On the inductive blockwise Alperin weight condition for classical groups
aa r X i v : . [ m a t h . R T ] A ug On the inductive blockwise Alperin weight conditionfor classical groups
Zhicheng Feng a,b , Zhenye Li c and Jiping Zhang d a School of Mathematics and Physics, University of Science and Technology Beijing, Beijing100083, China b FB Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany c Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academyof Sciences, Beijing 100190, China d Beijing International Center for Mathematical Research Lmam, The School of MathematicalSciences, Peking University, Beijing 100871, China
Abstract
Recently, there has been substantial progress on the Alperin weight conjecture. As astep to establish the Alperin weight conjecture for all finite groups, we prove the induc-tive blockwise Alperin weight condition for simple groups of classical type under someadditional assumption.
Keywords
Alperin weight conjecture, inductive blockwise Alperin weight condition, classi-cal groups, unipotent blocks
On the 1986 Arcata conference on representations of finite groups, J. L. Alperin put forwardhis famous conjecture, which is now called the
Alperin weight conjecture . To state it, let G be afinite group and ℓ a prime, B an ℓ -block of G . As usual, we denote by Irr( B ) and IBr ℓ ( B ) the setsof ordinary irreducible characters and irreducible ℓ -Brauer characters of B respectively. For an ℓ -subgroup R of G and ϕ ∈ Irr( N G ( R )), the pair ( R , ϕ ) is called an ℓ -weight if R ⊆ ker ϕ is of ℓ -defect zero viewed as a character of N G ( R ) / R . Note that R is necessarily an ℓ -radical subgroupof G for any ℓ -weight ( R , ϕ ). An ℓ -weight ( R , ϕ ) is called a B-weight if bl ℓ ( ϕ ) G = B , wherebl ℓ ( ϕ ) is the ℓ -block of N G ( R ) containing ϕ . We denote by W ℓ ( B ) the set of all G -conjugacyclasses of B -weights so that the Alperin weight conjecture can be stated as follows. Conjecture 1.1 (Alperin, [1]) . Let G be a finite group, ℓ a prime. If B is an ℓ -block of G, then |W ℓ ( B ) | = | IBr ℓ ( B ) | . Supported by SFB TRR 195 and NSFC (No. 11631001).
Email addresses : [email protected] (Z. Feng) , [email protected] (Z. Li), [email protected] (J. Zhang).
1o far substantial progress has been achieved for the Alperin weight conjecture. Specifically,it was shown to hold for finite groups of Lie type in defining characteristic by Cabanes [14], forsymmetric groups and general linear groups by Alperin and Fong [2], and for certain groups ofclassical type by An [3] and [4].Even though the Alperin weight conjecture was subsequently checked for several furtherfamilies of finite groups, it has not been possible so far to find a general proof for arbitraryfinite groups. A reduction theorem for the blockfree version of the Alperin weight conjecturewas obtained by Navarro and Tiep [48] in 2011. Soon afterwards, Sp¨ath [54] refined this resultto achieve a reduction theorem for the blockwise version of Alperin’s weight conjecture; if allfinite (quasi-)simple groups satisfy the so-called inductive blockwise Alperin weight (iBAW)condition, then the Alperin weight conjecture 1.1 holds for any finite group.To the present, the (iBAW) condition has been verified for some cases, such as simple alter-nating groups, many of the sporadic groups, simple groups of Lie type in the defining charac-teristic, Suzuki groups and Ree groups, simple groups of type G and D , some cases of blocksof groups of type A ; see for instance [8], [13], [17], [21], [38], [42], [52] and [54]. Unfortu-nately, it still seems a far way to deal with the case of simple groups of Lie type in non-definingcharacteristic in general.In this paper, we consider the classical groups. As a first step to verify the (iBAW) con-dition, we need to establish a blockwise equivariant bijection between ℓ -Brauer characters and ℓ -weights. In fact, An [4] has essentially given such a bijection. In this paper, we first con-sider the groups of type B and prove the bijection given in [4] for SO n + ( q ) is equivariantunder the field automorphism (see Theorem 4.9). From this, we obtain a blockwise Aut( S )-equivariant bijection between IBr ℓ ( S ) and W ℓ ( S ) under some assumption (see Theorem 4.13),where S = Ω n + ( q ). Our first main result about non-faithful blocks of groups of type B is thefollowing. Theorem 1.2.
Let X = Spin n + ( q ) with q = p f odd and n ≥ , ℓ , p an odd prime and B an ℓ -block of X. Assume further f is odd, ℓ is linear and B dominates some ℓ -block of Ω n + ( q ) .Then the inductive blockwise Alperin weight (iBAW) condition (cf. Definition 2.10) holds forB. Recall that an odd prime ℓ not dividing q is called linear (for q ) if the multiplicative orderof q modulo ℓ is odd. In this paper, the assumption that ℓ is linear is always to ensure theunitriangular shape of decomposition matrices, which is due to Gruber and Hiss [27]. It is anopen problem to show that decomposition matrices of finite groups of Lie type in non-definingcharacteristic have unitriangular shape (see for example [43, Problem 4.8]). If this is true, thenthe assumption that ℓ is linear can be removed from the main results of this paper.For groups of type C , we first verify the Alperin weight conjecture 1.1 for every ℓ -blockof Sp n ( q ) when both ℓ and q are odd, and then we prove the (iBAW) condition for the simplegroup of symplectic type and a linear prime if the outer automorphism group is cyclic, whichcan be stated as follows. Theorem 1.3.
