Robinson’s conjecture for classical groups

Zhicheng Feng; Conghui Li; Yanjun Liu; Gunter Malle; Jiping Zhang

doi:10.1515/jgth-2018-0177

Publicly Available Published by De Gruyter April 6, 2019

Robinson’s conjecture for classical groups

Zhicheng Feng , Conghui Li , Yanjun Liu , Gunter Malle and Jiping Zhang

From the journal Journal of Group Theory

https://doi.org/10.1515/jgth-2018-0177

Abstract

Robinson’s conjecture states that the height of any irreducible ordinary character in a block of a finite group is bounded by the size of the central quotient of a defect group. This conjecture had been reduced to quasi-simple groups by Murai. The case of odd primes was settled completely in our predecessor paper. Here we investigate the 2-blocks of finite quasi-simple classical groups.

1 Introduction

The arithmetic nature of the irreducible character degrees of a finite group has been a fruitful area of research since the very beginnings of the subject. Several deep conjectures in character theory concern the p-parts occurring in character degrees, like, for example, the McKay conjecture and the Dade conjecture. In 1996, Geoffrey Robinson [13] proposed an extension of Richard Brauer’s famous height zero conjecture from 1955, bounding the maximal power of a prime p dividing the degree of an irreducible character of a finite group G in terms of invariants of its p-block.

Conjecture (Robinson).

Let G be a finite group, p a prime and χ∈Irr⁡(G) lying in a p-block of G with defect group D. Then

(RC)pdef⁡(χ)≥|Z⁢(D)|

with equality if and only if D is abelian.

Here the p-defect of an irreducible character χ∈Irr⁡(G) is the integer def⁡(χ) such that |G|p=pdef⁡(χ)⁢χ⁢(1)p, where np denotes the p-part of an integer n.

We recently succeeded in showing this conjecture for all primes p≥3 (see [8]) based on Murai’s reduction of (RC) to blocks of quasi-simple groups [8, Theorem 2.3]. Here we prove (RC) for the 2-blocks of finite quasi-simple classical groups in odd characteristic.

Theorem 1.

The 2-blocks of covering groups of finite simple linear, unitary, symplectic and orthogonal groups do not provide minimal counterexamples to Robinson’s conjecture.

Thus, by the results in [8], in order to complete the proof of Robinson’s conjecture in full generality, it only remains to deal with the so-called isolated 2-blocks of quasi-simple groups of exceptional Lie type in odd characteristic.

After some preparations on 2-blocks of the general linear and unitary groups, we treat the principal 2-blocks of the special linear and unitary groups in Section 3 and the remaining finite quasi-simple classical groups in Section 4; the proof of Theorem 1 is then achieved in Section 5.

Throughout the paper, we let ν denote the exponential valuation associated to 2, normalised so that ν⁢(2)=1. For a finite group H, we write ν⁢(H) for ν⁢(|H|).

2 The general linear and unitary groups

2.1 Some notation and background

Assume q=pf is a power of a prime p. Let 𝔽¯ be an algebraic closure of the finite field 𝔽p. As usual, 𝐆=GLn⁡(𝔽¯) denotes the group of all invertible n×n matrices over 𝔽¯. Let γ:𝐆→𝐆 be the map sending A to (A-1)t, where t denotes the transpose of matrices. If Fp is the Frobenius map of 𝔽¯ and Fq=(Fp)f, then, for η∈{±1}, F=γ1-η2⁢Fq induces a Steinberg endomorphism of 𝐆 with the finite group of fixed points 𝐆F=GLn⁡(η⁢q). Recall that GLn⁡(-q) denotes the general unitary group

GUn⁡(q)={A∈GLn⁡(q2)∣Fq⁢(A)t⁢A=In},

where In is the identity matrix of degree n. We will use the analogous notation SLn⁡(η⁢q) for SLn⁡(q) or SUn⁡(q). Denote 𝔽=𝔽η⁢q for 𝔽q or 𝔽q2, depending on η=±1.

Let Irr⁡(𝔽⁢[X]) be the set of monic irreducible polynomials in 𝔽⁢[X] different from X. Denote

ℱ1:={f∈Irr(𝔽q2[X])∣f=f~},

ℱ2:={ff~∣f∈Irr(𝔽q2[X]),f≠f~},

where ⋅~ is the permutation of Irr⁡(𝔽q2⁢[X]) of order 2 mapping

f⁢(X)=Xm+am-1⁢Xm-1+⋯+a0

to Fq⁢(a0-1⁢Xm⁢f⁢(X-1)). Let

ℱ={Irr⁡(𝔽⁢[X])if⁢η=1,ℱ1∪ℱ2if⁢η=-1.

As introduced in [9, § 1], the polynomials in ℱ serve as the “elementary divisors” of semisimple elements of GLn⁡(η⁢q). Given a semisimple element s of G=GLn⁡(η⁢q), let ∏f∈ℱsf be the primary decomposition of s and ∏f∈ℱLf the corresponding decomposition of the Levi subgroup L:=CG(s). Then we have n=∑f∈ℱmf⁢df, where mf is the multiplicity of f in the characteristic polynomial of s and df denotes the degree of f∈ℱ. Notice that Lf is isomorphic to GLmf⁡((η⁢q)df). Correspondingly, the Weyl group W of L can be decomposed as ∏fWf, where Wf is the Weyl group of Lf.

From now on, we assume p is an odd prime. Let a:=ν(q2-1)-1. We give some elementary lemmas, which will be needed in the sequel.

Lemma 2.1.

Let d=2α⁢m, where m is odd and α≥0.

If ∣⁡4(q-η), then ν⁢((η⁢q)d-1)=a+α and ν⁢((η⁢q)d+1)=1.
Suppose ∣⁡4(q+η). Then, if α≥1,
ν⁢((η⁢q)d-1)=a+α 𝑎𝑛𝑑 ν⁢((η⁢q)d+1)=1,
while, if α=0, ν⁢((η⁢q)m+1)=a and ν⁢((η⁢q)m-1)=1.

Note that ∣⁡c((η⁢q)df-1) if c is the multiplicative order of some root of f∈ℱ.

Lemma 2.2.

Let f∈F. Suppose that f has a root of order 2m, where m≥1. Then df=2m′, where

m′={m-a𝑖𝑓⁢m>a,1∣⁡𝑖𝑓⁢ 4(q+η)⁢𝑎𝑛𝑑⁢ 1<m≤a,0𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒.

Proof.

By definition, we have ν⁢((η⁢q)2m′-1)≥m, so 2m divides (η⁢q)2m′-1. However, since df is the minimal integer such that ∣⁡2m((η⁢q)df-1), we conclude that ∣⁡df2m′.

We now show that 2m′∣df. If ∣⁡4(q-η), then ν⁢((η⁢q)df-1)=a+ν⁢(df). Thus ∣⁡2m′df, and the assertion follows. So we assume ∣⁡4(q+η). Then we fall into the following three cases: m>a, 1<m≤a, m=1. The proof is completed by a case-by-case checking. ∎

Lemma 2.3.

Let a≥2, and let n=2b1+⋯+2bt be the 2-adic expansion of a positive integer n with b1<⋯<bt. Suppose n=∑i≥0si⁢2i, where all si are non-negative integers. Then

∑isi≥t,
a⁢(∑isi-t)+∑i≥0i⁢si≥∑i=1tbi,
if n≥4 and s0≠0, then ∑isi-t≥b1.

In all three cases, equality occurs if and only if n=∑isi⁢2i is the 2-adic expansion.

Let i0=min⁡{i⁢∣si>⁢0} and k=min⁡{a,i0}. Then
a⁢(∑isi-t)+∑ii⁢si>k+1
unless n=∑isi⁢2i is the 2 -adic expansion and either n=3 or n=2h with h≤a.

Proof.

