Groups with few p 0 -character degrees in the principal block

Groups with few p ⁰ -character degrees in the principal block

Eugenio Giannelli

, Noelia Rizo

,

Benjamin Sambale

and A. A. Schaeffer Fry

October 13, 2020

Abstract

Let p≥5 be a prime and letGbe a finite group. We prove that Gis p-solvable of p-length at most2if there are at most two distinctp⁰-character degrees in the principalp-block ofG. This generalizes a theorem of Isaacs–Smith as well as a recent result of three of the present authors.

Keywords:p⁰-character degrees; principal block AMS classification:20C15, 20C30, 20C33

1 Introduction

LetGbe a finite group. If all non-linear irreducible characters ofGhave degree divisible by a primep, thenG has a normalp-complement by a theorem of Thompson [Tho70, Theorem 1] (see also [Isa06, Corollary 12.2]).

Moreover, Berkovich [Ber95, Proposition 9 and the subsequent remark] has shown that G is solvable in this situation. This result was extended in Kazarin–Berkovich [KB99] to the case where Ghas at most one non- linear character ofp⁰-degree. In a recent paper [GRS], three of the present authors proved more generally that G is solvable ofp-length at most 2 wheneverp≥5 and |{χ(1) : χ ∈Irr_p⁰(G)}| ≤ 2 where Irr_p⁰(G)is the set of irreducible characters of G of p⁰-degree. This has solved Problem 1 in [KB99, p. 588] and Problem 5.3 in [Nav16].

In the present paper we generalize our theorem to blocks. This is motivated by a result of Isaacs and Smith [IS76, Corollary 3] who showed thatGhas a normalp-complement if and only if all non-linear characters in the principal p-block ofGhave degree divisible byp. The following is our main theorem.

Theorem A. Let B0 be the principal block of a finite group G with respect to a prime p ≥5. Suppose that

|{χ(1) :χ∈Irr_p⁰(B₀)}| ≤2. ThenG/O_p⁰(G)is solvable andO^pp0pp0(G) = 1. In particular,Gisp-solvable.

As usual we defineO^pp0(G) := O^p0(O^p(G))and so on. It is easy to construct groups of p-length2satisfying the hypothesis of Theorem A (e. g. G= (C₅⁵oC11)oC5 with p= 5). In contrast to the main theorem of [GRS]

we cannot conclude further that G is solvable since every p⁰-group satisfies the assumption of Theorem A.

Furthermore, the examples given in [GRS] show that Theorem A does not extend top∈ {2,3}. We also like to mention a conjecture by Malle and Navarro [MN11], which generalizes the result of Isaacs and Smith to arbitrary

∗Dipartimento di Matematica e Informatica, U. Dini, Viale Morgagni 67/a, Firenze, Italy, eugenio.giannelli@unifi.it

†Dipartimento di Matematica e Informatica, U. Dini, Viale Morgagni 67/a, Firenze, Italy, noelia.rizocarrion@unifi.it

‡Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Leibniz Universität Hannover, Germany, sambale@math.uni- hannover.de

§Department of Mathematical and Computer Sciences, Metropolitan State University of Denver, Denver, CO 80217, USA, as- chaef6@msudenver.edu

blocks. More precisely, they conjectured that ap-blockB ofGis nilpotent if and only if all height0 characters inB have the same degree. We do not know if our main result admits a version for arbitrary blocks.

The proof of Theorem A relies on the classification of finite simple groups. In the next section we reduce Theorem A to a statement about simple groups (Proposition 2.1 below), which is proved case-by-case in the following two sections. We care to remark that in the case of alternating groups, Proposition 2.1 is deduced as a consequence of a more general statement giving a lower bound for the number of (extendable) p⁰-character degrees in any block of maximal defect. This is Proposition 3.5 below, which we believe is of independent interest.

2 Reduction to simple groups

The following proposition about simple groups will be proven in the next two sections.

Proposition 2.1. Let S be a finite non-abelian simple group of order divisible by a prime p≥5. (i) IfS 6=PΩ⁺₈(q), then there existα, β∈Irr(S) with the following properties:

• α6= 16=β,

• α(1) andβ(1)are not divisible by p,

• for everyS≤T ≤Aut(S),αextends to a character in the principal block ofT,

• β lies in the principal block ofS and isP-invariant for some Sylow p-subgroup P of Aut(S),

• β(1)-α(1).

(ii) IfS =PΩ⁺₈(q), then there existα, β∈Irr(S) with the following properties:

• α6= 16=β,

• α(1) andβ(1)are not divisible by p,

• α(1)>2β(1),

• for everyS≤T ≤Aut(S)there existα,ˆ βˆ∈Irr(Aut(T))in the principal block such thatαˆS∈ {α,2α}

andβˆS ∈ {β,2β}.

We make use of the following results.

Lemma 2.2 (Murai [Mur94, Lemma 4.3]). Let N EG be finite groups with principal p-blocks B_N and B_G respectively. Suppose that ψ ∈ Irr_p⁰(B_N) is invariant under a Sylow p-subgroup of G. Then there exists a character χ∈Irr_p⁰(B_G)lying overψ.

Lemma 2.3. Let χ, ψ∈Irr(B0)whereB0 is the principalp-block ofG. Suppose that p-χ(1)andχψ∈Irr(G). Thenχψ∈Irr(B0).

Proof. Clearly,ψ∈Irr(B0). Hence by [Nav98, Corollary 3.25], we have [χψ,1]⁰= [χ, ψ]⁰6= 0.

The claim follows from [Nav98, Theorem 3.19].

Now we are in a position to reduce Theorem A to simple groups.

Theorem 2.4. If Proposition 2.1 holds, then Theorem A holds.

Proof. Letp, G and B0 be as in Theorem A. Suppose first that Gis p-solvable. Let N := Op⁰(G). Then, by [Nav98, Theorem 10.20], Irr(B0) = Irr(G/N). It follows from [GRS, Theorem A] that G/N is solvable and O^pp0pp0(G/N) = 1. In particular,O^pp0p(G)N/N is ap⁰-group. SinceN is ap⁰-group, this impliesO^pp0pp0(G) = 1.