Let q be a power of an odd prime, n ≥ , ℓ , p an odd prime. Then the Alperinweight conjecture 1.1 holds for every ℓ -block of Sp n ( q ) . Theorem 1.4.
Let q = p f be a power of an odd prime p, ℓ , p an odd prime and n ≥ . Assumethat f is odd and ℓ is linear. Then the inductive blockwise Alperin weight (iBAW) condition (cf.Definition 2.11) holds for the simple group PSp n ( q ) and the prime ℓ . In order to prove Theorem 1.4, we need a parametrization of ℓ -blocks of Sp n ( q ), whichmay be of independent interest; see Theorem 5.10. In order to do this, we make use of both the2arametrization of ℓ -blocks of CSp n ( q ) from Fong–Srinivasan [24] , and the label of ℓ -blocks ofan arbitrary finite group of Lie type from Cabanes–Enguehard [16] for ℓ ≥ ℓ -Brauer characters and ℓ -weights of Sp n ( q ) which is equivariant under the actionof automorphisms (see Theorem 5.16).We should mention that Conghui Li has proved independently Theorem 5.16 in [37] withdi ff erent methods.In addition, we also determined a similar parametrization of ℓ -blocks for SO ± n ( q ), which isdescribed in Appendix B. From this, if a hypothesis for the action of GO ± n ( q ) on the charactersof SO ± n ( q ) is true, then the Alperin weight conjecture 1.1 holds for every ℓ -block of SO ± n ( q )when both ℓ and q are odd; see Theorem B.6.Is there an analogue of Jordan decomposition for weights of finite groups of Lie type?Malle proposed this problem in [43, Problem 4.9]. Furthermore, following Kessar–Malle [33, § § i.e. , quasi-isolated blocks. In this sense, the unipotent blocks would play afundamental and important role when considering the (iBAW) condition for finite quasi-simplegroups of Lie type. In [21], the author verified the (iBAW) condition for unipotent blocks ofgroups of type A , untwisted or twisted, under some additional assumption on the prime involved.Considering classical type, the following is our main result for unipotent blocks. Theorem 1.5.
Assume that both ℓ and q are odd such that ℓ ∤ q. Suppose that one of thefollowing holds.(i) X ∈ { Spin n + ( q ) , Sp n ( q ) } with n ≥ .(ii) X = Spin − n ( q ) with n ≥ .(iii) X = Spin + n ( q ) with n > and ℓ is linear.Then the inductive blockwise Alperin weight (iBAW) condition (cf. Definition 2.10) holds forevery unipotent ℓ -block of X. This paper is built up as follows. In section §
2, we introduce the general notation and statethe (iBAW) condition. In section §
3, the action of automorphisms on the weights of classicalgroups for a special case is considered. Then we prove Theorem 4.9 and prove Theorem 1.5for type B in section §
4. In section §
5, we give a classification for blocks of symplectic groupsand then prove Theorem 1.4 and prove Theorem 1.5 for type C . Finally, the (iBAW) conditionfor unipotent blocks of classical groups of type D and D are verified in section §
