Groups with few p ′ -character degrees in the principal block
Eugenio Giannelli, Noelia Rizo, Benjamin Sambale, A. A. Schaeffer Fry
aa r X i v : . [ m a t h . R T ] A p r Groups with few p ′ -character degreesin the principal block Eugenio Giannelli ∗ , Noelia Rizo † ,Benjamin Sambale ‡ and A. A. Schaeffer Fry § April 23, 2020
Abstract
Let p ≥ be a prime and let G be a finite group. We prove that G is p -solvable of p -length at most ifthere are at most two distinct p ′ -character degrees in the principal p -block of G . This generalizes a theoremof Isaacs–Smith as well as a recent result of three of the present authors. Keywords: p ′ -character degrees; principal block AMS classification:
Let G be a finite group. If all non-linear irreducible characters of G have degree divisible by a prime p , then G has a normal p -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 thissituation. This result was extended in Kazarin–Berkovich [KB99] to the case where G has at most one non-linear character of p ′ -degree. In a recent paper [GRS], three of the present authors proved more generally that G is solvable of p -length at most whenever p ≥ and |{ χ (1) : χ ∈ Irr p ′ ( G ) }| ≤ where Irr p ′ ( G ) is the setof irreducible characters of G of p ′ -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 that G has a normal p -complement if and only if all non-linear characters in the principal p -block of G have degree divisible by p . The following is our main theorem. Theorem A.
Let B be the principal block of a finite group G with respect to a prime p ≥ . Suppose that |{ χ (1) : χ ∈ Irr p ′ ( B ) }| ≤ . Then G/ O p ′ ( G ) is solvable and O pp ′ pp ′ ( G ) = 1 . In particular, G is p -solvable. As usual we define O pp ′ ( G ) := O p ′ (O p ( G )) and so on. It is easy to construct groups of p -length satisfying thehypothesis of Theorem A (e. g. G = ( C ⋊ C ) ⋊ C with p = 5 ). In contrast to the main theorem of [GRS]we cannot conclude further that G is solvable since every p ′ -group satisfies the assumption of Theorem A.Furthermore, the examples given in [GRS] show that Theorem A does not extend to p ∈ { , } . We also like tomention a conjecture by Malle and Navarro [MN11], which generalizes the result of Isaacs and Smith to arbitraryblocks. More precisely, they conjectured that a p -block B of G is nilpotent if and only if all height charactersin B have the same degree. We do not know if our main result admits a version for arbitrary blocks. ∗ 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,[email protected] § Department of Mathematical and Computer Sciences, Metropolitan State University of Denver, Denver, CO 80217, USA,[email protected] p ′ -characterdegrees in any block of maximal defect. This is Proposition 3.5 below, which we believe is of independentinterest. 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 ≥ .(i) If S = P Ω +8 ( q ) , then there exist α, β ∈ Irr( S ) with the following properties: • α = 1 = β , • α (1) and β (1) are not divisible by p , • for every S ≤ T ≤ Aut( S ) , α extends to a character in the principal block of T , • β lies in the principal block of S and is P -invariant for some Sylow p -subgroup P of Aut( S ) , • β (1) ∤ α (1) .(ii) If S = P Ω +8 ( q ) , then there exist α, β ∈ Irr( S ) with the following properties: • α = 1 = β , • α (1) and β (1) are not divisible by p , • α (1) > β (1) , • for every S ≤ T ≤ Aut( S ) there exist ˆ α, ˆ β ∈ Irr(Aut( T )) in the principal block such that ˆ α S ∈ { α, α } and ˆ β S ∈ { β, β } . We make use of the following results.
Lemma 2.2 (Murai [Mur94, Lemma 4.3]) . Let N E G be finite groups with principal p -blocks B N and B G respectively. Suppose that ψ ∈ Irr p ′ ( B N ) is invariant under a Sylow p -subgroup of G . Then there exists acharacter χ ∈ Irr p ′ ( B G ) lying over ψ . Lemma 2.3.
Let χ, ψ ∈ Irr( B ) where B is the principal p -block of G . Suppose that p ∤ χ (1) and χψ ∈ Irr( G ) .Then χψ ∈ Irr( B ) .Proof. Clearly, ψ ∈ Irr( B ) . Hence by [Nav98, Corollary 3.25], we have [ χψ, = [ χ, ψ ] = 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. roof. Let p , G and B be as in Theorem A. Suppose first that G is p -solvable. Let N := O p ′ ( G ) . Then, by[Nav98, Theorem 10.20], Irr( B ) = Irr( G/N ) . It follows from [GRS, Theorem A] that G/N is solvable and O pp ′ pp ′ ( G/N ) = 1 . In particular, O pp ′ p ( G ) N/N is a p ′ -group. Since N is a p ′ -group, this implies O pp ′ pp ′ ( G ) = 1 .Thus, it suffices to show that G is p -solvable. Let N be a minimal normal subgroup of G . Since the principalblock of G/N lies in B , we may assume that G/N is p -solvable by induction on | G | . If N is a p -group or a p ′ -group, then we are done. Therefore, by way of contradiction, we assume that N = S × . . . × S t with isomorphic non-abelian simple groups S := S ∼ = . . . ∼ = S t of order divisible by p . Since N is the uniqueminimal normal subgroup, C G ( N ) = 1 . Moreover, G permutes S , . . . , S t transitively by conjugation. Case 1: S = P Ω +8 ( q ) .Let α, β ∈ Irr( S ) as in Proposition 2.1. We may regard α as a character of S C G ( S ) , since S C G ( S ) / C G ( S ) ∼ = S/ Z( S ) = S . As such it extends to a character ˆ α in the principal block of N G ( S ) , because N G ( S ) / C G ( S ) ≤ Aut( S ) . Let M := N G ( S ) ∩ . . . ∩ N G ( S t ) E G . Since the principal block of N G ( S ) covers the principal block B M of M , the restriction ˆ α M lies in B M . Now by [Nav18, Corollary 10.5], the tensor induction ψ := ˆ α ⊗ G isan irreducible