On Héthelyi-Külshammer's conjecture for principal blocks
aa r X i v : . [ m a t h . R T ] F e b ON H ´ETHELYI-K ¨ULSHAMMER’S CONJECTUREFOR PRINCIPAL BLOCKS
NGUYEN NGOC HUNG AND A. A. SCHAEFFER FRY
Abstract.
We prove that the number of irreducible ordinary characters in theprincipal p -block of a finite group G of order divisible by p is always at least 2 √ p − p ′ -degree irreducible characters of finite groups, earlier works of Brou´e-Malle-Michel[BMM93] and Cabanes-Enguehard [CE94, CE04] on the distribution of charactersinto unipotent blocks and e -Harish-Chandra series of finite reductive groups, andknown cases of the Alperin-McKay conjecture. Introduction
Bounding the number k ( G ) of conjugacy classes of a finite group G in terms of acertain invariant associated to G is a fundamental problem in group representationtheory. Let p be a prime dividing the order of G . As observed by Pyber, a result ofBrauer [Bra42] on groups G of order divisible by p but not by p implies that k ( G ) ≥ √ p − k ( B ) of ordinary irreducible characters in a block. It is not surprising thatthis problem is closely related to the previous one on bounding k ( G ); for instance,the p -solvable case of the celebrated Brauer’s k ( B )-conjecture [Bra63, Problem 20],which asserts that k ( B ) is bounded above by the order of a defect group for B , wasknown to be equivalent to the coprime k ( GV )-problem, which in turn was eventuallysolved in 2004 [GMRS04, Sch07]. While there have been a number of results on upper Mathematics Subject Classification.
Primary 20C20, 20C15, 20C33, 20D06.
Key words and phrases. finite groups, principal blocks, characters, H´ethelyi-K¨ulshammer conjec-ture, Alperin-McKay conjecture.The first author thanks Gunter Malle for several stimulating discussions on the relation betweenthe relative Weyl groups of e -split Levi subgroups and maximal tori. The second author is partiallysupported by the National Science Foundation under Grant No. DMS-1801156. bounds for k ( B ) [BF59, Rob04, Sam17, Mal17], not much has been done on lowerbounds.In the proof of k ( G ) ≥ √ p − perhaps it is even true that k ( B ) ≥ √ p − for every p -block B of positive defect, where k ( B ) denotes the number of irreducibleordinary characters in B ”. Of course, H´ethelyi and K¨ulshammer were aware of blocksof defect zero, which have a unique irreducible ordinary character (whose degree hasthe same p -part as the order of the group) and a unique irreducible Brauer characteras well, see [Nav98, Theorem 3.18].The main aim of this paper is to confirm H´ethelyi-K¨ulshammer’s conjecture forprincipal blocks. Theorem 1.1.
Let G be a finite group and p a prime such that p | | G | . Let B ( G ) denote the principal p -block of G . Then k ( B ( G )) ≥ √ p − . Problem 21 in Brauer’s famous list [Bra63] asks whether there exists a function f ( q ) on prime powers q such that f ( q ) → ∞ for q → ∞ and that k ( B ) ≥ f ( p d ( B ) ) forevery p -block B of defect d ( B ) >
0. Our Theorem 1.1 provides an affirmative answerto this question for principal blocks of bounded defect.Building upon the ideas in [Mar16] and the subsequent paper [MM16] of Malle andMar´oti on bounding the number of p ′ -degree irreducible characters in a finite group,we observe that H´ethelyi-K¨ulshammer’s conjecture for principal blocks is essentiallya problem on bounding the number of irreducible ordinary characters in principalblocks of almost simple groups, as well as bounding the number of orbits of irreduciblecharacters in principal blocks of simple groups under the action of their automorphismgroups. Theorem 1.2.
Let S be a non-abelian simple group and p a prime such that p | | S | .Let G be an almost simple group with socle S such that p ∤ | G/S | . Then (i) k ( B ( G )) ≥ √ p − . Moreover, k ( B ( G )) > √ p − if S does not havecyclic Sylow p -subgroups. (ii) Assume furthermore that p ≥ and S does not have cyclic Sylow p -subgroups.Then the number of Aut( S ) -orbits on Irr( B ( S )) is at least p − / . As we will explain in the next section, Theorem 1.1 is a consequence of [Mar16]and the well-known Alperin-McKay conjecture, which asserts that the number of ir-reducible characters of height 0 in a block B of a finite group G coincides with thenumber of irreducible characters of height 0 in the Brauer correspondent of B of thenormalizer of a defect subgroup for B in G . We take advantage of the recent ad-vances on the conjecture in the proof of our results, particularly the fact that Sp¨ath’sinductive Alperin-McKay conditions hold for all p -blocks with cyclic defect groups[Spa13, KSp16]. This explains why simple groups with cyclic Sylow p -subgroups areexcluded in Theorem 1.2(ii). Additionally, we take advantage of recent results on N H´ETHELYI-K ¨ULSHAMMER’S CONJECTURE FOR PRINCIPAL BLOCKS 3 the possible structure of defect groups of principal blocks with few ordinary charac-ters [KS20, RSV21], and this explains why the smaller values of p are excluded inTheorem 1.2(ii).Theorem 1.2 turns out to be straightforward for alternating groups or groups of Lietype in characteristic p , but highly nontrivial for groups of Lie type in characteristicnot equal to p . We make use of Cabanes-Enguehard’s results on unipotent blocks[CE04] to prove that the so-called semisimple characters of S all fall into the princi-pal p -block B ( S ) in a certain nice situation. This and results of Brou´e-Malle-Michel[BMM93] and Cabanes-Enguehard [CE94] on the compatibility between the distri-butions of unipotent characters into unipotent blocks and e -Harish-Chandra seriesallow us to obtain a general bound for the number of Aut( S )-orbits of characters inIrr( B ( S )) in terms of certain data associated to S , for S a simple group of Lie type,see Theorem 8.3. We hope this result will be useful in other purposes.The next result classifies groups for which k ( B ( G )) is minimal in the sense ofTheorem 1.1. Theorem 1.3.
Let G be a finite group and p a prime. Let P be a Sylow p -subgroupof G and B denote the principal p -block of G . Then k ( B ) = 2 √ p − if and onlyif √ p − ∈ N and N G ( P ) / O p ′ ( N G ( P )) is isomorphic to the Frobenius group C p ⋊ C √ p − . We remark that, in the situation of Theorem 1.3, the number of p ′ -degree irre-ducible characters in B ( G ) is also equal to 2 √ p −
1. In general, if a p -block B ofa finite group has an abelian defect group, then every ordinary irreducible characterof B has height zero. This is the ‘if direction’ of Brauer’s height-zero conjecture,which is now known to be true, thanks to the work of Kessar and Malle [KM13].Theorem 1.1 therefore implies that if P ∈ Syl p ( G ) is abelian and nontrivial then k ( B ( G )) ≥ √ p −
1, where k ( B ) denotes the number of height zero ordinary irre-ducible characters of a block B .Theorems 1.1 and 1.3 are useful in the study of principal blocks with few heightzero ordinary irreducible characters. In fact, using them, we are able to show in aforthcoming paper [HSF21] that k ( B ( G )) = 3 if and only if P ∼ = C , and that k ( B ( G )) = 4 if and only if | P/P ′ | = 4 or P ∼ = C and N G ( P ) / O p ′ ( N G ( P )) isisomorphic to the dihedral group D . These results have been known only in thecase p ≤
3, see [NST18, Theorems A and C].The paper is organized as follows. In Section 2, we recall some known results on theAlperin-Mckay conjecture and prove that our results follow when all the non-abeliancomposition factors of G have cyclic Sylow p -subgroups. We also prove Theorem 1.2for the sporadic simple groups and groups of Lie type defined in characteristic p inSection 2. The alternating groups are treated in Section 3. Section 4 takes careof the case when the Sylow p -subgroups of S are non-abelian. Sections 6, 7, and 9 N. N. HUNG AND A. A. SCHAEFFER FRY are devoted to proving Theorem 1.2 for simple groups of Lie type defined in char-acteristics different from p . To do so, in Section 5, we provide some background onsemisimple characters of finite reductive groups and prove that those associated toconjugacy classes of p -elements belong to the principal p -block in a certain situation,and in Section 8 we obtain a bound for the number of Aut( S )-orbits of characters inIrr( B ( S )). Finally, we finish the proofs of Theorems 1.1 and 1.3 in Section 10.2. Some first observations
In this section we make some observations toward the proofs of the main results.2.1.
The Alperin-McKay conjecture.