Thus, it suffices to show that Gis p-solvable. LetN be a minimal normal subgroup of G. Since the principal block of G/N lies in B₀, we may assume that G/N is p-solvable by induction on |G|. If N is a p-group or a p⁰-group, then we are done. Therefore, by way of contradiction, we assume that

N=S1×. . .×St

with isomorphic non-abelian simple groups S :=S1 ∼=. . . ∼=St of order divisible by p. Since N is the unique minimal normal subgroup,CG(N) = 1. Moreover,GpermutesS1, . . . , Sttransitively by conjugation.

Case 1:S6=PΩ⁺₈(q).

Letα, β∈Irr(S)as in Proposition 2.1. We may regard αas a character of SCG(S), sinceSCG(S)/CG(S)∼= S/Z(S) = S. As such it extends to a character αˆ in the principal block of NG(S), because NG(S)/CG(S)≤ Aut(S). Let M := NG(S1)∩. . .∩NG(St)EG. Since the principal block of NG(S)covers the principal block BM of M, the restriction αˆM lies in BM. Now by [Nav18, Corollary 10.5], the tensor induction ψ := ˆα^⊗G is an irreducible character of Gwith p⁰-degree ψ(1) =α(1)^t. Letx₁, . . . , x_t ∈Gbe representatives for the right cosets ofN_G(S)in Gsuch thatS₁^xi =S_i. Then forg∈M we obtain

ψ(g) =

t

Y

i=1

ˆ α^xi(g)

from [Nav18, Lemma 10.4]. In particular, ψ_N =α^x1×. . .×α^xt ∈Irr(N) and thereforeψ_M ∈Irr(M) as well.

Sinceαˆ_M lies inB_M, so doesαˆ^x_Mⁱ. Hence, by Lemma 2.3 alsoψ_M = ˆα^x_M¹. . .αˆ^x_M^t lies inB_M.

LetQbe a Sylowp-subgroup ofM. ThenQ∩Siis a Sylowp-subgroup ofSi. It follows thatCG(Q)⊆CG(Q∩Si)⊆ NG(Si)fori= 1, . . . , tand thereforeCG(Q)⊆M. Hence, the Brauer correspondentB_M^G is defined (see [Nav98, Theorem 4.14]) and equalsB0 by Brauer’s third main theorem. Every block B ofGcoveringBM has a defect group containingQby [Nav98, Theorem 9.26]. Hence by [Nav98, Lemma 9.20],B is regular with respect toN and therefore B =B0 by [Nav98, Theorem 9.19]. Thus,B0 is the only block ofG coveringBM. This implies ψ ∈ Irrp⁰(B0). Since the trivial character in B0 has degree 1, d := ψ(1) = α(1)^t is the unique non-trivial p⁰-character degree inB0 by hypothesis.

Now we work withβ. LetP be a Sylowp-subgroup ofGsuch thatβ is invariant underNP(S). Without loss of generality, let{S1, . . . , S_r} be aP-orbit. Lety_i ∈P such thatS₁^yi =S_i fori= 1, . . . , r. Then β_i :=β^yi lies in the principal block ofS_i. By Lemma 2.3,β₁×. . .×β_rlies in the principal block ofN. Moreover, ifβ_i^y ∈Irr(S_j) for somey ∈P, theny_iyy⁻¹_j ∈N_P(S). Sinceβ isN_P(S)-invariant, it follows that β_i^y =β^{yiyy−1j yj} =β^yj =β_j. This shows that {β1, . . . , β_r} is P-orbit and β₁×. . .×β_r is P-invariant. If r < t, then we consider β_r+1 :=

β^xr+1 ∈Irr(S_r+1). By Sylow’s theorem, we can assume after conjugation insideN_G(S_r+1)thatβ_r+1isN_P(S_r+1)- invariant. Now we can form theP-orbit ofβ_r+1 to obtain anotherP-invariant characterβ_r+1×. . .×β_s∈Irr(N) in the principal block ofN. We repeat this with everyP-orbit and eventually get aP N-invariant character

τ:=β1×. . .×βt∈Irr(N)

in the principal block of N. Since o(τ) = 1 and gcd(τ(1),|P N/N|) = 1, τ extends to P N (see [Isa06, Corollary 8.16]). By Lemma 2.2, there exists some χ ∈ Irrp⁰(B0) such that τ is a constituent of χN. Since 1 6=β(1)^t =τ(1) | χ(1), it follows that χ(1) = d = ψ(1). But then β(1)^t | ψ(1) = α(1)^t and β(1) | α(1), a contradiction to the choice ofαandβ.

Case 2:S=PΩ⁺₈(q).

Letα, β ∈Irr(S)andα,ˆ βˆ∈Irr(N_G(S))as in Proposition 2.1. Since the principal block ofN_G(S)coversB_M, ˆ

α_M is the sum of at most two irreducible characters in B_M. Ifα₁∈Irr(B_M)is one of those summands, then α^x₁¹. . . α^x₁^t restricts to α^x1×. . .×α^xt ∈Irr(N).¹ Hence, by Lemma 2.3, α^x₁¹. . . α^x₁^t lies inB_M. As in Case 1 we see that( ˆα^⊗G)_M is a sum of irreducible characters inB_M. Moreover, ( ˆα^⊗G)_N =d(α^x1 ×. . .×α^xt) where

1Miquel Martínez pointed out that this is not always the case. A workaround (due to G. Navarro) will appear in a forthcoming paper of Martínez.

d∈ {1,2^t}. SinceB0 is the only block ofGcoveringBM, all irreducible constituents ofαˆ^⊗G lie inB0. We may choose such a constituentχofp⁰-degree. ThenχN =e(α^x1×. . .×α^xt)for some integere≤d≤2^t. Similarly, we choose a constituentψofβˆ^⊗G withp⁰-degree. Then by Proposition 2.1 we derive the contradiction

α(1)^t>2^tβ(1)^t≥ψ(1) =χ(1)≥α(1)^t.