character of G with p ′ -degree ψ (1) = α (1) t . Let x , . . . , x t ∈ G be representatives for the rightcosets of N G ( S ) in G such that S x i = S i . Then for g ∈ M we obtain ψ ( g ) = t Y i =1 ˆ α x i ( g ) from [Nav18, Lemma 10.4]. In particular, ψ N = α x × . . . × α x t ∈ Irr( N ) and therefore ψ M ∈ Irr( M ) as well.Since ˆ α M lies in B M , so does ˆ α x i M . Hence, by Lemma 2.3 also ψ M = ˆ α x M . . . ˆ α x t M lies in B M .Let Q be a Sylow p -subgroup of M . Then Q ∩ S i is a Sylow p -subgroup of S i . It follows that C G ( Q ) ⊆ C G ( Q ∩ S i ) ⊆ N G ( S i ) for i = 1 , . . . , t and therefore C G ( Q ) ⊆ M . Hence, the Brauer correspondent B GM is defined (see [Nav98,Theorem 4.14]) and equals B by Brauer’s third main theorem. Every block B of G covering B M has a defectgroup containing Q by [Nav98, Theorem 9.26]. Hence by [Nav98, Lemma 9.20], B is regular with respect to N and therefore B = B by [Nav98, Theorem 9.19]. Thus, B is the only block of G covering B M . This implies ψ ∈ Irr p ′ ( B ) . Since the trivial character in B has degree , d := ψ (1) = α (1) t is the unique non-trivial p ′ -character degree in B by hypothesis.Now we work with β . Let P be a Sylow p -subgroup of G such that β is invariant under N P ( S ) . Without loss ofgenerality, let { S , . . . , S r } be a P -orbit. Let y i ∈ P such that S y i = S i for i = 1 , . . . , r . Then β i := β y i lies inthe principal block of S i . By Lemma 2.3, β × . . . × β r lies in the principal block of N . Moreover, if β yi ∈ Irr( S j ) for some y ∈ P , then y i yy − j ∈ N P ( S ) . Since β is N P ( S ) -invariant, it follows that β yi = β y i yy − j y j = β y j = β j .This shows that { β , . . . , β r } is P -orbit and β × . . . × β r is P -invariant. If r < t , then we consider β r +1 := β x r +1 ∈ Irr( S r +1 ) . By Sylow’s theorem, we can assume after conjugation inside N G ( S r +1 ) that β r +1 is N P ( S r +1 ) -invariant. Now we can form the P -orbit of β r +1 to obtain another P -invariant character β r +1 × . . . × β s ∈ Irr( N ) in the principal block of N . We repeat this with every P -orbit and eventually get a P N -invariant character τ := β × . . . × β t ∈ 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 ∈ Irr p ′ ( B ) such that τ is a constituent of χ N . Since = 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) , acontradiction to the choice of α and β . Case 2: S = P Ω +8 ( q ) .Let α, β ∈ Irr( S ) and ˆ α, ˆ β ∈ Irr(N G ( S )) as in Proposition 2.1. Since the principal block of N G ( S ) covers B M , ˆ α M is the sum of at most two irreducible characters in B M . If α ∈ Irr( B M ) is one of those summands, then α x . . . α x t restricts to α x × . . . × α x t ∈ Irr( N ) . Hence, by Lemma 2.3, α x . . . α x t lies in B M . As in Case 1we see that (ˆ α ⊗ G ) M is a sum of irreducible characters in B M . Moreover, (ˆ α ⊗ G ) N = d ( α x × . . . × α x t ) where d ∈ { , t } . Since B is the only block of G covering B M , all irreducible constituents of ˆ α ⊗ G lie in B . We maychoose such a constituent χ of p ′ -degree. Then χ N = e ( α x × . . . × α x t ) for some integer e ≤ d ≤ t . Similarly,we choose a constituent ψ of ˆ β ⊗ G with p ′ -degree. Then by Proposition 2.1 we derive the contradiction α (1) t > t β (1) t ≥ ψ (1) = χ (1) ≥ α (1) t . Alternating groups
This section is devoted to proving Proposition 2.1 for the alternating groups S = A n where n ≥ . It iswell-known that Aut( S ) ∼ = S n is the symmetric group unless n = 6 .Given n ∈ N we let P ( n ) be the set of partitions of n . Let λ = ( λ , . . . , λ k ) ∈ P ( n ) . Adopting the notationof [Ols93, Chapter 1] we let ℓ ( λ ) = k denote the number of parts of λ , and Y ( λ ) be the Young diagram of λ .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 thenthe q -core C q ( λ ) of λ is the partition obtained from λ by successively removing all hooks of length q (usuallycalled q -hooks). We denote by H q ( λ ) the subset of nodes of Y ( λ ) having associated hook-length divisible by q .A partition γ is called a q -core if H q ( λ ) = ∅ .The set Irr( S n ) is naturally in bijection with P ( n ) . Given λ ∈ P ( n ) we let χ λ be the corresponding irreduciblecharacter of S n . 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 thesame p -block of S n if and only if C p ( λ ) = C p ( µ ) . If γ is a p -core partition then we denote by B ( S n , γ ) thecorresponding p -block of S n . We use the notation λ ⊢ p ′ n to say that χ λ has degree coprime to p .The following result follows from [Mac71] and it will be extremely useful for our purposes. Lemma 3.1.
Let p be a prime and let n be a natural number with p -adic expansion n = P kj =0 a j p j . Let λ be apartition of n . Then λ ⊢ p ′ n if and only if |H p k ( λ ) | = a k and C p k ( λ ) ⊢ p ′ n − a k p k . A straightforward consequence of Lemma 3.1 is that
Irr p ′ ( B ( S n , γ )) = ∅ if and only if | γ | < p .For λ ∈ P ( n ) , we denote by λ ′ its conjugate partition. From [JK81, 2.5.7] we know that ψ λ := ( χ λ ) A n isirreducible if and only if λ = λ ′ . In this case χ λ and χ λ ′ are all the extensions of ψ λ to S n . Let λ, µ be non-self-conjugate partitions of n . Then ψ λ and ψ µ lie in the same p -block of A n if and only if C p ( λ ) ∈ { C p ( µ ) , C p ( µ ) ′ } .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 same p -block. We denote by B ( n ; γ ) the p -block of A n labeled by γ .In order to show that Proposition 2.1 holds for alternating groups, we introduce the following conventions. Notation 1.