The well-known Alperin-McKay (AM) con-jecture predicts that the number of irreducible characters of height zero in a block B of a finite group G coincides with the number of irreducible characters of height zeroin the Brauer correspondent of B of the normalizer of a defect subgroup of B in G .For the principal blocks, the conjecture is equivalent to k p ′ ( B ( G )) = k p ′ ( B ( N G ( P ))) , where P is a Sylow p -subgroup of G and k p ′ ( B ( G )) denotes the number of p ′ -degreeirreducible ordinary characters in B ( G ).On the other hand, if p | | G | , we have k p ′ ( B ( N G ( P ))) ≥ k p ′ ( B ( N G ( P ) /P ′ ))= k ( B ( N G ( P ) /P ′ ))= k ( B (( N G ( P ) /P ′ ) / O p ′ ( N G ( P ) /P ′ )))= k (( N G ( P ) /P ′ ) / O p ′ ( N G ( P ) /P ′ )) ≥ p p − , where the first inequality follows from [Nav98, p. 137], the first equality followsfrom the fact that every irreducible ordinary character of N G ( P ) /P ′ has p ′ -degree,the last two equalities follow from [Nav98, Theorem 9.9] and Fong’s theorem (see[Nav98, Theorem 10.20]), and the last inequality follows from [Mar16]. Therefore,if the AM conjecture holds for G and p , then the number of p ′ -degree irreducibleordinary characters in B ( G ) is bounded by 2 √ p − G, p ). We now prove that the same is true for Theorem 1.3. Note that the “if”implication of this theorem is clear. Assume that the AM conjecture holds for B ( G )and k ( B ( G )) = 2 √ p − p such that √ p − ∈ N . Then, as seenabove, we have2 p p − k ( B ( G )) ≥ k (( N G ( P ) /P ′ ) / O p ′ ( N G ( P ) /P ′ )) ≥ p p − , N H´ETHELYI-K ¨ULSHAMMER’S CONJECTURE FOR PRINCIPAL BLOCKS 5 implying k (( N G ( P ) /P ′ ) / O p ′ ( N G ( P ) /P ′ )) = 2 p p − , and thus ( N G ( P ) /P ′ ) / O p ′ ( N G ( P ) /P ′ ) is isomorphic to the Frobenius group C p ⋊ C √ p − , by [Mar16, Theorem 1]. In particular, P/P ′ ∼ = C p , implying that P ∼ = C p ,and hence it follows that N G ( P ) / O p ′ ( N G ( P )) is isomorphic to the Frobenius group C p ⋊ C √ p − , as wanted.The AM conjecture is known to be true when G has a cyclic Sylow p -subgroup byDade’s theory [Dad66]. In fact, by [Spa13, KSp16], the so-called inductive Alperin-McKay conditions are satisfied for all blocks with cyclic defect groups. Therefore, wehave: Lemma 2.1 (Koshitani-Sp¨ath) . Let p be a prime. Assume that all the non-abeliancomposition factors of a finite group G have cyclic Sylow p -subgroups. Then theAlperin-Mckay conjecture holds for G and p , and thus Theorems 1.1, 1.2(i), and 1.3hold for G and p . Note that the linear groups PSL ( q ), the Suzuki groups B (2 f +1 ) and the Reegroups G (3 f +1 ) all have cyclic Sylow p -subgroups for p different from the definingcharacteristic of the group. So Theorem 1.2 automatically follows from Lemma 2.1for these groups in characteristic not equal to p .2.2. Small blocks.
Blocks with a small number of ordinary characters have beenstudied significantly in the literature. In particular, the possible structure of defectgroups of principal blocks with at most 5 ordinary irreducible characters are nowknown, see [Bra82, Bel90, KS20, RSV21]. Using these results, we can easily confirmour results for p ≤
7. For instance, to prove Theorems 1.1 and 1.2 for p = 7 it isenough to assume that k ( B ( G )) ≤
4, but by going through the list of possible defectgroups of B ( G ), we then have Syl p ( G ) ∈ { , C , C , C × C , C , C } , which cannothappen. To prove Theorem 1.3 for p < p = 5 and k ( B ( G )) = 4then P = C ; and if p = 2 and k ( B ( G )) = 2 then P = C , both of which cases P iscyclic, and thus the result of Subsection 2.1 applies.Therefore we will assume from now on that p ≥
11, unless stated otherwise.2.3.
Sporadic and the Tits groups.
We remark that Theorem 1.2 can be con-firmed directly using [Atl85, Atl95] or [GAP] for sporadic simple groups and the Titsgroup. Therefore, we are left with the alternating groups and groups of Lie type,which will be treated in the subsequent sections.2.4.
Groups of Lie type in characteristic p . Let S be a simple group of Lietype defined over the field of q = p f elements. According to results of Dagger andHumphreys on defect groups of finite reductive groups in defining characteristic, see[Cab18, Proposition 1.18 and Theorem 3.3] for instance, S has only two p -blocks. N. N. HUNG AND A. A. SCHAEFFER FRY
Namely, the only non-principal block is a defect-zero block containing only the Stein-berg character of S . Therefore, k ( B ( S )) = k ( S ) − . Let G be a simple algebraic group of simply connected type and let F be a Steinbergendomorphism on G such that S = X/ Z ( X ), where X = G F . Assume that the rankof G is r . By a result of Steinberg (see [FG12, Theorem 3.1]), X has at least q r semisimple conjugacy classes, and thus k ( X ) > q r . It follows that k ( B ( S )) ≥ (cid:20) q r | Z ( X ) | − (cid:21) , which yields k ( B ( S )) ≥ q r / | Z ( X ) | . Using the values of | Z ( X ) | and | Out( S ) | availablein [Atl85, p. xvi], it is straightforward to check that q r / | Z ( X ) | ≥ √ p − | Out( S ) | ,proving Theorem 1.2 for the relevant S and p .3. Alternating groups
In this section we prove Theorem 1.2 for the alternating groups. The backgroundon block theory of symmetric and alternating groups can be found in [Ols93] forinstance.The ordinary irreducible characters of S n are naturally labeled by partitions of n .Two characters are in the same p -block if and only if their corresponding partitionshave the same p -cores, which are obtained from the partitions by successive removalsof rim p -hooks until no p -hook is left. Therefore, p -blocks of S n are in one-to-onecorrespondence with p -cores of partitions of n .Let B be a p -block of S n . The number k ( B ) of ordinary irreducible charactersin B turns out to depend only on p and the so-called weight of B , which is definedto be w ( B ) := ( n − | µ | ) /p , where µ is the p -core corresponding to B under theaforementioned correspondence. In fact, k ( B ) = k ( p, w ( B )) := Σ ( w ,w ...,w p − ) π ( w ) π ( w ) · · · π ( w p − ) , where ( w , w ..., w p − ) runs through all p -tuples of non-negative integers such that w ( B ) = Σ p − i =0 w i and π ( x ) is the number of partitions of x , see [Ols93, Proposition11.4]. Note that k ( p, w ( B )) is precisely the number of p -tuples of partitions of w ( B ).For the principal block B ( S n ) of S n , we have w ( B ( S n )) = [ n/p ], which is at least1 by the assumption p | | S | . It follows that k ( B ( S n )) ≥ k ( p,
1) = p ≥ p p − . Moreover, according to [Ols92, Proposition 2.8], when p is odd and e B is a blockof A n covered by B , then B and e B have the same number of irreducible ordinarycharacters (and indeed the same number of irreducible Brauer characters as well). In N H´ETHELYI-K ¨ULSHAMMER’S CONJECTURE FOR PRINCIPAL BLOCKS 7 particular, when p is odd, we have k ( B ( A n )) = k ( B ( S n )) ≥ √ p −
1, which provesTheorem 1.2(i) for the alternating groups.For part (ii) of Theorem 1.2, recall that p ≥
11, and thus n ≥
11 and Aut( S ) = S n .The number of S n -orbits on Irr( B ( A n )) is at least k ( B ( A n )) /
2, which in turn is atleast 1 + ( p − / p + 1) / > p − / , and this proves Theorem 1.2(ii) for thealternating groups.4. Groups of Lie type: the non-abelian p -Sylow case In this section, we let G be a simple algebraic group of adjoint type and F aSteinberg endomorphism on G such that S ∼ = [ G , G ] where G := G F . Let q = ℓ f with ℓ = p be the absolute value of all eigenvalues of F on the character group of an F -stable maximal torus of G . Recall that we are assuming p ≥ S of Lie type in characteristicdifferent from p such that the Sylow p -subgroups of G are non-abelian.In that case, there are then more than one d ∈ N such that p | Φ d ( q ) with Φ d dividing the order polynomial of ( G , F ). Here, as usual, Φ d denotes the d th cyclo-tomic polynomial. (In fact, if there a unique such d , then a Sylow p -subgroup of G iscontained in a Sylow d -torus of G , and hence is abelian, see [MT11, Theorem 25.14].)Let e p ( q ) denote the multiplicative order of q modulo p . Note that, by [MT11,Lemma 25.13], p | Φ d ( q ) if and only if d = e p ( q ) p i for some i ≥