6. In addition,we consider the Alperin weight conjecture for special orthogonal groups in even-dimension inAppendices.
Let G be a finite group. Concerning the block and character theory of G we mainly followthe notation of [47], where for sets of ℓ -Brauer characters or ℓ -blocks we add a subscript toindicate the corresponding prime ℓ ( e.g. IBr ℓ ( G ), Bl ℓ ( G )). We denote the restriction of χ ∈ Irr( G ) ∪ IBr ℓ ( G ) to some subgroup H ≤ G by Res GH χ , while Ind GH ψ denotes the character induced3rom ψ ∈ Irr( H ) ∪ IBr ℓ ( H ) to G . For N E G we sometimes identify the characters of G / N withthe characters of G whose kernel contains N .The cardinality of a set, or the order of a finite group, X , is denoted by | X | . If a group A actson a finite set X , we denote by A x the stabilizer of x ∈ X in A , analogously we denote by A X ′ thesetwise stabilizer of X ′ ⊆ X .Let ℓ be a prime. If A acts on a finite group G by automorphisms, then there is a naturalaction of A on Irr( G ) ∪ IBr ℓ ( G ) given by a − χ ( g ) = χ a ( g ) = χ ( g a − ) for every g ∈ G , a ∈ A and χ ∈ Irr( G ) ∪ IBr ℓ ( G ). For P ≤ G and χ ∈ Irr( G ) ∪ IBr ℓ ( G ), we denote by A P ,χ the stabilizer of χ in A P .Let χ ∈ Irr( G ), we denote by χ ◦ the restriction of χ to the set of all ℓ ′ -elements of G . Let Y ⊆ IBr ℓ ( G ). A subset X ⊆ Irr( G ) is called a basic set of Y if { χ ◦ | χ ∈ X } is a Z -basis of Z Y . If Y = IBr ℓ ( B ) for some ℓ -block B of G , then we also say X a basic set of B .Let O denote the ring of algebraic integers in C . Following [47, §
2] we fix a maximalideal M of O containing the ideal ℓ O . Then by [47, Lemma 2.1] the field F : = O / M is analgebraic closure of its prime field F ℓ of characteristic ℓ , and we denote by ∗ : O → F thenatural epimorphism. Let χ ∈ Irr( G ). Then the central character associated to χ is the algebrahomomorphism ω χ : Z ( C G ) → C , C ω χ ( ˆ C ) = | C | χ ( x ) χ (1) , where C is a conjugacy class of G ,ˆ C = P x ∈ C x and x ∈ C . This yields an algebra homomorphism λ χ : Z ( F G ) → F such that λ χ ( ˆ C ) = ω χ ( ˆ C ) ∗ for a conjugacy class C of G . Then for χ, ψ ∈ Irr( G ), they are in the same ℓ -block of G if and only if λ χ = λ ψ . Let B be an ℓ -block of G , then we define λ B = λ χ for χ ∈ Irr( B ). For a prime ℓ and ϕ ∈ Irr( G ) ∪ IBr ℓ ( G ), we denote by bl ℓ ( ϕ ) the ℓ -block of G containing ϕ .A subgroup R ≤ G is ℓ -radical if R = O ℓ ( N G ( R )). We also say that R is an ℓ -radical subgroupof G . We denote by Rad ℓ ( G ) the set of ℓ -radical subgroups of G . Furthermore, Rad ℓ ( G ) / ∼ G denotes a G -transversal of radical ℓ -subgroup of G .We denote the set of all G -conjugacy classes of ℓ -weights of G by W ℓ ( G ) while W ℓ ( B )denotes the set of all G -conjugacy classes of B -weights for an ℓ -block B of G .The following lemma is elementary. Lemma 2.1.
Let G be a finite group, Z a central subgroup of G and π : G → ¯ G = G / Z bethe canonical homomorphism. Suppose that ¯ B is an ℓ -block of ¯ G which is dominated by the ℓ -block B of G. Let ( ¯ R , ¯ ϕ ) be a ¯ B-weight and let R = O ℓ ( π − ( ¯ R )) and ϕ the inflation of ¯ ϕ fromN ¯ G ( ¯ R ) = N G ( R ) to N G ( R ) . Then ( R , ϕ ) is a B-weight. Lemma 2.2.
Let G be a finite group, Z a central ℓ ′ -subgroup of G and ¯ G = G / Z. Then there isa bijection Θ : Rad ℓ ( G ) → Rad ℓ ( ¯ G ) given by R ¯ R with inverse given by Q / Z
7→ O ℓ ( Q ) .Moreover, Θ induces a bijection between Rad ℓ ( G ) / ∼ G and Rad ℓ ( ¯ G ) / ∼ ¯ G .Proof. This follows by [48, Lem. 2.3 (c)]. (cid:3)
Note that, we have N ¯ G ( ¯ R ) = N G ( R ) in Lemma 2.2. Lemma 2.3.
Keep the hypothesis and notation of Lemma 2.2.(i) If ( ¯ R , ¯ ϕ ) is an ℓ -weight of ¯ G, then ( R , ϕ ) is an ℓ -weight of G, where R = Θ − ( ¯ R ) and ϕ isthe inflation of ¯ ϕ to N G ( R ) .(ii) If ( R , ϕ ) is an ℓ -weight of G such that Z ≤ ker ϕ and ¯ R = Θ ( R ) , then ( ¯ R , ¯ ϕ ) is an ℓ -weightof ¯ G, where ¯ ϕ is the character of N ¯ G ( ¯ R ) whose inflation is ϕ . iii) Let ¯ B be an ℓ -block of ¯ G and B an ℓ -block of G dominating ¯ B. Then the map W ℓ ( B ) →W ℓ ( ¯ B ) given by ( R , ϕ ) ( ¯ R , ¯ ϕ ) is a bijection.Proof. Both (i) and (ii) are obvious. For (iii), by (i), (ii) and Lemma 2.2, it su ffi ces to show that Z ≤ ker ϕ holds for any ( R , ϕ ) ∈ W ℓ ( B ). Let b = bl ℓ ( ϕ ), then b G = B . Thus λ b ( z ) = λ B ( z ) forall z ∈ Z . By [47, Thm. (9.9)(c)], Z ≤ ker χ for all χ ∈ Irr( B ). Hence ω χ ( z ) = χ ( z ) /χ (1) = λ B ( z ) = z ∈ Z . Thus λ b ( z ) =
1, i.e. ω ϕ ( z ) ∗ = z ∈ Z . However, ω ϕ ( z )is an ℓ ′ -root of unity and then by [47, Lem. (2.1)], 1 = ω ϕ ( z ) = ϕ ( z ) /ϕ (1) for all z ∈ Z . Hence Z ≤ ker ϕ , as desired. (cid:3) Lemma 2.4.