By considering ν⁢(n), we have i0:=min{i∣si>0}≤b1. Then (a) and (b) follow by applying the inductive hypothesis to

n-2i0=(si0-1)⁢2i0+∑i>i0si⁢2i=2i0+⋯+2b1-1+∑i=2t2bi.

We prove (c). If b1=0, it is just (a). Thus we assume b1≠0. Then

n-1=1+⋯+2b1-1+2b2+⋯+2bt.

Since s0≠0, we get n-1=(s0-1)+∑i≠0si⁢2i, and thus the assertion follows from (a).

For (d), a⁢∑i(si-t)+∑ii⁢si>k follows from (b). Now note that, since a≥2, by an argument similar to the first paragraph of the proof of [8, Lemma 6.5], if n=∑isi⁢2i is not the 2-adic expansion, then

a⁢(∑isi-t)+∑ii⁢si≥a+∑ii⁢si>k+1,

as claimed. Now we assume that n=∑isi⁢2i is the 2-adic expansion. If t>1, then ∑ibi>b2+1≥k+1 unless n=3. ∎

2.2 Sylow 2-subgroups of GLn⁡(η⁢q)

Assume ∣⁡4(q-η). Then, by Lemma 2.1, ν⁢(q-η)=a. Let R+0 be a Sylow 2-subgroup of GL1⁡(η⁢q). For b≥1, let R+b be the Sylow 2-subgroup R+0≀Xb of GL2b⁡(η⁢q), where Xb denotes a Sylow 2-subgroup of the group of permutation matrices of degree 2b, which is the iterated wreath product C2≀⋯≀C2 of b copies of a cyclic group C2 of order 2. In particular, ν⁢(R+b)=a⁢2b+ν⁢(2b!).

On the other hand, when ∣⁡4(q+η), we denote by R-1 the Sylow 2-subgroup of GL2⁡(η⁢q) as defined in [5], isomorphic to the semi-dihedral group of order 2a+2. For b≥2, we denote R-b=R-1≀Xb-1≤GL2b⁡(η⁢q), where Xb is as defined above. For convenience, we set R-0:={±1}, the Sylow 2-subgroup of GL1⁡(η⁢q).

We write the 2-adic expansion of n as n=2b1+⋯+2bt with b1<⋯<bt. Recall some facts on 2-blocks and the structure of Sylow 2-subgroups of GLn⁡(η⁢q); for this, recall that, according to Lusztig, the set of irreducible characters of a finite reductive group G is partitioned into Lusztig seriesℰ⁢(G,s) labelled by semisimple elements s of the Langlands dual group G*, up to conjugation (see, e.g., [7, § 13]). The elements of ℰ⁢(G,1) are the unipotent characters of G. Lusztig’s Jordan decomposition states that ℰ⁢(G,s) is in bijection with ℰ⁢(CG*⁢(s),1) whenever s has connected centraliser in the ambient algebraic group in such a way that character degrees differ by a factor of |G*:CG*(s)|p′. For s a semisimple 2′-element, one sets

ℰ2(G,s):=⋃tℰ(G,st),

where the union runs over 2-elements t∈CG*⁢(s). This is known to be a union of 2-blocks of G. For s=1, this contains the trivial character and hence the principal 2-block of G.

Theorem 2.4.

Let G=GLn⁡(η⁢q), q odd, and ϵ∈{±} such that q≡ϵ⁢η⁢(mod⁡4).

The only unipotent 2 -block of G is the principal block ℰ2⁢(G,1).
A Sylow 2 -subgroup of G is given by
R=∏i=1tRϵbi.
Thus R is abelian if and only if n=1.
The centre of R is given by
Z⁢(R)=∏i=1tRϵ0⊗I2bi.
In particular, ν⁢(Z⁢(R))=t⁢ν⁢(q-η).

Proof.

Part (a) was shown by Broué [2], and (b) follows from [5]. From this, (c) can easily be derived. ∎

2.3 Robinson’s conjecture for GLn⁡(η⁢q)

Now we prove Robinson’s conjecture for the principal 2-block of GLn⁡(η⁢q).

Lemma 2.5.

Let χ be a unipotent character of GLn⁡(η⁢q). Then def⁡(χ)≥n⁢ν⁢(q-η).

Proof.

For G=GLn⁡(q), all unipotent characters lie in the principal series. Thus their degree polynomials are not divisible by q-1 (see [4, § 13.7]), while the order polynomial of G is divisible by (q-1)n, so we obtain the stated bound. The claim for GUn⁡(q) follows as the order polynomial as well as the degree polynomials of unipotent characters are obtained by replacing q by -q in those for GLn⁡(q). ∎

Proposition 2.6.

Let G=GLn⁡(η⁢q), n≥2. Let B be the principal 2-block of G, and let R be a Sylow 2-subgroup of G. Then def⁡(χ)>ν⁢(Z⁢(R)) for any χ∈Irr⁡(B).

Proof.

There is a semisimple 2-element s∈G such that χ∈ℰ⁢(G,s) by Theorem 2.4 (a). For b≥0, define ℱb={f∈ℱ∣df=2b}. Let ℱ⁢(s)⊂ℱ be the set of elementary divisors of s, and set ℱb(s):=ℱ(s)∩ℱb. Then, by Lemma 2.2, we have ℱ⁢(s)=∐bℱb⁢(s) and thus

n=∑f∈ℱmf⁢df=∑b≥0∑f∈ℱbmf⁢2b.

Let ψ be the unipotent character of L:=CG(s) in Jordan correspondence with χ. Let ∏f∈ℱ⁢(s)Lf, ⊠f∈ℱ⁢(s)ψf be the respective decompositions of L, ψ, corresponding to the primary decomposition of s. By the degree formula for the Jordan decomposition of characters, we have

ν(χ(1))=ν(G:L)+ν(ψ(1))

and so, as ν⁢(L)=∑f∈ℱ⁢(s)ν⁢(Lf),

def⁡(χ)=def⁡(ψ)=∑f∈ℱ⁢(s)def⁡(ψf).

As before, write n=2b1+⋯+2bt for the 2-adic expansion of n.

Let first ∣⁡4(q-η), so ν⁢(q-η)=a. By Lemmas 2.1 and 2.5,

def⁡(ψf)≥mf⁢(a+b) for⁢f∈ℱb⁢(s).

It follows by Lemma 2.3 that

(2.1)def⁡(χ)≥a⁢∑f∈ℱmf+∑bb⁢∑f∈ℱbmf≥a⁢t+∑i=1tbi>a⁢t,

as n>1. By Theorem 2.4 (c), we know that ν⁢(Z⁢(R))=a⁢t, whence the claim in this case.

Now assume ∣⁡4(q+η). Here, by Lemma 2.3 (a),

def⁡(χ)≥∑f∈ℱ0mf+∑b≥1∑f∈ℱbmf⁢(a+b)=∑f∈ℱmf+∑b≥1∑f∈ℱbmf⁢(a+b-1)≥t+∑b≥1∑f∈ℱbmf⁢(a+b-1),

with equality in the last line if and only if n=∑b∑f∈ℱbmf⁢2b is the 2-adic expansion of n. Furthermore, as a≥2, the last sum is zero only when ℱb⁢(s)=∅ for b≥1. But, in this case, n=1, which was excluded, so we conclude by Theorem 2.4 (c). ∎

We note the following for later use.

Remark 2.7.

In the notation of Proposition 2.6, assume ∣⁡4(q+η). The proof of Proposition 2.6 shows that n=2 if def⁡(χ)-ν⁢(Z⁢(R))=1 and n is even.

3 The groups SLn⁡(η⁢q)

3.1 Robinson’s conjecture for SLn⁡(η⁢q), n odd

The case of SLn⁡(η⁢q) with n odd is considerably easier than the even degree case.

Proposition 3.1.

Let H=SLn⁡(η⁢q) with n≥3 odd, and let Z≤Z⁢(H). Then Robinson’s conjecture holds for the unipotent 2-blocks of H/Z.

Proof.