0. Therefore, as thereis more than one d ∈ N such that p | Φ d ( q ), we must have p | d for some d ∈ N suchthat Φ d divides the order polynomial of ( G , F ). The fact that p ≥
11 then rules outthe cases when G is of exceptional type and thus we are left with only the classicaltypes. That is, G = PGL n ( q ), PGU n ( q ), SO n +1 ( q ), PCSp n ( q ), or P(CO ± n ( q )) .For G = PGL n ( q ) or PGU n ( q ), we define e to be the smallest positive integer suchthat p | ( q e − ( ǫ ) e ) ( ǫ = 1 for linear groups and ǫ = − e = e p ( q ) when G = PGL n ( q ) or G = PGU n ( q ) and 4 | e p ( q ), e = e p ( q ) / G = PGU n ( q ) and 2 | e p ( q ) but 4 ∤ e p ( q ), and e = 2 e p ( q ) when G = PGU n ( q ) and2 ∤ e p ( q ). For G = SO n +1 ( q ), PCSp n ( q ), or P(CO ± n ( q )) , we define e to be thesmallest positive integer such that p | ( q e ± e = e p ( q ) when e p ( q ) is oddand e = e p ( q ) / e p ( q ) is even.Let n = we + m where 0 ≤ m < e . We claim that p ≤ w . To see this, firstassume that G = PGL n ( q ). Then, as mentioned above, ep ≤ n , which implies that ep < ( w + 1) e , and thus p ≤ w . Next, assume that G = SO n +1 ( q ), PCSp n ( q ), orP(CO ± n ( q )) . If e = e p ( q ) is odd, then since p | ( q e −
1) and gcd( q e − , q i + 1) ≤ i ∈ N , we have p | ( q j −
1) for some e < j ≤ n , and it follows that ep ≤ n ,implying p ≤ w . On the other hand, if 2 e = e p ( q ) is even then 2 ep = e p ( q ) p ≤ n < w + 1) e , which also implies that p ≤ w . Finally, assume G = PGU n ( q ). The case4 | e p ( q ) is argued as in the case S = PGL n ( q ); the case 2 | e p ( q ) but 4 ∤ e p ( q ) isargued as in the case S = SO n +1 ( q ) and 2 | e p ( q ). For the last case 2 ∤ e p ( q ), wehave ep/ e p ( q ) p , and in order for Φ e p ( q ) p dividing the generic order of | PGU n ( q ) | , N. N. HUNG AND A. A. SCHAEFFER FRY e p ( q ) p ≤ n/
2, and hence it follows that ep ≤ n , which also implies that p ≤ w . Theclaim is fully proved.By [BMM93, Theorem 3.2] and [CE94, Main Theorem], the number of unipotentcharacters of G in the principal block B ( G ) is equal to k ( W e ) - the number ofirreducible complex characters of the relative Weyl group W e of a Sylow e p ( q )-torusof G . This W e is the wreath product C e ≀ S w when G is of type A and is a subgroupof index 1 or 2 of C e ≀ S w when G is of type B , C , or D , see [MM16, Proposition5.5 and its proof]. In any case we have that the number of unipotent characters inIrr( B ( G )) is at least k ( S w ) / π ( w ) /
2, which in turns is at least π ( p ) / p ≤ w .Since every unipotent character of G restricts irreducibly to S , it follows that thenumber of unipotent characters in Irr( B ( S )) is at least π ( p ) / S )-orbit of length at most 3. (In fact, everyAut( S )-orbit on unipotent characters of S has length 1 or 2, except when S = P Ω +8 ( q )whose the graph automorphism of order 3 produces two orbits of length 3.) Therefore,together with the conclusion of the previous paragraph, we deduce that the numberof Aut( S )-orbits on Irr( B ( S )) is at least π ( p ) /
6. This bound is greater than 2 √ p − p ≥
11, as required.5.
Semisimple characters and principal blocks
Before continuing with our proof of Theorem 1.2 for groups of Lie type, we recallsome background on certain characters known as semisimple characters and showhow they fall into the principal block in a certain situation. Background on charactertheory of finite reductive groups can be found in [Car85, CE04, DM91]. Let G be aconnected reductive group defined over F q and F an associated Frobenius endomor-phism on G . Let G ∗ be an algebraic group with a Frobenius endomorphism which,for simplicity, we denote by the same F , such that ( G , F ) is in duality to ( G ∗ , F ).Let t be a semisimple element of ( G ∗ ) F . The rational Lusztig series E ( G F , ( t ))associated to the ( G ∗ ) F -conjugacy class ( t ) of t is defined to be the set of irreduciblecharacters of G F occurring in some Deligne-Lusztig character R GT θ , where T is an F -stable maximal torus of G and θ ∈ Irr( T F ) such that ( T , θ ) corresponds in dualityto a pair ( T ∗ , s ) with s ∈ T ∗ ∩ ( t ). Here we recall that there is a one-to-one dualitycorrespondence between G F -conjugacy classes of pairs ( T , θ ), where T is an F -stablemaximal torus of G and θ ∈ Irr( T F ), and the ( G ∗ ) F -conjugacy classes of pairs ( T ∗ , s ),where T ∗ is dual to T and s ∈ ( T ∗ ) F .We continue to let t be a semisimple element of ( G ∗ ) F and assume furthermore that C G ∗ ( t ) is a Levi subgroup of G ∗ . Let G ( t ) be a Levi subgroup of G in duality with C G ∗ ( t ) and P be a parabolic subgroup of G for which G ( t ) is the Levi complement.The twisted induction R GG ( t ) ⊆ P and the multiplication by b t , a certain linear characterof Irr( G ( t ) F ) naturally defined by t (see [CE04, (8.19)]), then induce a bijection N H´ETHELYI-K ¨ULSHAMMER’S CONJECTURE FOR PRINCIPAL BLOCKS 9 between the Lusztig series E ( G ( t ) F ,
1) and E ( G F , ( t )), see [CE04, Proposition 8.26and Theorem 8.27] or [DM91, Theorem 13.25 and Proposition 13.30]. In fact, foreach λ ∈ E ( G ( t ) F , ε G ε G ( t ) R GG ( t ) ⊆ P ( b tλ ) ∈ E ( G F , ( t )) , where ε G := ( − σ ( G ) with σ ( G ) the F q -rank of G . Taking λ to be trivial, we havethe character χ ( t ) := ε G ε G ( t ) R GG ( t ) ⊆ P ( b t G ( t ) F ) ∈ E ( G F , ( t )) , which is often referred to as a semisimple character of G F , of degree χ ( t ) (1) = | ( G ∗ ) F : C G ∗ F ( t ) | ℓ ′ , where ℓ is the defining characteristic of G , see [DM91, Theorem 13.23].By [CE04, Theorem 9.12], every element of B ( G F ) lies in a Lusztig series E ( G F , ( t ))where t is a p -element of G ∗ F . Hence one might ask which such t indeed producesemisimple characters that contribute to the principal block. We will see in thefollowing theorem that in a certain nice situation which is indeed enough for ourpurpose, the centralizer C G ∗ ( t ) is a Levi subgroup of G ∗ , and thus the semisimplecharacter χ ( t ) associated to ( t ) is well-defined and belongs to B ( G F ).In the following, we recall that a prime p is good for G if it does not divide thecoefficients of the highest root of the root system associated to G . Theorem 5.1.
Let ( G , F ) be a connected reductive group defined over F q . Let p bea good prime for G and not dividing q . Let t be a p -element of G ∗ F . If C G ∗ ( t ) is connected, then the semisimple character χ ( t ) ∈ Irr( G F ) belongs to the principal p -block of G F . In particular, if Z ( G ) is connected, the character χ ( t ) belongs to theprincipal block of G F for every p -element t ∈ G ∗ F .Proof. Assume that C G ∗ ( t ) is connected. Since p is good for G , C G ∗ ( t ) is then aLevi subgroup of G ∗ , see [CE04, Proposition 13.16]. Define a Levi subgroup G ( t ) andparabolic subgroup P of G as above. Since χ ( t ) = ε G ε G ( t ) R GG ( t ) ⊆ P (cid:0)b t G ( t ) F (cid:1) , [CE04,Theorem 21.13] implies that all the irreducible constituents of R GG ( t ) ⊆ P ( G ( t ) F ) areunipotent characters of G F which are in the same p -block as χ ( t ) . As the trivialcharacter is obviously a constituent of R GG ( t ) ⊆ P ( G ( t ) F ), we deduce that χ ( t ) ∈ B ( G F ).The second statement of the theorem immediately follows from the first, as if Z ( G )is connected then the centralizer of every semisimple element of G ∗ is connected, see[DM91, Lemma 13.14]. (cid:3) Linear and Unitary Groups
In this section, we let S = PSL ǫn ( q ), where p ∤ q and ǫ ∈ {± } . Here PSL ǫn ( q ) :=PSL n ( q ) in the case ǫ = 1 and PSU n ( q ) in the case ǫ = −
1, and analogous for SL ǫn ( q ),GL ǫn ( q ), and PGL ǫn ( q ). We further let q := q if ǫ = 1 and q := q if ǫ = −
1. Note that with our notation, SL ǫn ( q ) and GL ǫn ( q ) are naturally subgroups of SL n ( q ) andGL n ( q ), respectively. Proposition 6.1.