Let B be a p -block of A n . We let cd extp ′ ( B ) be the set of degrees of irreducible characters in B ofdegree coprime to p that extend to an irreducible character of S n . Moreover, when S is a subset of P ( n ) we let cd( S ) = { χ λ (1) | λ ∈ S } . Observe that if B is the principal p -block of A n and ψ λ lies in B and extends to S n , then one of the twoextensions of ψ λ lies in the principal p -block of S n . This is explained in [Ols90]. Even if in this article we aremainly interested in studying the principal block, in Proposition 3.5 below we are going to compute an explicitlower bound for | cd extp ′ ( B ( n, γ )) | , for any p -core γ such that | γ | < p .Given γ = ( γ , . . . , γ ℓ ) ⊢ n and natural numbers x and y , we denote by γ ⋆ ( x, y ) the partition of n + x + y defined by γ ⋆ ( x, y ) = ( γ + x, γ , . . . , γ ℓ , y ) . We start by proving a technical lemma that will be useful later in this section.
Lemma 3.2.
Let p be a prime, let m, n, w ∈ N be such that m < p and n = m + pw . Let γ ⊢ m and let a ∈ N be such that ⌊ w +12 ⌋ + 1 ≤ a ≤ w . Setting λ = γ ⋆ ( ap, ( w − a ) p ) and µ = γ ⋆ (( a − p, ( w − a + 1) p ) , we havethat χ λ (1) < χ µ (1) .Proof. For ν ⊢ n we let π ( ν ) := Q h ij ( ν ) be the product of the hook-lengths in ν . From the hook length formula[JK81, 2.3.21] it follows that χ ν (1) π ( ν ) = n ! . We let h i = h i ( γ ) and h j = h j ( γ ) for all i ∈ { , . . . , γ } and all j ∈ { , . . . , ℓ ( γ ) } . It follows that π ( λ ) = ( ap )! · (( w − a ) p )! · γ Y i =2 ( h i + ap ) · ℓ ( γ ) Y i =2 ( h i + ( w − a ) p ) · b γ · ( h ( γ ) + pw ) , b γ is the product of the hook lengths h ij ( γ ) for all i, j ≥ . Similarly π ( µ ) = (( a − p )! · (( w − a + 1) p )! · γ Y i =2 ( h i + ( a − p ) · ℓ ( γ ) Y i =2 ( h i + ( w − a + 1) p ) · b γ · ( h ( γ ) + pw ) . It follows that π ( λ ) /π ( µ ) = A · B · C , where A = p Y i =1 ( a − p + i ( w − a ) p + i , B = γ Y i =2 h i + aph i + ( a − p , and C = ℓ ( γ ) Y i =2 h i + ( w − a ) ph i + ( w − a + 1) p . We remark that we always regard empty products as equal to . We observe that B ≥ . Since a − ≥ w − a + 1 by hypothesis, it is clear that A > . Hence, if ℓ ( γ ) = 1 then C = 1 and clearly A · B · C > . Suppose that ℓ ( γ ) ≥ . Then observe that p > | γ | > h > h > · · · > h ℓ ( γ ) ≥ . Hence for all i ∈ { , . . . , ℓ ( γ ) } we have that ( a − p + h i ( w − a ) p + h i is one of the factors appearing in A . Moreover ( a − p + h i ( w − a ) p + h i · h i + ( w − a ) ph i + ( w − a + 1) p ≥ , since a − ≥ w − a + 1 . We conclude that A · B · C ≥ A · C > and therefore that χ λ (1) < χ µ (1) . Definition 3.3.
Let p be a prime and n = wp + m , for some m < p . Let γ be a p -core partition of m . We let H ( n ; γ ) be the subset of P ( n ) defined by H ( n ; γ ) = { λ ⊢ p ′ n | C p ( λ ) = γ, λ = γ ⋆ ( a, n − m − a ) } . We also set Ω( n ; γ ) = { λ ∈ H ( n ; γ ) | λ > ( λ ′ ) } . Lemma 3.4.
Let n = P ki =0 a i p i be the p -adic expansion of n , with a k = 0 . If γ ⊢ a , then | cd(Ω( n ; γ )) | = | Ω( n ; γ ) | ≥ ⌊ a k + 12 ⌋ · k − Y i =1 ( a i + 1) . Proof.
Let λ = γ ⋆ ( x, n − a − x ) , for some ≤ x ≤ n − a . Let x = P ti =0 b i p i be the p -adic expansion of x . Bydefinition of H ( n ; γ ) we have that λ ∈ H ( n ; γ ) if and only if λ ⊢ p ′ n and C p ( λ ) = γ . In turn, this is equivalent tohave that p divides x (and n − a − x ) so that C p ( λ ) = γ and by Lemma 3.1 to have that b = 0 and ≤ b i ≤ a i for all i ≥ . It follows that |H ( n ; γ ) | = Q ki =1 ( a i + 1) . Moreover, if b k ≥ ⌊ a k / ⌋ + 1 , then certainly λ > ( λ ′ ) and therefore λ ∈ Ω( n ; γ ) . It follows that | Ω( n ; γ ) | ≥ ⌊ a k + 12 ⌋ · k − Y i =1 ( a i + 1) . We conclude by observing that Lemma 3.2 implies that given λ, µ ∈ Ω( n ; γ ) we have that χ λ (1) = χ µ (1) andhence 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 ) A n is irreducible and lies in B ( n ; γ ) . Asexplained in Notation 1 above, cd extp ′ ( B ( n ; γ )) denotes the set of degrees of irreducible characters of B ( n ; γ ) ofdegree coprime to p that extend to B ( S n ; γ ) .In the following proposition we are able to establish a lower bound for the number of extendable p ′ -characterdegrees lying in any given p -block of A n . We believe this statement of independent interest from the topic ofthis article. Proposition 3.5.
Let n = P ki =0 a i p i be the p -adic expansion of n , with a k = 0 . Let γ ⊢ a , then | cd extp ′ ( B ( n ; γ )) | ≥ ⌊ a k + 12 ⌋ · k − Y i =1 ( a i + 1) . roof. By definition, for every partition λ ∈ Ω( n ; γ ) we have that ( χ λ ) A n is a p ′ -degree character that lies in B ( n ; γ ) and extends to χ λ in B ( S n ; γ ) . The statement now follows from Lemma 3.4. Proposition 3.6.