Keep the hypothesis and notation of Lemma 2.2. Let σ ∈ Aut( G ) such that σ stabilizes Z, and ¯ σ the automorphism of ¯ G induced by σ . Let ¯ R i ∈ Rad ℓ ( ¯ G ) and R i = Θ − ( ¯ R i ) for i = , . If ¯ σ ( ¯ R ) = ¯ R , then σ ( R ) = R .Proof. By the assumption, σ ( R ) Z = R Z . Then σ ( R ) = R since it is the unique Sylow ℓ -subgroup of σ ( R ) Z = R Z , as stated. (cid:3) Lemma 2.5.
Keep the hypothesis and notation of Lemma 2.2 and Lemma 2.3 (iii). Let A be asubgroup of
Aut( G ) such that A stabilizes Z, and ¯ A the subgroup of
Aut( ¯ G ) induced by A. Ifthere is an ¯ A-equivariant bijection between
IBr ℓ ( ¯ B ) and W ℓ ( ¯ B ) , then there is an A-equivariantbijection between IBr ℓ ( B ) and W ℓ ( B ) .Proof. This follows by Lemma 2.3 (iii) and Lemma 2.4 and the fact that IBr ℓ ( B ) = IBr ℓ ( ¯ B ) (seefor example [47, Thm. (9.9)]). (cid:3) Let G be a finite group, χ ∈ Irr( G ), ℓ a prime, and θ a linear character of G . Then θχ isan irreducible character of G and the map χ θχ is a permutation on Irr( G ). Moreover, thispermutation respects ℓ -blocks. Now we let B be an ℓ -block of G and assume that θ is of ℓ ′ -order.Then by [21, Lem. 2.4], there is an ℓ -block θ ⊗ B of G , such that Irr( θ ⊗ B ) = { θχ | χ ∈ Irr( B ) } .Moreover, IBr ℓ ( θ ⊗ B ) = { θ ◦ φ | φ ∈ IBr ℓ ( B ) } . Lemma 2.6.
Let N be a normal subgroup of a finite group G such that G / N is cyclic andG / NZ ( G ) is an ℓ ′ -group and let b be an ℓ -block of N. Suppose that there are m ℓ -blocks of Gcovering b, where m = | G / N | ℓ ′ . Then the following statements hold.(i) Res GN : IBr ℓ ( B ) → IBr ℓ ( b ) is bijective for any ℓ -block B of G covering b.(ii) Let A be a subgroup of Aut( G ) stabilizing N. Suppose that B is an ℓ -block of G coveringb such that B is A-invariant. If φ ∈ IBr ℓ ( b ) which is A-invariant, then there is an extension ˜ φ ∈ IBr ℓ ( B ) of φ such that ˜ φ is A-invariant.Proof. Let φ ∈ IBr ℓ ( b ) and ˜ φ ∈ IBr ℓ ( G | φ ). Since G / N is cyclic, by Cli ff ord theory, eachirreducible ℓ -Brauer character covering φ has the form ˜ φτ with τ ∈ IBr ℓ ( G / N ). Then | IBr ℓ ( G | φ ) | ≤ m . Now there are m ℓ -blocks of G covering b , so | IBr ℓ ( G | φ ) | = m . By Cli ff ord theory, G / G φ is an ℓ -group. Since G / NZ ( G ) is an ℓ ′ -group, we have G = G φ , and then ˜ φ is an extensionof φ . Thus (i) follows easily.For (ii), let ˜ φ ∈ IBr ℓ ( B ) be the extension of φ . Since φ is A -invariant and B is A -invarianttoo, we get that ˜ φ a ∈ IBr ℓ ( B ) is also an extension of φ for any a ∈ A . By the uniqueness of ˜ φ ,we have ˜ φ a = ˜ φ . Then ˜ φ is A -invariant. (cid:3) Lemma 2.7.
Let K be a subgroup of a finite group G and b an ℓ -block of K, and θ a linearcharacter of G of ℓ ′ -order. Assume that both b G and ((Res GK θ ) ⊗ b ) G are defined. Then ((Res GK θ ) ⊗ b ) G = θ ⊗ b G . roof. Let B = b G . Then λ B ( ˆ C ) = λ b ( [ C ∩ K ) and λ ((Res GK θ ) ⊗ b ) G ( ˆ C ) = λ (Res GK θ ) ⊗ b ( [ C ∩ K ), for anyconjugacy class C of G . It is easy to check that λ θ ⊗ B ( ˆ C ) = θ ( x ) ∗ λ b ( [ C ∩ K ) and λ (Res GK θ ) ⊗ b ( [ C ∩ K ) = θ ( x ) ∗ λ b ( [ C ∩ K ) for x ∈ C . Thus λ θ ⊗ B = λ ((Res GK θ ) ⊗ b ) G and then ((Res GK θ ) ⊗ b ) G = θ ⊗ B . (cid:3) By Lemma 2.7 we have the following result immediately.