3 Alternating groups

This section is devoted to proving Proposition 2.1 for the alternating groups S = An where n ≥ 5. It is well-known thatAut(S)∼=Sn is the symmetric group unlessn= 6.

Given n ∈ N we let P(n) be the set of partitions of n. Let λ= (λ₁, . . . , λ_k) ∈ P(n). Adopting the notation of [Ols93, Chapter 1] we let `(λ) =k denote the number of parts of λ, andY(λ) be the Young diagram ofλ.

Given a node(i, j)∈ Y(λ)we denote by h_ij(λ) the length of the hook corresponding to (i, j). If q ∈N then theq-core C_q(λ) ofλis the partition obtained from λby successively removing all hooks of lengthq (usually calledq-hooks). We denote byH^q(λ)the subset of nodes ofY(λ)having associated hook-length divisible byq.

A partition γis called aq-core ifH^q(λ) =∅.

The setIrr(Sn)is naturally in bijection withP(n). Givenλ∈ P(n)we letχ^λbe the corresponding irreducible character of Sn. Let p be a prime and λ, µ ∈ P(n). By [JK81, 6.1.21] we know that χ^λ and χ^µ lie in the same p-block of Sn if and only if Cp(λ) = Cp(µ). If γ is ap-core partition then we denote by B(Sn, γ) the correspondingp-block ofSn. We use the notationλ`p⁰ nto say thatχ^λhas degree coprime top.

The following result follows from [Mac71] and it will be extremely useful for our purposes.

Lemma 3.1. Letpbe a prime and letnbe a natural number with p-adic expansionn=Pk

j=0ajp^j. Letλbe a partition ofn. Thenλ`p<sup>0</sup> nif and only if |H<sup>pk</sup>(λ)|=ak andC<sub>p</sub>k(λ)`p⁰ n−akp^k.

A straightforward consequence of Lemma 3.1 is thatIrrp⁰(B(Sn, γ))6=∅if and only if|γ|< p.

For λ ∈ P(n), we denote by λ⁰ its conjugate partition. From [JK81, 2.5.7] we know that ψ^λ := (χ^λ)_An is irreducible if and only ifλ6=λ⁰. In this caseχ^λ andχ^λ0 are all the extensions ofψ^λ toS_n. Letλ, µbe non-self- conjugate partitions ofn. Thenψ^λ andψ^µlie in the samep-block ofA_n if and only ifC_p(λ)∈ {Cp(µ), C_p(µ)⁰}.

It follows that also p-blocks of A_n can be labeled by p-core partitions, by keeping in mind that conjugated p-cores label the samep-block. We denote byB(n;γ)thep-block of A_n labeled byγ.

In order to show that Proposition 2.1 holds for alternating groups, we introduce the following conventions.

Notation 1. LetB be ap-block of An. We letcd^ext_p0 (B)be the set of degrees of irreducible characters in B of degree coprime topthat extend to an irreducible character ofSn. Moreover, whenS is a subset ofP(n)we let cd(S) ={χ^λ(1)|λ∈S}.

Observe that if B is the principal p-block of An and ψ^λ lies in B and extends to Sn, then one of the two extensions of ψ^λ lies in the principal p-block ofSn. This is explained in [Ols90]. Even if in this article we are mainly interested in studying the principal block, in Proposition 3.5 below we are going to compute an explicit lower bound for|cd^ext_p0 (B(n, γ))|, for any p-coreγsuch that|γ|< p.

Given γ = (γ₁, . . . , γ<sub>`</sub>) ` n and natural numbersx and y, we denote by γ ?(x, y)the partition of n+x+y defined by

γ ?(x, y) = (γ₁+x, γ₂, . . . , γ_`,1^y).

We start by proving a technical lemma that will be useful later in this section.

Lemma 3.2. Let pbe a prime, let m, n, w∈Nbe such thatm < p andn=m+pw. Let γ`mand leta∈N be such that b^w+1₂ c+ 1≤a≤w. Setting λ=γ ?(ap,(w−a)p)and µ=γ ?((a−1)p,(w−a+ 1)p), we have that χ^λ(1)< χ^µ(1).

Proof. Forν`nwe letπ(ν) :=Qhij(ν)be the product of the hook-lengths inν. From the hook length formula [JK81, 2.3.21] it follows thatχ<sup>ν</sup>(1)π(ν) =n!. We leth<sup>i</sup>=h1i(γ)andhj=hj1(γ)for alli∈ {1, . . . , γ1} and all j∈ {1, . . . , `(γ)}. It follows that

π(λ) = (ap)!·((w−a)p)!·

γ₁

Y

i=2

(hⁱ+ap)·

`(γ)

Y

i=2

(hi+ (w−a)p)·bγ·(h11(γ) +pw),

wherebγ is the product of the hook lengthsh_ij(γ)for alli, j≥2. Similarly π(µ) = ((a−1)p)!·((w−a+ 1)p)!·

γ₁

Y

i=2

(hⁱ+ (a−1)p)·

`(γ)

Y

i=2

(hi+ (w−a+ 1)p)·bγ·(h11(γ) +pw).

It follows thatπ(λ)/π(µ) =A·B·C, where A=

p

Y

i=1

(a−1)p+i (w−a)p+i, B=

γ1

Y

i=2

hⁱ+ap

hⁱ+ (a−1)p, and C=

`(γ)

Y

i=2

hi+ (w−a)p h_i+ (w−a+ 1)p.

We remark that we always regard empty products as equal to1. We observe thatB≥1. Sincea−1≥w−a+ 1 by hypothesis, it is clear that A > 1. Hence, if `(γ) = 1 thenC = 1 and clearlyA·B·C >1. Suppose that

`(γ)≥2. Then observe thatp >|γ|> h2> h3>· · ·> h<sub>`(γ)</sub>≥1. Hence for alli∈ {2, . . . , `(γ)} we have that

(a−1)p+hi

(w−a)p+hi is one of the factors appearing inA. Moreover (a−1)p+hi

(w−a)p+h_i · hi+ (w−a)p h_i+ (w−a+ 1)p ≥1,

sincea−1≥w−a+ 1. We conclude thatA·B·C≥A·C >1 and therefore thatχ^λ(1)< χ^µ(1).