Let n ≥ be a natural number and p > be a prime. Then Proposition 2.1 holds for A n . Inparticular if n ≥ then | cd extp ′ ( B ( A n )) | ≥ .Proof. Direct verification proves that Proposition 2.1 holds for A and A . Suppose that n ≥ and that n = a + P ki =1 a i p n i is the p -adic expansion of n , with a i = 0 for all i ≥ and with n < n < · · · < 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 B ( A n ) are P -invariant. Thus we just need to show that | cd extp ′ ( B ( A n )) | ≥ . From Proposition 3.5, we deduce that | cd extp ′ ( B ( A n )) | ≥ , whenever k ≥ . Suppose that k ≤ . If a ≤ then Irr p ′ ( B ( A n )) = Irr p ′ ( A n ) and thestatement follows from [GRS, Proposition 3.5]. Hence we can assume that a ≥ and consider λ, µ ∈ P ( n ) tobe defined as follows. λ = ( a , n − a ) , and µ = ( a , , n − a − ) . It is clear that both ( χ λ ) A n and ( χ µ ) A n lie in the principal p -block of A n and extend to the principal p -block of S n , to χ λ and χ µ respectively. Moreover λ and µ label characters of degree coprime to p by Lemma 3.1. Usingthe hook-length formula we verify that χ ( n ) (1) < χ λ (1) < χ µ (1) . The proof is complete.
Proposition 4.1.
Proposition 2.1 holds for all sporadic simple groups S and the Tits group F (2) ′ .Proof. Recall that | Aut( S ) : S | ≤ . Hence, we may take a p ′ -character ˆ α in the principal block of Aut( S ) suchthat α := ˆ α S = 1 is irreducible. For β we can choose any non-trivial p ′ -character in the principal block of S .Now it can be checked with GAP [GAP18] that there are choices such that β (1) ∤ α (1) .Now we consider simple groups S of Lie type, by which we mean groups of the form G/ Z( G ) , where G = G F is the set of fixed points of a simple simply connected algebraic group under a Steinberg morphism F . In thecase where Z( G ) is trivial, we define e G = G , and otherwise we let G ֒ → e G be a regular embedding, as in [CE04,Section 15], so that Z( e G ) is connected, [ e G , e G ] = [ G , G ] , and G is normal in e G := e G F . We write e S for the group e G/ Z( e G ) , so Aut( S ) may be viewed as generated by e S and graph and field automorphisms.Recall that the set Irr( e G ) can be partitioned into so-called Lusztig series E ( e G, s ) , where s is a semisimple elementof the dual group e G ∗ , up to conjugacy. Each series E ( e G, s ) has a unique character of degree [ e G ∗ : C e G ∗ ( s )] q ′ ,where F q is the field over which G is defined, called a semisimple character. Further, the characters in the series E ( e G, are called unipotent characters, and for a prime p , any p -block containing a unipotent character is calleda unipotent block.When G is type A n − (that is, in the case of linear and unitary groups), we will use the notation P SL ǫn ( q ) todenote P SL n ( q ) for ǫ = 1 and P SU n ( q ) for ǫ = − , and similar for GL ǫn ( q ) and SL ǫn ( q ) . Similarly, A ǫn − ( q ) willdenote the untwisted case A n − ( q ) when ǫ = 1 and the twisted case A n − ( q ) when ǫ = − . We also remarkthat the group P Ω +2 n ( q ) corresponds to D n ( q ) and P Ω − n ( q ) corresponds to D n ( 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 over F q , where q is a power of p > not in thefollowing list: P SL ( q ) , P SL ǫ ( q ) , or P Sp ( q ) . Then there exist two non-trivial characters χ , χ ∈ Irr p ′ ( B ( S )) such that χ (1) = χ (1) and: • If S = P Ω +8 ( q ) , then for every S ≤ T ≤ Aut( S ) , each of χ and χ extend to a character in the principal p -block of T . • If S = P Ω +8 ( q ) , then χ (1) > χ (1) and for every S ≤ T ≤ Aut( S ) , for i = 1 , , there exist b χ i in theprincipal p -block of T such that b χ i | S ∈ { χ i , χ i } . roof. In the proof of [GRS, Proposition 4.3], it is shown that there exist two characters χ , χ ∈ Irr p ′ ( e S ) that restrict irreducibly to S , extend to characters of Aut( S ) , have different degrees, and are obtained fromcharacters of e G trivial on Z( e G ) . Now, since Irr p ′ ( e G ) = Irr p ′ ( B ( e G )) (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 e S ,and their restrictions are members of the principal block of S .Now, let S ≤ T ≤ Aut( S ) . Then for i = 1 , , χ i | T ∩ e S is in the principal block of T ∩ e S , since B ( e S ) covers aunique block of T ∩ e S . Note that by [Nav98, Theorem 9.4], there must be a character of B ( T ) lying above χ i | T ∩ e S . If S = P Ω +8 ( q ) , we have Aut( S ) / e S is abelian, and hence every character of T lying above χ i | T ∩ e S is anextension, completing the proof in this case.If S = P Ω +8 ( q ) , then Aut( S ) / e S is of the form S × C , where C is cyclic. Then the character b χ i in B ( T ) lyingabove χ i | T ∩ e S must be such that b χ i | S ∈ { χ i , χ i } , as desired. Switching the roles of the semisimple elements s and s constructed in [GRS, Proposition 4.3], we further see that the characters have been constructed to satisfy χ (1) > χ (1) , since the centralizers of s and s have types A × T and A × T with T and T appropriatetori, and | C G ∗ ( s ) | p ′ < | C G ∗ ( s ) | p ′ .The following handles the exceptional cases left by Proposition 4.2. Proposition 4.3.