Let S = PSL ǫn ( q ) and let p ∤ q be a prime. Then Theorem 1.2holds for any almost simple group A with socle S and p ∤ | A/S | .Proof. With the results of the previous sections, we may assume n ≥ p ≥
11, and q is a power of some prime different from p .Write S = PSL ǫn ( q ), G = PGL ǫn ( q ), e G = GL ǫn ( q ), and G = SL ǫn ( q ). Then we have G = [ e G, e G ], S = G/ Z ( G ), and G = e G/ Z ( e G ). From Section 4, we may assume thatSylow p -subgroups of G are abelian, which implies that there is a unique e such that p | Φ e ( q ) and Φ e divides the generic order polynomial of G . Here e must be e p ( q ), themultiplicative order of q modulo p . Note that this also forces p ∤ n by again appealingto [MT11, Lemma 25.13].We will further define e := e p ( q ) and e ′ as follows: e ′ := (cid:26) e if ǫ = 1 or if ǫ = − p | q e − ( − e e if ǫ = − p | q e + ( − e . To prove Theorem 1.2, our aim is to show that when a Sylow p -subgroup of S isnot cyclic, then the number of Aut( S )-orbits on Irr( B ( S )) is larger than 2 √ p − p ∤ gcd( n, q − ǫ ) = | Z ( G ) | , the irreducible characters in the principalblock of S are the same as that of G , under inflation (see [Nav98, Theorem 9.9]).Similarly, if e ′ >
1, then p ∤ ( q − ǫ ) = | Z ( e G ) | and an analogous statement holds for G and e G . Hence, we begin by studying B ( e G ), which will be sufficient for our purposesin the case e ′ > n = we ′ + m with 0 ≤ m < e ′ . Set p a := ( q e − p ≥ p . The case p ≤ w wastreated in Section 4, so we assume that p > w . Note that by [MO83, Theorem 1.9], B ( e G ) and B (GL ǫwe ′ ( q )) have the same number of ordinary irreducible characters, sowe may assume that n = we ′ . (Note that the action of Aut( S ) is analogous as well.)Let F ( p, a ) denote the set of monic polynomials over F q in the set F defined in[FS82] whose roots have p -power order in F × q at most p a . Note that |F ( p, a ) | =1 + ( p a − /e ′ , see [MO83, p. 211].The conjugacy classes ( t ) := t e G of p -elements in e G are parameterized by p -weightvectors of w , which are functions w := w ( t ) : F ( p, a ) → Z ≥ such that w = P g ∈F ( p,a ) w ( g ). The characteristic polynomial of elements in ( t ) is( x − e ′ w ( x − Y x − = g ∈F ( p,a ) g w ( g ) , and the centralizer of t is C e G ( t ) = GL ǫe ′ w ( x − ( q ) × Y x − = g ∈F ( p,a ) GL η w ( g ) ( q e ′ ) N H´ETHELYI-K ¨ULSHAMMER’S CONJECTURE FOR PRINCIPAL BLOCKS 11 where η = ǫ unless ǫ = − e ′ = 2 e , in which case η = 1.Each character in the Lusztig series E ( e G, t ) is labeled by χ t,ψ where ψ is a unipo-tent character of C e G ( t ). So ψ = Q g ∈F ( p,a ) ψ g where ψ g is a unipotent character ofGL η w ( g ) ( q e ′ ) if g = x − e ′ w ( x − ( q ) if g = x −
1. Note that there is acanonical correspondence between unipotent characters of GL ± x ( q ) and partitions of x , so we may view ψ g as a partition of w ( g ) when g = x − e ′ w ( x −
1) when g = x −
1. Further, by [FS82, Theorem (7A)], the characters of B ( e G ) are exactlythose χ t,ψ satisfying t is a p -element and the partition ψ x − has trivial e ′ -core.By [Ols84, Proposition 6], k ( B ( e G )) = k (cid:18) e ′ + p a − e ′ , w (cid:19) , where k ( x, y ) is as defined in Section 3 above. This number is at least(6.1) e ′ + p a − e ′ ≥ p p a − ≥ p p − . But, recall that we wish to show that there are at least 2 √ p − B ( S ))under Aut( S ).Now, by taking t = 1, the number of unipotent characters in B ( e G ) is precisely k ( e ′ , w ). Note that k ( e ′ , w ) ≥ k ( e ′ ,
1) = e ′ , and that further k ( e ′ , w ) ≥ e ′ if w ≥ e ′ , w ) = (1 , S )-invariant. So we have at least e ′ Aut( S )-orbits of unipotent characters in B ( G ), andhence of B ( S ), since restriction yields a bijection between unipotent characters of S and G .Let e G := GL n ( F q ) so that e G = e G F . Since Z ( e G ) is connected, [CS13, Theorem 3.1]yields that the “Jordan decomposition” ψ t,ψ ↔ ( t, ψ ) can be chosen to be Aut( S )-equivariant. Since ψ is a unipotent character of a product of groups of the formGL ± x ( q d ), which are invariant under automorphisms as discussed above, it followsthat the orbit of χ t,ψ is completely determined by the action of Aut( S ) on the class( t ).Now, recall that the e G -class of t is completely determined by its eigenvalues. Let | t | = p c and note that c ≤ a . By viewing t as an element 1 × Q x − = g ∈F ( p,a ) ζ g of Z ( C e G ( t )) ∼ = C q − ǫ × Y x − = g ∈F ( p,a ) C q e ′ − η we see that for α ∈ Aut( S ), the eigenvalues of t α are those of t raised to some power ηq e for some η ∈ {± } and some q such that q is a power of q . This implies thatthe Aut( S )-orbit of ( t ) has size at most p c − e ′ ≤ p a − e ′ .Now, the Sylow p -subgroup P of e G is of the form C wp a ≤ ( F × q e ) w . Then if w = 1, P iscyclic, and hence we may assume that w ≥
2. In this case, we have at least (cid:0) p a − e ′ (cid:1) choices for ( t ) = (1), and hence at least (cid:0) p a − e ′ (cid:1) non-unipotent characters in B ( e G )by taking ψ x − to be trivial. This gives at least p a − e ′ distinct orbits of non-unipotentcharacters, and hence more than 2 √ p − B ( G ) under Aut( S )when e ′ >
1, by (6.1) with 2 e ′ rather than e ′ . This completes the proof of Theorem1.2 for S in the case e ′ > e ′ = 1, so w = n ≥ p > w .Consider the elements t of e G whose eigenvalues are of the form { ζ , ξ, ( ζ ξ ) − , , . . . , } with ζ and ξ p -elements of C q − ǫ ≤ F × q . Note that each member of E ( e G, t ) lies in theprincipal block of e G and that t lies in G = [ e G, e G ]. Further, t cannot be conjugateto tz for any nontrivial z ∈ Z ( e G ), since such a z would have determinant 1 and p -power order, contradicting p > n . Then using [RSV21, Lemma 1.4] and [SFT21,Proposition 2.6], each character in such a E ( e G, t ) is irreducible on restriction to G ,yielding at least ( p a − non-unipotent members of B ( G ). Since the Aut( S )-orbits ofsuch characters are again of size at most p a −
1, this yields at least 2 + p a − distinctorbits, which is larger than 2 √ p −
1. This completes the proof of Theorem 1.2 in thecase that S = PSL ǫn ( q ). (cid:3) Symplectic and Orthogonal Groups
In this section, we consider the simple groups coming from orthogonal and sym-plectic groups. That is, simple groups of Lie type B n , C n , D n , and D n . We let ǫ ∈ {± } , and let PΩ ǫ n ( q ) denote the simple group of Lie type D n ( q ) for ǫ = 1 andof type D n ( q ) for ǫ = − Proposition 7.1.