Definition 3.3. Letpbe a prime and n=wp+m, for somem < p. Letγ be ap-core partition of m. We let H(n;γ)be the subset ofP(n)defined by

H(n;γ) ={λ`_p⁰n| C_p(λ) =γ, λ=γ ?(a, n−m−a)}.

We also setΩ(n;γ) ={λ∈H(n;γ)| λ₁>(λ⁰)₁}.

Lemma 3.4. Let n=Pk

i=0a_ipⁱ be thep-adic expansion ofn, with a_k6= 0. If γ`a₀, then

|cd(Ω(n;γ))|=|Ω(n;γ)| ≥ ba_k+ 1 2 c ·

k−1

Y

i=1

(ai+ 1).

Proof. Letλ=γ ?(x, n−a0−x), for some0≤x≤n−a0. Letx=Pt

i=0bipⁱ be thep-adic expansion ofx. By definition ofH(n;γ)we have thatλ∈H(n;γ)if and only ifλ`p⁰ nandCp(λ) =γ. In turn, this is equivalent to have thatpdividesx(andn−a0−x) so thatCp(λ) =γand by Lemma 3.1 to have thatb0= 0and0≤bi≤ai

for alli≥1. It follows that |H(n;γ)|=Qk

i=1(ai+ 1). Moreover, if bk ≥ bak/2c+ 1, then certainlyλ1>(λ⁰)1

and thereforeλ∈Ω(n;γ). It follows that

|Ω(n;γ)| ≥ bak+ 1 2 c ·

k−1

Y

i=1

(ai+ 1).

We conclude by observing that Lemma 3.2 implies that given λ, µ∈Ω(n;γ)we have that χ^λ(1)6=χ^µ(1) and hence that|cd(Ω(n;γ))|=|Ω(n;γ)|.

Given λ ∈ Ω(n;γ) we have that χ^λ lies in B(S_n;γ) and that (χ^λ)_An is irreducible and lies in B(n;γ). As explained in Notation 1 above,cd^ext_p0 (B(n;γ))denotes the set of degrees of irreducible characters of B(n;γ)of degree coprime topthat extend toB(Sn;γ).

In the following proposition we are able to establish a lower bound for the number of extendable p⁰-character degrees lying in any given p-block of An. We believe this statement of independent interest from the topic of this article.

Proposition 3.5. Let n=Pk

i=0aipⁱ be thep-adic expansion ofn, with ak6= 0. Let γ`a0, then

|cd^ext_p0 (B(n;γ))| ≥ ba_k+ 1 2 c ·

k−1

Y

i=1

(a_i+ 1).

Proof. By definition, for every partition λ∈Ω(n;γ)we have that (χ^λ)A_n is a p⁰-degree character that lies in B(n;γ)and extends toχ^λ inB(Sn;γ). The statement now follows from Lemma 3.4.

Proposition 3.6. Letn≥5 be a natural number andp >3 be a prime. Then Proposition 2.1 holds forAn. In particular ifn≥7 then|cd^ext_p0 (B₀(A_n))| ≥3.

Proof. Direct verification proves that Proposition 2.1 holds for A₅ and A₆. Suppose that n ≥ 7 and that n = a₀+Pk

i=1a_ipⁿⁱ is the p-adic expansion of n, with a_i 6= 0 for all i ≥ 1 and with n₁ < n₂ < · · · < n_k. Since p is odd, for P ∈ Syl_p(S_n) we have that P ≤A_n and hence that all irreducible characters in B₀(A_n) are P-invariant. Thus we just need to show that|cd^ext_p0 (B₀(A_n))| ≥3. From Proposition 3.5, we deduce that

|cd^ext_p0 (B0(An))| ≥ 3, wheneverk ≥3. Suppose thatk ≤2. If a0 ≤1 then Irrp⁰(B0(An)) = Irrp⁰(An)and the statement follows from [GRS, Proposition 3.5]. Hence we can assume thata0≥2 and considerλ, µ∈ P(n) to be defined as follows.

λ= (a0,1^n−a0), and µ= (a0,2,1^n−a0−2).

It is clear that both(χ^λ)A_n and(χ^µ)A_n lie in the principalp-block ofAnand extend to the principalp-block of Sn, toχ^λ andχ^µ respectively. Moreoverλandµlabel characters of degree coprime topby Lemma 3.1. Using the hook-length formula we verify that1 =χ⁽ⁿ⁾(1)< χ^λ(1)< χ^µ(1).The proof is complete.

4 Sporadic groups and groups of Lie type

Proposition 4.1. Proposition 2.1 holds for all sporadic simple groupsS and the Tits group²F4(2)⁰.

Proof. Recall that|Aut(S) :S| ≤2. Hence, we may take ap⁰-characterαˆ in the principal block ofAut(S)such that α:= ˆαS 6= 1 is irreducible. For β we can choose any non-trivial p⁰-character in the principal block of S.

Now it can be checked with GAP [GAP18] that there are choices such thatβ(1)-α(1).

Now we consider simple groupsS of Lie type, by which we mean groups of the formG/Z(G), whereG=G^F is the set of fixed points of a simple simply connected algebraic group under a Steinberg morphism F. In the case whereZ(G)is trivial, we defineGe=G, and otherwise we letG,→Ge be a regular embedding, as in [CE04, Section 15], so thatZ(G)e is connected,[G,e G] = [G,e G], andGis normal inGe:=Ge^F. We writeSefor the group G/e Z(G), soe Aut(S)may be viewed as generated bySeand graph and field automorphisms.