Set G=GLn⁡(η⁢q). Since n is odd, H∩𝐎2⁢(Z⁢(G))=1, so

G=H1×𝐎2⁢(Z⁢(G)) with⁢H1=𝐎2⁢(G).

Obviously, all irreducible characters of G restrict irreducibly to H1, and the Sylow 2-subgroups of G are the direct products of Sylow 2-subgroups of H1 with 𝐎2⁢(Z⁢(G)). Thus, if b1 is a 2-block of H1 and B is the 2-block of G covering b1, then Robinson’s conjecture holds for b1 if and only if it holds for B.

Furthermore, |H1:H| is odd, and so, for χ∈Irr⁡(H1), all constituents of χ|H have the same defect as χ. Thus, if b is a 2-block of H and b1 is a 2-block of H1 covering b, then Robinson’s conjecture holds for b if and only if it holds for b1.

Finally, |Z⁢(H)|=gcd⁡(n,q-1) is odd. Thus, if b¯ is a 2-block of H/Z, where Z≤Z⁢(H), and b is the 2-block of H dominating b¯, then Robinson’s conjecture holds for b¯ if and only if it holds for b. The claim thus follows from Proposition 2.6. ∎

3.2 Sylow 2-subgroups of SLn⁡(η⁢q) for n even

For n≥2 even, we first determine the centres of Sylow 2-subgroups. Denote c=ν⁢(q-η), and, for b≥0, set c(b):=max{c-b,0}. Note that

c={a∣⁡if⁢ 4(q-η),1∣⁡if⁢ 4(q+η).

For a subgroup H≤GLn⁡(𝔽), we set

𝒟(H):={det(A)∣A∈H}≤𝔽×.

Lemma 3.2.

Let b≥0. In the notation of Theorem 2.4, we have

|𝒟⁢(Rϵb)|=2c 𝑎𝑛𝑑 |𝒟⁢(Z⁢(Rϵb))|=2c⁢(b).

Proof.

If ∣⁡4(q-η), then R+b contains diag⁡(ζ,1,…,1) with ζ∈𝔽× of order o⁢(ζ)=2c, so |𝒟⁢(R+b)|=2c. Then |𝒟⁢(Z⁢(R+b))|=2c⁢(b) follows from

𝒟⁢(Z⁢(R+b))={x2b∣x∈𝒟⁢(R+0)}.

For ϵ=-, this follows from the fact that R-1 is conjugate to 〈(ζζη⁢q),(11)〉 for ζ∈𝔽× of order o⁢(ζ)=2a+1. ∎

Lemma 3.3.

Let G=GLn⁡(η⁢q) and H=SLn⁡(η⁢q) with n even. Suppose that R is a Sylow 2-subgroup of G as in Theorem 2.4 (b). Write Q=R∩H. Then |D⁢(R)|=2c. Furthermore,

|𝒟(Z(R))|=2c⁢(b1) 𝑎𝑛𝑑 ν(R:QZ(R))=c-c(b1)=min{c,b1}.

Proof.

This follows easily from Lemma 3.2. ∎

When ∣⁡4(q-η), the Weyl group part of R may not be contained in SLn⁡(η⁢q). To circumvent this problem, for b≥1, we will use R+b=R+0≀Xb≅R+1≀Xb-1 and call Xb-1 the Weyl part and (R+1)2b-1 the base group of R+b. Observe that, with this convention, the Weyl part of R+b is contained in SLn⁡(η⁢q).

Lemma 3.4.

Let G=GLn⁡(η⁢q) and H=SLn⁡(η⁢q) with n even. Suppose that R is as in Theorem 2.4 (b). Let R0,i be the base group and RW,i=Xbi-1 the Weyl part of Rϵbi. Write R0=R0,1×⋯×R0,t and RW=RW,1×⋯×RW,t. Denote Q=R∩H. Then

Q=(R0∩H)⋊RW 𝑎𝑛𝑑 Z⁢(Q)=Z⁢(R)∩H.

More specifically, if R=Rϵ1, then Q is isomorphic to the generalised quaternion group of order 2a+1, and so Z⁢(Q)={±I2}.

Proof.

The Weyl part RW,i is generated by direct products of tensor products of matrices of the form (𝟎I2I2𝟎), whence RW,i≤H. Hence RW≤H, and so Q=(R0∩H)⋊RW.

By Lemma 3.2, 𝒟⁢(Rϵ0)=𝒟⁢(Rϵ1). If t≥2, then the projections pi:Q→Rϵbi are surjective for all i. Hence Z⁢(Q)≤Z⁢(R), and so Z⁢(Q)≤Z⁢(R)∩H. Since the converse inclusion is obvious, we get Z⁢(Q)=Z⁢(R)∩H in this case.

Now we may assume R=Rϵb+1 with b≥0. If b=0, the assertion is obvious since Q is isomorphic to the generalised quaternion group of order 2a+1. So we may assume b>0. Let (A1,…,A2b)⁢τ be an element of Z⁢(Q) with τ∈Xb and (A1,…,A2b)∈R0. Let B be any element of Rϵ1 with det⁡(B)=±1 so that (B,…,B)∈Q. Now

[(A1,…,A2b)⁢τ,(B,…,B)]=1.

Hence (A1,…,A2b)∈Z⁢(R). Moreover, since [(A1,…,A2b)⁢τ,π]=1 for any π∈Xb and Xb transitively permutes the Ai, we have A1=…=A2b. It follows that τ∈Z⁢(Q) and τ=1, and so Z⁢(Q)=Z⁢(R)∩H, completing the proof. ∎

3.3 Robinson’s conjecture for SLn⁡(η⁢q), n even

Now we prove Robinson’s conjecture for unipotent 2-blocks of SLn⁡(η⁢q) with even n.

Proposition 3.5.

Let H=SLn⁡(η⁢q) with n≥2 even, and let Q be a Sylow 2-subgroup of H. Then, for every θ∈E2⁢(H,1), we have def⁡(θ)>ν⁢(Z⁢(Q)).

Proof.

Let s∈H* be a 2-element such that θ∈ℰ⁢(H,s). Let G=GLn⁡(η⁢q), and let χ∈ℰ2⁢(G,1) lie above θ. By Theorem 2.4 (a), χ lies in the principal 2-block of G, so θ lies in the principal 2-block of H. Let R be a Sylow 2-subgroup of G as described in Theorem 2.4 (b) and with R∩H=Q.

Now we have

def⁡(θ)=ν⁢(H)-ν⁢(θ⁢(1))=def⁡(χ)-ν⁢(q-η)+ν⁢(χ⁢(1)/θ⁢(1))

and

ν⁢(Z⁢(Q))=ν(Z(R))-ν(R:Q)+ν(R:QZ(R))=ν⁢(Z⁢(R))-ν⁢(q-η)+min⁡{c,b1}

by Lemmas 3.3 and 3.4.

We first let ∣⁡4(q-η). Then def⁡(χ)-ν⁢(Z⁢(R))≥∑i=1tbi by (2.1), so

def⁡(θ)≥ν⁢(Z⁢(R))-ν⁢(q-η)+∑i=1tbi+ν⁢(χ⁢(1)/θ⁢(1))=ν⁢(Z⁢(Q))-min⁡{c,b1}+∑i=1tbi+ν⁢(χ⁢(1)/θ⁢(1))≥ν⁢(Z⁢(Q))

with equality only when t=1, c≥b1=:b, and so n=2b. In that case, there is exactly one element f in ℱ⁢(s), and mf⁢(s)=1, df=2b=n. Therefore, in order to complete the proof, it suffices to show that ν⁢(χ⁢(1)/θ⁢(1))>0. To do this, we compute the number m of irreducible constituents of χ|H according to the method of Denoncin [6]. Let ζ be a root of f in 𝔽¯. Since G/H is cyclic and isomorphic to Z⁢(G), it follows by [6, Proposition 3.5] that

m=|{z∈Z⁢(G)∣z⁢s⁢is⁢G⁢-conjugate to⁢s}|.