Let q be a power of a prime different from p and let S = PSp n ( q ) with n ≥ , PΩ n +1 ( q ) with n ≥ , or PΩ ǫ n ( q ) with n ≥ . Then Theorem 1.2 holdsfor any almost simple group A with socle S and p ∤ | A/S | .Proof. With the results of the previous sections, we may again assume that p ≥ p -subgroup of S is abelian, but not cyclic.Let H be the corresponding symplectic or special orthogonal group Sp n ( q ), SO n +1 ( q ),or SO ǫ n ( q ) and let ( H , F ) be the corresponding simple algebraic group and Frobeniusendomorphism so that H = H F . Let G = G F be the corresponding group of simplyconnected type, so that G = H in the symplectic case or G is the appropriate spingroup in the orthogonal cases. Further, let ( H ∗ , F ) and ( G ∗ , F ) be dual to ( H , F )and ( G , F ), respectively, and H ∗ = H ∗ F and G ∗ = G ∗ F .Define ¯ H to be the group GO ǫ n ( q ) in the case S = PΩ ǫ n ( q ), and ¯ H := H otherwise.We also let Ω be the unique subgroup of index 2 in H for the orthogonal cases when q is odd, and let Ω = H otherwise, so that Ω / Z (Ω) = S = G/ Z ( G ) and Ω ✁ ¯ H . Notethat B ( S ) can be identified with B (Ω) or with B ( G ), by [Nav98, Theorem 9.9].Now, let e := e p ( q ) / gcd( e p ( q ) ,
2) and write n = we + m with 0 ≤ m < e . FromSection 4, we may again assume w < p . To obtain our result, we will rely on the N H´ETHELYI-K ¨ULSHAMMER’S CONJECTURE FOR PRINCIPAL BLOCKS 13 case of linear groups and will use some of the ideas of the arguments used in [Mal17,Propositions 5.4 and 5.5], which provides an analogue in this situation to the resultsof Michler and Olsson discussed above. Namely, [Mal17, Propositions 5.4 and 5.5]tells us k ( B ( ¯ H )) = k (cid:18) e + p a − e , w (cid:19) , where p a = ( q e − p . Note that again, this number is at least 2 √ p − w ≥ k ( B ( A )). Inmost cases, we will again show that the number of Aut( S )-orbits of characters in B ( S ) is at least 2 √ p − w = 1, a Sylow p -subgroup of Ω, G , H , or ¯ H (recall p ≥
11) is cyclic, so wemay assume by Lemma 2.1 that w ≥
2. Note that the unipotent characters of H are irreducible on restriction to Ω. Arguing as in the first paragraph of the proofof [RSV21, Lemma 3.10], if S = D ( q ) nor Sp (2 f ) with f odd, then the number ofAut( S )-orbits of unipotent characters in B ( H ), and hence B ( S ), is k (2 e, w ). Notethat k (2 e, w ) > e since w ≥ B ( H ) and B ( G ) lie in Lusztig series indexed by p -elements t of H ∗ , respectively G ∗ , by [CE04, Theorem 9.12]. Note that centralizers of odd-orderelements of H ∗ and of G ∗ are always connected (see e.g. [MT11, Exercise (20.16)])and that every odd p is good for H and G , so that χ ( t ) lies in B ( H ), respectively B ( G ), for every p -element t of H ∗ , respectively G ∗ , by Theorem 5.1. Further, notethat the action on χ ( t ) under a graph-field automorphism of H is determined by theaction of a corresponding graph-field automorphism on ( t ), by [NTT08, Corollary2.8]. (See also (8.1) below.)Now let G ֒ → e G be a regular embedding as in [CE04, 15.1] and let e G := e G F . Thenthe action of e G on G induces all diagonal automorphisms of S . Now, since C G ∗ ( t )is connected for any p -element t ∈ G ∗ , we have every character in E ( G, ( t )) extendsto a character in e G . (Indeed, since e G/G is abelian and restrictions from e G to G are multiplicity-free, the number of characters lying below a given e χ ∈ Irr( e G ) is thenumber of β ∈ Irr( e G/G ) such that e χβ = e χ , as noted in [RSV21, Lemma 1.4]. Hence[Bon05, Corollary 2.8] and [SFT21, Proposition 2.6] yields the claim.) Therefore,each member of B ( S ) is invariant under diagonal automorphisms.First consider the case H = SO n +1 ( q ) or Sp n ( q ), so H ∗ = Sp n ( q ) or SO n +1 ( q ),respectively. Note that Aut( S ) /S in this case is generated by field automorphisms,which also act on H , along with a diagonal or graph automorphism of order at most2. If H = SO n +1 ( q ), then GL n ( q ) may be embedded into H ∗ = Sp n ( q ) in a naturalway (namely, block diagonally as the set of matrices of the form ( A, A − T ) for A ∈ GL n ( q )), and the conjugacy class of t is again determined by its eigenvalues. Arguingas in the case of SL n ( q ) above and noting that every eigenvalue of t must have the same multiplicity as its inverse, we then have at least p a − e distinct orbits of non-unipotentcharacters in B ( H ) under the field automorphisms, and hence at least p a − e orbitsin B ( S ) under Aut( S ). This gives more than 4 e + p a − e orbits in Irr( B ( S )) underAut( S ), which proves Theorem 1.2 in this case using (6.1).If H = Sp n ( q ), by [GH91, Theorem 4.2], there is a bijection between classes of p -elements of H and H ∗ , and we note that field automorphisms act analogously onthe p -elements of H and H ∗ . Then the above again yields the result in this case aslong as S = Sp (2 f ) with f odd.If S = Sp (2 f ) with f odd, then we must have e = 1 and w = 2. Here [Mal08,Theorem 2.5] tells us that there is a pair of unipotent characters permuted by theexceptional graph automorphism, leaving k (2 , − B ( S ) under Aut( S ). In this case, arguing as before and considering the action ofthe graph automorphism gives at least 4 + p a − orbits in B ( S ) under Aut( S ), whichis at least 2( p − / . Hence part (ii) of Theorem 1.2 holds. So let S ≤ A ≤ Aut( S ),and we wish to show that B ( A ) contains more than 2 √ p − S ) /S is cyclic. Let X := SC A ( P ) for P ∈ Syl p ( S ). Then A/X is cyclic,say of size b , and B ( A ) is the unique block covering B ( X ) by [Nav98, (9.19) and(9.20)]. Note that since at least 3 of the unipotent characters of S are A -invariant,we have at least 3 b characters in B ( A ) lying above unipotent characters. Further,since the automorphisms corresponding to those in X stabilize p -classes in G ∗ , thearguments above give at least · (cid:0) p a − (cid:1) members of B ( X ) lying above semisimplecharacters of S , and hence there are at least ( p a − b members of B ( A ) lying abovesemisimple characters of S . Note then that the size of B ( A ) is at least 3 b + ( p a − b ,which is larger than 2 √ p −
1, completing the proof in this case.Now, suppose we are in the case that ¯ H = GO ǫ n ( q ). Note that the action of ¯ H/H induces a graph automorphism of order 2 in the case ǫ = 1, and that Aut( S ) /S isgenerated by a group of diagonal automorphisms of size at most 4, along with graphand field automorphisms. Further, note that the action of H on Ω induces a diagonalautomorphism of order 2 on S . We may embed ¯ H in SO n +1 ( q ), and by [Mal17,Proof of Proposition 5.5], the classes of p -elements t with Lusztig series contributingto B ( ¯ H ) are parametrised exactly as in the case of SO n +1 ( q ) above.Assume that ( n, ǫ ) = (4 , χ ( t ) of H with t ∈ H ∗ p -elements, we may conclude that the number of orbits of non-unipotent characters in B ( S ) under the Aut( S ) is at least p a − e . This yields at least k (2 e, w ) + p a − e orbits in Irr( B ( S )) under Aut( S ). Hence we have the number ofAut( S ) orbits in B ( S ) is strictly larger than 4 e + p a − e , completing Theorem 1.2again in this case using (6.1).Finally, suppose S = D ( q ) = PΩ +8 ( q ) so ¯ H = GO +8 ( q ). In this case, the graphautomorphisms generate a group of size 6, and a triality graph automorphism of N H´ETHELYI-K ¨ULSHAMMER’S CONJECTURE FOR PRINCIPAL BLOCKS 15 order 3 permutes two triples of unipotent characters (see [Mal08, Theorem 2.5]).Since w ≥
2, we have ( e, w ) ∈ { (1 , , (2 , } . The arguments above give at least k (2 e, w ) − p a − e distinct Aut( S )-orbits in Irr( B ( S )). Since k (2 ,
4) = 20, by againapplying (6.1), we may assume e = 2 = w . In this case, we have k (2 e, w ) − p a − e =10 + p a − > p − / , so Theorem 1.2(ii) is proved in this case.Now, let S ≤ A ≤ Aut( S ), let Γ be the subgroup of Aut( S ) generated by inner,diagonal, and graph automorphisms, and let X := (Γ ∩ A ) C A ( P ). Then A/X iscyclic, and by [Nav98, (9.19) and (9.20)], B ( A ) is the unique block covering B ( X ).Let b := | A/X | . Now, the arguments above give at least · · (cid:0) p − (cid:1) members of B ( X ) lying above semisimple characters of S , since members of C A ( P ) correspondto automorphisms stabilizing classes of p -elements of G ∗ , and hence there are at least ( p − b members of B ( A ) lying above semisimple characters of S . Further, there areat least 10 characters in B ( X ) lying above unipotent characters in B ( S ). Sinceunipotent characters extend to their inertia groups and are invariant under fieldautomorphisms (see [Mal08, Theorems 2.4 and 2.5]), this gives at least 10 b elementsof B ( A ) lying above unipotent characters of S . Together, this gives k ( B ( A )) ≥ b + ( p − b > √ p −
1, proving part (i) of Theorem 1.2. (cid:3) A general bound for the number of
Aut( S ) -orbits on Irr( B ( S ))The aim of this section is to obtain a general bound for the number of Aut( S )-orbits on irreducible ordinary characters in the principal block of S , for S a simplegroup of Lie type.Building off of Theorem 5.1, we will show that the principal block of S contains many irreducible semisimple characters. By controlling the length of Aut( S )-orbitson these characters, we are able to bound below the number of Aut( S )-orbits onIrr( B ( S )). The bound turns out to be enough to prove Theorem 1.2 for groups ofexceptional types, at least in the case when the Sylow p -subgroups of the group ofinner and diagonal automorphisms of S are abelian but non-cyclic, which is preciselythe case we need after Sections 2.1 and 4.8.1. Specific setup for our purpose.