Recall that the setIrr(G)e can be partitioned into so-called Lusztig seriesE(G, s), wheree sis a semisimple element of the dual group Ge^∗, up to conjugacy. Each seriesE(G, s)e has a unique character of degree [Ge^∗ : C

Ge^∗(s)]_q⁰, whereFq is the field over whichGis defined, called a semisimple character. Further, the characters in the series E(G,e 1)are called unipotent characters, and for a primep, anyp-block containing a unipotent character is called a unipotent block.

WhenGis typeA_n−1 (that is, in the case of linear and unitary groups), we will use the notation P SL_n(q)to denoteP SL_n(q)for= 1andP SU_n(q)for=−1, and similar forGL_n(q)andSL_n(q). Similarly,A_n−1(q)will denote the untwisted case A_n−1(q) when = 1and the twisted case ²A_n−1(q)when = −1. We also remark that the groupPΩ⁺_2n(q)corresponds toDn(q)and PΩ⁻_2n(q)corresponds to²Dn(q).

The following result settles Proposition 2.1 for most simple groups in defining characteristic.

Proposition 4.2. Let S be a simple group of Lie type defined overFq, whereq is a power of p >3 not in the following list:P SL2(q),P SL₃(q), orP Sp4(q). Then there exist two non-trivial charactersχ1, χ2∈Irrp⁰(B0(S)) such that χ1(1)6=χ2(1)and:

• If S6=PΩ⁺₈(q), then for everyS≤T ≤Aut(S), each ofχ1 andχ2 extend to a character in the principal p-block ofT.

• If S =PΩ⁺₈(q), then χ1(1) >2χ2(1) and for every S ≤T ≤Aut(S), for i= 1,2, there exist χbi in the principalp-block of T such thatχbi|S∈ {χi,2χi}.

Proof. In the proof of [GRS, Proposition 4.3], it is shown that there exist two characters χ₁, χ₂ ∈ Irr_p⁰(S)e that restrict irreducibly to S, extend to characters of Aut(S), have different degrees, and are obtained from characters ofGetrivial onZ(G). Now, sincee Irrp⁰(G) = Irre p⁰(B0(G))e (using, for example, [CE04, 6.18, 6.14, 6.15]) and using [CE04, Lemma 17.2], we see that in fact these characters are members of the principal block of S,e and their restrictions are members of the principal block ofS.

Now, letS ≤T ≤ Aut(S). Then for i = 1,2, χ_i|_T∩

Se is in the principal block of T∩S, sincee B₀(S)e covers a unique block of T ∩S. Note that by [Nav98, Theorem 9.4], there must be a character ofe B₀(T)lying above χ_i|_T∩

Se. IfS6=PΩ⁺₈(q), we haveAut(S)/Seis abelian, and hence every character ofT lying aboveχ_i|_T∩

Seis an extension, completing the proof in this case.

IfS=PΩ⁺₈(q), thenAut(S)/Seis of the formS3×C, whereC is cyclic. Then the characterχbi in B0(T)lying aboveχi|_T∩eS must be such that χbi|S ∈ {χi,2χi}, as desired. Switching the roles of the semisimple elementss1

ands2constructed in [GRS, Proposition 4.3], we further see that the characters have been constructed to satisfy χ1(1)>2χ2(1), since the centralizers ofs1 ands2 have typesA1×T1andA³₁×T2 withT1andT2appropriate tori, and 2|CG^∗(s1)|p⁰ <|CG^∗(s2)|p⁰.

The following handles the exceptional cases left by Proposition 4.2.

Proposition 4.3. Let S be one ofP SL₂(q),P SL₃(q), orP Sp₄(q), whereqis a power of a prime p >3. Then there exist two non-trivial characters χ₁, χ₂ ∈ Irr_p⁰(B₀(S)) such that χ₂(1) - χ₁(1); χ₂ is invariant under a Sylowp-subgroup ofAut(S); and for every S≤T ≤Aut(S),χ₁ extends to a character in the principal p-block of T.

Proof. In this case, charactersχ1 andχ2 are constructed in the proof of [GRS, Lemma 4.4] that satisfy all of the needed properties, except possibly the property that for every S≤T ≤Aut(S), χ1 extends to a character in the principal block ofT. However,χ1is again constructed from a character ofGetrivial onZ(G)e that restricts irreducibly toG. Hence since againAut(S)/Seis abelian, the proof is complete arguing as in the second paragraph of Proposition 4.2.

For the remainder of the section, we consider the case of non-defining characteristic. That is, we assumep >3 is a prime andS is a simple group of Lie type defined in characteristic different thanp.

Proposition 4.4. Let p > 3 be a prime and let S be a simple group of Lie type defined over F_q, where q is a power of a prime different than p and S is not in the following list: P SL₂(q), P SL₃(q) with p | (q+),

2B2(2^2a+1)with p|(2^2a+1−1), or²G2(3^2a+1) withp|(3^2a+1−1). Then there exist two non-trivial characters χ1, χ2∈Irrp⁰(B0(S))such thatχ1(1)6=χ2(1) and:

• If S6=PΩ⁺₈(q), then for everyS≤T ≤Aut(S), each ofχ1 andχ2 extend to a character in the principal p-block ofT.

• If S =PΩ⁺₈(q), then χ1(1) >2χ2(1) and for every S ≤T ≤Aut(S), for i= 1,2, there exist χbi in the principalp-block of T such thatχbi|S∈ {χi,2χi}.

Proof. We adapt our proof of [GRS, Proposition 4.5], ensuring that we may choose unipotent characters ofp⁰- degree satisfying the principal block conditions required here. That is, we will exhibit unipotent characters ofGe with different degree (and in the case ofPΩ⁺₈(q), satisfyingχ1(1)>2χ2(1)) that are contained inIrrp⁰(B0(G)),e which as unipotent characters must be trivial on Z(G)e and restrict irreducibly toG. Then the restriction lies in B0(G), since B0(G)e covers a unique block of G, and by [CE04, Lemma 17.2], the resulting characters of Se andS=G/Z(G)also lie in the principal blocks. By [Mal08, Theorems 2.4 and 2.5], every unipotent character extends to its inertia group inAut(S), and except for some specifically stated exceptions, the inertia group is all ofAut(S). Then arguing as in Proposition 4.2, the required properties will hold for eachS≤T ≤Aut(S).

