we have the following.
′ ′
Lemma 3.4. Let s G⋆ be a semisimple element and let b,b Z be integers. If γ Gal(Q /Q) is an
automorphism sucht∈hatγ(ζ) = ζℓb forallℓ′-rootsofunityandγ′(ζ∈) = ζb′ forallℓ-rootsofu∈nity,then|G| (G,s)γ =
E
(G,sbℓ′sℓℓb).
E ′
Proof. Assume (T,θ) (G,F). Then by thecharacter formula for RG(θ) [Car93, 7.2.8], and thefact that
∈ C T
Green functions are integral valued, we easily deduce that RG(θ)γ = RG(θγ). In particular, as γ is an
T T
isometry we have (G,T,θ)γ = (G,T,θγ). Now, if Π([T,θ]) = [T⋆,s], then it is an easy consequence
E E
of the description of the map Π, see [DM91, 13], and the definition of γ that Π([T,θγ]) = [T⋆,sbℓ′sℓℓb].
§ ′
In particular this shows that (G,s)γ (G,sbℓ′sℓℓb). An almost identical argument shows that (G,s)
E ⊆ E ′ E ⊆
(G,sbℓ′sℓℓb)γ−1 (G,t) for somesemisimple element t G⋆. However,bythedisjointnessoftherational
E ′ ⊆ E ∈
series we must have equality which proves the lemma. (cid:4)
3.5. For any irreducible character χ Irr(G) we denote by ωχ : Z(G) Q×ℓ the central character
∈ →
determined by χ. This is a linear character defined by ω (z) = χ(z)/χ(1) for any z Z(G). The
χ
∈
following will prove to be useful later; it follows from [Bon06, 11.1(d)].
Lemma 3.6. For any two irreducible characters χ, ψ (G,s) we have ω = ω . In particular, if γ
χ ψ
∈ E ∈
Gal(Q|G|/Q) is an automorphism and E(G,s)γ = E(G,s) then ωχγ = ωχγ = ωχ for all χ ∈ E(G,s).
3.7. TryingtounderstandtheactionoftheGalois groupontheelementsofarationalLusztigseriesis,
in general, a difficult problem. However, in this section we will deal with two special cases. To describe
thesecases weneedtointroducesomenotation. For s G⋆ a semisimpleelement,we denoteby T⋆ 6 G⋆
∈ s
⋆ ⋆
a fixed F -stable maximal torus containing s; note that we then have T is contained in the centraliser
s
⋆ ⋆
CG⋆(s). We denote by W◦(s) = NCG◦⋆(s)(Ts)/Ts the Weyl group of the connected centraliser with respect
to this maximal torus. For each w ∈ W◦(s) we choose an F⋆-stable maximal torus T⋆s,w = gT⋆s 6 CG◦⋆(s),
5
where g ∈ CG◦⋆(s) is an element such that g−1F⋆(g) ∈ NCG◦⋆(s)(T⋆s) represents w. By [Bon06, 15.11] there
then exists a sign such that
1
ρs = ± W (s) ∑ RGT⋆s,w(s),
| ◦ | w∈W◦(s)
isacharacterof G. Each irreducibleconstituentofthischaracteriscontainedintherationalLusztigseries
(G,s) and is a semisimple character. Recall that a character is called semisimple if it is contained in the
E
Alvis–Curtis dual of a Gelfand–Graev character, see [DM91, 8.8, 14.39]. We first consider theaction ofthe
Galois group on these characters.
Proposition 3.8. Let γ be as in Lemma3.4 and assume s G⋆ is a semisimple element such that (G,s)γ =
∈ E
(G,s), then the following hold:
E
(a) ρ is fixed by γ,
s
(b) every semisimple character contained in (G,s) is fixed by γ if every Gelfand–Graev character of G is fixed
E
by γ.