From now on we will work with the followingsetup: G is a simple algebraic group of adjoint type defined over F q and F a Frobeniusendomorphism on G such that S = [ G , G ] with G = G F . Let ( G ∗ , F ∗ ) be the dualpair of ( G , F ) and for simplicity we will use the same notation F for F ∗ , and thus G ∗ is a simple algebraic group of simply connected type and S = G ∗ / Z ( G ∗ ), where G ∗ := ( G ∗ ) F .Theorem 5.1 has the following consequence. Lemma 8.1.
Assume the above notation. Let p be a good prime for G not dividing q . For every p -element t of G ∗ , the semisimple character χ ( t ) ∈ E ( G , ( t )) belongs tothe principal block of G . Proof.
Since G ∗ is of simply connected type, the centralizer C G ∗ ( t ) is connected forevery semisimple element t ∈ G ∗ , by [MT11, Exercise 20.16]. The lemma followsfrom Theorem 5.1. (cid:3) Orbits of semisimple characters.
Knowing that the semisimple characters χ ( t ) ∈ Irr( G ) associated to G ∗ -conjugacy classes of p -elements all belong to B ( G ),we now wish to control the number of orbits of the action of the automorphism groupAut( S ) on these characters. By a result of Bonnaf´e [NTT08, § α ∈ Aut( G ), which in our situation will be a product of a field automorphismand a graph automorphism. It is easy to see that α then can be extended to a bijectivemorphism α : G → G such that α commutes with F . This α induces a bijectivemorphism α ∗ : G ∗ → G ∗ which commutes with the dual of F . The restriction of α ∗ to G ∗ , which we denote by α ∗ , is now an automorphism of G ∗ . Recall that α ∈ Aut( G )induces a natural action on Irr( G ) by χ α := χ ◦ α − . By [NTT08, § α maps theLusztig series E ( G , ( t )) of G associated to ( t ) to the series E ( G , ( α ∗ ( t ))) associatedto ( α ∗ ( t )). Consequently, if C G ∗ ( t ) is connected, then C G ∗ ( α ∗ ( t )) is also connected,and(8.1) χ ( t ) α = χ ( α ∗ ( t )) , which means that an automorphism of G maps a semisimple character associated toa conjugacy class ( t ) (of G ∗ ) to the semisimple character associated to ( α ∗ ( t )).Due to Section 4 and Subsection 2.1, we may assume that the Sylow p -subgroupsof G are abelian but not cyclic. Therefore, G is not of type B or G . Assume fora moment that G is not of type F as well. Then there is a unique positive integer e such that p | Φ e ( q ) and Φ e divides the generic order of G . (Recall that Φ e denotesthe e th cyclotomic polynomial.) This e then must be the multiplicative order of q modulo p , which means that p | ( q e −
1) but p ∤ ( q i −
1) for every 0 < i < e . In thecase G is of type F , we use Φ ± ( q ) := q ± √ q + 1, and what we discuss below stillholds with slight modification.Let Φ e ( q ) = p a m where gcd( p, m ) = 1 and Φ k e e the precise power of Φ e dividingthe generic order of G . We will use k for k e for convenience if e is not specified. ASylow e -torus of G ∗ has order Φ e ( q ) k and contains a Sylow p -subgroup of G ∗ . Sylow p -subgroups of G ∗ (and G ) are then isomorphic to C p a × C p a × · · · × C p a | {z } k times . Assume that q = ℓ f where ℓ is the defining characteristic of S . Lemma 8.2.
Assume the above notation. Let α be a field automorphism of G oforder f , and thus h α i is the group of field automorphisms of G . Each α -orbit on N H´ETHELYI-K ¨ULSHAMMER’S CONJECTURE FOR PRINCIPAL BLOCKS 17 semisimple characters χ ( t ) ∈ Irr( G ) associated to conjugacy classes of p -elements( p = ℓ ) has length at most min { f, p a − p a − } .Proof. Let α ∗ be an automorphism of G ∗ constructed from α by the process describedabove. For simplicity we use α for α ∗ . By (8.1) and since | α | = f , it is enough toshow that each α -orbit on G ∗ -conjugacy classes of (semisimple) p -elements of G ∗ isat most p a − p a − .Let t ∈ G ∗ be a p -element. Note that each element in G ∗ conjugate to t under G ∗ is automatically conjugate to t under G ∗ , by [DM91, (3.25)] and the fact that C G ∗ ( t ) is connected. Let t be conjugate to h α ( λ ) · · · h α n ( λ n ), where the h α i arethe coroots corresponding to a set of fundamental roots with respect to a maximaltorus T ∗ of G ∗ and n is the rank of G ∗ . Since G ∗ is simply connected, note that( t , . . . , t n ) h α ( t ) · · · h α n ( t n ) is an isomorphism from ( F × q ) n to T ∗ (see [GLS94,Theorem 1.12.5]).Now, if λ = λ i for some 1 ≤ i ≤ n , then λ p a = 1, since | t | | p a . Recall that ℓ = p ,and thus ℓ p a − p a − ≡
1( mod p a ) by Euler’s totient theorem. It follows that λ ℓ pa − pa − = λ , which yields that the α -orbit on ( t ) is contained in { ( t ) , ( α ( t )) , ..., ( α p a − p a − − ( t )) } ,as desired. (cid:3) A bound for the number of
Aut( S ) -orbits on Irr( B ( S )) . Let T e be an F -stable maximal torus containing a Sylow e -torus of G ∗ , and thus T e contains aSylow p -subgroup P of G ∗ . Let W ( T e ) := N G ∗ ( T e ) / T Fe be the relative Weyl groupof T e . It is well-known that fusion of semisimple elements in a maximal torus iscontrolled by its relative Weyl group (see [MT11, Exercise 20.12] or [MM16, p. 6]).Therefore, the number of conjugacy classes of (nontrivial) p -elements of G ∗ is at least | P | − | W ( T e ) | = p ak − | W ( T e ) | . Note that χ ( t ) belongs to the Lusztig series E ( G , ( t )) defined by the conjugacy class( t ) and the Lusztig series are disjoint, and so two semisimple characters χ ( t ) and χ ( t ) are equal if and only if t and t are conjugate in G ∗ . Therefore, using Lemma 8.1, wededuce that, when p is a good prime for G and not dividing q ,(8.2) | Irr ss ( B ( G )) | ≥ p ak − | W ( T e ) | , where Irr ss ( B ( G )) denotes the set of (nontrivial) semisimple characters (associatedto p -elements of G ∗ ) in B ( G ). Let n ( X, Y ) denote the number of X -orbits on a set Y . Using Lemma 8.2, we then have n (Aut( S ) , Irr ss ( B ( G ))) ≥ p ak − g min { f, p a − p a − }| W ( T e ) |≥ p k − g ( p − | W ( T e ) | , where g is the order of the group of graph automorphisms of S . Let d := | G /S | –the order of the group of diagonal automorphisms of S and viewing the irreducibleconstituents of the restrictions of semisimple characters of G to S as semisimplecharacters of S , we now have(8.3) n (Aut( S ) , Irr ss ( B ( S ))) ≥ p k − dg ( p − | W ( T e ) | . We note that values of d, f , and g for various families of simple groups are known,see [Atl85, p. xvi] for instance.We now turn to unipotent characters in the principal block B ( S ). Brou´e, Malle,and Michel [BMM93, Theorem 3.2] partitioned the set E ( G ∗ ,
1) of unipotent char-acters of G ∗ into e -Harish-Chandra series associated to e -cuspidal pairs of G ∗ , andfurthermore obtained one-to-one correspondences between e -Harish-Chandra seriesand the irreducible characters of the relative Weyl groups of the e -cuspidal pairsdefining these series. Cabanes and Enguehard [CE04, Theorem 21.7] then proved thecompatibility between Brou´e-Malle-Michel’s partition of unipotent characters of G ∗ by e -Harish-Chandra series and the partition of unipotent characters by unipotentblocks. These results imply that the number of unipotent characters in B ( S ) (and B ( G ∗ ) as well) is the same as the number of conjugacy classes of the relative Weylgroup W ( L e ) of the centralizer L e := C G ∗ ( S e ) of a Sylow e -torus S e of G ∗ . Here wenote that L e is a minimal e -split Levi subgroup of G ∗ , and L e = S e if S e happens tobe a maximal torus of G ∗ , since every maximal torus is equal to its centralizer in aconnected reductive group.By the aforementioned result of Lusztig ([Mal08, Theorem 2.5]), every unipotentcharacter of a simple group of Lie type lies in a Aut( S )-orbit of length at most 3.In fact, every unipotent character of S is Aut( S )-invariant, except in the followingcases:(1) S = P Ω +2 n ( q ) ( n even), the graph automorphism of order 2 has o ( S ) orbitsof length 2, where o ( S ) is the number of degenerate symbols of defect 0 andrank n parameterizing unipotent characters of S (see [Car85, p. 471]).(2) S = P Ω +8 ( q ), the graph automorphism of order 3 has o ( S ) = 2 orbits oflength 3, each of which contains one pair of characters parameterized by onedegenerate symbol of defect 0 and rank 2 in (1).(3) S = Sp (2 f ) ( f odd), the graph automorphism of order 2 has o ( S ) = 1 orbitof length 2. N H´ETHELYI-K ¨ULSHAMMER’S CONJECTURE FOR PRINCIPAL BLOCKS 19 (4) S = G (3 f ) ( f odd), the graph automorphism of order 2 has o ( S ) = 1 orbitof length 2 on unipotent characters.(5) S = F (2 f ) ( f odd), the graph automorphism of order 2 has o ( S ) = 8 orbitsof length 2 on unipotent characters.Combining this with the bound (8.3), we obtain: Theorem 8.3.
