On Self-Normalising Sylow 2-Subgroups in Type A
aa r X i v : . [ m a t h . R T ] J a n On Self-Normalising Sylow -Subgroups in Type A Amanda Schaeffer Fry and Jay Taylor
Abstract.
Navarro has conjectured a necessary and sufficient condition for a finite group G tohave a self-normalising Sylow 2-subgroup, which is given in terms of the ordinary irreduciblecharacters of G . The first-named author has reduced the proof of this conjecture to showing thatcertain related statements hold when G is quasisimple. In this article we show that these conditionsare satisfied when G / Z ( G ) is PSL n ( q ) , PSU n ( q ) , or a simple group of Lie type defined over a finitefield of characteristic 2. Introduction
For any integer n > Q n the n th cyclotomic field, obtained from the rationals Q by adjoining a primitive n th root of unity. In [SF16], the first-named author began an investigation intothe following conjecture. Conjecture 1.2 (Navarro).
Let G be a finite group and let σ ∈ Gal ( Q | G | / Q ) be an automorphism fixing -roots ofunity and squaring ′ -roots of unity. Then G has a self-normalising Sylow -subgroup if and only if every ordinaryirreducible character of G with odd degree is fixed by σ . This statement would be an immediate consequence of the Galois-McKay conjecture, which is arefinement of the well-known McKay conjecture due to Navarro, see [Nav04, Conjecture A]. For a finitegroup G we denote by Irr ( G ) the set of ordinary irreducible characters and given a prime ℓ we denote byIrr ℓ ′ ( G ) ⊆ Irr ( G ) those irreducible characters whose degree is coprime to ℓ . The Galois-McKay conjecturethen posits that for any finite group G , prime ℓ , and Sylow ℓ -subgroup P G , there should exist abijection between Irr ℓ ′ ( G ) and Irr ℓ ′ ( N G ( P )) , as predicted by the McKay conjecture, which behaves nicelywith respect to the action of certain elements of the Galois group. While the McKay conjecture has been reduced to proving certain inductive statements for simplegroups in [IMN07], and even recently proven for ℓ = ℓ = ℓ =
2. We hope that some of theobservations made in the course of proving Conjecture 1.2 will be useful in working with an eventualreduction for Galois-McKay for ℓ =
2. We also remark that the corresponding weak form for odd ℓ hasbeen proven in [NTT07]. The main result of [SF16] is a reduction of Conjecture 1.2 to certain inductive statements forsimple groups, which we recall below in Section 2, and the verification of these statements for somesimple groups. The goal of this work is to extend and simplify the proofs there in order to completethe verification for simple groups of Lie type in characteristic 2 and simple groups of type A in allcharacteristics. Specifically we prove the following. Theorem A.
Assume G is a simple and simply connected algebraic group defined over K = F p , an algebraicclosure of the finite field F p of prime order p > , and let F : G → G be a Frobenius endomorphism of G . If eitherp = or G = SL n ( K ) , then whenever the quotient G F / Z ( G F ) is simple, it is SN2S-Good. One of our key tools used in the proof of Theorem A is Kawanaka’s generalised Gelfand–Graevrepresentations (GGGRs). These are a family of characters which have already shown themselves tobe remarkably useful for deducing the action of automorphisms of a finite reductive group on the set ofirreducible characters, see [CS15] and [Tay16a]. One of the reasons why they are so useful is that the imageof a GGGR under an automorphism of the group is again a GGGR and the resulting GGGR can be easilydescribed. Here we show that the same holds for certain Galois automorphisms, see Proposition 4.10.The statement holds whenever the GGGRs are defined and may be of independent interest.
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 oddcharacteristic are SN2S-Good. In this situation one should be able to employ the Harish-Chandra tech-niques used by Malle and Sp ¨ath in [MS15] to solve the McKay Conjecture for ℓ =
2. However, this isultimately quite different from our line of argument here and will be considered elsewhere.
We now outline the structure of the paper. In Section 2, we discuss the reduction of Conjecture 1.2to simple groups proved in [SF16]. In Section 3, we introduce some general notation regarding finitereductive groups and the action of the Galois group on Lusztig series under specific conditions. InSection 4, we continue this discussion by introducing generalized Gelfand-Graev characters and theirbehavior under the action of the Galois group. Sections 5 and 6 are dedicated to proving Theorem A inthe case that p =
2. In the remaining sections, we prove Theorem A for G = SL n ( K ) . The Reduction Statements for Simple Groups
In [SF16] it was shown that Conjecture 1.2 holds for any finite group if every finite simple groupis SN2S-Good. The notion of being SN2S-Good is comprised of two conditions. One condition is on thesimple group itself and the second is on its quasisimple covering groups. Before stating these conditions,we introduce some notation.
Let G be a finite group. We will denote by Aut ( G ) the automorphism group of G . If Q Aut ( G ) is any subgroup then we denote by GQ the semidirect product of Q acting on G . As in theintroduction, Irr ( G ) denotes the set of ordinary irreducible characters of G and Irr ℓ ′ ( G ) ⊆ Irr ( G ) is the setof those irreducible characters whose degree is coprime to ℓ , where ℓ is a prime. The set of all Sylow ℓ -subgroups of G will be denoted by Syl ℓ ( G ) . If H G is a subgroup of G and χ ∈ Irr ( H ) is an irreduciblecharacter then we denote by Irr ( G | χ ) the set of all irreducible characters ψ ∈ Irr ( G ) whose restrictionRes GH ( ψ ) to H contains χ as an irreducible constituent; we say that ψ covers χ . Moreover, for any element g ∈ G we denote by g χ the irreducible character of g H = gHg − defined by g χ ( h ) = χ ( g − hg ) for all h ∈ g H . We will write Irr ℓ ′ ( G | χ ) for the intersection Irr ( G | χ ) ∩ Irr ℓ ′ ( G ) .From this point forward σ ∈ Gal ( Q | G | / Q ) will denote the Galois automorphism fixing2-roots of unity and squaring 2 ′ -roots of unity, c.f., Conjecture 1.2. Condition 2.3.
Let G be a finite quasisimple group with centre Z G and Q Aut ( G ) a 2-group.Assume there exists a Q -invariant Sylow 2-subgroup P / Z ∈ Syl ( G / Z ) such that C N G ( P ) / P ( Q ) =
1. Thenfor any Q -invariant and σ -fixed λ ∈ Irr ( Z ) , we have χ σ = χ for any Q -invariant χ ∈ Irr ′ ( G | λ ) . Condition 2.4.
Let S be a finite nonabelian simple group and Q Aut ( S ) a 2-group. Assume P ∈ Syl ( S ) is a Q -invariant Sylow then if every Q -invariant χ ∈ Irr ′ ( S ) is fixed by σ we have C N S ( P ) / P ( Q ) = Definition 2.5.
Let S be a finite non-abelian simple group. We say S is SN2S-Good if Condition 2.4 holdsfor S and Condition 2.3 holds for any quasisimple group G satisfying G / Z ∼ = S . We end this section with some remarks concerning the above conditions. Firstly, if P G and Q Aut ( G ) are as in Condition 2.3 then the condition that C N G ( P ) / P ( Q ) = GQ / Z havinga self-normalising Sylow 2-subgroup, see [NTT07, Lemma 2.1(ii)]. Secondly, assume G is quasisimple withsimple quotient S = G / Z and let ˆ G be a universal perfect central extension, or Schur cover, of S . It is easilychecked that if Condition 2.3 holds for ˆ G then it holds for G . Indeed, as ˆ G is a Schur cover there exists asurjective homomorphism ˆ G → G with central kernel. This induces an injective map Irr ( G ) → Irr ( ˆ G ) anda surjective homomorphism Aut ( ˆ G ) → 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, hasbeen proven in [NT15]. However, for the purposes of this paper, we work with our stronger condition. Galois Automorphisms and Lusztig Series
From this point forward we denote by K = F p an algebraic closure of the finite field ofprime order p . Moreover, ℓ denotes a prime. We introduce here the basic setup that will be used throughout this article. Inparticular, G will be a connected reductive algebraic group defined over K and F : G → G will be aFrobenius endomorphism admitting an F q -rational structure G = G F . Moreover, we denote by ι : G ֒ → e G a regular embedding, in the sense of [Lus88, § e G will again be denotedby F and e G = e G F will be the resulting finite reductive group.We assume fixed pairs ( G ⋆ , F ⋆ ) and ( e G ⋆ , F ⋆ ) dual to ( G , F ) and ( e G , F ) respectively. As before we set G ⋆ = G ⋆ F ⋆ and e G ⋆ = e G ⋆ F ⋆ . We now choose an F -stable maximal torus T G and a dual F ⋆ -stablemaximal torus T ⋆ G ⋆ . The group e T : = ι ( T ) Z ( e G ) is then an F -stable maximal torus of e G . Recallthat the regular embedding ι induces a surjective homomorphism ι ⋆ : e G ⋆ → G ⋆ which is defined over F q . If e T ⋆ e G ⋆ is a torus dual to e T then ι ⋆ ( e T ⋆ ) = T ⋆ and ι ⋆ is unique up to composing with an innerautomorphism affected by an element of e T ⋆ . We will denote by C ( G , F ) the set of all pairs ( T , θ ) consisting of an F -stable maximal torus T G and an irreducible character θ ∈ Irr ( T F ) . Note we have an action of G on ∇ ( G , F ) defined by g · ( T , θ ) = ( g T , g θ ) ; we write C ( G , F ) / G for the orbits under this action and [ T , θ ] for the orbit containing ( T , θ ) . Dually, we denote by S ( G ⋆ , F ⋆ ) the set of all pairs ( T ⋆ , s ) consisting of an F ⋆ -stable maximal torus T ⋆ G ⋆ and a semisimple element s ∈ T ⋆ F ⋆ . Again we have an action of G ⋆ on S ( G ⋆ , F ⋆ ) defined by g · ( T ⋆ , s ) = ( g T ⋆ , g s ) , and we write S ( G ⋆ , F ⋆ ) / G ⋆ for the corresponding orbits and [ T ⋆ , s ] for the orbitcontaining ( T ⋆ , s ) . By [DL76, 5.21.3], see also [DM91, 13.13], we have a bijection Π : C ( G , F ) / G → S ( G ⋆ , F ⋆ ) / G ⋆ between these orbits. Note that this bijection depends on the choice of a group isomorphism ı : ( Q / Z ) p ′ → K × and an injective group homomorphism : Q / Z ֒ → Q × ℓ , so we implicitly assume that such homomor-phisms have been chosen. For any semisimple element s ∈ G ⋆ we denote by C ( G , F , s ) ⊆ C ( G , F ) the set of all pairs ( T , θ ) such that Π ([ T , θ ]) = [ T ⋆ , t ] and t is G ⋆ -conjugate to s . Now, to each pair ( T , θ ) ∈ C ( G , F ) , there isa corresponding Deligne–Lusztig character R GT ( θ ) , and we denote by E ( G , T , θ ) the set { χ ∈ Irr ( G ) |h χ , R GT ( θ ) i G = } of its irreducible constituents. Note we will sometimes also write R GT ⋆ ( s ) for R GT ( θ ) when Π ([ T , θ ]) = [ T ⋆ , s ] . The union E ( G , s ) = [ ( T , θ ) ∈C ( G , F , s ) E ( G , T , θ ) is, by definition, a rational Lusztig series. The set of all irreducible characters is then a disjoint unionIrr ( G ) = S E ( G , s ) , where we run over all G ⋆ -conjugacy classes of semisimple elements, see [Bon06, 11.8].If H is a finite group and x ∈ H is an element then we denote by x ℓ , resp., x ℓ ′ , the ℓ -part, resp., ℓ ′ -part, of x = x ℓ x ℓ ′ = x ℓ ′ x ℓ . 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 | G | / Q ) is anautomorphism such that γ ( ζ ) = ζ ℓ b for all ℓ ′ -roots of unity and γ ( ζ ) = ζ b ′ for all ℓ -roots of unity, then E ( G , s ) γ = E ( G , s b ′ ℓ s ℓ b ℓ ′ ) . Proof.