To see that the unipotent characters exhibited are indeed of p⁰-degree, it will often be useful to recall that q^s−1 = Q

m|sΦm and note that p| Φm if and only if m = dpⁱ for some non-negative integer i, where Φm

denotes them-th cyclotomic polynomial inqand dis the order ofq modulop. Further,p² divides Φ_m only if m=d. (This is [Mal07, Lemma 5.2].)

First, we consider groups of exceptional type. IfSis one of²G₂(3^2a+1)or²B₂(2^2a+1)but not one of the exceptions of the statement, then the unipotent characters mentioned in the proof of [GRS, Proposition 4.5] work here, since by [H90, Proposition 3.2], respectively [B79, Section 2], there is a unique unipotent block of maximal defect. If S is ²F4(2^2a+1), then by [Mal90, Bemerkung 1], there is again a unique unipotent block of maximal defect unless p|(2^2a+1−1), in which case the principal block contains the Steinberg character and two more unipotent characters of p⁰-degree. Hence we are also done in this case. If S = ³D4(q), then there is either a unique unipotent block of maximal defect or the principal block contains the Steinberg character and one other unipotent character of p⁰-degree, using [DM87, Propositions 5.6 and 5.8], so we are similarly finished in this case.

Now letS be one ofG2(q), F4(q), E6(q),²E6(q), E7(q),or E8(q). Letdbe the order of qmodulop. Using [E00, Theorem A], we have the unipotent blocks ofGe are indexed by conjugacy classes of pairs(L, λ)forLa d-split Levi subgroup and λa d-cuspidal unipotent character. In particular, the characters in the d-Harish-Chandra series indexed by such an(L, λ)all lie in the same block ofG. Further, [Mal07, Corollary 6.6] then yields that ife a unipotent character in the series indexed by(L, λ)hasp⁰-degree, thenLis the centralizer of a Sylowd-torus.

Now, using this and [BMM93, Theorem 5.1], we see that either such anLis a maximal torus (yielding a unique block containing unipotent characters ofp⁰ degree, and hence we are done using [GRS, Proposition 4.5] again) or we may use the decompositions in [BMM93, Table 2] to see there are at least two non-trivial unipotent characters in the principal block with different degrees relatively prime top. (For an example of the argument in the latter situation, considerE8(q)in the cased= 7. Then Line 58 of [BMM93, Table 2] shows that the trivial character and the unipotent charactersφ8,91andφ400,7in the notation of [Ca85, Section 13.9], which have degree q⁹¹Φ²₄Φ8Φ12Φ20Φ24and ¹₂q⁶Φ⁴₂Φ²₅Φ²₆Φ8Φ²₁₀Φ14Φ15Φ18Φ20Φ24Φ30, respectively, lie in the samed-Harish-Chandra series, and hence the same block. Sincep|Φ7 andp6= 2, we see these two non-trivial character degrees arep⁰ and distinct.)

We are left to consider the classical groups, in which case the unipotent characters of Ge are parametrized by certain partitions or symbols. By a symbol of rank n, we mean a pair of partitions ^λ_µ^{1 λ2 ···λa}

1 µ2···µb

= ^λ_µ , where λ₁ < λ₂ <· · · < λ_a, µ₁ < µ₂ <· · · < µ_b, λ₁ and µ₁ are not both 0, and n =P

iλ_i+P

jµ_j − b ^a+b−1₂ ² c.

(The symbol ^λ_µ

is equivalent to ^µ_λ

, and ifλ1and µ1are both 0, the symbol is equivalent to ^λ_µ^{2−1···λa−1}

2−1···µ_b−1

.) The defect of a symbol is |b−a|. Given an integere, an e-hook is a pair of non-negative integers(x, y) with y−x=e,x6∈λ (resp.µ), andy ∈λ(resp.µ). The e-core of a symbol is obtained by successively removing e-hooks, which means replacingy byxin λ(resp.µ) and then replacing the result with an equivalent symbol satisfying that λ1 and µ1 are not both 0. An e-cohook is defined similarly, except that x6∈ λand y ∈ µ (or x6∈µ and y ∈λ), and thee-cocore is obtained by removing e-cohooks, which means removing y from µ and addingxto λ(resp. removingy fromλand adding xto µ), and again replacing the result with an equivalent symbol satisfying thatλ1 andµ1are not both0.

Tables 1 through 4 describe two unipotent characters for each classical type satisfying the properties described in the first paragraph and not in the list of exceptions of [Mal08, Theorem 2.5]. For each type, we include a brief discussion, but we remark that a more complete description of the degrees of such characters and the partitions and symbols can be found in [Ca85, Section 13.8], and a more complete discussion of their distribution into blocks may be found in [FS82, FS89]. We will include the details for type A_n−1 in this respect, and note that the other types have similar arguments.

Types A_n−1 and ²A_n−1. Here Ge = GL_n(q). In this case, lete be the order of q modulop. The unipotent characters are in bijection with partitions of n, and two such characters are in the same block if and only if they have the same e-core. In particular, the trivial character is given by the partition (n), which has e-core (r), where 0 ≤r < eis the remainder when n:= me+r is divided by e. Table 1 lists the desired unipotent characters in this case whenn≥4. Indeed, consider the case= 1. The partitions listed have e-core(r), and hence the corresponding characters are in the principal block and it suffices to show that they havep⁰-degree.