Let S be a simple group of Lie type. Let p be a good prime for S anddifferent from the defining characteristic of S . Assume that Sylow p -subgroups of thegroup of inner and diagonal automorphisms of S are abelian. Let k, d, f, g , T e , and L e as above. Let n ( S ) denote the number of Aut( S ) -orbits on irreducible ordinarycharacters in B ( S ) . Then n ( S ) ≥ k ( W ( L e )) + p k − dg ( p − | W ( T e ) | , except possibly the above cases (1) , (3) , (4) , and (5) in which the bound is lower by thenumber o ( S ) of orbits of length on unipotent characters and case (2) in which thebound is lower by . We remark that when e is regular for G ∗ , which means that the centralizer C G ∗ ( S e )of the Sylow e -torus S e is a maximal torus of G ∗ , the maximal torus T e containing S e can be chosen to be precisely L e = C G ∗ ( S e ). When e is not regular, W ( T e ) is alwaysbigger than W ( L e ). However, for exceptional types, e being non-regular happensonly when G is of type E and e ∈ { , , , , } , see [BMM93, Table 3]. We thankG. Malle for pointing out these facts to us.We also remark that when the Sylow p -subgroups of the group of inner and diagonalautomorphisms of S are furthermore noncyclic, then k ≥
2, and, away from thoseexceptions, we have a rougher bound(8.4) n ( S ) ≥ k ( W ( L e )) + p + 1 dg | W ( T e ) | , but turns out to be sufficient for our purpose in most cases.9. Groups of exceptional types
In this section we prove Theorem 1.2 for S being of exceptional type. This isachieved by considering each type case by case, with the help of Theorem 8.3.We keep all the notation in Section 8. In particular, the underlying field of S hasorder q = ℓ f . By Section 2.1, we may assume that ℓ = p ≥
11. This assumptionon p guarantees that Sylow p -subgroups of G are abelian. Recall also that e isthe multiplicative order of q modulo p , p a = Φ e ( q ) p , and Φ ke = Φ k e e is the precisepower of Φ e dividing the generic order of G . By Section 2.1, we may assume thatthe Sylow p -subgroups of S are not cyclic, and thus k e ≥
2. Also, S e is a Sylow e -torus of a simple algebraic group G ∗ of simply connected type associated with a Frobenius endomorphism F such that S = G ∗ / Z ( G ∗ ) and G ∗ := G ∗ F , and T e isan F -stable maximal torus of G ∗ containing S e . As mentioned already, we choose T e = L e := C G ∗ ( S e ) if e is regular for G ∗ . This indeed is the case for all types andall e , except the single case of type E and e = 4. The relative Weyl groups W ( L e )are always finite complex reflection groups, and we will follow the notation for thesegroups in [Ben76]. Relative Weyl groups for various L e are available in [BMM93,Tables 1 and 3]. The structure of Out( S ) is available in [GLS94, Theorem 2.5.12].We will use these data freely without further notice.It turns out that Theorem 8.3 is sufficient to prove Theorem 1.2 whenever k e ≥ k e = 2, Theorem 8.3 is also sufficient for Theorem 1.2(ii). Wehave to work harder, though, to achieve Theorem 1.2(i) in the case k e = 2 for sometypes. Proposition 9.1.
Theorem 1.2 holds for simple groups of exceptional types.Proof. (1) S = G ( q ) and S = F ( q ) :First we consider S = G ( q ) (so S = G ) with q >
2. Then e ∈ { , } and k = k = 2. Also, the Sylow e -tori are maximal tori, and their relative Weyl groupsare the dihedral group D . The bound (8.4) implies that n ( S ) ≥ p + 1) /
12 for q = 3 f with odd f , and n ( S ) ≥ p + 1) /
12 otherwise. In any case it follows that n ( S ) > p − / , proving Theorem 1.2(ii) for G ( q ).Note that Aut( S ) is a cyclic extension of S . First assume that q = 3 f with odd f or G does not contain the graph automorphism of S . In particular, every unipo-tent character of S is extendible to G . Let H := h S, C G ( P ) i , where P is a Sylow p -subgroup of G (and S as well by the assumption p ∤ | G/S | ). Since P C G ( P ) iscontained in H , B ( H ) is covered by a unique block of G , which is B ( G ). It followsthat, each unipotent character in B ( S ) extends to an irreducible character in B ( H ),which in turns lies under | G/H | irreducible characters in B ( G ). Therefore, the num-ber of irreducible characters in B ( G ) lying over unipotent characters of S is at least k ( D ) | G/H | = 6 | G/H | . When q = 3 f with odd f and G does contain the non-trivial graph automorphism, similar arguments yield that the number of irreduciblecharacters in B ( G ) lying over unipotent characters of S is at least 5 | G/H | .On the other hand, each G -orbit on semisimple characters (associated to p -elements)of S now has length at most | G/H | by (8.1) and the fact that H = h S, C G ( P ) i fixesevery conjugacy class of p -elements of S . Therefore, the bound (8.2) yields n ( G, Irr ss ( B ( S ))) ≥ p − | G/H | . This and the conclusion of the last paragraph imply that k ( B ( G )) ≥ | G/H | + p − | G/H | ≥ r p − , N H´ETHELYI-K ¨ULSHAMMER’S CONJECTURE FOR PRINCIPAL BLOCKS 21 which in turns implies the desired bound k ( B ( G )) > √ p − p ≥ S = F ( q ), we have e ∈ { , } for which k e = 4, or e ∈ { , , } for which k e = 2. Therefore all the Sylow e -tori are maximal tori, and their relative Weylgroups are G = GO +4 (3) for e = 1 , G = SL (3) × C for e = 3 ,
6; and G = C . S for e = 4. Now we just follow along similar arguments as above to prove the theoremfor this type. (2) S = F ( q ) with q = 2 n +1 ≥ and S = D ( q ) : These two types are treated in a fairly similar way as for G . Note that Out( S )here is always cyclic. First let S = F ( q ). Then e ∈ { , , + , − } and k e = 2for all e . All the Sylow e -tori are maximal. The relative Weyl groups of these toriare D , G = GL (3), G = C . S and G for e = 1 , , + , and 4 − , respectively.One can now easily check the inequality n ( S ) ≥ p − / , using (8.4). The bound k ( B ( G )) > √ p − G .Now let S = D ( q ). Then e ∈ { , , , } and k e = 2 for all e . For e ∈ { , } , aSylow e -torus is maximal with the relative Weyl group G = SL (3). For e = 1 or2, Sylow e -tori of S are not maximal anymore but are contained in maximal tori oforders Φ ( q )Φ ( q ) and Φ ( q )Φ ( q ), respectively. The relative Weyl groups of thesetori are both isomorphic to D . Now the routine estimates are applied to achievethe required bounds. (3) S = E ( q ) and S = E ( q ) : These two types are approached similarly and we will provide details only for E .Then e = 1 for which k e = 6, or e = 2 for which k e = 4, or e = 3 for which k e = 3,or e ∈ { , } for which k e = 2.Assume e = 1. Then S is a maximal torus and its Weyl group is G = SO (3).Theorem 8.3 then implies that n ( S ) ≥ k (SO (3)) + p − p − | SO (3) | = 25 + p − p − > p p − , proving both parts of Theorem 1.2 in this case. The case e ∈ { , } is similar. Wenote that S is a maximal torus with the relative Weyl group G = 3 . SL (3), anda maximal torus containing a Sylow 2-torus has relative Weyl group G .Assume e = 4. Then a maximal torus containing a Sylow 4-torus of E ( q ) sc hasorder Φ ( q )Φ ( q ) and its relative Weyl group is G = C . S , whose order is 96 andclass number is 16. Now the bound (8.4) yields n ( S ) > p − / , proving part (ii)of the theorem.We need to to do more to obtain part (i) in this case. In fact, when 2 √ p − ≤ p ≤
65, we have n ( S ) > ≥ √ p −
1, which proves part (i) aswell. So let us assume that p > S ) is a semidirect product C (3 ,q − ⋊ ( C f × C ), which may notbe abelian but every unipotent character of S is still fully extendible to Aut( S ) by [Mal08, Theorems 2.4 and 2.5]. As before, let G be the extension of S by diagonalautomorphisms. Similar to the proof for type G , let H := h G ∩ G , C G ( P ) i , where P is a Sylow p -subgroup of S . Each unipotent character in B ( S ) then lies under atleast | Irr(