Corollary 2.8.
Let G be a finite group, B an ℓ -block and ( R , ϕ ) a B-weight. Suppose that θ is alinear character of G with ℓ ′ -order. Then ( R , (Res GN G ( R ) θ ) ϕ ) is a ( θ ⊗ B ) -weight. We will make use of the following result.
Lemma 2.9.
Let A be a finite group, G a normal subgroup of A and B an ℓ -block of G. Supposethat there exists a basic set X ⊆ Irr( B ) of B such that the corresponding decomposition matrixis unitriangular with respect to a suitable order. If every χ ∈ X extends to A χ , then every φ ∈ IBr ℓ ( B ) extends to A φ .Proof. By [17, Lem. 7.5], there exists an A -equivariant bijection D : X → IBr ℓ ( B ) such that χ ◦ = D ( χ ) + P φ ∈ IBr ℓ ( B ) \{ D ( χ ) } d φ φ with d φ ∈ Z ≥ . In particular, A χ = A D ( χ ) . Now let ˜ χ be an extensionof χ to A χ and let ˜ φ be an irreducible constituent of ˜ χ ◦ such that ˜ φ ∈ IBr ℓ ( A χ | D ( χ )). Since D ( χ ) is A χ -invariant, by Cli ff ord theory, we know that ˜ φ is an extension of D ( χ ) to A D ( χ ) . (cid:3) For a finite group H and a prime ℓ , we denote by dz ℓ ( H ) the set of ℓ -defect zero irreduciblecharacters of H . If Q is a radical ℓ -subgroup of H and B an ℓ -block of H , then we define the setdz ℓ ( N H ( Q ) / Q , B ) : = { χ ∈ dz ℓ ( N H ( Q ) / Q ) | bl ℓ ( χ ) H = B } , where we regard χ as an irreducible character of N G ( Q ) containing Q in its kernel when consid-ering the induced ℓ -block bl ℓ ( χ ) H .There are several versions of the (iBAW) condition. Apart from the original version given in[54, Def. 4.1], there is also a version treating only blocks with defect groups involved in certainsets of ℓ -groups [54, Def. 5.17], or a version handling single blocks [36, Def. 3.2]. We shallconsider the inductive condition for a single block here. Definition 2.10 ([36, Def. 3.2]) . Let ℓ be a prime, S a finite non-abelian simple group and X the universal ℓ ′ -covering group of S . Let B be an ℓ -block of X . We say the inductive blockwiseAlperin weight (iBAW) condition holds for B if the following statements hold.(i) There exist subsets IBr ℓ ( B | Q ) ⊆ IBr ℓ ( B ) for Q ∈ Rad ℓ ( X ) with the following properties.(1) IBr ℓ ( B | Q ) a = IBr ℓ ( B | Q a ) for every Q ∈ Rad ℓ ( X ), a ∈ Aut( X ) B ,(2) IBr ℓ ( B ) = ˙ S Q ∈ Rad ℓ ( X ) / ∼ X IBr ℓ ( B | Q ).(ii) For every Q ∈ Rad ℓ ( X ) there exists a bijection Ω XQ : IBr ℓ ( B | Q ) → dz ℓ ( N X ( Q ) / Q , B )such that Ω XQ ( φ ) a = Ω XQ a ( φ a ) for every φ ∈ IBr ℓ ( B | Q ) and a ∈ Aut( X ) B .6iii) For every Q ∈ Rad ℓ ( X ) and every φ ∈ IBr ℓ ( B | Q ) there exist a finite group A : = A ( φ, Q )and ˜ φ ∈ IBr ℓ ( A ) and ˜ φ ′ ∈ IBr ℓ ( N A ( Q )), where we use the notation Q : = QZ / Z and Z : = Z ( X ) ∩ ker( φ ) , with the following properties.(1) for X : = X / Z the group A satisfies X E A , A / C A ( X ) (cid:27) Aut( X ) φ , C A ( X ) = Z ( A ) and ℓ ∤ | Z ( A ) | ,(2) ˜ φ ∈ IBr ℓ ( A ) is an extension of the ℓ -Brauer character of X associated with φ ,(3) ˜ φ ′ ∈ IBr ℓ ( N A ( Q )) is an extension of the ℓ -Brauer character of N X ( Q ) associated withthe inflation of Ω XQ ( φ ) ◦ ∈ IBr ℓ ( N X ( Q ) / Q ) to N X ( Q ),(4) bl ℓ (Res AJ ( ˜ φ )) = bl ℓ (Res N A ( Q ) N J ( Q ) ( ˜ φ ′ )) J for every subgroup J satisfying X ≤ J ≤ A . Definition 2.11.
Let ℓ be a prime, S a finite non-abelian simple group and X the universal ℓ ′ -covering group of S . We say that the inductive blockwise Alperin weight (iBAW) conditionholds for S and the prime ℓ if the (iBAW) condition holds for every ℓ -block of X . Lemma 2.12.