Proof. If (G,s)γ = (G,s), then we have s is G⋆-conjugate to sbℓ′sℓℓb. From the arguments in the proof
E E ′
of Lemma 3.4 it is clear that, under this assumption, we have RGT⋆ (s)γ = RGT⋆ (s) so clearly ρs is fixed
s,w s,w
by γ. Now, if Γ is a Gelfand–Graev character of G and D denotes Alvis–Curtis duality, see [DM91,
G
8.8], then there exists a unique irreducible constituent χ of ρ such that D (Γ),χ = 0, see [Bon06,
s G G
h i 6
15.11]. Certainly we have χγ is both a constituent of ργ = ρ and D (Γ)γ. From the definition of D , and
s s G G
the character formula for Harish-Chandra induction/restriction [DM91, 4.5], it is not difficult to see that
D (Γ)γ = D (Γγ). Hence, if Γγ = Γ then we must have χγ is a constituent of D (Γ); but this implies
G G G
χγ = χ by the uniqueness. (cid:4)
3.9. The next case we wish to consider is that of GL (K). First, we introduce some notation that
n
⋆ ⋆
will be useful later. Specifically, let s G be a semisimple element. Then the Frobenius F induces
∈
⋆ ⋆ ⋆
an automorphism F : W (s) W (s) because T is assumed to be F -stable. We denote by W (s) the
◦ → ◦ s ◦
semidirect product W◦(s)⋊ F⋆ and for any class function f : W◦(s) Qℓ we define a correesponding
h i →
class function e
RGf (s) = W1(s) ∑ f(wF⋆)RGT⋆s,w(s)
| ◦ | w∈W◦(s)
of G. With this we can prove the following.
Proposition 3.10. Assume G is GL (K), γ is as in Lemma3.4, and s G⋆ is a semisimple element such that
n
∈
(G,s)γ = (G,s). Then every χ (G,s) is fixed by γ.
E E ∈ E
Proof. By [Lus84, 3.2, 4.23] every irreducible character in the Lusztig series (G,s) is of the form RG(s)
E f
where f : W◦(s) Qℓ is a rational valued irreducible character, see also [DM91, 13.25(ii), 15.4]. The
→ §
statement neow follows immediately from the fact that each RGT⋆ (s) is fixed by γ, c.f., the proof of
s,w
Proposition 3.8. (cid:4)
6
4. GGGRs and Galois Automorphisms
Inthissection,and inthissectiononly,weassumethat p isagoodprimefor G andthat
G is a proximate algebraic group in the sense of [Tay16b, 2.10]. Recall that this means
some (any) simply connected covering of the derived subgroupof G is seperable.
4.1. To any unipotent element u G Kawanaka has defined a corresponding generalised Gelfand–
∈
Graev representation (GGGR) of G which we denote Γ , see [Kaw85; Tay16b]. If u is a regular unipotent
u
element then Γ is a Gelfand–Graev character. Moreover, we have Γ = Γ for any g G. In this
u gug−1 u ∈
sectionwewishtodeterminetheeffectofσontheGGGRsof G;forthiswemustrecalltheirconstruction.
Let g denote the Lie algebra of G and let g, resp., G, denote the nilpotent cone of g, resp., the
N ⊆ U ⊆
unipotent variety of G. The Frobenius endomorphism F : G G induces a corresponding Frobenius
→
endomorphism F : g g on the Lie algebra. We have F( ) = and F( ) = .
→ U U N N
q
4.2. Let G denote the set K 0 viewed as a multiplicative algebraic group and let X(G) =
m
\ { }
Hom(G ,G) be the set of all cocharacters of G. Let F : G G denote the Frobenius endomorphism
m q m m
q → q
given by F (k) = kq, with q as in 3.1. Then for any λ X(G) we define a new cocharacter F λ X(G)
q
∈ · ∈
by setting
(F λ)(k) = F(λ(F 1(k)))
· q−
q q
for all k G . We denoteby X(G)F X(G) the set of all cocharacters λ satisfying F λ = λ.
m
∈ ⊆ ·
q
4.3. To each cocharacter λ X(G) we have a corresponding parabolic subgroup P(λ) 6 G with
∈
unipotent radical U(λ) 6 P(λ) and Levi complement L(λ) = C (λ(G )), see [Spr09, 3.2.15, 8.4.5]. The
G m
group G acts on g via the adjoint representationAd : G GL(g). Through Ad we have each cocharacter
→
λ defines a Z-grading g = g(λ,i) on the Lie algebra. For any i > 0 we have u(λ,i) = g(λ,j) is
i Z j>i
L∈ L
a subalgebraoftheLie algebraof U(λ) and it is theLie algebraofa closedconnectedsubgroup U(λ,i) 6
U(λ) which is normal in P(λ). The group L(λ) preserves each weight space g(λ,i) and we denote by
q
g(λ,2) g(λ,2) the unique open dense orbit of L(λ) acting on g(λ,2). Note that if λ X(G)F then
reg
⊆ ∈
the subgroups P(λ), U(λ), U(λ,i), and L(λ) are all F-stable and we set P(λ) = P(λ)F, U(λ) = U(λ)F,
U(λ,i) = U(λ,i)F, and L(λ) = L(λ)F.