Now

m=|{τ∈𝔽×∣z=diag⁡(τ,…,τ)∈Z⁢(G),τ⁢ζ⁢is a root of⁢f⁢in⁢𝔽¯}|=|{τ∈𝐎2⁢(𝔽×)∣τ⁢ζ⁢is a root of⁢f⁢in⁢𝔽¯}|.

Notice that the roots of f are ζ,ζη⁢q,…,ζ(η⁢q)n-1. Since n=2b, we may write

ζ(η⁢q)n2=ξ⁢ζ with⁢ξ=ζ(η⁢q)2b-1-1≠1.

We claim ξ2=1. Indeed, ζ has multiplicative order 2a+b. In addition, we have ν⁢((η⁢q)2b-1)=ν⁢((η⁢q)2b-1-1)+1 by Lemma 2.1. Hence ξ has multiplicative order 2. Thus ν⁢(m)>0, and we have ν⁢(χ⁢(1)/θ⁢(1))>0 by Clifford theory.

Now we assume ∣⁡4(q+η). If Z⁢(R)≰H, then ν(R:QZ(R))=0, and hence the claim holds by Proposition 2.6. So we may assume Z⁢(R)≤H. Then we get ν(R:QZ(R))=1, and so the claim holds if def⁡(χ)-ν⁢(Z⁢(R))≥2. By Remark 2.7, we have def⁡(χ)-ν⁢(Z⁢(R))=1 only when n=2. But, in this case, |Z⁢(Q)|=2, and so Robinson’s conjecture holds by an old result of Brauer (see [8, Lemma 3.1]). ∎

3.4 Robinson’s conjecture for central quotients of SLn⁡(η⁢q), n even

In this section, we investigate unipotent 2-blocks of SLn⁡(η⁢q)/Z for odd q, where Z≤Z⁢(SLn⁡(η⁢q)). Throughout, n≥2 is even. As before, we first determine the centres of Sylow 2-subgroups of central quotient groups of SLn⁡(η⁢q).

Lemma 3.6.

Keep the notation of Lemma 3.4; in particular, n is even. We write Q¯=Q/Z and Z⁢(Q¯)=Z0/Z for 1≠Z≤O2⁢(Z⁢(H)).

If t≥2, then Z0=Z⁢(Q).
Assume R=Rϵb.
1. If b=1, then Q/Z is isomorphic to the dihedral group of order 2a.
2. If b>1, then Z0=〈Z⁢(Q),I2b⊗diag⁡(1,-1)〉.

Thus, if Q¯ is not abelian, then Z⁢(Q¯)=Z⁢(Q)/Z unless either R=Rϵ1 with a>2 or R=Rϵb with b>1, in which cases we have

ν⁢(Z⁢(Q¯))=ν⁢(Z⁢(Q))-ν⁢(Z)+1.

Proof.

Recall that |𝒟⁢(R-1)|=2 and 𝒟⁢(R+0)=𝒟⁢(R+1) by Lemma 3.2.

(a) Suppose t≥2. We first claim that Z0⊆R0, where R0 is the base group of R as in Lemma 3.4. Otherwise, let (A1,A2,…)⁢τ be an element of Z0, where 1≠τ∈RW and Ai is an element of R0,i for each i; here, R0,i is the base subgroup of Rϵbi as in Lemma 3.4. Since R has more than one component, arguing as in the last paragraph of the proof of Lemma 3.4, we have (A1,A2,…)∈Z⁢(R). We may take a non-trivial orbit of 〈τ〉, say i1,i2=τ-1⁢(i1),…, and a j outside this orbit (such j exists since t≥2). In addition, take (B1,B2,…)∈Q, where Bi=1 for all i except that Bi1 and Bj satisfy that the order of Bi1 is greater than 2 and det⁡(Bi1)⁢det⁡(Bj)=1. Then direct calculation shows that

[(A1,A2,…)⁢τ,(B1,B2,…)]∉Z,

which is a contradiction. Thus Z0⊆R0, as claimed.

Now let A=(A1,…,At)∈Z0, where Ai∈R0,i. Then, obviously, each component of A is in the centre of the corresponding component of Q, and thus Z0=Z⁢(Q).

(b) The statement for b=1 is well known. So assume b≥2. Then a slight modification of the argument in (a) shows that Z0⊆R0, and then the claim follows as there. ∎

Proposition 3.7.

Let H=SLn⁡(η⁢q) with n≥2 even and 1≠Z≤O2⁢(Z⁢(H)). If a Sylow 2-subgroup Q¯ of H/Z is non-abelian, then, for every ϕ in the principal 2-block of H/Z, we have def⁡(ϕ)>ν⁢(Z⁢(Q¯)).

Proof.

Write θ for the inflation of ϕ to H; then θ lies in the principal 2-block of H. It is easy to see that def⁡(ϕ)=def⁡(θ)-ν⁢(Z). Let G=GLn⁡(η⁢q), and let R be a Sylow 2-subgroup of G and Q=R∩H, a Sylow 2-subgroup of H. Let χ∈ℰ⁢(G,s) with s∈R be such that θ is an irreducible constituent of χ|H.

We first let ∣⁡4(q-η). If t≥2, then, by Lemma 3.6,

ν⁢(Z⁢(Q¯))=ν⁢(Z⁢(Q))-ν⁢(Z).

Hence the result follows from Proposition 3.5. So now we may assume t=1, that is, R=R+b. Since Q¯ is non-abelian, we have b>0. Now Lemma 3.6 yields ν⁢(Z⁢(Q¯))=ν⁢(Z⁢(Q))-ν⁢(Z)+1.

By the proof of Proposition 3.5, def⁡(θ)>ν⁢(Z⁢(Q))+1 unless

n=∑f∈ℱb⁢(s)mf⁢2b

is the 2-adic expansion and b≤a. Hence ℱb⁢(s)={f}, mf=1 and o⁢(s)=2a+b. Notice that L=CG⁢(s) is in fact a Coxeter torus of G, and s generates the Sylow 2-subgroup of L and hence of H*/[H*,H*]. But then all characters in ℰ⁢(H,s) are faithful on 𝐎2⁢(Z⁢(H)). However, this contradicts the obvious fact that

Z≤ker⁡(θ)=ker⁡(χ).

This achieves the proof when ∣⁡4(q-η).

Now assume ∣⁡4(q+η). As before, if ν⁢(Z⁢(Q¯))=ν⁢(Z⁢(Q))-ν⁢(Z), then the assertion follows from Proposition 3.5. So, by Lemma 3.6, it remains to consider the case ν⁢(Z⁢(Q¯))=ν⁢(Z⁢(Q))-ν⁢(Z)+1, i.e., either R=R-1 with a>2 or R=R-b with b≥2. Here |Z⁢(Q¯)|=2 by Lemma 3.6, and so Robinson’s conjecture holds by [8, Lemma 3.1]. ∎

4 Principal 2-blocks of quasi-simple groups of classical type

Let 𝐆 be a simple algebraic group of symplectic or orthogonal type over an algebraically closed field of odd characteristic and F a Frobenius endomorphism of 𝐆 with respect to an 𝔽q-rational structure, and denote G=𝐆F. So, in particular, q is an odd prime power. Throughout, we will fix the prime ℓ=2, with respect to which defects will be considered. As for the linear and unitary groups, we will need three pieces of information: defects of unipotent characters, centralisers of 2-elements and the centres of Sylow 2-subgroups.

4.1 Unipotent characters of classical groups

Here we determine lower bounds on defects of unipotent characters of classical groups. Observe that, by Lusztig’s results, the classification and degrees of unipotent characters are insensitive to the isogeny type (see [4, § 13.7]), so, for our purposes, we will not need to specify these here.

Lemma 4.1.