Let ℓ be a prime, S a finite non-abelian simple group and X the universal ℓ ′ -covering group of S . Let B be an ℓ -block of X. If there is an Aut( X ) B -equivariant bijectionbetween IBr ℓ ( B ) and W ℓ ( B ) , then there are natural defined sets IBr ℓ ( B | Q ) and bijections Ω Q ( X ) such that (i) and (ii) of Definition 2.10 holds for B.Proof. This is [52, Lem. 2.10]. (cid:3)
Corollary 2.13.
Let ℓ be a prime, S a finite non-abelian simple group such that Aut( S ) / S iscyclic and X the universal ℓ ′ -covering group of S . Let B be an ℓ -block of X. If there is an Aut( X ) B -equivariant bijection between IBr ℓ ( B ) and W ℓ ( B ) , then the (iBAW) condition holdsfor B.Proof. By the proof of [54, Lem. 6.1], it su ffi ces to prove (i) and (ii) of Definition 2.10 (fordetails, see [52, Rmk. 2.7]), which follows by Lemma 2.12. (cid:3) By [54, Thm. C], the (iBAW) condition holds for every finite non-abelian simple group ofLie type and its defining characteristic. From this, we only consider non-defining characteristicfor classical groups in this paper.
We will need to view some finite classical groups as the groups of fixed points under someFrobenius endomorphisms of certain connected reductive algebraic groups. Let q be a power ofa prime p and let F q be the field of q elements. Also let F q be the algebraic closure of the field F q . Algebraic groups are usually denoted by boldface letters. Suppose that G is a connectedreductive algebraic group over F q and F : G → G a Frobenius endomorphism endowing G with an F q -structure. The group of rational points G F is finite. Let G ∗ be dual to G withcorresponding Frobenius endomorphism also denoted F .Let ℓ be a prime number di ff erent from p . For a semisimple ℓ ′ -element s of G ∗ F , we denoteby E ℓ ( G F , s ) the union of the Lusztig series E ( G F , st ), where t runs through semisimple ℓ -elements of G ∗ F commuting with s . By [12], the set E ℓ ( G F , s ) is a union of ℓ -blocks of G F .7lso, we denote by E ( G F , ℓ ′ ) the set of irreducible characters of G F lying in a Lusztig series E ( G F , s ), where s ∈ G ∗ F is a semisimple ℓ ′ -element. Considering the elements of E ( G F , ℓ ′ ) as abasic set is the main argument of [26] with the assumption that ℓ is good and Z ( G ) is connected.It was generalized in [25, Thm. A], which can be stated as follows. Theorem 2.14.
Let ℓ be a prime good for G and not dividing the defining characteristic of G .Assume that ℓ does not divide ( Z ( G ) / Z ◦ ( G )) F (the largest quotient of Z ( G ) on which F actstrivially). Let s ∈ G ∗ F be a semisimple ℓ ′ -element. Then E ( G F , s ) form a basic set of E ℓ ( G F , s ) . In this paper, any algebraic group G involved is of classical type and the prime ℓ is alwaysodd. Thus the hypothesis of Theorem 2.14 is always satisfied.Let d be a positive integer. We will make use of the terminology of Sylow d -theory (see forinstance [10] and [11]). For an F -stable maximal torus T of G , denote by T d its Sylow d -torus.An F -stable Levi subgroup L of G is called d-split if L = C G ( Z ◦ ( L ) d ), and ζ ∈ Irr( L F ) is called d-cuspidal if ∗ R LM ⊆ P ( ζ ) = d -split Levi subgroups M < L and any parabolicsubgroup P of L containing M as Levi complement.Let s ∈ G ∗ F be semisimple. Following [32, Def. 2.1], we say χ ∈ E ( G F , s ) is d-Jordan-cuspidal if • Z ◦ ( C ◦ G ∗ ( s )) d = Z ◦ ( G ∗ ) d , and • χ corresponds under Jordan decomposition (cf. [40, Prop. 5.1]) to the C G ∗ ( s ) F -orbit of a d -cuspidal unipotent character of C ◦ G ∗ ( s ) F .If L is a d -split Levi subgroup of G and ζ ∈ Irr( L F ) is d -Jordan-cuspidal, then ( L , ζ ) is called a d-Jordan-cuspidal pair of G .Now we define an integer e = e ( q , ℓ ) for odd prime ℓ , which is denoted by “ e ” in [32] (inthis paper, we will use “ e ” for another integer, see § e = e ( q , ℓ ) = multiplicative order of q modulo ℓ. (2.1)The paper [16] gave a label for arbitrary ℓ -blocks of finite groups of Lie type