Assume ( T , θ ) ∈ C ( G , F ) . Then by the character formula for R GT ( θ ) [Car93, 7.2.8], and the fact thatGreen functions are integral valued, we easily deduce that R GT ( θ ) γ = R GT ( θ γ ) . In particular, as γ is anisometry we have E ( G , T , θ ) γ = E ( G , T , θ γ ) . Now, if Π ([ T , θ ]) = [ T ⋆ , s ] , then it is an easy consequenceof the description of the map Π , see [DM91, § γ that Π ([ T , θ γ ]) = [ T ⋆ , s b ′ ℓ s ℓ b ℓ ′ ] .In particular this shows that E ( G , s ) γ ⊆ E ( G , s b ′ ℓ s ℓ b ℓ ′ ) . An almost identical argument shows that E ( G , s ) ⊆E ( G , s b ′ ℓ s ℓ b ℓ ′ ) γ − ⊆ E ( G , t ) for some semisimple element t ∈ G ⋆ . However, by the disjointness of the rationalseries we must have equality which proves the lemma. (cid:4) For any irreducible character χ ∈ Irr ( G ) we denote by ω χ : Z ( G ) → Q × ℓ the central characterdetermined by χ . This is a linear character defined by ω χ ( z ) = χ ( z ) / χ ( ) for any z ∈ Z ( G ) . Thefollowing will prove to be useful later; it follows from [Bon06, 11.1(d)]. Lemma 3.6.
For any two irreducible characters χ , ψ ∈ E ( G , s ) we have ω χ = ω ψ . In particular, if γ ∈ Gal ( Q | G | / Q ) is an automorphism and E ( G , s ) γ = E ( G , s ) then ω γχ = ω χ γ = ω χ for all χ ∈ E ( G , s ) . Trying to understand the action of the Galois group on the elements of a rational Lusztig series is,in general, a difficult problem. However, in this section we will deal with two special cases. To describethese cases we need to introduce some notation. For s ∈ G ⋆ a semisimple element, we denote by T ⋆ s G ⋆ a fixed F ⋆ -stable maximal torus containing s ; note that we then have T ⋆ s is contained in the centraliser C G ⋆ ( s ) . We denote by W ◦ ( s ) = N C ◦ G ⋆ ( s ) ( T ⋆ s ) / T ⋆ s the Weyl group of the connected centraliser with respectto this maximal torus. For each w ∈ W ◦ ( s ) we choose an F ⋆ -stable maximal torus T ⋆ s , w = g T ⋆ s C ◦ G ⋆ ( s ) ,where g ∈ C ◦ G ⋆ ( s ) is an element such that g − F ⋆ ( g ) ∈ N C ◦ G ⋆ ( s ) ( T ⋆ s ) represents w . By [Bon06, 15.11] therethen exists a sign such that ρ s = ± | W ◦ ( s ) | ∑ w ∈ W ◦ ( s ) R GT ⋆ s , w ( s ) ,is a character of G . Each irreducible constituent of this character is contained in the rational Lusztig series E ( G , s ) and is a semisimple character. Recall that a character is called semisimple if it is contained in theAlvis–Curtis dual of a Gelfand–Graev character, see [DM91, 8.8, 14.39]. We first consider the action of theGalois group on these characters. Proposition 3.8.
Let γ be as in Lemma 3.4 and assume s ∈ G ⋆ is a semisimple element such that E ( G , s ) γ = E ( G , s ) , then the following hold:(a) ρ s is fixed by γ ,(b) every semisimple character contained in E ( G , s ) is fixed by γ if every Gelfand–Graev character of G is fixedby γ . Proof. If E ( G , s ) γ = E ( G , s ) , then we have s is G ⋆ -conjugate to s b ′ ℓ s ℓ b ℓ ′ . From the arguments in the proofof Lemma 3.4 it is clear that, under this assumption, we have R GT ⋆ s , w ( s ) γ = R GT ⋆ s , w ( s ) so clearly ρ s is fixedby γ . Now, if Γ is a Gelfand–Graev character of G and D G denotes Alvis–Curtis duality, see [DM91,8.8], then there exists a unique irreducible constituent χ of ρ s such that h D G ( Γ ) , χ i G =
0, see [Bon06,15.11]. Certainly we have χ γ is both a constituent of ρ γ s = ρ s and D G ( Γ ) γ . From the definition of D G , andthe character formula for Harish-Chandra induction/restriction [DM91, 4.5], it is not difficult to see that D G ( Γ ) γ = D G ( Γ γ ) . Hence, if Γ γ = Γ then we must have χ γ is a constituent of D G ( Γ ) ; but this implies χ γ = χ by the uniqueness. (cid:4) The next case we wish to consider is that of GL n ( K ) . First, we introduce some notation thatwill be useful later. Specifically, let s ∈ G ⋆ be a semisimple element. Then the Frobenius F ⋆ inducesan automorphism F ⋆ : W ◦ ( s ) → W ◦ ( s ) because T ⋆ s is assumed to be F ⋆ -stable. We denote by e W ◦ ( s ) thesemidirect product W ◦ ( s ) ⋊ h F ⋆ i and for any class function f : e W ◦ ( s ) → Q ℓ we define a correspondingclass function R Gf ( s ) = | W ◦ ( s ) | ∑ w ∈ W ◦ ( s ) f ( wF ⋆ ) R GT ⋆ s , w ( s ) of G . With this we can prove the following. Proposition 3.10.
Assume G is GL n ( K ) , γ is as in Lemma 3.4, and s ∈ G ⋆ is a semisimple element such that E ( G , s ) γ = E ( G , s ) . Then every χ ∈ E ( G , s ) is fixed by γ . Proof.
By [Lus84, 3.2, 4.23] every irreducible character in the Lusztig series E ( G , s ) is of the form R Gf ( s ) where f : e W ◦ ( s ) → Q ℓ is a rational valued irreducible character, see also [DM91, 13.25(ii), § R GT ⋆ s , w ( s ) is fixed by γ , c.f., the proof ofProposition 3.8. (cid:4) GGGRs and Galois Automorphisms
In this section, and in this section only, we assume that p is a good prime for G and that G is a proximate algebraic group in the sense of [Tay16b, 2.10]. Recall that this meanssome (any) simply connected covering of the derived subgroup of G is seperable. To any unipotent element u ∈ G Kawanaka has defined a corresponding generalised Gelfand–Graev representation (GGGR) of G which we denote Γ u , see [Kaw85; Tay16b]. If u is a regular unipotentelement then Γ u is a Gelfand–Graev character. Moreover, we have Γ gug − = Γ u for any g ∈ G . In thissection we wish to determine the effect of σ on the GGGRs of G ; for this we must recall their construction.Let g denote the Lie algebra of G and let N ⊆ g , resp., U ⊆ G , denote the nilpotent cone of g , resp., theunipotent variety of G . The Frobenius endomorphism F : G → G induces a corresponding Frobeniusendomorphism F : g → g on the Lie algebra. We have F ( U ) = U and F ( N ) = N . Let G m denote the set K \ { } viewed as a multiplicative algebraic group and let q X ( G ) = Hom ( G m , G ) be the set of all cocharacters of G . Let F q : G m → G m denote the Frobenius endomorphismgiven by F q ( k ) = k q , with q as in 3.1. Then for any λ ∈ q X ( G ) we define a new cocharacter F · λ ∈ q X ( G ) by setting ( F · λ )( k ) = F ( λ ( F − q ( k ))) for all k ∈ G m . We denote by q X ( G ) F ⊆ q X ( G ) the set of all cocharacters λ satisfying F · λ = λ . To each cocharacter λ ∈ q X ( G ) we have a corresponding parabolic subgroup P ( λ ) G withunipotent radical U ( λ ) P ( λ ) and Levi complement L ( λ ) = C G ( λ ( G m )) , see [Spr09, 3.2.15, 8.4.5]. Thegroup G acts on g via the adjoint representation Ad : G → GL ( g ) . Through Ad we have each cocharacter λ defines a Z -grading g = L i ∈ Z g ( λ , i ) on the Lie algebra. For any i > u ( λ , i ) = L j > i g ( λ , j ) isa subalgebra of the Lie algebra of U ( λ ) and it is the Lie algebra of a closed connected subgroup U ( λ , i ) U ( λ ) which is normal in P ( λ ) . The group L ( λ ) preserves each weight space g ( λ , i ) and we denote by g ( λ , 2 ) reg ⊆ g ( λ , 2 ) the unique open dense orbit of L ( λ ) acting on g ( λ , 2 ) . Note that if λ ∈ q X ( G ) F thenthe 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 . The action of G on g preserves N and the action of G on itself by conjugation preserves U ; wedenote the resulting sets of orbits by N / G and O / G . Recall that each nilpotent orbit O ∈ N / G is of theform O = ( Ad G ) g ( λ , 2 ) reg for some λ ∈ q X ( G ) , see [Tay16b, 3.22]. Moreover, if O is F -stable then we mayassume that λ ∈ q X ( G ) F , see [Tay16b, 3.25]. Following [Tay16b, § §
5] we assume a chosen G -equivariantisomorphism of varieties φ spr : U → N which commutes with F and whose restriction to each U ( λ ) is aKawanaka isomorphism. In particular, the map φ spr satisfies the following two properties:(K1) φ spr ( U ( λ , 2 )) ⊆ u ( λ , 2 ) ,(K2) φ spr ( uv ) − φ spr ( u ) − φ spr ( v ) ∈ u ( λ , 3 ) for any u , v ∈ U ( λ , 2 ) .Note also that φ spr induces a bijection U / G → N / G . Before introducing the GGGRs we consider thefollowing lemmas, which were not covered in [Tay16b]. Lemma 4.5.