Let χ be a cuspidal unipotent character of a finite group G of classical type. Let 2b be the precise power of 2 dividing q+1. Then

def⁡(χ)={(b+1)⁢nfor types⁢Bn⁢(q),Cn⁢(q),(b+1)⁢n-1for types⁢Dn⁢(q),Dn2⁢(q).

Proof.

We discuss the various types individually. If G=Bn⁢(q) or Cn⁢(q), then, by [4, Section 13.7], we have n=s2+s for some s≥1, and, according to the formula given there,

def⁡(χ)=s+b⁢∑i=1s2⁢i+∑i=1s(2⁢i-1)

=s+b⁢(s2+s)+s2

=(b+1)⁢n.

If G=Dn⁢(q) or Dn2⁢(q), then we have n=s2 for some integer s (which is even in the first case, odd in the second) and

def⁡(χ)=s-1+b⁢∑i=1s(2⁢i-1)+∑i=1s-12⁢i

=s-1+b⁢s2+s2-s

=(b+1)⁢n-1.∎

Lemma 4.2.

Let χ be a unipotent character of a finite classical group G.

If G is of type Bn⁢(q),Cn⁢(q) with n≥1 or of type Dn⁢(q),Dn2⁢(q) with n≥2, then def⁡(χ)>n.
If G is of type Dn⁢(q2) or Dn2⁢(q2) with n≥2, then def⁡(χ)≥2⁢n-1, with equality only possibly when n is a square.

Proof.

Assume that χ lies in the Harish-Chandra series of the cuspidal unipotent character λ of a Levi subgroup L≤G for an F-stable Levi subgroup 𝐋 of an F-stable parabolic subgroup of 𝐆. Let 2a, 2b be the precise power of 2 dividing q-1, q+1, respectively. By Lusztig’s classification (see [4, § 13.7]), then [𝐋,𝐋] is simple of the same classical type as 𝐆; hence λ is as considered in Lemma 4.1. Moreover, χ⁢(1) divides the degree |G:L|p′λ(1) of the Harish-Chandra induced character RLG⁢(λ) (as can be seen, for example, from [4, Theorem 10.11.5]). Using 𝐋=[𝐋,𝐋]⁢Z∘⁢(𝐋), we conclude that 2def⁡(χ)≥2def⁡(λ)⁢|Z∘⁢(𝐋)F|2. Now first assume that G is of type Bn⁢(q) or Cn⁢(q), and so L has type Bu⁢(q) or Cu⁢(q) for some u≥1. Then |Z∘⁢(𝐋)F|=(q-1)n-u, and so

def⁡(χ)≥a⁢(n-u)+(b+1)⁢u

by Lemma 4.1. This is linear in u and hence at least as big as the minimum of its values at u=0 and u=n. Hence it is larger than n unless u=0 and a=1. In the latter case, χ lies in the principal series. Our claim follows if χ⁢(1) is not divisible by (q+1)n. If it is divisible by (q+1)n, then χ is 2-cuspidal, and, with the Ennola duality, we obtain def⁡(χ)≥(a+1)⁢n>n from Lemma 4.1. If G has type Dn⁢(q) or Dn2⁢(q), then L has type Du⁢(q) or Du2⁢(q), giving

def⁡(χ)≥a⁢(n-u)+(b+1)⁢u-1,

and we conclude as before. For types Dn⁢(q2) or Dn2⁢(q2), we similarly obtain def⁡(χ)≥3⁢n-1 when u=0 (as ∣⁡8(q2-1)) and def⁡(χ)≥(b+1)⁢n-1=2⁢n-1 when u=n is a square. ∎

4.2 Centralisers of semisimple elements in classical groups

We will make use of the primary decomposition of semisimple elements in classical groups. We follow the notation of [10]. As in Section 2, let Irr⁡(𝔽q⁢[X]) be the set of non-constant monic irreducible polynomials in 𝔽q⁢[X] different from X. For f∈Irr⁡(𝔽q⁢[X]), let f* be the polynomial in Irr⁡(𝔽q⁢[X]) whose roots (in 𝔽¯q) are the inverses of the roots of f. Denote

ℱ0:={X-1,X+1},

ℱ+:={f∈Irr(𝔽q[X])∣f∉ℱ0,f=f*},

ℱ-:={ff*∣f∈Irr(𝔽q[X]),f≠f*},

ℱ:=ℱ0∪ℱ+∪ℱ-.

For any integer b≥1, we also set ℱb:={f∈ℱ∣df=2b}. Here, and later on, df denotes the degree of f∈ℱ. Define the reduced degreeδf of f∈ℱ by

δf={dfif⁢f∈ℱ0,12⁢dfif⁢f∈ℱ+∪ℱ-.

Since the polynomials in ℱ+∪ℱ- have even degree, δf is an integer. In addition, we introduce a sign εf for f∈ℱ+∪ℱ- defined by

εf={-1if⁢f∈ℱ+,1if⁢f∈ℱ-.

Let V be a finite-dimensional symplectic or orthogonal space over 𝔽q with q odd and I⁢(V)=Sp⁡(V) or GO⁡(V), respectively. Given a semisimple element s∈I⁢(V), there exist unique orthogonal decompositions V=⊕f∈ℱVf and s=∏f∈ℱsf, where the Vf are non-degenerate subspaces of V, sf∈I⁢(Vf) and sf has minimal polynomial f. The above decomposition of s is often called the primary decomposition of s. Correspondingly, the centraliser of s in I⁢(V) has a decomposition CI⁢(V)⁢(s)=∏fCf⁢(s) with

Cf(s):=CI⁢(Vf)(sf)={I⁢(Vf)if⁢f∈ℱ0,GLmf⁡(εf⁢qδf)if⁢f∈ℱ+∪ℱ-,

where mf is the multiplicity of f in the characteristic polynomial of sf (and of s).

Now let V be an orthogonal space over 𝔽q and I∘⁢(V)=SO⁡(V). Let s∈I⁢(V). Then s∈I∘⁢(V) if and only if mX+1⁢(s) is even. For more details, see [10, § 1]. The following is elementary; see also Lemma 2.2.

Lemma 4.3.

Let a:=ν(q2-1)-1. If f∈F+∪F- has a root (in F¯q) of order 2m for some positive integer m, then

δf={1𝑖𝑓⁢m≤a,2m-a𝑖𝑓⁢m>a.

Moreover, f∈F- unless ∣⁡4(q+1) and m≤a.

4.3 Symplectic groups

Lemma 4.4.

Let G=Sp2⁢n⁡(q) with n≥2 and q odd, and let n=2b1+⋯+2bt be the 2-adic expansion. Then def⁡(χ)≥t+2 for every character χ in the principal 2-block of G.

Proof.

As recalled earlier, the principal 2-block lies in ℰ2⁢(G,1). So let s∈G* be a 2-element such that χ∈ℰ⁢(G,s), and let ψ~∈ℰ⁢(CG*⁢(s),1) denote the Jordan correspondent of χ (see, for example, [3, Corollary 15.14]). Then, by the degree formula for Jordan decomposition, we have

def⁡(χ)=ν⁢(|G|)-ν⁢(χ⁢(1))=ν⁢(|CG*⁢(s)|)-ν⁢(ψ~⁢(1)).

Let ψ be a unipotent character of CG*∘⁢(s) below ψ~. Then, by Clifford theory, ψ~⁢(1)/ψ⁢(1) divides |CG*⁢(s):CG*∘⁢(s)|, so

def⁡(χ)=ν⁢(|CG*⁢(s)|)-ν⁢(ψ~⁢(1))≥ν⁢(|CG*∘⁢(s)|)-ν⁢(ψ⁢(1))=def⁡(ψ).

Thus we need to discuss the defects of unipotent characters of CG*∘⁢(s). By our preliminary observations, CG*∘⁢(s) is isogenous to (and hence has the same unipotent characters as) a product of certain orthogonal, linear and unitary groups, which we will now investigate in detail.