for ℓ ≥ G F -conjugacy classes of e -Jordan-cuspidal pairs ( L , ζ ) of G such that ζ ∈ E ( L F , ℓ ′ ), is a labeling set of the ℓ -blocks of G F .By [6, Thm.], the Mackey formula holds for groups of classical type, hence the Lusztiginduction R GL ⊆ P is independent of the ambient parabolic subgroup P in this paper. So throughoutthis paper we always omit the parabolic subgroups when considering Lusztig inductions. From now on we always assume that p is an odd prime, q = p f with a positive integer f ,and ℓ is an odd prime number di ff erent from p . We will consider some classical groups over F q ,such as (conformal) symplectic or orthogonal groups. For the definitions of the classical groupsappearing in this paper, we refer to [34, § ff ersfrom the notation used in this paper in the way that in [34] the conformal orthogonal groups aredenoted by GO ± n ( q ) and the general orthogonal groups are written as O ± n ( q ), while in this paper(follows the notation of [46]) the conformal orthogonal groups are denoted by CO ± n ( q ) and thegeneral orthogonal groups are written as GO ± n ( q ).We follow mainly the notation from [24] and [4]. Let V be a finite-dimensional symplecticor orthogonal space over the field F q . We denote by I ( V ) the group of isometries of V , I ( V )8he subgroups of I ( V ) of determinant 1, and η ( V ) = ± V if V is orthogonal. Forsimplicity, we set η ( V ) = V is symplectic. Furthermore, we identify 1, − + , − respectively when considering the type of spaces and groups. Obviously, I ( V ) = I ( V ) = Sp( V )if V is a symplectic space and I ( V ) = GO( V ), I ( V ) = SO( V ) if V is an orthogonal space.We recall that there exists a set F of polynomials serving as elementary divisors for allsemisimple elements of each of these groups. We denote by Irr( F q [ x ]) the set of all monicirreducible polynomials over the field F q . For each ∆ in Irr( F q [ x ]) \ { x } , we define ∆ ∗ be thepolynomial in Irr( F q [ x ]) whose roots are the inverses of the roots of ∆ . Now, we denote by F = { x − , x + } , F = n ∆ ∈ Irr( F q [ x ]) | ∆ < F , ∆ , x , ∆ = ∆ ∗ o , F = n ∆∆ ∗ | ∆ ∈ Irr( F q [ x ]) \ F , ∆ , x , ∆ , ∆ ∗ o . Let F = F ∪ F ∪ F . Given Γ ∈ F , denote by d Γ its degree and by δ Γ its reduced degree defined by δ Γ = ( d Γ if Γ ∈ F ; d Γ if Γ ∈ F ∪ F . Since the polynomials in F ∪ F have even degree, δ Γ is an integer. In addition, we mention asign ε Γ for Γ ∈ F ∪ F defined by ε Γ = ( − Γ ∈ F ;1 if Γ ∈ F . Given a semisimple element s ∈ I ( V ), there exists a unique orthogonal decomposition V = X Γ V Γ ( s ) , s = Y Γ s ( Γ ) , (2.2)where the V Γ ( s ) are non-degenerate subspaces of V , s ( Γ ) ∈ I ( V Γ ( s )), and s ( Γ ) has minimalpolynomial Γ . The decomposition (2.2) is called the primary decomposition of s in I ( V ). Let m Γ ( s ) be the multiplicity of Γ in s ( Γ ). If m Γ ( s ) ,
0, then we say Γ is an elementary divisor of s . Then the centralizer of s in I ( V ) has a decomposition C I ( V ) ( s ) = Q Γ C Γ ( s ), where C Γ ( s ) = C I ( V Γ ( s )) ( s ( Γ )). Moreover, by [24, (1.13)], C Γ ( s ) = ( I ( V Γ ( s )) if Γ ∈ F ;GL m Γ ( s ) ( ε Γ q δ Γ ) if Γ ∈ F ∪ F . Here, GL m ( − q ) means GU m ( q ). Note that C Γ ( s ) ≤ I ( V Γ ( s )) for Γ ∈ F ∪ F .Let η Γ ( s ) be the type of V Γ ( s ). Here η Γ ( s ) = Γ ∈ F if V is symplectic. By [24,(1.12)], the multiplicity and type functions Γ m Γ ( s ), Γ η Γ ( s ) satisfy the following relationsdim V = X Γ d Γ m Γ ( s ) ,η ( V ) = ( − q − m x − ( s ) m x + ( s ) Y Γ η Γ ( s ) ,η Γ ( s ) = ε m Γ ( s ) Γ for Γ ∈ F ∪ F . (2.3)Conversely, if Γ m Γ ( s ), Γ η Γ are functions from F to N , {± } respectively satisfying (2.3),then there exists a semisimple element s of I ( V ) with these functions as multiplicity and type9unctions. Moreover, two semisimple elements s and s ′ of I ( V ) are I ( V )-conjugate if and onlyif m Γ ( s ) = m Γ ( s ′ ) and η Γ ( s ) = η Γ ( s ′ ) for all Γ ∈ F .Now assume that V is orthogonal. A