For each cocharacter λ ∈ q X ( G ) we have φ spr ( U ( λ , 2 )) = u ( λ , 2 ) . Proof. As φ spr is an isomorphism we have φ spr ( U ( λ , 2 )) is a closed subset of the same dimension as u ( λ , 2 ) . As u ( λ , 2 ) is irreducible we must have φ spr ( U ( λ , 2 )) = u ( λ , 2 ) . (cid:4) Lemma 4.6.
Assume
O ∈ U / G is such that φ spr ( O ) = ( Ad G ) g ( λ , 2 ) reg for some cocharacter λ ∈ q X ( G ) . Then O ∩ U ( λ , 2 ) is an open dense subset of U ( λ , 2 ) and is a single P ( λ ) -conjugacy class. Proof.
Choose an element e ∈ g ( λ , 2 ) reg and let u ∈ U be the unique unipotent element satisfying φ spr ( u ) = e . By Lemma 4.5 we have u ∈ U ( λ , 2 ) so the P ( λ ) -conjugacy class O P ( λ ) containing u iscontained in O ∩ U ( λ , 2 ) ⊆ U ( λ , 2 ) . We thus clearly have a corresponding sequence of closed sets O P ( λ ) ⊆ O ∩ U ( λ , 2 ) ⊆ O ∩ U ( λ , 2 ) ⊆ U ( λ , 2 ) .According to [Tay16b, 3.22(ii.b)] we have φ spr ( O P ( λ ) ) = ( Ad P ( λ )) e = u ( λ , 2 ) . As φ spr is an isomorphismit follows from Lemma 4.5 that O P ( λ ) = U ( λ , 2 ) so all of these containments above must be equalities.This certainly shows O ∩ U ( λ , 2 ) is dense and as O is open in O we have the intersection is also open.Let v ∈ O ∩ U ( λ , 2 ) be another element in the intersection and denote by O ′ ⊆ O ∩ U ( λ , 2 ) the P ( λ ) -conjugacy class containing v . As v is G -conjugate to u we have dim C G ( v ) = dim C G ( u ) sodim O ′ = dim P ( λ ) − dim C P ( λ ) ( v ) > dim P ( λ ) − dim C G ( u ) = dim U ( λ , 2 ) ,where the last equality follows from [Tay16b, 3.22(ii)]. As O ′ ⊆ U ( λ , 2 ) we must have dim O ′ = dim U ( λ , 2 ) so O ′ = U ( λ , 2 ) , because U ( λ , 2 ) is irreducible, and O ′ is also a dense open subset of U ( λ , 2 ) . Again, as U ( λ , 2 ) is irreducible this implies O P ( λ ) ∩ O ′ = ∅ which shows O P ( λ ) = O ∩ U ( λ , 2 ) . (cid:4) Corollary 4.7.
Let u ∈ U F be a rational unipotent element and let O ∈ U / G be the F-stable class containing u.If λ ∈ q X ( G ) F is such that φ spr ( O ) = ( Ad G ) g ( λ , 2 ) reg then any element contained in O ∩ U ( λ , 2 ) is of the form hl u with h ∈ U ( λ ) and l ∈ L ( λ ) . Proof.
Assume v ∈ O ∩ U ( λ , 2 ) , so by Lemma 4.6 there exists an element g ∈ P ( λ ) such that v = g u .As F ( v ) = v we must have g − F ( g ) ∈ C P ( λ ) ( u ) . If we set A P ( λ ) ( u ) = C P ( λ ) ( u ) / C ◦ P ( λ ) ( u ) then the map g u g − F ( g ) C ◦ P ( λ ) ( u ) induces a bijection between the P ( λ ) -conjugacy classes contained in O ∩ U ( λ , 2 ) =( O ∩ U ( λ , 2 )) F and the F -conjugacy classes of A P ( λ ) ( u ) , see [Gec03, 4.3.5]. If A L ( λ ) ( u ) = C L ( λ ) ( u ) / C ◦ L ( λ ) ( u ) then it’s known that the embedding C L ( λ ) ( u ) ֒ → C P ( λ ) ( u ) induces an isomorphism A L ( λ ) ( u ) → A P ( λ ) ( u ) .Indeed, arguing as in the proof of [Tay16b, 3.22] we obtain from [Pre03, 2.3] that C P ( λ ) ( u ) = C L ( λ ) ( u ) ⋉ C U ( λ ) ( u ) from which the statement follows immediately. Applying the Lang–Steinberg theorem to theconnected group L ( λ ) there exists an element l ∈ L ( λ ) such that l − F ( l ) C ◦ P ( λ ) ( u ) = g − F ( g ) C ◦ P ( λ ) ( u ) .We therefore have l u and v are P ( λ ) conjugate. As P ( λ ) = U ( λ ) ⋊ L ( λ ) the statement follows. (cid:4) 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 q -oppositionautomorphism † : g → g , see [Tay16b, 5.1] for the definition. Moreover, we assume χ q : F + q → Q × ℓ is acharacter of the finite field F q viewed as an additive group. Let u ∈ U F be a rational unipotent elementand let λ ∈ q X ( G ) F be a cocharacter such that e = φ spr ( u ) ∈ g ( λ , 2 ) reg . Following [Tay16b, 5.10] we definea linear character ϕ u : U ( λ , 2 ) → Q ℓ by setting ϕ u ( x ) = χ q ( κ ( e † , φ spr ( x ))) .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 Γ u = [ U ( λ , 1 ) : U ( λ , 2 )] − Ind GU ( λ ,2 ) ( ϕ u ) .is a character of G known as a generalised Gelfand–Graev representation (GGGR). Proposition 4.10.
Let γ ∈ Gal ( Q | G | / Q ) be a Galois automorphism such that γ ( ζ ) = ζ n for all p-roots of unity,where n ∈ Z is an integer coprime to p. Then for any unipotent element u ∈ U F we have Γ γ u = Γ u n . Proof.
We assume e and λ are as in 4.8. Let O ∈ U / G be the class containing u . As n is coprime to p we have u and u n generate the same cyclic subgroup of G so u n ∈ O by [LS12, Corollary 3]. Now clearly u n ∈ O ∩ U ( λ , 2 ) so by Corollary 4.7 there exist elements h ∈ U ( λ ) and l ∈ L ( λ ) such that u n = hl u . Wethus have φ spr ( u n ) = φ spr ( hl u ) = ( Ad hl ) e .By property (K2) above we have φ spr ( u n ) ≡ ne ( mod u ( λ , 3 )) . As φ spr ( u n ) = ( Ad hl ) e and h ∈ U ( λ ) we conclude from [McN04, Lemma 10] that ( Ad l ) e ≡ ne ( mod u ( λ , 3 )) .However, as L ( λ ) preserves each weight space we have ( Ad l ) e ∈ g ( λ , 2 ) so it must be that ( Ad l ) e = ne .As mentioned in 4.1 we have Γ u n = Γ hl u = Γ l u so it is sufficient to show that Γ γ u = Γ l u . Clearly φ spr ( l u ) =( Ad l ) e ∈ g ( λ , 2 ) reg so it is sufficient from the definition of the GGGR to show that ϕ γ u = ϕ l u .As F + q is an abelian p -group and χ q : F + q → Q ℓ is a homomorphism it is clear that χ q ( a ) γ = χ q ( na ) for any a ∈ F + q . Now, for any x ∈ U ( λ , 2 ) we thus have ϕ γ u ( x ) = χ q ( n κ ( e † , φ spr ( x ))) = χ q ( κ (( ne ) † , φ spr ( x ))) = ϕ l u ( x ) as desired. (cid:4) Condition 2.3 when p = In this section and the following section we assume that p = In [SF16, § G satisfies Condition 2.3 in most cases where G is aquasisimple group. The purpose of this section is to complete this work to show that all quasisimplegroups of Lie type in characteristic 2 satisfy Condition 2.3. We will do this using a general statementwhich describes precisely which odd degree characters of G are fixed by σ . Note the techniques andideas we use here are a synthesis of those already used in [SF16]. When q > q = Lemma 5.2 (Malle, [Mal07, 6.8]).
Assume either that q > or the Dynkin diagram of G is simply laced then theonly odd degree unipotent character is the trivial character. Proposition 5.3.
An odd degree character χ ∈ E ( G , s ) is σ -fixed if and only if s is G ⋆ -conjugate to s . Proof.