G/H ) | = | G/H | irreducible characters in B ( G ). (Here we note that G/H is abelian.) Thus, the number of irreducible characters in B ( G ) lying over unipotentcharacters of S is at least 16 | G/H | .As in Subsection 8.3, here we have | Irr ss ( B ( G )) | ≥ p − | W ( T ) | = p − . Let Irr ss ( B ( S )) be the set of restrictions of characters in Irr ss ( B ( G )) to S . Notethat these restrictions are irreducible as the sesisimple elements of G ∗ associated tothese sesisimple characters are p -elements whose orders are coprime to | Z ( G ∗ ) | =gcd(3 , q − χ ( t ) and χ ( t ) to S are the same,then ( t ) = ( t z ) for some z ∈ Z ( G ∗ ) (see [Tie15, Proposition 5.1]), which happensonly when z is trivial since t and t are p -elements. It follows that | Irr ss ( B ( S )) | = | Irr ss ( B ( G )) | ≥ p − . Now, each H -orbit of conjugacy classes of p -elements of S has length at most gcd(3 , q − ≤
3. Therefore, each G -orbit of semisimple characters in B ( S ) has length at most3 | G/H | , and it follows that the number of irreducible characters in B ( G ) lyingover sesisimple characters in B ( S ) is at least ( p − / (288 | G/H | ). Together withthe bound of 16 | G/H | for the number of irreducible characters in B ( G ) lying overunipotent characters of S , we deduce that k ( B ( G )) ≥ | G/H | + p − | G/H | ≥ r p − , and thus, when p >
65, the desired bound k ( B ( G )) > √ p − e = 6 can be argued in a similar way, with notice that a maximaltorus containing a Sylow 6-torus of E ( q ) sc has order Φ ( q )Φ ( q ) and its relative Weylgroup is G = SL (3) × C , whose order is 72 and class number is 21. (4) S = E ( q ) : Then e ∈ { , } for which k e = 7, or e ∈ { , } for which k e = 3, or e = 4 forwhich k e = 2. When k e >
2, the bound 8.4 again is sufficient to achieve the desiredbound n ( S ) > √ p −
1. In fact, even for the case k e = 2, we have n ( S ) ≥ p − / .So it remains to prove Theorem 1.2(i) for e = 4, in which case e is not regular andthe relative Weyl group of the minimal e -split Levi subgroup L e = S e .A is G (see[BMM93, Table 1]). As A has eight classes of maximal tori (two in each factor A ),all of which have Weyl group C , the relative Weyl group of every maximal torus T e N H´ETHELYI-K ¨ULSHAMMER’S CONJECTURE FOR PRINCIPAL BLOCKS 23 containing S e is of the form G .C . The estimates are now similar to those in thecase e = 4 of the type E . (5) S = E ( q ) : Then e ∈ { , } for which k e = 8, or e ∈ { , , } for which k e = 4, or e ∈{ , , , } for which k e = 2. The standard approach as above works for all e with k e > e ∈ { , } . Then a Sylow e -torus of S is maximal and its relativeWeyl group is G ∼ = SL (5) × C . A similar proof to the case of type G yields k ( B ( G )) ≥ p p − / √ p − p ≥ k ( B ( G )) ≥ > √ p − p , and thusthe desired bound holds for all p . Finally, the case e ∈ { , } is entirely similar, withnotice that the relative Weyl groups of Sylow e -tori are G = C . S and G = C . S for e = 8 and 12, respectively. (cid:3) Theorem 1.2 is now completely proved.10.
Proof of Theorems 1.1 and 1.3
We are now ready to prove the main results.
Proof of Theorems 1.1 and 1.3.
First we remark that the ‘if’ implication of Theo-rem 1.3 is clear, and moreover, we are done if the Sylow p -subgroups of G are cyclic,thanks to Subsection 2.1.Let ( G, p ) be a counterexample to either Theorem 1.1 or the ‘only if’ implicationof Theorem 1.3 with | G | minimal. Let N be a minimal normal subgroup of G . Inparticular, Sylow p -subgroups of G are not cyclic and k ( B ( G )) ≤ √ p − p | | G/N | . Then, since Irr( B ( G/N )) ⊆ Irr( B ( G )) and by theminimality of | G | , we have2 p p − ≥ k ( B ( G )) ≥ k ( B ( G/N )) ≥ p p − , and thus k ( B ( G )) = k ( B ( G/N )) = 2 p p − . The minimality of G again then implies that G/N is isomorphic to the Frobeniusgroup C p ⋊ C √ p − . It follows that p | | N | , and thus there exists a nontrivial irreduciblecharacter θ ∈ Irr( B ( N )). As B ( G ) covers B ( N ), there is some χ ∈ Irr( B ( G )) lyingover θ , implying that k ( B ( G )) > k ( B ( G/N )), a contradiction.So we must have p ∤ | G/N | , and it follows that p | | N | . This in fact also yieldsthat N is a unique minimal normal subgroup of G . Assume first that N is abelian.We then have that G is p -solvable, and hence Fong’s theorem (see [Nav98, Theorem10.20]) implies that k ( B ( G )) = k ( B ( G/ O p ′ ( G ))) = k ( G/ O p ′ ( G )) , which is greater than 2 √ p − N ∼ = S × S × · · · × S k , a direct product of k ∈ N copiesof a non-abelian simple group S . If S has cyclic Sylow p -subgroups, then G is not acounterexample for Theorem 1.1 by Lemma 2.1, and furthermore, Sylow p -subgroupsof G are abelian, implying that k ( B ( G )) ≥ k ( N G ( P ) / O p ′ ( N G ( P ))) > p p − P ∈ Syl p ( G ) isnot cyclic.So the Sylow p -subgroups of S are not cyclic. Let n be the number of N G ( S ) /N -orbits on Irr( B ( S )). By Theorem 1.2(ii), we have n ≥ p − / . Therefore, if k ≥
2, the number of G -orbits on Irr( B ( N )) = Q ki =1 Irr( B ( S i )) is at least n ( n + 1) / ≥ p − / (2( p − / + 1) / > √ p −
1, and it follows that k ( B ( G )) > √ p − N = S and G is then an almost simple group with socle S . Furthermore, p ∤ | G/S | . But such a group G cannot be a counterexample byTheorem 1.2(i). The proof is complete. (cid:3) With Theorem 1.1 in mind, it follows that for any p -block B for a finite group suchthat B shares its invariants with the the principal block of some finite group H (oreven just satisfying k ( B ) = k ( B ( H ))), we have k ( B ) ≥ √ p −
1. In particular, wemay record the following:
Corollary 10.1.
Let G be one of the classical groups GL n ( q ) , GU n ( q ) , Sp n ( q ) , SO n +1 ( q ) , or GO ± n ( q ) . Let p be a prime dividing | G | and not dividing q . Then forany p -block B of G with positive defect, we have k ( B ) ≥ √ p − .Proof. If p = 2, then the statement is clear, so we assume p is odd. First, if G =GL n ( q ) or GU n ( q ), the statement follows immediately from Theorem 1.1 and [MO83,Theorem (1.9)], which states that B has the same block invariants as the principalblock of a product of lower-rank general linear and unitary groups.Now suppose that G is Sp n ( q ), SO n +1 ( q ), or GO ± n ( q ). If B is a unipotent block,then by [Mal17, Proposition 5.4 and 5.5], B has the same block invariants as anappropriate general linear group. (In the case GO ± n ( q ), we define a unipotent blockto be one lying above a unipotent block of SO ± n ( q ).) Hence the statement holds if B is a unipotent block.Now, the block B determines a class of semisimple p ′ -elements ( s ) of the dual group G ∗ (see [CE04, Theorem 9.12]) such that B contains some member of E ( G, ( s )). By[E08, Th´eor`eme 1.6], there exists a group G ( s ) dual to C G ∗ ( s ) such that k ( B ) = k ( b )for some unipotent block b of G ( s ). Now, in the cases under consideration, C G ∗ ( s ) and G ( s ) are direct products of lower-rank classical groups of the types being consideredhere, completing the proof. (cid:3) N H´ETHELYI-K ¨ULSHAMMER’S CONJECTURE FOR PRINCIPAL BLOCKS 25
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Department of Mathematics, The University of Akron, Akron, OH 44325, USA
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