Sincep-q, we need only consider the part of the degree relatively prime toq, which are listed following [Ca85, Section 13.8]. If e= 1, then since p >3, the characterχ1 in the cases of line 1 or line 2 has p⁰-degree, since (q^d−1)/(q−1) is divisible by pin this case if and only if dis divisible by p. Hence, forχ1, we may assume e6= 1. Consider line 3 of Table 1 in this case. Sinceme+kis not divisible byefor1≤k < e, we see(q^me+k−1) contains no factors of the form Φ_epi. Hence we see (q^me+1−1)· · ·(qⁿ−1) is not divisible by p. Similarly, if r+ 1 6=e, then (q^me−r−1−1) is not divisible by p. If r+ 1 = e, then (q^me−e−1)/(q^e−1) is divisible by p only if p|(m−1), so that(q^me−e−1) has factors of the formΦ_epi with i≥1. Hence the character listed in line 3 has p⁰-degree, given the stated conditions, and similar for lines 6 and 7. Line 5 refers to the Steinberg character, which is certainly ofp⁰-degree. So, consider the characters in lines 4 and 8, of degree Qe

i=1 qⁿ⁻ⁱ−1

qⁱ−1 , withp|(m−1). IfpdividesQe

i=1 qⁿ⁻ⁱ−1

qⁱ−1 , thenp|(q^n−r−1)/(q^e−1) = (q^me−1)/(q^e−1), and hencep|m, a contradiction. The argument is similar in the case=−1.

Finally, if n= 3 and p - (q+), then note that e = 1 or 3, r <2, and the characters listed in Table 1 still satisfy our conditions. (In this case, the two characters are the Steinberg character and the unipotent character of degreeq(q+).)

TypesBn and Cn. Here the unipotent characters ofGeare in bijection with symbols of ranknand odd defect.

In this case, we let e be the order of q² modulo p. Then two symbols are in the same block if and only if they have the samee-core, respectively e-cocore, ifp| q^e−1, respectively p| q^e+ 1. The trivial character is represented by the symbol ⁿ_∅

, which has e-core and e-cocore ^r_∅

, where 0 ≤ r < e is the remainder when n:=me+r is divided bye. Table 2 lists the desired unipotent characters in this case, as long as n6= 2 or q is not an odd power of 2. When n= 2 andq is an odd power of 2, we havee = 1or 2, so we may still take the Steinberg character forχ2, but the the characters listed forχ1are not necessarily fixed by the exceptional graph automorphism (see [Mal08, Theorem 2.5(c)]). Here we may instead take the character indexed by ^{0 2}₁ of degree(q+ 1)²/2whenp|(q−1), and otherwise we use the character of degree(q−1)²/2indexed by ^{0 1 2}_∅

. Type Dn and ²Dn. In this case the unipotent characters of Ge are in bijection with symbols of rank n and defect0 (mod 4), respectively2 (mod 4)in caseD_n, respectively²D_n. Again, letebe the order ofq²modulop, and letn=me+rwhere0≤r < eis the remainder whennis divided bye. The block distribution is described the same way as for typesB_n andC_n.

For typeD_n(q), the trivial character is represented by the symbol ⁿ₀

, which hase-core ^r₀

ife-n and ^∅_∅ if e|n. It hase-cocore ^r₀

ifmis even ande-n; ⁰_∅^r

ifmis odd ande-n; ^∅_∅

ifmis even ande|n; and ₀^e if mis odd and e|n. Table 3 lists the desired unipotent characters as long as n≥5. (In some cases, more than two characters are listed.) We remark that ifn=e, then it must be thatp|(q^e−1).

ForD4(q) = PΩ⁺₈(q), note that1 ≤e≤3 and that p|(q²+ 1)when e= 2. Then the Steinberg character of degree q¹², labeled by ^{0 1 2 3}_{1 2 3 4}

may be taken for χ1. For χ2, we take the character labeled by ³₁

, of degree q(q² + 1)² when e = 1 or 3, and ^{1 3}_{0 2}

of degree ¹₂q³(q+ 1)³(q³+ 1) when e = 2. In either case, we have χ1(1)>2χ2(1).

For type²Dn(q), the trivial character is represented by the symbol ⁰_∅ⁿ

, which hase-core ⁰_∅^r

whene-nand

0e

∅

ife|n. The e-cocore is ⁰_∅^r

ife-nandm is even, ^r₀

ife-nandm is odd, ^e₀

ife|nand mis even, and ^∅_∅

ife|nandmis odd. Table 4 lists the desired unipotent characters in this case.

Proposition 4.5. Let p > 3 be a prime and let q be a power of a prime different than p. Let S be one of P SL2(q), P SL₃(q) with p|(q+),²B2(2^2a+1) with p|(2^2a+1−1), or ²G2(3^2a+1)with p|(3^2a+1−1). Then there exist two non-trivial characters χ1, χ2 ∈ Irrp⁰(B0(S)) such that χ2(1) - χ1(1); χ2 is invariant under a Sylowp-subgroup ofAut(S); and for every S≤T ≤Aut(S),χ₁ extends to a character in the principal p-block of T.

Table 1: Some unipotent characters inIrr_p⁰(B₀(S))for type A_n−1(q)withn≥4 andp-q

Additional condition on

Partition χ(1)q⁰

n=me+r,r < e

χ1

e= 1andp|(n−1) (2, n−2) ^(qn−n)(q

n−3−ⁿ⁻³) (q−)(q²−1)

e= 1andp-(n−1) (1, n−1) ^qn−1_q−⁻ⁿ⁻¹

16=e6=r+ 1orp-(m−1) (r+ 1, me−1) ^{(qme+1−me+1)(qme+2−me+2)···(qn−n)(q}

me−r−1−^me−r−1) (q−)(q²−1)···(q^r+1−^r+1)

16=e=r+ 1andp|(m−1) (1^r+1, me−1) Qe i=1

(qⁿ⁻ⁱ−ⁿ⁻ⁱ) (qⁱ−ⁱ)

χ2

r <2 (1ⁿ) 1

r≥2,e6=r+ 1orm≥2, and

e6=r+ 2orp-(m−1) (1, r+ 1, me−2) ^{(qme+1−me+1)(qme+2−me+2)···(qn−n)(qme−r−2−me−r−2)(qme−1−me−1)}