Here we have G*=SO2⁢n+1⁡(q). Let V be the underlying space of G*. Let s=∏fsf be the primary decomposition, and let V=⊕fVf be the corresponding orthogonal decomposition of V. Then

CG*∘⁢(s)=∏f∈ℱCf with⁢Cf=CI∘⁢(Vf)⁢(sf).

Observe that m2:=mX+1 is even; m1:=mX-1≥1 must be odd since df is even for f∈ℱ+∪ℱ-. Write ℱb(s):={f∈ℱb∣mf>0}. By Lemma 4.3, df must be a power of 2 if mf≠0.

Now 2⁢n+1=m1+m2+∑fmf⁢df, and thus

(4.1)n=m1-12+m22+∑b≥1∑f∈ℱbmf⁢2b-1.

Then, by Lemma 2.3 (a),

(4.2)t≤m1-12+m22+∑f∈ℱ∖ℱ0mf,

and equality holds if and only if (4.1) is the 2-adic expansion of n.

Write ψ=⊠fψf, where ψf is a unipotent character of Cf. By Lemma 4.2 (a),

(4.3)def⁡(ψX-1)≥m1-12

with equality only when m1=1 and, also by Lemma 4.2 (a), if m2>0, then

(4.4)def⁡(ψX+1)≥m22

with equality only when m2=2. If f∈⋃b>1ℱb⁢(s), then δf=2b-1≥2 is a power of 2 by Lemma 4.3 and Cf≅GLmf⁡(qδf). By Lemma 2.5,

(4.5)def⁡(ψf)≥ν⁢(qδf-1)⁢mf≥(b+1)⁢mf≥3⁢mf≥mf+2.

If f∈ℱ1⁢(s), then ∣⁡4(q+1), δf=1 and Cf≅GLmf⁡(-q), so, by Lemma 2.5,

(4.6)def⁡(ψf)≥mf⁢ν⁢(q+1)≥2⁢mf≥mf+1.

By (4.2), (4.3), (4.4), (4.5) and (4.6), we can now compare def⁡(χ) with t+2. If ⋃b>1ℱb⁢(s)≠∅, then def⁡(χ)≥t+2 by (4.5). So now assume ℱb⁢(s)=∅ for b>1. Then n=m1-12+m22+∑f∈ℱ1⁢(s)mf, and hence equality does not hold in (4.2). If n≥4, then n-t≥3, and then def⁡(χ)≥t+3 holds. On the other hand, when n=2,3, then n-t=1. If ℱ1⁢(s)≠∅, then def⁡(χ)≥t+2 holds by (4.6). So we assume ℱ1⁢(s)=∅. If m1>1, then the result follows from (4.3). Hence we may further assume m1=1. Then n=m22, and def⁡(χ)≥t+2 holds by (4.4). This completes the proof. ∎

Let W be a Sylow 2-subgroup of Sp2⁡(q)=SL2⁡(q). Clearly, ν⁢(Z⁢(W))=1. For any positive integer b, we let Wb=W≀Xb, where Xb is a Sylow 2-subgroup of the symmetric group of degree 2b (i.e., Xb is isomorphic to C2≀⋯≀C2 with b factors). Then ν⁢(Z⁢(Wb))=1.

Now we let n=2b1+⋯+2bt, with 0≤b1<⋯<bt, be the 2-adic expansion of n. Then, by [5, Theorem 1], a Sylow 2-subgroup R of G=Sp2⁢n⁡(q) is isomorphic to Wb1×⋯×Wbt, and thus ν⁢(Z⁢(R))=t. The next result follows by a similar proof as for Lemma 3.6

Lemma 4.5.

In the notation above, let

S=G/Z⁢(G),R∈Syl2⁡(G) 𝑎𝑛𝑑 Q=R/Z⁢(G)∈Syl2⁡(S).

Let Z0≤R be such that Z0/Z⁢(G)=Z⁢(Q). Then Z0=Z⁢(R) if t≥2, and ν⁢(Z0)=ν⁢(Z⁢(R))+1 if t=1.

From Lemmas 4.4 and 4.5, we immediately deduce the following result.

Proposition 4.6.

Let G=Sp2⁢n⁡(q) with n≥2 and q odd, and let Z≤Z⁢(G). Then (RC) holds for the principal 2-block of G/Z.

4.4 Odd-dimensional orthogonal groups

Lemma 4.7.

Let G=SO2⁢n+1⁡(q), n≥3 and q odd, and let n=2b1+⋯+2bt be the 2-adic expansion. Then def⁡(χ)≥t+2 for every character χ in the principal 2-block of G.

Proof.

Let s∈G*≅Sp2⁢n⁡(q) be a semisimple 2-element such that χ∈ℰ⁢(G,s). Let ψ∈ℰ⁢(CG*⁢(s),1) be a unipotent character in Jordan correspondence with χ. As in the proof of Lemma 4.4, we have def⁡(χ)≥def⁡(ψ).

Let V be the underlying space of G*, s=∏fsf, V=⊕fVf and

CG*⁢(s)=∏f∈ℱ⁢(s)Cf with⁢Cf=CI⁢(Vf)⁢(sf).

Keep the notation used in the proof of Lemma 4.4. Then m1 and m2 are both even. Also, n=m12+m22+∑b≥1∑f∈ℱbmf⁢2b-1, and so

t≤m12+m22+∑f∈ℱ∖ℱ0mf.

By Lemma 4.2 (a), def⁡(ψi)≥mi2 for i=1,2, with equality only when mi≤2. For f∈ℱ⁢(s)∖ℱ0, the formulas for def⁡(ψf) are the same as in (4.5) and (4.6).

Now first assume n≥4. If ℱb⁢(s)=∅ for b>1, then

n=m12+m22+∑f∈ℱ1mf.

Hence def⁡(χ)≥n≥t+3 as n≥4. So now assume ⋃b>1ℱb⁢(s)≠∅. Also, by (4.5), it is easy to see that the claim holds unless |⋃b>1ℱb⁢(s)|=1 and mf=1 for the unique polynomial f in ⋃b>1ℱb⁢(s). If ℱ1⁢(s)≠∅, then, by (4.6), we get the result. Hence ℱ1⁢(s)=∅, and then ℱ⁢(s)∖ℱ0={f}. Again by (4.5), def⁡(ψf)≥mf+2, so n=m12+m22+δf must be the 2-adic expansion, and then m1+m2≤2. Hence δf≥n-1≥3, and then δf≥4 since it is a power of 2. Thus, by (4.5), def⁡(ψf)≥4⁢mf, and so def⁡(χ)≥t+3, completing the proof in case n≥4.

Now assume n=3. By Lemma 4.2 (a), for i∈{1,2}, if mi>0, then

(4.7)def⁡(ψi)>mi2.

Similar to the proof of Lemma 4.4, if ⋃b>1ℱb⁢(s)≠∅, then, by (4.5), we have def⁡(χ)≥t+2. So we may assume ℱb⁢(s)=∅ for b>1, and then

n=3=m12+m22+∑f∈ℱ1mf.

Note that here n-t=1. Moreover, by (4.7), we may assume m1=m2=0, and then ℱ⁢(s)=ℱ1⁢(s). Thus def⁡(χ)≥t+2 follows from (4.6), and we are done. ∎

To discuss the Sylow 2-subgroups, let W be a Sylow 2-subgroup of GO2η⁡(q) with q≡η⁢1⁢(mod⁡4), where η∈{±}. Then W is isomorphic to the dihedral group of order 2a+1, where a=ν⁢(q2-1)-1, and so ν⁢(Z⁢(W))=1. For any positive integer b, we let Wb=W≀Xb, where Xb is defined as in Section 2.2. Then ν⁢(Z⁢(Wb))=1.

Now let n=2b1+⋯+2bt be the 2-adic expansion of n, and let

ι:GO2⁢nη⁡(q)→SO2⁢n+1⁡(q),A↦diag⁡(A,det⁡(A))

be an embedding of GO2⁢nη⁡(q) into SO2⁢n+1⁡(q) with qn≡η⁢1⁢(mod⁡4). Then, by [5, Theorem 2], the image of Wb1×⋯×Wbt under ι is a Sylow 2-subgroup of SO2⁢n+1⁡(q).