Let χ ∈ E ( G , s ) be an irreducible character of G of odd degree and choose an irreducible character e χ ∈ Irr ( e G | χ ) covering χ . By [Lus88, Proposition 10] the restriction Res e GG ( e χ ) is multiplicity free so e χ ( ) =[ e G : I e G ( χ )] χ ( ) , where G I e G ( χ ) is the inertia group of χ . The order of the quotient e G / G is coprime to p , hence so is [ e G : I e G ( χ )] . This implies e χ ( ) is odd.Now, assume e χ is contained in the Lusztig series E ( e G , e s ) then by [Lus84, 4.23] there exists a bijection Ψ e s : E ( e G , e s ) → E ( C e G ⋆ ( e s ) , 1 ) such that e χ ( ) = [ e G : C e G ⋆ ( e s )] p ′ Ψ e s ( e χ )( ) , see also [DM91, 13.23, 13.24]. As e χ ( ) is odd we must therefore have that Ψ s ( e χ )( ) is also odd.Let us assume, for the moment, that q >
2. Then according to Lemma 5.2, there is only one unipotentcharacter of C e G ⋆ ( e s ) of odd degree, namely the trivial character. Consequently, this implies that E ( e G , e s ) contains a unique character of odd degree and so e χ must be the unique semisimple character contained inthis series, see [Car93, 8.4.8]. The character χ must therefore also be semisimple. Now any Gelfand–Graevcharacter of G is obtained by inducing a linear character from a Sylow p -subgroup of G . As p = σ -fixed so χ σ = χ by Proposition 3.8.We now assume that q =
2. If the Dynkin diagram of C e G ⋆ ( e s ) is simply laced then we may applythe previous argument; so assume this is not the case. The Dynkin diagram of G must then also havea component which is not simply laced. This corresponds to a semisimple subgroup of G which has atrivial centre so splits off as a direct factor. With this it is clear that we need only consider the case where G is simple of type B n , C n , F , or G .Let F ⊆
Irr ( W ◦ ( s )) be an F ⋆ -stable family of characters of the Weyl group of C G ⋆ ( s ) = C ◦ G ⋆ ( s ) . Foreach F ⋆ -fixed character in F we choose one of its extensions to e W ◦ ( s ) which is defined over Q , c.f., [Lus84,3.2], and denote by e F ⊆
Irr ( e W ◦ ( s )) the resulting set of extensions. According to [Lus84, 4.23] there is aunique family such that h χ , R Gf ( s ) i G = f ∈ e F , c.f., 3.9. Now as each f ∈ e F is rational valuedwe see that h χ , R Gf ( s ) i G = h χ σ , R Gf ( s ) σ i G = h χ σ , R Gf ( s ) i G If G is of type B n or C n then these multiplicities uniquely determine the character χ so we must have χ = χ σ in these cases, see [DM90, 6.3]. This statement is not true in general when G is of type G or F . However, comparing the tables of unipotent characters in [Car93, § (cid:4) From now until the end of this article we assume that G is simple and simply connected. If G is perfect then the quotient S = G / Z is a simple group of Lie type defined in characteristic2. We now wish to show that G satisfies Condition 2.3. With regards to this let Q Aut ( G ) be a 2-groupwhich stabilises a Sylow P ∈ Syl ( G ) . The normaliser B = N G ( P ) is a Borel subgroup of G , c.f., [CE04,2.29(i)], because p =
2. We may clearly replace P and Q by any G -conjugate so we may assume that B contains our fixed maximal torus T , c.f., 3.1. In particular, we have B = P ⋊ T so N G ( P ) / P ∼ = T . Notethat as Q stabilises P it also stabilises B and hence also T . We will denote by B G an F -stable Borelsubgroup such that B = B F .0 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 canhappen when q = T is the trivial subgroup, c.f., [Car93, 3.6.7]. The following shows that T is degenerate only when q = Lemma 5.6.
The maximal torus T is non-degenerate if and only if q > or G is A n ( ) with n > . Proof.
To show that T is non-degenerate we must show that for any root α ∈ Φ ⊆ X ( T ) there exists anelement t ∈ T such that α ( t ) = G is of type A n ( q ) with n >
2. We may assume that G = SL n + ( K ) and T B are the subgroups of diagonal matrices and upper triangular matrices respectively. Moreover,we assume that F = F q ◦ φ = φ ◦ F q where F q : G → G is the Frobenius endomorphism raising eachmatrix entry to the power q and φ : G → G is the automorphism defined by φ ( x ) = ( x − T ) n , where n ∈ N G ( T ) is the permutation matrix representing the longest element in the symmetric group. Forany 1 i n we consider the usual homomorphisms ε i : T → K × and q ε i : K × → T such that {± ε i ∓ ε j | i < j n + } is the set of roots and {± q ε i ∓ q ε j | i < j n + } is the set of coroots.Given an element ζ ∈ F × q K × and an integer 1 i n + t i ( ζ ) = ( q ε i − q q ε n + − i )( ζ ) ∈ T .Now assume α = ε i − ε j with 1 i < j n +
1. If j = n + − i then we have α ( t i ( ζ )) = ζ and if j = n + − i then we have α ( t i ( ζ − ) t n + − j ( ζ )) = ζ q − . Thus, as we can clearly choose ζ F × q we see that T is always non-degenerate. With this case dealt with we may assume that G is not of type A n ( q ) with n > h− , −i : X ( T ) × q X ( T ) the usual perfect pairing between the character andcocharacter groups of T . Let τ : Φ → Φ and q τ : q Φ → q Φ be the permutation of the roots and corootsinduced by F . Given α ∈ Φ we denote by k > q τ k ( q α ) = q α . Given an element ζ ∈ F × q k K × we define a corresponding element t α ( ζ ) ∈ T by setting t α ( ζ ) = q α ( ζ ) · q τ ( q α )( ζ q ) · · · q τ k − ( q α )( ζ q k − ) As we assume that G is not of type A n ( q ) we have by [Spr09, 10.3.2(iii)] that h α , q τ i ( q α ) i = i k − α ( t α ( ζ )) = ζ h α , q α i ζ q h α , q τ ( q α ) i · · · ζ q k − h α , q τ k − ( q α ) i = ζ .Hence, if F × q k contains a non-trivial element then we have the torus is non-degenerate. This is the case if q > q =
2. If F is split then we have T = { } by [Car93, 3.6.7], so certainly the torusis degenerate in this case. Finally, it is an easy exercise with root systems to show that T is degeneratewhen G is D n ( ) ( n > D ( ) , or E ( ) . We leave the details to the reader. (cid:4) As G is simply connected, any automorphism of G can be obtained by restricting a bijectivemorphism of G which commutes with F . Now recall that, with respect to T and B , we have thenotions of a graph, field, and diagonal automorphism, see [Ste68, Theorem 30, pg. 158]. In particular, the1automorphism x x of K determines a bijective morphism of G that generates the cyclic subgroupof all field automorphisms. We refer to this automorphism as a generating field automorphism . Nowany ϕ ∈ Aut ( G ) can be written as a product αβγδ where α is an inner automorphism, β is a fieldautomorphism, γ is a graph automorphism, and δ is a diagonal automorphism. We note, however, thatgraph automorphisms are omitted when F is twisted, see [Ste68, Theorem 36, pg. 195]. With these notionsin place we have the following relating to Condition 2.3. Lemma 5.8.
Keep the notation and assumptions of 5.4 and furthermore assume that T is nondegenerate. Then wehave C N G ( P ) / P ( Q ) ∼ = C T ( Q ) = { } if and only if Q contains a generating field automorphism. Proof.
Rephrasing, we have C T ( Q ) = { t ∈ T | ϕ ( t ) = t for all ϕ ∈ Q } . Now assume ϕ ∈ Q . Then,as above, we write ϕ as a product αβγδ . Firstly, by definition, we have δ acts trivially on T so we mayassume ϕ = αβγ . By assumption ϕ stabilises B and T , c.f., 5.4, which implies that α stabilises B and T because β and γ do by definition. This implies that α is affected by an element of B because N G ( B ) = B .As T is non-degenerate we have by [Car93, 3.6.7] that N G ( T ) = N G ( T ) F so N B ( T ) = T , hence α actstrivially on T . We may thus assume that ϕ = βγ .Assume F is twisted, so that ϕ = β and for any t ∈ T we have ϕ ( t ) = t a for some a ∈ N . Identifying T with a direct product F × q m × · · · × F × q mk we see that ϕ has a non-trivial fixed point if and only if a > F is split. Then we may identify T with a direct product F × q × · · · × F × q such that γ permutesfactors and β acts as a 2-power map. If γ is non-trivial then ϕ will have a non-trivial fixed point, so weare reduced to the previous case. (cid:4) Proposition 5.9.
Assume G is perfect, so the quotient G / Z is simple. Then any quasisimple group whose simplequotient is isomorphic to G / Z satisfies Condition 2.3.
Proof.
We will assume that G has a trivial Schur multiplier because the remaining cases were dealt within [SF16, §
4] using explicit computations with
GAP . This means G is a Schur cover and it suffices to showthat Condition 2.3 holds for G , c.f., 2.6.We start with the assumption that the maximal torus T is non-degenerate. Let Q and P ∈ Syl ( G ) beas in 5.4 such that C N G ( P ) / P ( Q ) = { } . Then Q contains a generating field automorphism ϕ by Lemma 5.8.We will denote by χ ∈ E ( G , s ) a Q -invariant character of odd degree.The bijective morphism ϕ may be extended to a bijective morphism e ϕ : e G → e G by setting e ϕ ( z ) = z for any z ∈ Z ( e G ) . There then exists a dual bijective morphism ϕ ⋆ : e G ⋆ → e G ⋆ such that F ⋆ ◦ e ϕ ⋆ = e ϕ ⋆ ◦ F ⋆ and e ϕ ⋆ ( t ) = t for all t ∈ e T ⋆ . This map also descends to a homomorphism ϕ ⋆ : T ⋆ → T ⋆ defined by ϕ ⋆ ( t ) = ι ⋆ ( e ϕ ⋆ ( e t )) where e t ∈ e T ⋆ satisfies ι ⋆ ( e t ) = t . This map is well defined because Ker ( ι ⋆ ) = Z ( e G ⋆ ) ,which is preserved by e ϕ ⋆ .As the quotient e G / G is an abelian, hence solvable, 2 ′ -group and h e ϕ i Aut ( e G ) is a 2-group we haveby Glauberman’s Lemma [Isa06, 13.28] that there exists a character e χ ∈ Irr ( e G | χ ) covering χ which is fixedby e ϕ . Now, if e χ is contained in the Lusztig series E ( e G , e s ) then it is also contained in the Lusztig series E ( e G , e ϕ ⋆ ( e s )) by [NTT08, 2.4]. This implies e s and ϕ ⋆ ( e s ) are e G ⋆ -conjugate.There exists an element g ∈ e G ⋆ such that g e s ∈ e T ⋆ so g − e ϕ ⋆ ( g ) e ϕ ⋆ ( e s ) = e s , which means that e ϕ ⋆ ( e s ) is e G ⋆ -conjugate to e s . However, e G ⋆ -conjugacy is equivalent to e G ⋆ -conjugacy so this implies that e s is e G ⋆ -conjugate to e s . By [Bon06, 11.7] we have χ ∈ E ( G , s ) where s = ι ⋆ ( e s ) . Clearly we have s is G ⋆ -conjugateto s so every odd degree character in E ( G , s ) is σ -fixed by Proposition 5.3. This shows that Condition 2.3holds in this case.2Now consider the case where the maximal torus T is degenerate. By Lemma 5.6 we have q = G is not A n ( ) with n >
2. As we assumed that G has a trivial Schur multiplier we have G is not E ( ) andso G has a trivial centre. This implies G ∼ = G ⋆ is a finite simple group so the argument in [GMN04, Lemma2.4] shows that every semisimple element s ∈ G ⋆ is G ⋆ -conjugate to s . By Lemma 3.4 and Proposition 5.3we thus have every odd degree character of G is σ -fixed so Condition 2.3 holds in this case. (cid:4) Condition 2.4 when p = Assume G is perfect with centre Z , so the quotient S = G / Z is simple. We now wish to outlinea strategy for showing that S satisfies Condition 2.4. Firstly, we note that the homomorphism Aut ( G ) → Aut ( S ) induced by the natural surjection G → S is an isomorphism, see [GLS98, Theorem 2.5.14(d)]. Nowassume A = SQ = GQ / Z for some 2-group Q Aut ( S ) ∼ = Aut ( G ) . We wish to show that if A does nothave a self-normalising Sylow 2-subgroup, then there exists a character χ ∈ Irr ′ ( S ) which is A -invariantbut is not fixed by σ . We will construct such a character by finding a character e χ ∈ Irr ′ ( e G ) such that e χ σ = e χ and the restriction χ = Res e GG ( e χ ) ∈ Irr ( G ) is irreducible, Q -invariant, and has Z in its kernel. We’rethen done by viewing χ as a character of S . Let s be a semisimple element of e G ⋆ then there exists a unique semisimple character e χ s ∈ E ( e G , s ) of e G , which has degree e χ s ( ) = [ e G ⋆ : C e G ⋆ ( s )] p ′ . Recall that the number of irreducible constituentsof χ : = Res e GG ( e χ s ) is exactly the number of irreducible characters θ ∈ Irr ( e G / G ) satisfying e χ s θ = e χ s .Furthermore, we have Irr ( e G / G ) = { e χ t | t ∈ Z ( e G ⋆ ) } and E ( e G , s ) e χ t = E ( e G , st ) for such t ∈ Z ( e G ⋆ ) ,see [DM91, 13.30]. Hence we see that χ is irreducible if and only if s is not e G ⋆ -conjugate to st for anynontrivial t ∈ Z ( e G ⋆ ) . Moreover, if s ∈ [ e G ⋆ , e G ⋆ ] then e χ s is trivial on Z ( e G ) so Res e GG ( e χ s ) is trivial on Z ( G ) ,see [NT13, Lemma 4.4(ii)]. Note that, by construction, the character χ is fixed by all inner and diagonalautomorphisms of G . Assume now that s has odd order, so by Lemma 3.4 and [NTT08, Corollary 2.4], we have χ σ s = χ s and χ ψ s = χ ψ ⋆ ( s ) for any ψ ∈ Aut ( G ) , where ψ ⋆ : e G ⋆ → e G ⋆ is an automorphism dual to ψ . Hence to provethat Condition 2.4 holds it suffices to find an element s ∈ [ e G ⋆ , e G ⋆ ] such that the following hold:(S1) s has odd order and [ e G : C e G ( s )] p ′ is odd,(S2) s is not e G ⋆ -conjugate to s ,(S3) s is not e G ⋆ -conjugate to st for any t ∈ Z ( e G ⋆ ) ,(S4) s is G ⋆ -conjugate to ψ ⋆ ( s ) for any field or graph automorphism ψ ∈ Q .With this in place we may now complete the proof of A when p =
2. Indeed, we have already shownthat Condition 2.3 holds in Proposition 5.9 so it suffices to show that Condition 2.4 holds under thisassumption.