Let S be one of P SL ( q ) , P SL ǫ ( q ) , or P Sp ( q ) , where q is a power of a prime p > . Thenthere exist two non-trivial characters χ , χ ∈ Irr p ′ ( B ( S )) such that χ (1) ∤ χ (1) ; χ is invariant under aSylow p -subgroup of Aut( S ) ; and for every S ≤ T ≤ Aut( S ) , χ extends to a character in the principal p -blockof T .Proof. In this case, characters χ and χ are constructed in the proof of [GRS, Lemma 4.4] that satisfy all ofthe needed properties, except possibly the property that for every S ≤ T ≤ Aut( S ) , χ extends to a characterin the principal block of T . However, χ is again constructed from a character of e G trivial on Z( e G ) that restrictsirreducibly to G . Hence since again Aut( S ) / e S is abelian, the proof is complete arguing as in the second paragraphof Proposition 4.2.For the remainder of the section, we consider the case of non-defining characteristic. That is, we assume p > is a prime and S is a simple group of Lie type defined in characteristic different than p . Proposition 4.4.
Let p > be a prime and let S be a simple group of Lie type defined over F q , where q isa power of a prime different than p and S is not in the following list: P SL ( q ) , P SL ǫ ( q ) with p | ( q + ǫ ) , B (2 a +1 ) with p | (2 a +1 − , or G (3 a +1 ) with p | (3 a +1 − . Then there exist two non-trivial characters χ , χ ∈ Irr p ′ ( B ( S )) such that χ (1) = χ (1) and: • If S = P Ω +8 ( q ) , then for every S ≤ T ≤ Aut( S ) , each of χ and χ extend to a character in the principal p -block of T . • If S = P Ω +8 ( q ) , then χ (1) > χ (1) and for every S ≤ T ≤ Aut( S ) , for i = 1 , , there exist b χ i in theprincipal p -block of T such that b χ i | S ∈ { χ i , χ i } .Proof. We adapt our proof of [GRS, Proposition 4.5], ensuring that we may choose unipotent characters of p ′ -degree satisfying the principal block conditions required here. That is, we will exhibit unipotent characters of e G with different degree (and in the case of P Ω +8 ( q ) , satisfying χ (1) > χ (1) ) that are contained in Irr p ′ ( B ( e G )) ,which as unipotent characters must be trivial on Z( e G ) and restrict irreducibly to G . Then the restriction liesin B ( G ) , since B ( e G ) covers a unique block of G , and by [CE04, Lemma 17.2], the resulting characters of e S and S = G/ Z( G ) also lie in the principal blocks. By [Mal08, Theorems 2.4 and 2.5], every unipotent characterextends to its inertia group in Aut( S ) , and except for some specifically stated exceptions, the inertia group isall of Aut( S ) . Then arguing as in Proposition 4.2, the required properties will hold for each S ≤ T ≤ Aut( S ) .To see that the unipotent characters exhibited are indeed of p ′ -degree, it will often be useful to recall that q s − Q m | s Φ m and note that p | Φ m if and only if m = dp i for some non-negative integer i , where Φ m denotes the m -th cyclotomic polynomial in q and d is the order of q modulo p . Further, p divides Φ m only if m = d . (This is [Mal07, Lemma 5.2].) 7irst, we consider groups of exceptional type. If S is one of G (3 a +1 ) or B (2 a +1 ) but not one of the exceptionsof 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 maximaldefect. If S is F (2 a +1 ) , then by [Mal90, Bemerkung 1], there is again a unique unipotent block of maximaldefect unless p | (2 a +1 − , in which case the principal block contains the Steinberg character and two moreunipotent characters of p ′ -degree. Hence we are also done in this case. If S = D ( q ) , then there is either aunique unipotent block of maximal defect or the principal block contains the Steinberg character and one otherunipotent character of p ′ -degree, using [DM87, Propositions 5.6 and 5.8], so we are similarly finished in thiscase.Now let S be one of G ( q ) , F ( q ) , E ( q ) , E ( q ) , E ( q ) , or E ( q ) . Let d be the order of q modulo p . Using [E00,Theorem A], we have the unipotent blocks of e G are indexed by conjugacy classes of pairs ( L, λ ) for L a d -splitLevi subgroup and λ a d -cuspidal unipotent character. In particular, the characters in the d -Harish-Chandraseries indexed by such an ( L, λ ) all lie in the same block of e G . Further, [Mal07, Corollary 6.6] then yields that ifa unipotent character in the series indexed by ( L, λ ) has p ′ -degree, then L is the centralizer of a Sylow d -torus.Now, using this and [BMM93, Theorem 5.1], we see that either such an L is a maximal torus (yielding a uniqueblock containing unipotent characters of p ′ 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 unipotentcharacters in the principal block with different degrees relatively prime to p . (For an example of the argumentin the latter situation, consider E ( q ) in the case d = 7 . Then Line 58 of [BMM93, Table 2] shows that the trivialcharacter and the unipotent characters φ , and φ , in the notation of [Ca85, Section 13.9], which have degree q Φ Φ Φ Φ Φ and q Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ , respectively, lie in the same d -Harish-Chandraseries, and hence the same block. Since p | Φ and p = 2 , we see these two non-trivial character degrees are p ′ and distinct.)We are left to consider the classical groups, in which case the unipotent characters of e G are parametrized bycertain partitions or symbols. By a symbol of rank n , we mean a pair of partitions (cid:0) λ λ ··· λ a µ µ ··· µ b (cid:1) = (cid:0) λµ (cid:1) , where λ < λ < · · · < λ a , µ < µ < · · · < µ b , λ and µ are not both , and n = P i λ i + P j µ j − ⌊ (cid:0) a + b − (cid:1) ⌋ .