(q^r+2−^r+2)(q−)(q−)(q²−1)···(q^r−^r)

r≥2,m= 1,e=r+ 1 (1, e−1, e−1) ^(qe+2_(q−)(q^−e+2₂^)(q_{−1)···(q}^e+3−e−2^e+3−^{)···(qe−2n})⁻ⁿ⁾

r≥2,e=r+ 2,p|(m−1) (1^r+2, me−2) Qe i=1

(qⁿ⁻ⁱ−ⁿ⁻ⁱ) (qⁱ−ⁱ)

Table 2: Some unipotent characters inIrrp⁰(B0(S))for typesBn(q),Cn(q)withn≥2, p-q,(n, q)6= (2,2^2a+1)

Conditions on

Symbol χ(1)q⁰ (possibly excluding factors of ¹₂) n=me+r,r < e

χ1

p|(q^e−1) ⁰_me^r+1 (q^2(me+1)−1)···(q²ⁿ−1)(q^me−r−1+1)(q^me+1)(q^r+1−1) (q²−1)(q⁴−1)···(q^2(r+1)−1)

p|(q^e+ 1), modd ⁰_r+1^me (q^2(me+1)−1)···(q²ⁿ−1)(q^me−r−1+1)(q^me−1)(q^r+1+1) (q²−1)(q⁴−1)···(q^2(r+1)−1)

p|(q^e+ 1), meven ^r+1₀^me (q^2(me+1)−1)···(q²ⁿ−1)(q^me−r−1−1)(q^me+1)(q^r+1+1) (q²−1)(q⁴−1)···(q^2(r+1)−1)

χ2

e|n ^{0 1}_1···^···_n−1ⁿ⁻¹_nⁿ

1

p|(q^e−1), e-n, e6=r+ 1orp-(m−1) ^r+1₀^me (q^2(me+1)−1)···(q²ⁿ−1)(q^me−r−1−1)(q^me+1)(q^r+1+1) (q²−1)(q⁴−1)···(q^2(r+1)−1)

p|(q^e−1), e-n, e=r+ 1, p|(m−1) ^0me_e (q^2(me+1)−1)···(q²ⁿ−1)(q^me−e+1)(q^me−1)(q^e+1) (q²−1)(q⁴−1)···(q^2e−1)

p|(q^e+ 1), e-n, modd ^{0 1}_1r+2^me (q^2(me+1)−1)···(q²ⁿ−1)(q^me−r−2+1)(q^2(me−1)−1)(q^me−1) (q²−1)²(q²−1)(q⁴−1)···(q^2r−1)(q^r+2−1)

p|(q^e+ 1), e-n, meven ^1r+2_{0 1}^me (q^2(me+1)−1)···(q²ⁿ−1)(q^me−r−2−1)(q^2(me−1)−1)(q^me+1) (q²−1)²(q²−1)(q⁴−1)···(q^2r−1)(q^r+2−1)

Table 3: Some unipotent characters in Irrp⁰(B0(S))for typeDn(q)withn≥5,p-q

Conditions on

Symbol χ(1)_q0 (possibly excluding factors of ¹₂) n=me+r,r < e

χ1

e-n ^me_r (q^2(me+1)−1)···(q²⁽ⁿ⁻¹⁾−1)(qⁿ−1)(q^me−r+1)

(q²−1)(q⁴−1)···(q^2r−1)

p|(q^e−1),e|n; or

0 1· · · n−1 1· · · n

1 p|(q^e+ 1),e|n,meven; or

e|(n−1)

p|(q^e+ 1),16=e|n,modd ¹₀^n−e_e+1 (q^2(n−e+1)−1)···(q²⁽ⁿ⁻¹⁾−1)(qⁿ−1)(q^n−2e−1+1)(q^n−e+1)(q^n−e−1−1) (q−1)(q²−1)(q⁴−1)···(q^2(e−1)−1)(q^e−1)(q^e+1+1)

χ2

p|(q^e−1),e-n ₀¹_r+1^me (q^2(me+1)−1)···(q²⁽ⁿ⁻¹⁾−1)(qⁿ−1)(q^me−r−1+1)(q^me+1)(q^me−1−1) (q²−1)(q⁴−1)···(q^2(r−1)−1)(q^r−1)(q^r+1+1)(q−1)

e|n,e6= 1orp-(n−1), 1n 0 1

(q²⁽ⁿ⁻¹⁾−1)

(q²−1)

withp|(q^e−1)ormeven

p|(q²−1),p|(n−1) ⁿ⁻¹₁ (qⁿ−1)(qⁿ⁻²+1) q²−1

p|(q^e+ 1),e-n,meven ^r+1_{0 1}^me (q^2(me+1)−1)···(q²⁽ⁿ⁻¹⁾−1)(qⁿ−1)(q^me−r−1−1)(q^me+1)(q^me−1+1) (q²−1)(q⁴−1)···(q^2(r−1)−1)(q^r−1)(q^r+1−1)(q+1)

p|(q^e+ 1),e-n,modd ₁⁰_r+1^me (q^2(me+1)−1)···(q²⁽ⁿ⁻¹⁾−1)(qⁿ−1)(q^me−r−1+1)(q^me−1)(q^me−1+1) (q²−1)(q⁴−1)···(q^2(r−1)−1)(q^r+1)(q^r+1−1)(q−1)

p|(q^e+ 1),e|n,modd,p-(m−2) ^n−e_e (q^2(me−e+1)−1)···(q^2(me−1)−1)(q^me−1)(q^me−2e+1) (q²−1)(q⁴−1)···(q^2e−1)

p|(q^e+ 1),e|n,modd,p-(m−1) ^1n−e+1_0e (q^2(n−e+2)−1)···(q²⁽ⁿ⁻¹⁾−1)(qⁿ−1)(q^n−2e+1+1)(q^n−e+1+1)(q^n−e−1) (q−1)(q²−1)(q⁴−1)···(q^2(e−2)−1)(q^e−1−1)(q^e+1)