Proposition 6.4.
Assume p = . If G is perfect, then the finite simple group S = G / Z satisfies Condition 2.4.
Proof.
Assume q =
2. If F is split then T = { } so G ∼ = S and N S ( P ) / P ∼ = T = { } , so certainlyCondition 2.4 holds in this case. The cases A n ( ) , D n ( ) , D ( ) , and E ( ) are dealt with in [SF16] so3we may assume that q >
2. We will now prove the statement by finding a semisimple element s ∈ [ e G ⋆ , e G ⋆ ] satisfying the conditions outlined in 6.3. What follows is a synthesised version of the arguments in [SF16].We will denote by q Φ ⋆ ⊆ q X ( e T ⋆ ) the coroots of e G ⋆ with respect to e T ⋆ . Clearly for any q α ∈ q Φ ⋆ and ζ ∈ K × we have q α ( ζ ) ∈ [ e G ⋆ , e G ⋆ ] . We now choose a set of simple coroots q ∆ ⋆ = { q α , . . . , q α n } ⊆ q Φ ⋆ , whichcorresponds to choosing a Borel subgroup of e G ⋆ containing e T ⋆ . With this in place we fix a coroot q α = q α + · · · + q α n ∈ q Φ .Note this is always a coroot for any indecomposable root system, as is easily checked.As p = ( q α ) = { } for any coroot q α ∈ q Φ ⋆ , c.f., the proof of [Spr09, 7.3.5]. In particular,the map K × × · · · × K × → e T ⋆ defined by ( ζ , . . . , ζ n ) q α ( ζ ) · · · q α n ( ζ n ) (6.5)is an injective morphism of algebraic groups. The torus T is non-degenerate because we assume q > A has a self-normalising Sylow 2-subgroup if and only if Q contains a generating fieldautomorphism which we denote by ϕ , c.f., Lemma 5.8. Let us write q = p a for some integer a >
1. Byassumption, Q is a 2-group so it may contain any field automorphism of the form ϕ i where i > a such that a / i is a 2-power.With this in mind let us write a = t m with t > m > ψ = ϕ m if m > ϕ if m = ψ generates the subgroup of all field automorphisms that may possibly be contained in Q . Forthe moment we will assume that q >
4. We now fix an element ζ ∈ K × with the following properties:(i) if m > ζ = ζ − and ζ m − = m = ζ ∈ K × is an element of order 5.Now consider the corresponding element s = q α ( ζ ) ∈ e T ⋆ . One readily checks that if q > s is F ⋆ -fixed. Now assume q = w ∈ N e G ⋆ ( e T ⋆ ) an element representing thereflection of q α . If g ∈ e G ⋆ is an element such that g − F ⋆ ( g ) = ˙ w then clearly the conjugate s = g s is F ⋆ -fixed. Hence, in all cases we have defined a rational semisimple element s ∈ e T ⋆ contained in thederived subgroup [ e G ⋆ , e G ⋆ ] . We now show that the conditions (S1) to (S4) hold for s .(S1). As p = s and s are not e G ⋆ -conjugate, hence are not e G ⋆ -conjugate. If they were e G ⋆ -conjugate then s would be e G ⋆ -conjugate to s so by [Car93, 3.7.1] there would exist an element ˙ w ∈ N e G ⋆ ( e T ⋆ ) , representing w ∈ W e G ⋆ ( e T ⋆ ) : = N e G ⋆ ( e T ⋆ ) / e T ⋆ , such that ˙ w s = s . Assume w ( q α ) = a q α + · · · + a n q α n with a i ∈ Z then ˙ w s = q α ( ζ a ) · · · q α n ( ζ a n ) . Clearly w ( q α ) ∈ q Φ ⋆ is a coroot. Inspecting theindecomposable root systems one easily observes that one of the following is true: a i = ± ( a i , a j ) =( ± ± ) for some 1 i , j n . In particular, the condition ˙ w s = s implies that either ζ = ζ ± or ζ = ζ ± = ζ ± . From the choice of our element ζ one easily confirms that this is impossible, so s cannotbe e G ⋆ -conjugate to s .4(S3). We need only show that C G ⋆ ( s ) = g C G ⋆ ( s ) is connected, see [Bon05, 2.8(a)]. The argument usedabove shows that an element ˙ w ∈ N G ⋆ ( T ⋆ ) , representing w ∈ W G ⋆ ( T ⋆ ) , satisfies ˙ w s = s if and only if w ( q α ) = q α . The centraliser of q α in W G ⋆ ( T ⋆ ) is a parabolic subgroup, see [MT11, A.29], which impliesthat C G ⋆ ( s ) is connected by [DM91, 2.4]. We thus have C G ⋆ ( s ) is connected.(S4). Assume γ ∈ Q is a graph or field automorphism. As C G ⋆ ( s ) is connected we have γ ⋆ ( s ) is G ⋆ -conjugate to s if and only γ ⋆ ( s ) is G ⋆ -conjugate to s . Moreover, it is clear that γ ⋆ ( s ) is G ⋆ -conjugate to s if and only γ ⋆ ( s ) is G ⋆ -conjugate to s . Now certainly s is fixed by all graph automorphisms. If m > ψ ⋆ ( s ) = s m = s and if m = ψ ⋆ ( s ) = s = s − . However if ˙ w ∈ N G ⋆ ( T ⋆ ) represents the reflection of q α then ˙ w s = s − so we’re done. (cid:4) Sylow -Subgroups of GL ε n ( q ) From this point forward we assume that p is odd, G = SL n ( K ) , e G = GL n ( K ) , and ι is thenatural inclusion map. Moreover, we assume that G ⋆ = PGL n ( K ) , e G ⋆ = GL n ( K ) , and ι ⋆ is the natural projection. The Frobenius endomorphism F will be assumed to denoteeither the morphism F q or F q φ = φ F q , with the notation as in the proof of Lemma 5.6. Thereference tori T , T ⋆ , e T , e T ⋆ will be taken to be the maximal tori of diagonal matrices. Throughout we will adopt the following convention: The split group G F q , resp., twisted group G F q φ , which we continue to refer to as G , will be denoted by SL + n ( q ) , resp., SL − n ( q ) . To unify this we let ε denote ± ε n ( q ) to denote the two rational forms of SL n ( K ) . We also write GL ε n ( q ) andPGL ε n ( q ) to have the corresponding meanings. Furthermore we define q to be q if ε = q if ε = − ε n ( q ) SL n ( q ) , GL ε n ( q ) GL n ( q ) , and PGL ε n ( q ) PGL n ( q ) .Recall that in this setting, e G = e G ⋆ = GL ε n ( q ) , G ⋆ = PGL ε n ( q ) , and we write Z for the centre Z ( G ) ofSL ε n ( q ) . In this section we recall results of Carter–Fong on the Sylow 2-subgroups of GL ε n ( q ) . For this weintroduce the following notation. For r > S ε r ( q ) a Sylow 2-subgroup of GL ε r ( q ) .With this in place we have the following, see [CF64, Theorem 1, Theorem 4]. Theorem 7.3 (Carter–Fong).
Let n = r + · · · + r t , with r < · · · < r t , be an integer written in its -adicexpansion. If e P e G is a Sylow -subgroup of e G = GL ε n ( q ) then e P ∼ = ∏ ti = S ε r i ( q ) andN e G ( e P ) ∼ = e P × C ( q − ε ) ′ × · · · × C ( q − ε ) ′ (7.4) with t copies of the cyclic group C ( q − ε ) ′ . The group N e G ( e P ) can be described more explicitly. Firstly, the Sylow e P can be realised by em-bedding ∏ ti = S ε r i ( q ) ∏ ti = GL ε ri ( q ) block-diagonally in a natural way. Now for each 1 ≤ j ≤ t thecorresponding factor C ( q − ε ) ′ is embedded as the largest odd-order subgroup of the centre Z ( GL ε rj ( q )) . Inparticular, writing I k for the identity of GL k ( q ) , elements of N e G ( e P ) are of the form xz where x ∈ e P and z = t M i = λ j I rj = diag ( λ I r , . . . , λ t I rt ) (7.6)5with λ j ∈ C ( q − ε ) ′ ≤ F × q . In what follows, we will use the notation z = L tj = z j for this matrix with z j = λ j I rj for each 1 ≤ j ≤ t . We close this section with a result which will be used as part of the proof ofLemma 10.2. Lemma 7.7.