(The symbol (cid:0) λµ (cid:1) is equivalent to (cid:0) µλ (cid:1) , and if λ and µ are both , the symbol is equivalent to (cid:0) λ − ··· λ a − µ − ··· µ b − (cid:1) .)The defect of a symbol is | b − a | . Given an integer e , an e -hook is a pair of non-negative integers ( x, y ) with y − x = e , x λ (resp. µ ), and y ∈ λ (resp. µ ). The e -core of a symbol is obtained by successively removing e -hooks, which means replacing y by x in λ (resp. µ ) and then replacing the result with an equivalent symbolsatisfying that λ and µ are not both . An e -cohook is defined similarly, except that x λ and y ∈ µ (or x µ and y ∈ λ ), and the e -cocore is obtained by removing e -cohooks, which means removing y from µ andadding x to λ (resp. removing y from λ and adding x to µ ), and again replacing the result with an equivalentsymbol satisfying that λ and µ are not both .Tables 1 through 4 describe two unipotent characters for each classical type satisfying the properties describedin the first paragraph and not in the list of exceptions of [Mal08, Theorem 2.5]. For each type, we include a briefdiscussion, but we remark that a more complete description of the degrees of such characters and the partitionsand symbols can be found in [Ca85, Section 13.8], and a more complete discussion of their distribution intoblocks may be found in [FS82, FS89]. We will include the details for type A n − in this respect, and note thatthe other types have similar arguments. Types A n − and A n − . Here e G = GL ǫn ( q ) . In this case, let e be the order of ǫq modulo p . The unipotentcharacters are in bijection with partitions of n , and two such characters are in the same block if and only ifthey have the same e -core. In particular, the trivial character is given by the partition ( n ) , which has e -core ( r ) , where ≤ r < e is the remainder when n := me + r is divided by e . Table 1 lists the desired unipotentcharacters in this case when n ≥ . Indeed, consider the case ǫ = 1 . The partitions listed have e -core ( r ) , andhence the corresponding characters are in the principal block and it suffices to show that they have p ′ -degree.Since p ∤ q , we need only consider the part of the degree relatively prime to q , which are listed following [Ca85,Section 13.8]. If e = 1 , then since p > , the character χ in the cases of line 1 or line 2 has p ′ -degree, since ( q d − / ( q − is divisible by p in this case if and only if d is divisible by p . Hence, for χ , we may assume e = 1 . Consider line 3 of Table 1 in this case. Since me + k is not divisible by e for ≤ k < e , we see ( q me + k − contains no factors of the form Φ ep i . Hence we see ( q me +1 − · · · ( q n − is not divisible by p . Similarly, if r + 1 = e , then ( q me − r − − is not divisible by p . If r + 1 = e , then ( q me − e − / ( q e − is divisible by p only if p | ( m − , so that ( q me − e − has factors of the form Φ ep i with i ≥ . Hence the character listed in8able 1: Some unipotent characters in Irr p ′ ( B ( S )) for type A ǫn − ( q ) with n ≥ and p ∤ q Additional condition on Partition χ (1) q ′ n = me + r , r < eχ e = 1 and p | ( n −
1) (2 , n − ( q n − ǫ n )( q n − − ǫ n − )( q − ǫ )( q − e = 1 and p ∤ ( n −
1) (1 , n − q n − − ǫ n − q − ǫ = e = r + 1 or p ∤ ( m −
1) ( r + 1 , me − ( q me +1 − ǫ me +1 )( q me +2 − ǫ me +2 ) ··· ( q n − ǫ n )( q me − r − − ǫ me − r − )( q − ǫ )( q − ··· ( q r +1 − ǫ r +1 ) = e = r + 1 and p | ( m −
1) (1 r +1 , me − Q ei =1 ( q n − i − ǫ n − i )( q i − ǫ i ) χ r < n ) r ≥ , e = r + 1 or m ≥ , and e = r + 2 or p ∤ ( m −
1) (1 , r + 1 , me − ( q me +1 − ǫ me +1 )( q me +2 − ǫ me +2 ) ··· ( q n − ǫ n )( q me − r − − ǫ me − r − )( q me − − ǫ me − )( q r +2 − ǫ r +2 )( q − ǫ )( q − ǫ )( q − ··· ( q r − ǫ r ) r ≥ , m = 1 , e = r + 1 (1 , e − , e − ( q e +2 − ǫ e +2 )( q e +3 − ǫ e +3 ) ··· ( q n − ǫ n )( q − ǫ )( q − ··· ( q e − − ǫ e − ) r ≥ , e = r + 2 , p | ( m −
1) (1 r +2 , me − Q ei =1 ( q n − i − ǫ n − i )( q i − ǫ i ) line 3 has p ′ -degree, given the stated conditions, and similar for lines 6 and 7. Line 5 refers to the Steinbergcharacter, which is certainly of p ′ -degree. So, consider the characters in lines 4 and 8, of degree Q ei =1 q n − i − q i − ,with p | ( m − . If p divides Q ei =1 q n − i − q i − , then p | ( q n − r − / ( q e −
1) = ( q me − / ( q e − , and hence p | m , acontradiction. The argument is similar in the case ǫ = − .Finally, if n = 3 and p ∤ ( q + ǫ ) , then note that e = 1 or , r < , and the characters listed in Table 1 stillsatisfy our conditions. (In this case, the two characters are the Steinberg character and the unipotent characterof degree q ( q + ǫ ) .) Types B n and C n . Here the unipotent characters of e G are in bijection with symbols of rank n and odd defect.In this case, we let e be the order of q modulo p . Then two symbols are in the same block if and only ifthey have the same e -core, respectively e -cocore, if p | q e − , respectively p | q e + 1 . The trivial character isrepresented by the symbol (cid:0) n ∅ (cid:1) , which has e -core and e -cocore (cid:0) r ∅ (cid:1) , where ≤ r < e is the remainder when n := me + r is divided by e . Table 2 lists the desired unipotent characters in this case, as long as n = 2 or q is not an odd power of . When n = 2 and q is an odd power of , we have e = 1 or , so we may still takethe Steinberg character for χ , but the the characters listed for χ are not necessarily fixed by the exceptionalgraph automorphism (see [Mal08, Theorem 2.5(c)]). Here we may instead take the character indexed by (cid:0) (cid:1) of degree ( q + 1) / when p | ( q − , and otherwise we use the character of degree ( q − / indexed by (cid:0) ∅ (cid:1) . Type D n and D n . In this case the unipotent characters of e G are in bijection with symbols of rank n anddefect , respectively in case D n , respectively D n . Again, let e be the order of q modulo p ,and let n = me + r where ≤ r < e is the remainder when n is divided by e . The block distribution is describedthe same way as for types B n and C n .For type D n ( q ) , the trivial character is represented by the symbol (cid:0) n (cid:1) , which has e -core (cid:0) r (cid:1) if e ∤ n and (cid:0) ∅∅ (cid:1) if e | n . It has e -cocore (cid:0) r (cid:1) if m is even and e ∤ n ; (cid:0) r ∅ (cid:1) if m is odd and e ∤ n ; (cid:0) ∅∅ (cid:1) if m is even and e | n ; and (cid:0) e (cid:1) if m is odd and e | n . Table 3 lists the desired unipotent characters as long as n ≥ . (In some cases, more thantwo characters are listed.) We remark that if n = e , then it must be that p | ( q e − .For D ( q ) = P Ω +8 ( q ) , note that ≤ e ≤ and that p | ( q + 1) when e = 2 . Then the Steinberg character ofdegree q , labeled by (cid:0) (cid:1) may be taken for χ . For χ , we take the character labeled by (cid:0) (cid:1) , of degree q ( q + 1) when e = 1 or , and (cid:0) (cid:1) of degree q ( q + 1) ( q + 1) when e = 2 . In either case, we have χ (1) > χ (1) .For type D n ( q ) , the trivial character is represented by the symbol (cid:0) n ∅ (cid:1) , which has e -core (cid:0) r ∅ (cid:1) when e ∤ n and (cid:0) e ∅ (cid:1) if e | n . The e -cocore is (cid:0) r ∅ (cid:1) if e ∤ n and m is even, (cid:0) r (cid:1) if e ∤ n and m is odd, (cid:0) e (cid:1) if e | n and m is even,and (cid:0) ∅∅ (cid:1) if e | n and m is odd. Table 4 lists the desired unipotent characters in this case. Proposition 4.5.