4.4. The action of G on g preserves and the action of G on itself by conjugation preserves ; we
N U
denotethe resulting setsof orbits by /G and /G. Recall that each nilpotent orbit /G is of the
N q O O ∈ N
form = (AdG)g(λ,2) for some λ X(G), see[Tay16b, 3.22]. Moreover,if is F-stable thenwe may
reg
O q ∈ O
assume that λ X(G)F, see [Tay16b, 3.25]. Following [Tay16b, 4, 5] we assume a chosen G-equivariant
∈ § §
isomorphism of varieties φ : which commutes with F and whose restriction to each U(λ) is a
spr
U → N
Kawanaka isomorphism. In particular, the map φ satisfies the following two properties:
spr
(K1) φ (U(λ,2)) u(λ,2),
spr
⊆
(K2) φ (uv) φ (u) φ (v) u(λ,3) for any u,v U(λ,2).
spr spr spr
− − ∈ ∈
Note also that φ induces a bijection /G /G. Before introducing the GGGRs we consider the
spr
U → N
following lemmas, which were not coveredin [Tay16b].
7
q
Lemma 4.5. Foreach cocharacter λ X(G) wehave φ (U(λ,2)) = u(λ,2).
spr
∈
Proof. As φ is an isomorphism we have φ (U(λ,2)) is a closed subset of the same dimension as
spr spr
u(λ,2). As u(λ,2) is irreducible we must have φ (U(λ,2)) = u(λ,2). (cid:4)
spr
q
Lemma 4.6. Assume /G is such that φ ( ) = (AdG)g(λ,2) for some cocharacter λ X(G). Then
spr reg
O ∈ U O ∈
U(λ,2) is an open dense subset of U(λ,2) andis a single P(λ)-conjugacy class.
O∩
Proof. Choose an element e g(λ,2) and let u be the unique unipotent element satisfying
reg
∈ ∈ U
φ (u) = e. By Lemma 4.5 we have u U(λ,2) so the P(λ)-conjugacy class containing u is
spr ∈ OP(λ)
contained in U(λ,2) U(λ,2). We thus clearly have a correspondingsequence of closed sets
O∩ ⊆
U(λ,2) U(λ,2) U(λ,2).
OP(λ) ⊆ O∩ ⊆ O∩ ⊆
According to [Tay16b, 3.22(ii.b)] we have φ ( ) = (AdP(λ))e = u(λ,2). As φ is an isomorphism
spr OP(λ) spr
it follows from Lemma 4.5 that = U(λ,2) so all of these containments above must be equalities.
OP(λ)
This certainly shows U(λ,2) is dense and as is open in we have the intersection is also open.
O∩ O O
Let v U(λ,2) be another element in the intersection and denote by U(λ,2) the
′
∈ O ∩ O ⊆ O ∩
P(λ)-conjugacy class containing v. As v is G-conjugate to u we have dimC (v) = dimC (u) so
G G
dim = dimP(λ) dimC (v) > dimP(λ) dimC (u) = dimU(λ,2),
O′ − P(λ) − G
wherethelastequalityfollowsfrom[Tay16b,3.22(ii)].As U(λ,2)wemusthavedim = dimU(λ,2)
′ ′
O ⊆ O
so = U(λ,2), because U(λ,2) is irreducible, and is also a dense open subset of U(λ,2). Again, as
O′ O′
U(λ,2) is irreducible this implies = ∅ which shows = U(λ,2). (cid:4)
OP(λ)∩O′ 6 OP(λ) O∩
Corollary 4.7. Let u F be a rational unipotent element and let /G be the F-stable class containing u.
q ∈ U O ∈ U
If λ X(G)F is such that φ ( ) = (AdG)g(λ,2) then any element contained in U(λ,2) is of the form
spr reg
∈ O O∩
hlu with h U(λ) and l L(λ).
∈ ∈
Proof. Assume v U(λ,2), so by Lemma 4.6 there exists an element g P(λ) such that v = gu.