Let m = r + · · · + r t ∈ N , with r < · · · < r t , be an integer written in its -adic expansion.Then [ GL ε m ( q ) : GL ε m ( q ) ] = t where n denotes the -part of an integer n > . Proof.
As 2 m = r + + · · · + r t + is clearly the 2-adic expansion of 2 m , we have by Theorem 7.3 that [ GL ε m ( q ) : GL ε m ( q ) ] = t ∏ i = (cid:0) | S ε r i + ( q ) | / | S ε r i ( q ) | (cid:1) .According to [CF64, Eq. (4)] we have | S ε r + ( q ) | = | S ε r ( q ) | for any integer r >
0. From this the resultfollows immediately. (cid:4) Condition 2.4 for Type A Let e P be a Sylow 2-subgroup of e G , so that P = e P ∩ G is a Sylow 2-subgroup of G which is normalin e P . Then [Kon05, Theorem 1] yields that N e G ( P ) = e PC e G ( e P ) = N e G ( e P ) , (8.2)and hence we see that N G ( P ) = N G ( e P ) = N e G ( e P ) ∩ G . Now, if n is not a power of 2, write n = r + r + ... + r t (8.3)with t ≥ r > r > ... > r t ≥ n . We now wish to describe when thequotient GQ / Z , with Q Aut ( G ) a 2-group, has a self-normalising Sylow 2-group; thus allowing us toshow Condition 2.4 holds. The following gives a complete description of those subgroups Q with thisproperty. Lemma 8.4 (see [Kon05]).
A simple group
PSL ε n ( q ) has a self-normalising Sylow -subgroup if and only if oneof the following holds:(i) n = r for some r > ,(ii) n = r for any r > and ( q − ε ) ′ = ,(iii) n = r + r for some r > r > and ( q − ε ) ′ = ( n , q − ε ) ′ . Lemma 8.5.
Write q = p a and let Q Aut ( G ) be a -group. The quotient GQ / Z has a self-normalising Sylow -subgroup if and only if at least one of the following is satisfied:(1) G / Z has a self-normalising Sylow -subgroup; (2) Q contains a graph automorphism in case ε = or an involutary field automorphism in case ε = − , eitherof which we may identify as the map φ , up to inner and diagonal automorphisms;(3) ε = , a is a -power, ( p − ) ′ = , and Q contains a field automorphism of order a (which we identify withF p , up to inner and diagonal automorphisms);(4) ε = , a is a -power, p = , and Q contains a field automorphism of order a /2 (which we identify with F ,up to inner and diagonal automorphisms); or(5) ε = , n = r + r for integers r > r ≥ , ( p m − ) ′ = gcd ( n , p m − ) ′ for some m dividing a,and Q contains a field automorphism of order a / m (which we identify with F mp , up to inner and diagonalautomorphisms). Remark 8.6.
Since the involutary field automorphism F q induces the map φ on GU n ( q ) , condition (2) inthe case ε = − Q contains any field automorphism whose order is a power of 2. Proof (of Lemma 8.5).
Let P be a Sylow 2-subgroup of G stabilized by Q . Specifically, we may choose P as in the setup for (8.2).(I) First suppose that one of (1), (2), (3), (4), or (5) holds. Note that in case (1), the statement is certainlytrue, since then N G ( P ) = PZ , so C N G ( P ) / PZ ( Q ) =
1. Hence we may assume that G / Z does not have aself-normalising Sylow 2-subgroup and that Q contains an outer automorphism. Specifically, either Q contains a graph automorphism (in case ε =
1) or involutary field automorphism (in case ε = − φ on G , up to conjugation in e G ; or ε = Q contains a field automorphism, whichwe identify as F p m on G , up to conjugation in e G , for some m ≥
1. Write ϕ for the corresponding graph orfield automorphism, respectively. We will show that C N G ( P ) / PZ ( ϕ ) =
1. Write N : = N G ( P ) / PZ and let g denote the image of an element g ∈ N G ( P ) in N . Suppose g ∈ N G ( P ) satisfies that g ∈ C N ( ϕ ) . That is, g is fixed by ϕ .Write n as in (8.3), so that by (7.6) and (8.2) we have g = xz for some x ∈ P and z = L tj = λ j I rj asin (7.6) such that ∏ tj = λ rj j =
1. Then observing the action of ϕ on the 2 ′ -part of g , we see ϕ ( z ) = zy forsome y ∈ Z of odd order. Write y = η I n for some ( n , q − ε ) -root of unity η in F × q . Then since the blocksizes 2 r j are distinct, we must have that λ j η = λ − j or λ p m j , respectively, for each 1 ≤ j ≤ t .(IA) Hence if condition (2), (3), or (4) holds, then there is some integer c ≥ λ c j = η for each1 ≤ j ≤ t . (Recall that in situation (3) p − p m =
9, so p m − j there is a 2 c -root of unity ζ j satisfying λ = ζ j λ j , and wemay write z as the product of λ I n and a diagonal matrix d whose diagonal entries are 2-power roots ofunity. Further, since z has determinant 1, | d | has 2-power order, and the multiplicative order of λ is odd,it follows that λ I n ∈ Z and d ∈ P . We therefore see that g ∈ PZ , so that g =
1, yielding that in cases (2),(3), and (4), C N ( ϕ ) = t = ε =
1, and ( p m − ) ′ = gcd ( n , p m − ) ′ . Notethen that PSL n ( p m ) and PGL n ( p m ) have self-normalising Sylow 2-subgroups, see Lemma 8.4. Note that λ p m − = η = λ p m − , so that λ = ζλ , for some ( p m − ) -root of unity ζ in F × q .Then as an element of GL n ( q ) , we may write z as the product of the central element λ I n and adiagonal matrix d whose diagonal entries are ( p m − ) -roots of unity. In particular, d ∈ GL n ( p m ) is anelement centralising a Sylow 2-subgroup e P m contained in e P (see the constructions in [CF64]), where e P is aSylow 2-subgroup of GL n ( q ) such that P = e P ∩ G . Hence the image of z in G / Z ∼ = GZ ( e G ) / Z ( e G ) must be7trivial, since PGL n ( p m ) has a self-normalising Sylow 2-subgroup and z has odd order. Then again, g = C N ( ϕ ) = C N G ( P ) / PZ ( Q ) =
1, so that GQ / Z does not have a self-normalising Sylow 2-subgroup. We do this by exhibiting a nontrivial element of N : = N G ( P ) / PZ which is fixed by all possible elements of Q . Note that we may assume Q contains anouter automorphism.Since (1) does not hold, we see that neither n nor ( q − ε ) is a power of 2, see Lemma 8.4. Hencewriting n as in (8.3), we see t ≥ z = ⊕ tj = λ j I rj as in (7.6). Since (2) does not hold, Q does not contain φ up to conjugation in e G . Further, any diagonalautomorphism in Q is induced by the quotient group e P / P , and therefore is centralised by such a z byconstruction. Hence it suffices to exhibit a z such that ϕ ( z ) = z for each field automorphism ϕ containedin Q , the λ j for 1 ≤ j ≤ t are not all the same, and λ r · ... · λ rt t = Q contains no field automorphisms. (In particular, this is the case if ε = − λ j = j > λ be a primitive ( q − ε ) ′ root of unity in F × q and λ = λ b , where b ≡ − r − r ( mod ( q − ε ) ′ ) . (Note that this is possible since ( q − ε ) ′ = ( q − ε ) ′ .) Thenthe determinant of z is λ r · λ r = λ r b · λ r = λ − r ( r − r ) · λ r = λ − r + r = z ∈ G . Further, if t >
2, then the λ j are not all the same, so z is not central, and the proof iscomplete in this case.If t =
2, then since (1) does not hold, we know by Lemma 8.4 that gcd ( n , q − ε ) ′ = ( q − ε ) ′ , sogcd ( r − r + q − ε ) ′ = ( q − ε ) ′ . This yields that b ( q − ε ) ′ , so λ = λ , and z is again notcentral.(IIB) Now assume that ε = Q contains a field automorphism. Write q = p a . Without loss, wemay identify the generator of the subgroup of Q consisting of field automorphisms as F mp for some m | a .Further, it suffices to assume that F mp generates the largest 2-group of automorphisms possible withoutinducing conditions (3)-(5). Note that since (5) does not hold, we have ( p m − ) ′ = gcd ( n , p m − ) ′ if t = ( p − ) ′ =
1, then also ( p m − ) ′ =
1. If p − a are both powers of 2, thensince neither (3) nor (4) hold, we may assume that 2 | m when p = | m when p =
3. Then ( p m − ) ′ =
1, since both of p − p + p =
3, inwhich case 3 − p − a is not, then we may assume that m is divisible by a ′ , the odd part of a . Then ( p m − ) ′ is divisible by the a ′ ‘th cyclotomic polynomialevaluated at p , which is odd since a ′ and p are.Hence in all cases, we may assume p m − q replaced with p m , we may choose λ , λ ∈ F × p m so that z is non-central and lies in SL n ( p m ) . Hence z isfixed by the field automorphisms in Q and has the required form. (cid:4) Proposition 8.7.
If G is perfect then the simple group G / Z = PSL ε n ( q ) satisfies Condition 2.4. Proof.