Now we consider the simple group Ω2⁢n+1⁢(q)=[SO2⁢n+1⁡(q),SO2⁢n+1⁡(q)]. Let R≅Wb1×⋯×Wbt be a Sylow 2-subgroup of SO2⁢n+1⁡(q) as above, and let Q=R∩Ω2⁢n+1⁢(q). We give an explicit description of Q. Let

θ:GO2η⁡(q)→𝔽×/𝔽×2

be the spinor norm on GO2η⁡(q). Let W be the Sylow 2-subgroup of GO2η⁡(q) as before. Then W0=ker⁡θ|W is isomorphic to the dihedral group of order 2a. In particular, Z⁢(W)⊆W0. Denote by B=W×⋯×W the base subgroup of R. Then R=B⋊A with A=Xb1×⋯×Xbt, so Q=B0⋊A, where B0 is the subgroup of B consisting of all elements (w1,…,wn) satisfying θ⁢(w1)⁢⋯⁢θ⁢(wn)=1¯; see, for example, [14, § 4].

Proposition 4.8.

Let S=Ω2⁢n+1⁢(q) with n≥3 and q odd. Then (RC) holds for the principal 2-blocks of all covering groups of S.

Proof.

For Ω2⁢n+1⁢(q), we let R,Q be as above. Then Z⁢(Q)=Z⁢(R) by our description of Q. Then, by Lemma 4.7, (RC) holds for the principal 2-block of S.

For 2.S=Spin2⁢n+1⁡(q), the assertion follows directly from [8, Lemma 3.1] by the fact that the centre of a Sylow 2-subgroup of Spin2⁢n+1⁡(q) has order 2 by [11, Lemma 4.4]. ∎

4.5 Even-dimensional orthogonal groups

The simply connected group of type Dn is the spin group Spin2⁢n±⁡(q), and its dual group is the projective conformal special orthogonal group PCSO2⁢n±⁡(q), the quotient of the conformal special orthogonal group CSO2⁢n±⁡(q) modulo its central torus. We thus need some control on centralisers of 2-elements in conformal special orthogonal groups. Recall that q is an odd prime power.

Lemma 4.9.

Let s∈G*:=CSO2⁢n±(q) be a 2-element. Then CG*∘⁢(s) is a product of groups Dki⁢(ϵi⁢q) (two factors) or Dki⁢(ϵi⁢q2) (one factor) with groups of type GUki⁡(q) and GLki⁡(q2di) for suitable ki, di≥0 and ϵi∈{±1}, where GUki⁡(q) only occurs when ∣⁡4(q+1).

Proof.

Let s∈𝐆*=CSO2⁢n. Since CSO2⁢n=SO2⁢n⋅Z⁢(CSO2⁢n), we can write s=s1⁢z with elements s1∈SO2⁢n and z∈Z⁢(CSO2⁢n) a scalar matrix. Clearly, C𝐆*⁢(s)=C𝐆*⁢(s1). Note that s1 and z need not be F-stable, but as

SO2⁢n∩Z⁢(CSO2⁢n)={±1}

and both groups are F-stable, we have F⁢(s1)=±s1, and s1 is F2-stable. So C𝐆*⁢(s)F2 has the structure described above, corresponding to an orthogonal decomposition of 𝔽q22⁢n into the s1-eigenspaces. Now F permutes these eigenspaces according to whether F⁢(s1)=s1 or F⁢(s1)=-s1. Thus the two orthogonal factors of C𝐆*⁢(s)F2 are either fixed or permuted, and we obtain a collection of type A-factors, as claimed. See also [12, Lemma 2.5] for a more precise statement. ∎

Lemma 4.10.

Let G=Spin2⁢n±⁡(q), n≥4 and q odd, and let n=2b1+⋯+2bt be the 2-adic expansion with b1<…<bt. Then

def⁡(χ)≥max⁡{t+2,t+b1+1}

for every character χ in the principal 2-block of G.

Proof.

Let G=Spin2⁢n±⁡(q) with n≥4. Assume χ∈ℰ⁢(G,s) for some semisimple 2-element s∈G*=PCSO2⁢n±⁡(q). Let ψ∈ℰ⁢(CG*∘⁢(s),1) be a unipotent character in Jordan correspondence with χ. Then, as in the proof of Lemma 4.4, we have def⁡(χ)≥def⁡(ψ).

Let s~∈G~*:=CSO2⁢n±(q) be a preimage of s under the natural map. Then the structure of CG~*∘⁢(s~) is described in Lemma 4.9, and thus also the structure of CG*∘⁢(s). We write m12,m22 for the ranks of the two D-factors, or m02 if there is just one over 𝔽q2, and mf for the ranks of the various GL- and GU-factors with f∈ℱ∖ℱ0. Then

(4.8)n=m0+m12+m22+∑b≥1∑f∈ℱbmf⁢2b-1,

t≤m0+m12+m22+∑f∈ℱ∖ℱ0mf, def⁡(ψf)≥mf⁢(b+1) for every f∈ℱb, and def⁡(ψ0)≥m0-1 for the unipotent character ψ0 from a possible D-factor over 𝔽q2. Thus we have

def⁡(ψ)≥(m0-1)+m12+m22+∑b≥1∑f∈ℱb(b+1)⁢mf≥t-1+∑b≥1∑f∈ℱbb⁢mf

(where the summand (m0-1) is only present when m0>0). By the same argument as in Lemma 4.7, we obtain def⁡(χ)≥t+2 (noting that n≥4) and even def⁡(χ)≥t+3 when m0=0.

For the second bound, by (4.5) and (4.6) in the proof of Lemma 4.4, it suffices to show that

(m0-1)+m12+m22+∑b≥1∑f∈ℱbmf⁢(b+1)≥t+b1+1.

If m0=m1=m2=0, then, by Lemma 2.3 (b),

∑b≥1∑f∈ℱbmf⁢(b+1)-(t+b1+1)=2⁢(∑b≥1∑f∈ℱbmf-t)+∑b≥1∑f∈ℱb(b-1)⁢mf+t-(b1+1)≥∑i=1tbi+t-(b1+1)≥0.

If m1+m2>0 (and so m0=0), then, by Lemma 2.3 (c),

m12+m22+∑b≥1∑f∈ℱbmf⁢(b+1)≥t+b1+∑b≥1∑f∈ℱbmf⁢b≥t+b1.

Equality only holds when (4.8) is the 2-adic expansion of n. Hence m1+m2=2 and ℱ1⁢(s)=∅; then ⋃b≥1ℱb⁢(s)≠∅ since n≥4. Thus ∑b≥1∑f∈ℱbmf⁢b>0. So equality does not hold.

Finally, if m0>0 (and hence m1=m2=0), then the 2-adic expansion of n-1=m0-1+∑b,fmf⁢2b-1 has t+b1-1 terms, so, by Lemma 2.3 (a), we are done unless ∑b,fb⁢mf<2. But note that m0-1 is odd, so, unless m0=2, its 2-adic expansion has at least 2 terms. Thus either m0=2 or n=m0. Both cases are easily dealt with. ∎

Note that n≥4 implies either t+2≥4 or t+b1+1≥4, so def⁡(χ)≥t+3 in Lemma 4.10 when t=1.

We next discuss Sylow 2-subgroups. Let Wb be as in the previous subsection. First assume qn≡-η⁢1⁢(mod⁡4). Then we have SO2⁢nη⁡(q)=S×C2 with S=Ω2⁢nη⁢(q) simple, and Spin2⁢nη⁡(q)=2.S is the only proper covering group of S.

Lemma 4.11.