Let p > be a prime and let q be a power of a prime different than p . Let S be one of P SL ( q ) , P SL ǫ ( q ) with p | ( q + ǫ ) , B (2 a +1 ) with p | (2 a +1 − , or G (3 a +1 ) with p | (3 a +1 − . Thenthere exist two non-trivial characters χ , χ ∈ Irr p ′ ( B ( S )) such that χ (1) ∤ χ (1) ; χ is invariant under a Irr p ′ ( B ( S )) for types B n ( q ) , C n ( q ) with n ≥ , p ∤ q , ( n, q ) = (2 , a +1 ) Conditions on Symbol χ (1) q ′ (possibly excluding factors of ) n = me + r , r < eχ p | ( q e − (cid:0) r +1 me (cid:1) ( q me +1) − ··· ( q n − q me − r − +1)( q me +1)( q r +1 − q − q − ··· ( q r +1) − p | ( q e + 1) , m odd (cid:0) mer +1 (cid:1) ( q me +1) − ··· ( q n − q me − r − +1)( q me − q r +1 +1)( q − q − ··· ( q r +1) − p | ( q e + 1) , m even (cid:0) r +1 me (cid:1) ( q me +1) − ··· ( q n − q me − r − − q me +1)( q r +1 +1)( q − q − ··· ( q r +1) − χ e | n (cid:0) ··· n − n ··· n − n (cid:1) p | ( q e − , e ∤ n, e = r + 1 or p ∤ ( m − (cid:0) r +1 me (cid:1) ( q me +1) − ··· ( q n − q me − r − − q me +1)( q r +1 +1)( q − q − ··· ( q r +1) − p | ( q e − , e ∤ n, e = r + 1 , p | ( m − (cid:0) mee (cid:1) ( q me +1) − ··· ( q n − q me − e +1)( q me − q e +1)( q − q − ··· ( q e − p | ( q e + 1) , e ∤ n, m odd (cid:0) me r +2 (cid:1) ( q me +1) − ··· ( q n − q me − r − +1)( q me − − q me − q − ( q − q − ··· ( q r − q r +2 − p | ( q e + 1) , e ∤ n, m even (cid:0) r +2 me (cid:1) ( q me +1) − ··· ( q n − q me − r − − q me − − q me +1)( q − ( q − q − ··· ( q r − q r +2 − Table 3: Some unipotent characters in
Irr p ′ ( B ( S )) for type D n ( q ) with n ≥ , p ∤ q Conditions on Symbol χ (1) q ′ (possibly excluding factors of ) n = me + r , r < eχ e ∤ n (cid:0) mer (cid:1) ( q me +1) − ··· ( q n − − q n − q me − r +1)( q − q − ··· ( q r − p | ( q e − , e | n ; or (cid:0) · · · n − · · · n (cid:1) p | ( q e + 1) , e | n , m even; or e | ( n − p | ( q e + 1) , = e | n , m odd (cid:0) n − e e +1 (cid:1) ( q n − e +1) − ··· ( q n − − q n − q n − e − +1)( q n − e +1)( q n − e − − q − q − q − ··· ( q e − − q e − q e +1 +1) χ p | ( q e − , e ∤ n (cid:0) me r +1 (cid:1) ( q me +1) − ··· ( q n − − q n − q me − r − +1)( q me +1)( q me − − q − q − ··· ( q r − − q r − q r +1 +1)( q − e | n , e = 1 or p ∤ ( n − , (cid:0) n (cid:1) ( q n − − q − with p | ( q e − or m even p | ( q − , p | ( n − (cid:0) n − (cid:1) ( q n − q n − +1) q − p | ( q e + 1) , e ∤ n , m even (cid:0) r +1 me (cid:1) ( q me +1) − ··· ( q n − − q n − q me − r − − q me +1)( q me − +1)( q − q − ··· ( q r − − q r − q r +1 − q +1) p | ( q e + 1) , e ∤ n , m odd (cid:0) me r +1 (cid:1) ( q me +1) − ··· ( q n − − q n − q me − r − +1)( q me − q me − +1)( q − q − ··· ( q r − − q r +1)( q r +1 − q − p | ( q e + 1) , e | n , m odd, p ∤ ( m − (cid:0) n − ee (cid:1) ( q me − e +1) − ··· ( q me − − q me − q me − e +1)( q − q − ··· ( q e − p | ( q e + 1) , e | n , m odd, p ∤ ( m − (cid:0) n − e +10 e (cid:1) ( q n − e +2) − ··· ( q n − − q n − q n − e +1 +1)( q n − e +1 +1)( q n − e − q − q − q − ··· ( q e − − q e − − q e +1) Table 4: Some unipotent characters in
Irr p ′ ( B ( S )) for type D n ( q ) with n ≥ , p ∤ q Conditions on Symbol χ (1) q ′ (possibly excluding factors of ) n = me + r , r < eχ e ∤ n (cid:0) r me ∅ (cid:1) ( q me +1) − ··· ( q n − − q n +1)( q me − r − q − q − ··· ( q r − p | ( q e + 1) , = e | n , m odd (cid:0) n (cid:1) ( q n − − q − or p | ( q − , p ∤ ( n − p | ( q − , p | ( n − (cid:0) n − ∅ (cid:1) ( q n +1)( q n − − q − p | ( q e + 1) , = e | n , m even (cid:0) n − ee +1 (cid:1) ( q n − e +1) − ··· ( q n − − q n +1)( q n − e − +1)( q n − e − q n − e − − q +1)( q − q − ··· ( q e − − q e − q e +1 − p | ( q e − , = e | n (cid:0) e +1 n − e (cid:1) ( q n − e +1) − ··· ( q n − − q n +1)( q n − e − − q n − e +1)( q n − e − − q − q − q − ··· ( q e − − q e +1)( q e +1 − χ p | ( q e − , e ∤ n (cid:0) r +1 me (cid:1) ( q me +1) − ··· ( q n − − q n +1)( q me − r − +1)( q me +1)( q me − +1)( q − q − ··· ( q r − − q r +1)( q r +1 +1)( q +1) p | ( q e + 1) , e ∤ n , m even (cid:0) r +1 me (cid:1) ( q me +1) − ··· ( q n − − q n +1)( q me − r − − q me +1)( q me − − q − q − ··· ( q r − − q r +1)( q r +1 − q − p | ( q e + 1) , e ∤ n , m odd (cid:0) mer +1 (cid:1) ( q me +1) − ··· ( q n − − q n +1)( q me − r − +1)( q me − q me − − q − q − ··· ( q r − − q r − q r +1 − q +1) p | ( q e + 1) , e | n , m odd (cid:0) · · · n · · · n − (cid:1) or e | ( n − p | ( q e + 1) , = e | n , m even (cid:0) e +1 n − e (cid:1) ( q n − e +1) − ··· ( q n − − q n +1)( q n − e − − q n − e − q n − e − +1)( q − q − q − ··· ( q e − − q e − q e +1 +1) p | ( q e − , = e | n , p ∤ ( m − (cid:0) e n − e ∅ (cid:1) ( q n − e +1) − ··· ( q n − − q n +1)( q n − e − q − q − ··· ( q e − p | ( q e − , = e | n , p ∤ ( m − (cid:0) n − e +1 e (cid:1) ( q n − e +2) − ··· ( q n − − q n +1)( q n − e +1 +1)( q n − e +1 − q n − e − q +1)( q − q − ··· ( q e − − q e − − q e − ylow p -subgroup of Aut( S ) ; and for every S ≤ T ≤ Aut( S ) , χ extends to a character in the principal p -blockof T .Proof. First suppose S is P SL ( q ) or P SL ǫ ( q ) with p | ( q + ǫ ) . In these cases the order of q modulo p is or , and there is a unique unipotent block of maximal defect, so χ may still be taken to be the Steinbergcharacter. Let δ be an element of order p in F q . Write q = ℓ a , for some prime ℓ = p , and write a = p b c with p ∤ c . Then p | ℓ c − since the order of ℓ modulo p divides a , and hence c . Then δ is either fixed orinverted by F cℓ , where F ℓ is the generating field automorphism. In particular, since the semisimple classes of e G ∗ ∼ = GL ( q ) , resp. GL ǫ ( q ) , are determined by their eigenvalues, this means that a semisimple element s of e G ∗ with eigenvalues { δ, δ − } , respectively { δ, δ − , } is conjugate to its image under F cℓ . Thus the correspondingsemisimple character of e G is fixed by F cℓ , and hence a Sylow p -subgroup of Aut( S ) . Further, s satisfies (1)-(2)of [GRS, Section 4.1.1], that is, s is a member of [ e G ∗ , e G ∗ ] ∼ = SL ( q ) , resp. SL ǫ ( q ) , and is not conjugate to sz forany z ∈ Z( e G ∗ ) , since | δ | ≥ . Then this character is irreducible on G and trivial on the center. Further, it hasdegree ( q − η ) , where η ∈ {± } is such that p | ( q + η ) for P SL ( q ) , and degree q − ǫ for P SL ǫ ( q ) with p | ( q + ǫ ) .Since s is a p -element, the character lies in a unipotent block, and hence B ( e G ) , using [CE04, Theorem 9.12].Then as in the first paragraph of Proposition 4.4, the corresponding character of S lies in the principal block.It also has non-trivial degree prime to q , which therefore does not divide the degree of the Steinberg character.Hence this character satisfies our conditions.Now let S be B ( q ) with q = 2 a +1 and p | ( q − and write a + 1 = p b c with p ∤ c . Let s be such that γ s has order p | (2 c − , where γ has order q − . Then using [B79, Section 2] and arguing as in the case above,we see that a slight modification of the characters used in [GRS, Lemma 4.8] works here: we may take χ to bethe Steinberg character and χ to be the character χ ( s ) in CHEVIE notation.Finally, let S be G ( q ) with q = 3 a +1 and p | ( q − . Again write a + 1 = p b c with p ∤ c . Using [H90,Proposition 3.2], there is a unique unipotent block of maximal defect, so we may take χ again to be theSteinberg character. For χ , it follows from [H90, Proposition 4.1] and arguments as above that we may takethe character χ ( s ) in CHEVIE notation, where now s is such that γ s has order p | (3 c − and γ has order q − .Proposition 2.1 now follows from Propositions 3.6 and 4.1 through 4.5, completing the proof of Theorem A. Acknowledgments
The second author is partially supported by the Spanish Ministerio de Ciencia e Innovación PID2019-103854GB-I00 and FEDER funds. The third author is supported by the German Research Foundation (SA 2864/1-1and SA 2864/3-1). The fourth author is partially supported by a grant from the National Science Foundation(Award No. DMS-1801156). Part of this work was also completed while the second and fourth authors were inresidence at the Mathematical Sciences Research Institute in Berkeley, CA, during Summer 2019 under grantsfrom the National Security Agency (Award No. H98230-19-1-0119), The Lyda Hill Foundation, The McGovernFoundation, and Microsoft Research. The authors would also like to extend their gratitude to the anonymousreferee for their careful reading and comments about the manuscript.
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