By Proposition 6.4 our assumption that p is odd is not restrictive. We will argue this by proving thecontrapositive, as in Section 6. Specifically we assume A = SQ = GQ / Z ≤ Aut ( S ) is a group obtained byadjoining a 2-group Q of automorphisms to S . We wish to show that if A does not have a self-normalizing8Sylow 2-subgroup, i.e., Q is not as in (1) to (5) of Lemma 8.5, then there exists a character χ ∈ Irr ′ ( S ) which is A -invariant but not fixed by σ . We do this by finding a semisimple element s ∈ [ e G ⋆ , e G ⋆ ] = SL ε n ( q ) satisfying the conditions in 6.3.Since A has no self-normalising Sylow 2-subgroup, we see by Lemma 8.4 that neither n nor q − ε is apower of 2. Write n = r + r + ... + r t with t ≥ r > r > ... > r t ≥ n . From the discussion in Section 7, toensure that s has odd order and centralises a Sylow 2-subgroup of e G ⋆ , it suffices to choose a nonidentity s in the form s = L tj = λ j · I rj , as in (7.6).If s is non-central, then since the block sizes 2 r j are distinct, it follows that this choice of s is notconjugate in GL ε n ( q ) to s or st for any nontrivial t ∈ Z ( GL ε n ( q )) .We note further that A does not contain a graph automorphism, by Lemma 8.5. Hence it suffices toexhibit an s as above such that: ϕ ( s ) is conjugate to s for each field automorphism ϕ contained in A , the λ j for 1 ≤ j ≤ t are not all the same, and λ r · ... · λ rt t = s ∈ SL ε n ( q ) .Letting s be the element z obtained in parts (IIA) and (IIB) of the proof of Lemma 8.5, we see that theconjugacy class of s is fixed by A and has the required form. (cid:4) Covering Odd Degree Characters of SL ε n ( q ) We wish to show that G satisfies the hypotheses of Condition 2.3. As we already saw in the proofof Proposition 5.9 it is important to know that a σ -invariant odd degree character of G can be covered bya σ -invariant character of e G . Unfortunately, we cannot appeal to Glauberman’s Lemma as in the proof ofProposition 5.9. The following gives the desired covering result. Proposition 9.2.
Let S = G / Z and suppose Q Aut ( S ) is a -group such that GQ / Z has a self-normalisingSylow -subgroup. Assume λ ∈ Irr ( Z ) is σ -fixed and Q-invariant and let χ ∈ Irr ′ ( G | λ ) be Q-invariant. Thenthere exists an irreducible character e χ ∈ Irr ( e G | χ ) covering χ which is contained in a Lusztig series E ( e G , e s ) labelledby an element e s of -power order. In particular, we have e χ σ = e χ . Proof.
Let χ be as in the statement, so χ ∈ Irr ( G ) has odd degree. Then, in particular, χ lies in a series E ( G , s ) for some semisimple element s ∈ G ⋆ for which [ G ⋆ : C G ⋆ ( s )] p ′ is odd. This implies s centralises,hence normalises, a Sylow 2-subgroup of G ⋆ .Now, the characters e χ ∈ Irr ( e G | χ ) lying above χ are members of rational series of the form E ( e G , e s ) ,where e s ∈ e G ⋆ satisfies ι ⋆ ( e s ) = s , see [Bon06, Corollaire 9.7]. We will write e Z for the centre Z ( e G ⋆ ) anddenote by e P a Sylow 2-subgroup of e G ⋆ such that s centralises e P e Z / e Z . We aim to show that e s may be chosento have 2-power order. If this is the case then by Lemma 3.4 and Proposition 3.10 we must have e χ σ = e χ .First, suppose that Q is as in Lemma 8.5(1), so that S has a self-normalising Sylow 2-subgroup. ThenPGL ε n ( q ) also has a self-normalising Sylow 2-subgroup, so s must be contained in the Sylow 2-subgroup e P e Z / e Z of PGL ε n ( q ) . Let e s ′ = rz be a pre-image of s , where r ∈ e P and z ∈ e Z . Then noting that e s = e s ′ z − isanother pre image of s , the claim is proved in this case.Next, assume condition (2), (3), (4), or (5) of Lemma 8.5 holds. Then either Q contains a graphautomorphism (in case ε =
1) or involutary field automorphism (in case ε = − φ on e G ∼ = e G ⋆ ∼ = GL ε n ( q ) ; or ε = Q contains a field automorphism, which we identify as F p m on9 e G ∼ = e G ⋆ ∼ = GL n ( q ) , for some m ≥
1. By an abuse of notation, write ϕ for φ or F p m , respectively, on e G and e G ⋆ . Then χ ϕ = χ , and in particular, E ( G , s ) ϕ = E ( G , s ) , yielding that the class ( s ) is fixed by ϕ , by[NTT08, Corollary 2.4].Let e s ∈ GL ε n ( q ) be a pre-image of s . Then if x ∈ e P , we see e sx e s − ∈ x e Z . But further, e sx e s − has ordera power of 2, so must be contained in the unique Sylow 2-subgroup e P of e P e Z . We therefore see that e s normalises e P .Write n = r + r + ... + r t with r > r > ... > r t for the 2-adic expansion of n . From the discussionin Section 7, we may then choose e s in the form e s = s z , where s ∈ e P and z = L tj = λ j · I rj , as in (7.6).Further, since ϕ ( s ) is conjugate in PGL ε n ( q ) to s , we see that ϕ ( ˜ s ) is conjugate in GL ε n ( q ) to e sy for some y ∈ e Z . Then ϕ ( z ) is conjugate to zy ′ , where y ′ denotes the odd part of y . Let η ∈ F × q be such that y ′ = η I n . Then as in part (I) of the proof of Lemma 8.5, since the block sizes 2 r j are distinct, we musthave that these eigenvalues satisfy λ j η = λ − j or λ p m j , respectively, for each 1 ≤ j ≤ t .Hence if condition (2), (3), or (4) holds, arguing exactly as in part (IA) of the proof of Lemma 8.5 yieldsthat we may write z as the product of λ I n ∈ e Z and a diagonal matrix whose diagonal entries are 2-powerroots of unity. We may then replace e s with e s λ − , which is also a pre-image of s and has 2-power order,completing the proof in this case.Finally, assume condition (5) of Lemma 8.5 holds, so that t = ε =
1, and ( p m − ) ′ = gcd ( n , p m − ) ′ . Note then that PSL n ( p m ) and PGL n ( p m ) have self-normalising Sylow 2-subgroups. Further, bymultiplying by the central element λ − I n , we may assume that z = λ I r ⊕ I r , and η =
1. Therefore itmust be that λ p m − =
1, yielding that λ I r is an element of order dividing ( p m − ) ′ in Z ( GL r ( p m )) .That is, z is contained in the centraliser in GL n ( p m ) of a Sylow 2-subgroup e P m contained in e P (see theconstructions in [CF64]). Then the image of z in PGL n ( q ) is an element of odd order in PGL n ( p m ) centralising e P m e Z / e Z , which is self-normalising in PGL n ( p m ) . This yields that the image of z in PGL n ( q ) istrivial, so s is a 2-element. Arguing as in the case that Lemma 8.5(1) holds, the proof is complete. (cid:4) Condition 2.3 for Type A We wish to understand the effect of the Galois automorphism σ on the odd degree characters of G . For this we will use results of Navarro–Tiep on the extension of odd degree characters from G to e G .Specifically we have the following slight refinement of results from [NT15]. Lemma 10.2 (Navarro–Tiep).
Assume χ ∈ Irr ( G ) is an odd degree character and e χ ∈ Irr ( e G | χ ) covers χ . Thenone of the following holds:(a) e χ ( ) = χ ( ) ,(b) e χ ( ) = χ ( ) and n = r for some r > . Proof.
The case n = n >
2. Let us assume e χ ∈ E ( e G , e s ) thenaccording to [NT15, Lemma 4.5, Lemma 4.6] we have either (a) holds or the following holds • e χ ( ) = χ ( ) and C e G ( e s ) ∼ = GL ε m ( q ) with n = m .By Lusztig’s Jordan decomposition of characters we see that the index [ e G : C e G ( e s )] p ′ divides e χ ( ) , see[DM91, Remark 13.24], so [ e G : C e G ( e s )] divides e χ ( ) because p is odd. If m = r + · · · + r t is the 2-adic0expansion of m then we have [ e G : C e G ( e s )] = t by Lemma 7.7. However χ ( ) is odd so we must have t =
1, which proves the statement. (cid:4)
By Proposition 3.10 we know the effect of σ on the irreducible characters of e G . Hence whenan odd degree character of G extends to e G we can easily determine the effect of σ on such a character.Thus we are left with considering the second case of Lemma 10.2. For this case, we record the followingobservations. Lemma 10.4.
Assume χ ∈ Irr ( G ) is an irreducible character and e χ ∈ Irr ( e G | χ ) covers χ . If the G-conjugacy classof g ∈ G is invariant under conjugation by e G, then χ ( g ) = e χ ( ) e χ ( g ) / χ ( ) . In particular, we have χ ( g ) σ = χ ( g ) if and only if e χ ( g ) σ = e χ ( g ) . Proof.
This follows immediately from the fact that Res e GG ( e χ ) is multiplicity free. (cid:4) Lemma 10.5.
Recall our assumption that p is odd and let χ ∈ Irr ( G ) be an irreducible character. Then χ ( u ) σ = χ ( u ) for any unipotent element u ∈ G. Proof.
Let X be a complex representation affording χ . Then χ ( u ) is the sum ∑ χ ( ) i = λ i of eigenvalues λ i of the matrix X ( u ) and χ ( u ) is the sum ∑ χ ( ) i = λ i of eigenvalues of the matrix X ( u ) . Hence, since u is a2 ′ -element, each λ i is a 2 ′ -root of unity, so χ ( u ) σ = ∑ χ ( ) i = λ i = χ ( u ) . (cid:4) As we will see below, the case when n =
4, i.e., when G = SL ε ( q ) , will need to be treatedseparately with ad-hoc methods. In particular, we will need some knowledge of the Levi subgroup L = { diag ( A , B ) | A , B ∈ GL ( K ) and det ( B ) = det ( A ) − } G = SL ( K ) . (10.7)Note this subgroup is stable by the Frobenius endomorphism F . Let W = N G ( T ) / T be the Weyl groupof G with respect to T and let W L = N L ( T ) / T be the corresponding parabolic subgroup determinedby L .The section N G ( L ) / L of G is isomorphic to the section N W ( W L ) / W L of W , which has order 2. Iden-tifying W with S in the usual way, we have the non-trivial coset of N W ( W L ) / W L is represented by thepermutation (
1, 3 )(
2, 4 ) . Let n ∈ N G ( T ) be the permutation matrix representing (
1, 3 )(
2, 4 ) ; note thismatrix has determinant 1. If ı n : L → L denotes the conjugation map defined by ı n ( l ) = nln − then themap ı n F : L → L is a Frobenius endomorphism of L stabilising T . Now assume M = g L is an F -stable G -conjugate of L . After possibly replacing g by gl , for some l ∈ L , we may assume that conjugation by g identifies the pair ( M , F ) with either ( L , F ) or ( L , ı n F ) . With this we are ready to prove the followinglemmas. Lemma 10.8.
Assume n = so that G = SL ε ( q ) and recall that p is odd. If M = g L is an F-stable G -conjugateof L , then any rational unipotent element u ∈ M F is M F -conjugate to u . Proof.