∈ O ∩ ∈
As F(v) = v we must have g 1F(g) C (u). If we set A (u) = C (u)/C (u) then the map
− ∈ P(λ) P(λ) P(λ) P◦(λ)
gu g 1F(g)C (u) inducesabijectionbetweenthe P(λ)-conjugacyclassescontainedin U(λ,2) =
7→ − P◦(λ) O∩
( U(λ,2))F andthe F-conjugacyclassesof A (u),see[Gec03,4.3.5].If A (u) = C (u)/C (u)
O∩ P(λ) L(λ) L(λ) L◦(λ)
then it’s known that the embedding C (u) ֒ C (u) induces an isomorphism A (u) A (u).
L(λ) → P(λ) L(λ) → P(λ)
Indeed, arguing as in the proof of [Tay16b, 3.22] we obtain from [Pre03, 2.3] that C (u) = C (u)⋉
P(λ) L(λ)
C (u) from which the statement follows immediately. Applying the Lang–Steinberg theorem to the
U(λ)
connected group L(λ) there exists an element l L(λ) such that l 1F(l )C (u) = g 1F(g)C (u).
1 ∈ 1− 1 P◦(λ) − P◦(λ)
We thereforehave l1u and v are P(λ) conjugate. As P(λ) = U(λ)⋊L(λ) the statement follows. (cid:4)
4.8. We are now ready to introduce GGGRs. For this we assume a chosen G-invariant trace form
κ( , ) : g g K, which is not too degenerate in the sense of [Tay16b, 5.6], and an F -opposition
q
− − × →
automorphism † : g → g, see [Tay16b, 5.1] for the definition. Moreover, we assume χq : F+q → Q×ℓ is a
character of the finite field F viewed as an additive group. Let u F be a rational unipotent element
q
q ∈ U
and let λ X(G)F be a cocharacter such that e = φ (u) g(λ,2) . Following [Tay16b, 5.10] we define
spr reg
∈ ∈
8
a linear character ϕu : U(λ,2) Qℓ by setting
→
ϕ (x) = χ (κ(e†,φ (x))).
u q spr
With this we have the following definition of the GGGR Γ .
u
Definition 4.9. The index [U(λ,1) : U(λ,2)] is an even power of q and the class function
Γu = [U(λ,1) : U(λ,2)]−1/2IndUG(λ,2)(ϕu).
is a character of G known as a generalised Gelfand–Graev representation(GGGR).
Proposition 4.10. Let γ Gal(Q /Q) be a Galois automorphism such that γ(ζ) = ζn for all p-roots of unity,
where n Z is an integer∈coprime t|Go|p. Then for any unipotent element u F wehave Γγu = Γun.
∈ ∈ U
Proof. We assume e and λ are as in 4.8. Let /G be the class containing u. As n is coprime to p
O ∈ U
we have u and un generate the same cyclic subgroup of G so un by [LS12, Corollary 3]. Now clearly
∈ O
un U(λ,2) so by Corollary 4.7 there exist elements h U(λ) and l L(λ) such that un = hlu. We
∈ O∩ ∈ ∈
thus have φ (un) = φ (hlu) = (Adhl)e.
spr spr
By property (K2) above we have φ (un) ne (mod u(λ,3)). As φ (un) = (Adhl)e and h U(λ)
spr spr
≡ ∈
we conclude from [McN04, Lemma 10] that
(Adl)e ne (mod u(λ,3)).
≡
However, as L(λ) preserves each weight space we have (Adl)e g(λ,2) so it must be that (Adl)e = ne.
∈
As mentioned in 4.1 we have Γun = Γhlu = Γlu so it is sufficient to show that Γγu = Γlu. Clearly φspr(lu) =
(Adl)e g(λ,2) so it is sufficient from the definition of the GGGR to show that ϕγ = ϕ .
∈ reg u lu
As F+q is an abelian p-group and χq : F+q → Qℓ is a homomorphism it is clear that χq(a)γ = χq(na)
for any a F+. Now, for any x U(λ,2) we thus have
∈ q ∈
ϕγ(x) = χ (nκ(e†,φ (x))) = χ (κ((ne)†,φ (x))) = ϕ (x)
u q spr q spr lu
as desired. (cid:4)
5. Condition 2.3 when p = 2
In this section and the following section we assume that p = 2.