Let R be a Sylow 2-subgroup of Ω2⁢nη⁢(q) with qn≡-η⁢1⁢(mod⁡4), and let n-1=2b1+⋯+2br be the 2-adic expansion. Then ν⁢(Z⁢(R))=r.

Proof.

By [14, Theorem 7], a Sylow 2-subgroup of Ω2⁢nη⁢(q) is isomorphic to a Sylow 2-subgroup of GO2⁢n-2η′⁡(q) with qn-1≡η′⁢1⁢(mod⁡4), i.e., to Wb1×⋯×Wbr. The assertion follows. ∎

Proposition 4.12.

Let S=Ω2⁢nη⁢(q) with n≥4 and qn≡-η⁢1⁢(mod⁡4). Then (RC) holds for the principal 2-blocks of all covering groups of S.

Proof.

If qn≡-η⁢1⁢(mod⁡4), then, by Lemma 4.11, ν⁢(Z⁢(R))=r=t+b1-1 in the notation of Lemmas 4.10, 4.11. If χ is a character of G=2.S=Spin2⁢nη⁡(q) with Z⁢(G) in its kernel, then, considering it as a character of S, def⁡(χ)≥t+b1, which shows the claim. If R^ denotes a Sylow 2-subgroup of G, then clearly ν⁢(Z⁢(R^))≤r+1=t+b1. Again, the claim follows with Lemma 4.10. ∎

We now turn to the more difficult case qn≡η⁢1⁢(mod⁡4). Here we have

SO2⁢nη⁡(q)=2.S⁢.2 with⁢S=P⁢Ω2⁢nη⁡(q)⁢simple,

Spin2⁢nη⁡(q)={22.Sif⁢n⁢is even,4.Sif⁢n⁢is odd.

First note that a Sylow 2-subgroup R of GO2⁢nη⁡(q) is isomorphic to Wb1×⋯×Wbt, where n=2b1+⋯+2bt. Let Q=R∩SO2⁢nη⁡(q). Then ν⁢(Z⁢(Q))=t.

Lemma 4.13.

Let H=Ω2⁢nη⁢(q) with qn≡η⁢1⁢(mod⁡4), and keep all the notation above.

Let Q0=R∩H. Then ν⁢(Z⁢(Q0))=t.
Let Q¯0=Q0/Z⁢(H), a Sylow 2 -subgroup of P⁢Ω2⁢nη⁡(q), Z⁢(Q¯0)=Z0/Z⁢(H). Then Z0=Z⁢(Q0) if t>1, while, if t=1, ν⁢(Z0)=ν⁢(Z⁢(Q0))+1=2 and |Z⁢(Q¯0)|=2.

Proof.

(a) Let θ be the spinor norm on GO2η⁡(q). Write R=B⋊A, where B is the base subgroup, a direct product of copies of W and A=Xb1×⋯×Xbt. Then an element (w1,…,wn)⁢a with wi∈W and a∈A is in Q0 if and only if det⁡(w1)⁢⋯⁢det⁡(wn)=1 and θ⁢(w1)⁢⋯⁢θ⁢(wn)=1¯. Thus ν⁢(Z⁢(Q0))=t follows easily. Part (b) follows by a similar proof as for Lemma 3.6. ∎

Proposition 4.14.

Let S=P⁢Ω2⁢nη⁡(q) with n≥4 and qn≡η⁢1⁢(mod⁡4). Then (RC) holds for the principal 2-blocks of all covering groups of S.

Proof.

First assume that n is even. Then, by [8, Corollary 2.4], we just need to consider the three groups S, 2.S=Ω2⁢nη⁢(q) and 2′.S=HSpin2⁢nη⁡(q). (The two half-spin groups are isomorphic under the graph automorphism of order 2.) In the notation of Lemma 4.13, the centres of Sylow 2-subgroups R of these groups satisfy ν⁢(Z⁢(R))≤t, t, t+1, respectively, while, by Lemma 4.10, the defects of characters belonging to the principal 2-block of G that descend to these groups are at least t+1, t+2, t+2, respectively. So (RC) holds in all cases.

Now assume that n is odd. Then the groups to consider are S, 2.S=Ω2⁢nη⁢(q) and 4.S. Here again, the centres of Sylow 2-subgroups R satisfy ν⁢(Z⁢(R))≤t, t, t+1, while the defects are bounded below by t+1, t+2, t+3, respectively. ∎

5 Proof of the main result

Proof of Theorem 1.

Let B be a 2-block of a quasi-simple covering group G of a finite simple classical group S. If S is defined in characteristic 2, our claim is in [8, Theorem 5.1]. So S is a classical group in odd characteristic. By [8, Theorem 3.6], we may assume that G is not an exceptional covering group. Then G is one of the groups considered in the previous sections. According to [3, Theorem 21.14], then G has only one unipotent 2-block, the principal 2-block ℰ2⁢(G,1). This is not a counterexample to (RC) by Propositions 3.1, 3.5, 3.7, 4.6, 4.8, 4.12 and 4.14. If B is not unipotent, then Irr⁡(B)⊆ℰ2⁢(G,s) for some semisimple 2′-element 1≠s∈G*. If G is of symplectic or orthogonal type, then centralisers of non-trivial 2′-elements in G* are proper Levi subgroups. If G is special linear or unitary, then at least the connected components of these centralisers are proper Levi subgroups. In either case, by the reduction theorem of Bonnafé–Rouquier [1], then B is Morita equivalent to a 2-block of a strictly smaller group and thus cannot be a minimal counterexample to (RC) either. ∎

Communicated by Radha Kessar

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11631001

Award Identifier / Grant number: 11661042

Funding statement: The first and fourth author gratefully acknowledge financial support by SFB TRR 195, and the others by NSFC (No. 11631001). In addition, the third author gratefully acknowledges financial support by NSFC (No. 11661042) and the Key Laboratory of Mathematics and Its Applications of Peking University.

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Received: 2018-09-04

Revised: 2019-01-22

Published Online: 2019-04-06

Published in Print: 2019-07-01

Robinson’s conjecture for classical groups

Abstract

1 Introduction

Conjecture (Robinson).

Theorem 1.

2 The general linear and unitary groups

2.1 Some notation and background

Lemma 2.1.

Lemma 2.2.

Proof.

Lemma 2.3.

Proof.

2.2 Sylow 2-subgroups of GLn⁡(η⁢q)

Theorem 2.4.

Proof.

2.3 Robinson’s conjecture for GLn⁡(η⁢q)

Lemma 2.5.

Proof.

Proposition 2.6.

Proof.

Remark 2.7.

3 The groups SLn⁡(η⁢q)

3.1 Robinson’s conjecture for SLn⁡(η⁢q), n odd

Proposition 3.1.

Proof.

3.2 Sylow 2-subgroups of SLn⁡(η⁢q) for n even

Lemma 3.2.

Proof.

Lemma 3.3.

Proof.

Lemma 3.4.

Proof.

3.3 Robinson’s conjecture for SLn⁡(η⁢q), n even

Proposition 3.5.

Proof.

3.4 Robinson’s conjecture for central quotients of SLn⁡(η⁢q), n even

Lemma 3.6.

Proof.

Proposition 3.7.

Proof.

4 Principal 2-blocks of quasi-simple groups of classical type

4.1 Unipotent characters of classical groups

Lemma 4.1.

Proof.

Lemma 4.2.

Proof.

4.2 Centralisers of semisimple elements in classical groups

Lemma 4.3.

4.3 Symplectic groups

Lemma 4.4.

Proof.

Lemma 4.5.

Proposition 4.6.

4.4 Odd-dimensional orthogonal groups

Lemma 4.7.

Proof.

Proposition 4.8.

Proof.

4.5 Even-dimensional orthogonal groups

Lemma 4.9.

Proof.

Lemma 4.10.

Proof.

Lemma 4.11.

Proof.

Proposition 4.12.

Proof.

Lemma 4.13.

Proof.

Proposition 4.14.

Proof.

5 Proof of the main result

Proof of Theorem 1.

References

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