By 10.6 each pair ( M , F ) can be identified with the pair ( L , F ′ ) where F ′ denotes either F or nF ,hence it suffices to prove the statement for the pair ( L , F ′ ) . The unipotent conjugacy classes of L areparameterised by the Jordan normal form. Let O ⊆ L be a unipotent conjugacy class. Then O is invariantunder the map x x because the elements have the same Jordan normal form.Assume now that O is F ′ -stable and u ∈ O F ′ . As the component group C L ( u ) / C ◦ L ( u ) has order atmost | Z ( L ) / Z ◦ ( L ) | = O F ′ is a single L F ′ -conjugacy class or it’s a union of two such1classes, see [Gec03, 4.3.5]. Therefore, it suffices to show that for one element u ∈ O F ′ we have u is L F ′ -conjugate to u . Applying [Gec03, 4.3.5] it suffices to find an element t ∈ T such that t u = u and t − F ( t ) ∈ C ◦ T ( u ) C ◦ L ( u ) .Let J = (cid:2) (cid:3) and let u be one of the elements diag ( J , J ) , diag ( J , I ) , diag ( I , J ) , or diag ( I , I ) . Theseelements represent the unipotent conjugacy classes of L . Setting t = diag ( a , a , a − , 2 − a − ) , for some a ∈ G m , one easily checks that t u = u and t − F ′ ( t ) ∈ C ◦ T ( u ) as desired. (cid:4) Lemma 10.9.
Assume n = so that G = SL ε ( q ) and recall that p is odd. If u ∈ G is a non-regular unipotentelement then u is G-conjugate to u . In particular, we have Γ σ u = Γ u for any non-regular unipotent element u ∈ G. Proof. If u ∈ G is a unipotent element then u and u have the same Jordan normal form so they are G -conjugate. Each unipotent element has a connected centraliser unless u is either regular or conjugate todiag ( J , J ) with J = (cid:2) (cid:3) . The arguments used in the proof of Lemma 10.8 show the first statement. Thelast statement follows from Proposition 4.10. (cid:4) Remark 10.10.
As 2 is a generator of F × p we see from the proof of [TZ04, 6.7] that if u ∈ G = SL ε ( q ) isregular unipotent then we need not necessarily have u is G -conjugate to u . To understand the effect of σ on the odd degree characters of G we will need to be able todistinguish between odd degree characters which are contained in the same e G -orbit. To do this we willuse the GGGRs of G , see Section 4. In this direction we will need the following consequence of [TZ04]. Proposition 10.12.
Let Γ u be a GGGR of G = SL ε n ( q ) . Then the following hold.(a) For any g ∈ G we have Γ u ( g ) ∈ Q ( √ η p ) , where η ∈ {± } is such that p ≡ η ( mod 4 ) ,(b) if q is a square, n is odd, or n / ( n , q − ε ) is even then Γ u ( g ) ∈ Z for all g ∈ G.In particular, if q ≡ ± ( mod 8 ) , n is odd, or n / ( n , q − ε ) is even, then Γ σ = Γ . Proof.
This follows from [TZ04, Theorem 1.8, Lemma 2.6, and Theorem 10.10], together with the fact that Γ is a unipotently supported character of G . The last statement follows by noting that √ p σ = √ p if p ≡ ± ( mod 8 ) and √ p σ = −√ p if p ≡ ± ( mod 8 ) , since if q is not a square, then q ≡ p ( mod 8 ) . (cid:4) We are now in a position to prove the second part of Theorem A, thus concluding its proof.Namely, we need to show that the simple groups PSL ε n ( q ) are SN2S-Good. Note that Propositions 5.9and 6.4 show that PSL ε n ( q ) is SN2S-Good when q is even so our standing assumption that q is odd is notrestrictive. As we have already shown in Proposition 8.7 that Condition 2.4 holds for PSL ε n ( q ) , we needonly show that Condition 2.3 holds for the corresponding quasisimple groups. Parts of the argument aresimilar to that used in [SF16, Theorem 4.15, part (3)] but we include it here for completeness. Proposition 10.14.
Assume G is perfect so the quotient G / Z = PSL ε n ( q ) is simple. Then any quasisimple groupwhose simple quotient is isomorphic to PSL ε n ( q ) satisfies Condition 2.3. Proof.
We may assume that G has a trivial Schur multiplier because the case PSL ( ) ∼ = A was treatedin [SF16]. In this case G is a Schur cover of S and it suffices to show that G satisfies Condition 2.3, c.f., 2.6.Let Q and χ ∈ Irr ′ ( G ) be as in the hypothesis of Condition 2.3. By Proposition 9.2 there exists a σ -fixed irreducible character e χ ∈ Irr ( e G | χ ) covering χ . A well known result of Kawanaka assures that there2exists a unipotent element u ∈ G ⊆ e G whose corresponding GGGR e Γ u of e G satisfies h e Γ u , e χ i e G =
1, see[Kaw85, 3.2.18] or [Tay16b, 15.7]. By the construction of the GGGRs we have e Γ u = Ind e GG ( Γ u ) where Γ u isthe GGGR of G determined by u . Hence, applying Frobenius reciprocity we have the restriction Res e GG ( e χ ) contains a unique irreducible constituent χ ∈ Irr ( G ) satisfying h Γ u , χ i G =
1. Assume that Γ σ u = Γ u . Thenas e χ σ = e χ we have χ σ is also a constituent of Res e GG ( e χ ) satisfying h Γ u , χ σ i G =
1. The uniqueness of such acharacter forces χ σ = χ . By Clifford theory, we may write χ = χ g for some g ∈ e G , so χ σ = ( χ σ ) g = χ .If either q ≡ ± ( mod 8 ) , or n / ( n , q − ε ) is even, then by Proposition 10.12 we have each GGGR Γ u of G is σ -fixed so the above argument applies and Condition 2.3 holds for G . Thus we may assumethat q ≡ ± ( mod 8 ) and n / ( n , q − ε ) is odd. If χ extends to e G then Gallagher’s theorem implies thatRes e GG ( e χ ) = χ so χ σ = χ . We may therefore assume that χ does not extend to e G , so by Lemma 10.2 wemust have n = r for some r >
1. Now, as n / ( n , q − ε ) is odd, we must have n = r divides the 2-part ( q − ε ) of q − ε . But q − ε ≡ ± − ε ( mod 8 ) , which is either ± ( mod 8 ) or 4 ( mod 8 ) . Hence ( q − ε ) is either 2 or 4 so n is either 2 or 4.The case n = G = SL ε ( q ) with q ≡ ± ( mod 8 ) . By Lemma 10.9 we have Γ σ u = Γ u unless u is regular unipotent, so the above argumentshows that χ σ = χ unless χ is a regular character. Assume the σ -invariant character e χ ∈ Irr ( e G | χ ) covering χ is contained in the Lusztig series E ( e G , e s ) . Then by Proposition 9.2 we may assume e s is of 2-power order;in particular s is of 2-power order.We now aim to show that χ ( g ) σ = χ ( g ) for each g ∈ G , thus showing χ σ = χ . First, assume g is semisimple. Then as G is simply connected, we have C G ( g ) is connected. This easily implies thatthe G -conjugacy class containing g is invariant under conjugation by e G so χ ( g ) σ = χ ( g ) in this case byLemma 10.4. Next, assume g is unipotent, so by Lemma 10.5 we have χ ( g ) σ = χ ( g ) . If g is not a regularunipotent element then g and g are G -conjugate, c.f., Lemma 10.9, so again χ ( g ) σ = χ ( g ) .If g is regular unipotent then we claim χ ( g ) =
0, thus trivially χ ( g ) σ = χ ( g ) . By [DM91, Corollary14.38] we have χ ( g ) = D G ( χ ) does not occur as a constituent of any Gelfand–Graev character. Assumefor a contradiction that D G ( χ ) does occur in some Gelfand–Graev character. Then χ is both regular andsemisimple. This implies e χ is both regular and semisimple. However, by [Bon06, 15.6, 15.10] this can onlyhappen if the trivial and sign character of the Weyl group of C e G ( s ) coincide. Clearly this is not the case,so we must have χ ( g ) = g = g s g u = g u g s with g s = g u = C G ( g ) = C C G ( g s ) ( g u ) and the centraliser C G ( g s ) is a Levi subgroup of G .The subgroup C G ( g s ) is G -conjugate to a standard Levi subgroup of G so C G ( g s ) is isomorphic to eitherGL ( K ) , GL ( K ) × G m , or the subgroup L defined in (10.7). In the first two cases the centraliser of everyunipotent element is connected, which implies C G ( g ) is connected. As argued above we can concludefrom Lemma 10.4 that χ ( g ) σ = χ ( g ) .Thus we are left with the case where C G ( g s ) is G -conjugate to L . As is remarked in [Bon06, § χ ( g ) = ∗ R G C G ( g s ) ( χ )( g ) so we need only show that ∗ R G C G ( g s ) ( χ )( g ) σ = ∗ R G C G ( g s ) ( χ )( g ) . The classfunction ∗ R G C G ( g s ) ( χ ) is a Z -linear combination of irreducible characters, so it suffices to show that λ ( g ) σ = λ ( g ) for each irreducible constituent λ of ∗ R G C G ( g s ) ( χ ) .Assume λ is such a constituent. Then λ is a contained in a Lusztig series of C G ( g s ) labelled by asemisimple element which is G ⋆ -conjugate to s , see [Bon06]. As mentioned above, we have s is of 2-powerorder, hence so is any G ⋆ -conjugate of s . By Lemma 3.4 we thus have the Lusztig series containing λ is3 σ -invariant. If ω λ : Z ( C G ( g s )) → Q × ℓ is the central character of λ , then λ ( g ) = ω λ ( g s ) λ ( g u ) because g s ∈ Z ( C G ( g s )) . As the Lusztig series containing λ is σ -invariant, we have ω λ ( g s ) σ = ω λ ( g s ) by Lemma 3.6.Applying Lemma 10.5, we see that λ ( g u ) σ = λ ( g u ) . However, by Lemma 10.8 we must have λ ( g u ) = λ ( g u ) because C G ( g s ) is an F -stable G -conjugate of L . In particular, we have λ ( g ) σ = ω λ ( g s ) σ λ ( g u ) σ = ω λ ( g s ) λ ( g u ) = λ ( g ) as desired. (cid:4) Acknowledgements
The authors would like to thank the organizers of the 2015 workshop on “Representations of FiniteGroups” at Mathematisches Forschungsinstitut Oberwolfach, where this collaboration began. The first-named author was supported in part by a grant from the Simons Foundation (Award
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