5.1. In [SF16, 4.1] the first author showed that G satisfies Condition 2.3 in most cases where G is a
§
quasisimple group. The purpose of this section is to complete this work to show that all quasisimple
groups of Lie type in characteristic 2 satisfy Condition 2.3. We will do this using a general statement
which describes precisely which odd degree characters of G are fixed by σ. Note the techniques and
>
ideas we use hereare a synthesisofthosealready usedin [SF16]. When q 2 thesecharacters arealways
semisimple and we may apply Proposition 3.8, which generalises [SF16, 4.6]. When q = 2 not all odd
degree characters are semisimple and we must provide some additional ad-hoc arguments to deal with
these cases.
9
Lemma 5.2 (Malle, [Mal07, 6.8]). Assumeeither that q > 2ortheDynkindiagram ofG issimply lacedthenthe
only odd degree unipotent character isthe trivial character.
Proposition 5.3. An odd degree character χ (G,s) is σ-fixed if andonly if s is G⋆-conjugate to s2.
∈ E
Proof. Letχ (G,s) be an irreducible character of G ofodddegreeandchoosean irreducible character
∈ E
χ Irr(G χ) covering χ. By [Lus88, Proposition 10] the restriction ResG(χ) is multiplicity free so χ(1) =
∈ | Ge
[eG : I (χe)]χ(1), where G 6 I (χ) is the inertia group of χ. The order ofethe quotient G/G is copreime to
G G
pe, henece so is [G : I (χ)]. Thiseimplies χ(1) is odd. e
G
Now, assumee χeis contained in the Leusztig series (G,s) then by [Lus84, 4.23] there exists a bijection
E
Ψ : (G,s) (Ce (s),1)suchthatχ(1) = [G : C (s)]eΨe(χ)(1),seealso[DM91,13.23, 13.24]. Asχ(1)
s E → E G⋆ G⋆ p′ s
iseodd weeemust thereefoerehave that Ψe(χ)(1) isealsoeoded. e e e
s
>
Let us assume, for the moment, thateq 2. Then according to Lemma 5.2, thereis only one unipotent
character of C (s) of odd degree, namely the trivial character. Consequently, this implies that (G,s)
G⋆ E
containsauniqeueecharacterofodddegreeandsoχmustbetheuniquesemisimplecharactercontaineedien
thisseries,see[Car93,8.4.8]. Thecharacter χmustethereforealsobesemisimple. NowanyGelfand–Graev
character of G is obtained by inducing a linear character from a Sylow p-subgroup of G. As p = 2 this
implies all Gelfand–Graev characters are σ-fixed so χσ = χ by Proposition 3.8.
We now assume that q = 2. If the Dynkin diagram of C (s) is simply laced then we may apply
G⋆
the previous argument; so assume this is not the case. The Deynkein diagram of G must then also have
a component which is not simply laced. This corresponds to a semisimple subgroup of G which has a
trivial centre so splits off as a direct factor. With this it is clear that we needonly consider the case where
G is simple of type B , C , F , or G .
n n 4 2
⋆
Let F ⊆ Irr(W◦(s)) be an F -stable family of characters of the Weyl group of CG⋆(s) = CG◦⋆(s). For
each F⋆-fixedcharacterin wechooseoneofitsextensionstoW (s)whichisdefinedoverQ,c.f.,[Lus84,
◦
F
3.2], and denote by Irr(W (s)) the resulting set of extensieons. According to [Lus84, 4.23] there is a
◦
F ⊆
unique family such tehat χ,ReG(s) = 0 for some f , c.f., 3.9. Now as each f is rational valued
h f iG 6 ∈ F ∈ F
we see that e e
χ,RG(s) = χσ,RG(s)σ = χσ,RG(s)
h f iG h f iG h f iG
If G is of type B or C then these multiplicities uniquely determine the character χ so we must have
n n
χ = χσ in these cases, see [DM90, 6.3]. This statement is not true in general when G is of type G or
2
F . However, comparing the tables of unipotent characters in [Car93, 13.9] with [DM90, 6.3] we see the
4
§
statement still holds for those of odd degree. (cid:4)
FromnowuntiltheendofthisarticleweassumethatGissimpleandsimplyconnected.
5.4. If G is perfect then the quotient S = G/Z is a simple group of Lie type defined in characteristic
2. We now wish to show that G satisfies Condition 2.3. With regardsto this let Q 6 Aut(G) be a 2-group
which stabilises a Sylow P Syl (G). The normaliser B = N (P) is a Borel subgroup of G, c.f., [CE04,
∈ 2 0 G
2.29(i)], because p = 2. We may clearly replace P and Q by any G-conjugate so we may assume that B
0
contains our fixed maximal torus T , c.f., 3.1. In particular, we have B = P⋊T so N (P)/P = T . Note
0 0 0 G ∼ 0
that as Q stabilises P it also stabilises B and hence also T . We will denote by B 6 G an F-stable Borel
0 0 0
subgroup such that B = BF.
0 0
10
5.5. As we are working in characteristic 2, we have to be careful when dealing with small fields.
Namely we have to be mindful of degenerate tori, in the sense of [Car93, 3.6.1]. For instance, it can
happen when q = 2 that the torus T is the trivial subgroup, c.f., [Car93, 3.6.7]. The following shows that
0
T is degenerateonly when q = 2.
0
Lemma 5.6. Themaximal torus T isnon-degenerate if and only if q > 2 or G is 2A (2) with n > 2.
0 n
Proof. To show that T is non-degeneratewe must show that for any root α Φ X(T ) there exists an
0 0
∈ ⊆
element t T such that α(t) = 1.
0
∈ 6
WestartbytreatingthecasewhereGisoftype2A (q)withn >2. WemayassumethatG = SL (K)
n n+1
andT 6 B arethesubgroupsofdiagonalmatricesanduppertriangularmatricesrespectively. Moreover,
0 0
we assume that F = F φ = φ F where F : G G is the Frobenius endomorphism raising each
q q q
◦ ◦ →
matrix entry to the power q and φ : G G is the automorphism defined by φ(x) = (x−T)n0, where
→
n N (T ) is the permutation matrix representing the longest element in the symmetric group. For
0 G 0
∈
any 1 6 i 6 n we consider the usual homomorphisms ε : T K and qε : K T such that
i 0 × i × 0
→ →
ε ε 1 6 i < j 6 n+1 is the set of roots and qε qε 1 6 i < j 6 n+1 is the set of coroots.
i j i j
{± ∓ | } {± ∓ | }
Given an element ζ F 6 K and an integer 1 6 i 6 n+1 we define a correspondingelement
∈ ×q2 ×
t (ζ) = (qε qqε )(ζ) T .
i i n+2 i 0
− − ∈
Now assume α = ε ε with 1 6 i < j 6 n+1. If j = n+2 i then we have α(t (ζ)) = ζ2 and if
i j i
− −
j = n+2 i then we have α(t (ζ 1)t (ζ)) = ζq 1. Thus, as we can clearly choose ζ F we see that
6 − i − n+2−j − 6∈ ×q
T is always non-degenerate. With this case dealt with we may assume that G is not of type 2A (q) with
0 n
n > 2.
q
Now let us denote by , : X(T ) X(T ) the usual perfect pairing between the character and
0 0
cocharacter groups of T . hL−et−τi: Φ Φ×and τq : Φq Φq be the permutation of the roots and coroots
0
→ →
inducedby F. Given α Φwedenotebyk > 1thesmallestintegersuchthatτqk(qα) =qα. Givenanelement
∈
ζ F 6 K we define a correspondingelement t (ζ) T by setting
∈ ×qk × α ∈
tα(ζ) = qα(ζ) τq(qα)(ζq) τqk−1(qα)(ζqk−1)
· ···
As we assume that G is not of type 2A (q) we have by [Spr09, 10.3.2(iii)] that α,τqi(qα) = 0 for any
n
h i
1 6 i 6 k 1 and so
−
α(tα(ζ)) = ζhα,qαiζqhα,τq(qα)i ζqk−1hα,τqk−1(qα)i = ζ2.
···
Hence, if F contains a non-trivial element then we have the torus is non-degenerate. This is the case if
×qk
>
q 2.
Now assume that q = 2. If F is split then we have T = 1 by [Car93, 3.6.7], so certainly the torus
0
{ }
is degenerate in this case. Finally, it is an easy exercise with root systems to show that T is degenerate
0
when G is 2D (2) (n > 4), 3D (2), or 2E (2). We leave the details to the reader. (cid:4)
n 4 6
5.7. As G is simply connected, any automorphism of G can be obtained by restricting a bijective
morphism of G which commutes with F. Now recall that, with respect to T and B , we have the
0 0
notions of a graph, field, and diagonal automorphism, see [Ste68, Theorem30, pg. 158]. In particular, the