semisimple element s lies in I ( V ) if and only if m x + ( s )is even. If s ∈ I ( V ), then | C I ( V ) ( s ) : Y Γ C I ( V Γ ( s )) ( s ( Γ )) | = , and index 2 occurs if and only if m x − ( s ) and m x + ( s ) are both non-zero. For more details, see[24, § s ∈ I ( V ), we define Ψ Γ ( s ) to be the set of partitions of m Γ ( s ) if Γ ∈ F ∪ F . If Γ ∈ F , then Ψ Γ ( s ) is defined to be the set of symbols of rank [ m Γ ( s )2 ] such that • If V Γ ( s ) is symplectic or orthogonal of odd-dimension, then the symbols have odd defect. • If V Γ ( s ) is orthogonal of even-dimension and type + , then the symbols have defect divis-ible by 4. Moreover, degenerate symbols are counted twice. • If V Γ ( s ) is orthogonal of even-dimension and type − , then the symbols have defect con-gruent to 2 modulo 4.Let Ψ ( s ) = Y Γ Ψ Γ ( s ) . (2.4)Following [24, p. 132], we define an operate ′ on the sets Ψ ( s ) and Ψ Γ ( s ) as follows. Let µ Γ ∈ Ψ Γ ( s ). Then define ( µ Γ ) ′ = µ Γ if µ Γ is a partition or a non-degenerate symbol, and define( µ Γ ) ′ to be the other copy of µ Γ in Ψ Γ ( s ) if µ Γ is a degenerate symbol. If µ = Q Γ µ Γ ∈ Ψ ( s ), thenwe define µ ′ = Q Γ ( µ Γ ) ′ .In this paper, we let e be the multiplicative order of q modulo ℓ . Then e = e / gcd(2 , e ),where e is defined as in (2.1). We say the prime ℓ is linear if e is odd while ℓ is unitary if e is even.Let F ′ be the subset of F consisting of polynomials whose roots are of ℓ ′ -order. For Γ ∈F ′ , we define e Γ to be the multiplicative order of q or ε Γ q δ Γ modulo ℓ according as Γ ∈ F or Γ ∈ F ∪ F . Then e Γ = e for Γ ∈ F . Let s be a semisimple ℓ ′ -element of I ( V ) and µ = Q Γ µ Γ ∈ Ψ ( s ). Now we define the e Γ -core of µ Γ for every Γ ∈ F ′ . If Γ ∈ F ∪ F , the e Γ -core of µ Γ is defined in the usual way for partitions (see for example [50, § Γ ∈ F ,the e Γ -core of µ Γ is defined in [24, p. 159] which we state as follows. Let κ Γ be the symbolswhich is gotten by actually removing w Γ e Γ -hooks (or e Γ -cohooks, resp.) from µ Γ and there isno e Γ -hooks (or e Γ -cohooks, resp.) in κ Γ if ℓ is linear (or unitary, resp.). If κ Γ is degenerate and w Γ >
0, then both copies of κ Γ are considered as the e Γ -core of µ Γ ( i.e. , the e Γ -core of µ Γ isdefined to be the doubleton { κ Γ , κ ′ Γ } ). If κ Γ is degenerate and w Γ = i.e. , κ Γ = µ Γ ), then only κ Γ ,but not its copy, is the e Γ -core of µ Γ . If κ Γ is non-degenerate, then the e Γ -core of µ Γ is κ Γ .Note that the definition of e Γ -core of a symbol here (as in [24]) is the same with the definitionin [23, p. 307], and is slightly di ff erent from those used in [11], [49] and [50]. We follow [23]and [24] and say e Γ -core for both e Γ -core (when ℓ is linear) and e Γ -cocore (when ℓ is unitary) in[11], [49] and [50].Let s be a semisimple ℓ ′ -element. For Γ ∈ F , we define C Γ ( s ) the set of κ Γ such that thereexists µ Γ ∈ Ψ Γ ( s ) satisfying that κ Γ is an e Γ -core of µ Γ . Denote C ( s ) = Y Γ C Γ ( s ) . (2.5)10n particular, for each κ ∈ C ( s ), the cardinality of the set κ is 1, 2 or 4. We also define an operate ′ on the sets C ( s ) and C Γ ( s ) as follows. Define κ ′ Γ = κ Γ if κ Γ is a doubleton and define κ ′ Γ = κ Γ otherwise. For κ = Q Γ µ Γ ∈ C ( s ), we define κ ′ = Q Γ ( κ Γ ) ′ .For κ ∈ C ( s ), we let Ψ ( s , κ ) = Y Γ Ψ Γ ( s , κ ) , (2.6)where Ψ Γ ( s , κ ) : = { µ ∈ Ψ Γ ( s ) | κ Γ is the e Γ -core of µ Γ for every Γ ∈ F } .For integers d ≥ w ≥
0, we let P ( d , w ) = { ( µ , . . . , µ d ) | d X i = | µ i | = w } , (2.7)where µ , . . . , µ d are partitions.Now let P (2 e , w ) = { ( µ , . . . , µ e ) ∈ P (2 e , w ) | µ i = µ i + e for 1 ≤ i ≤ e } (2.8)and P (2 e , w ) = P (2 e , w ) \ P (2 e , w ). Then P (2 e , w ) is not empty if and only if w is even. Wedefine an equivalence relation on P (2 e , w ): for µ ( k ) = ( µ ( k )1 , . . . , µ ( k )2 e ) with k = , µ (1) ∼ µ (2) if and only if µ (1) i = µ (2) e + i and µ (2) i = µ (1) e + i for every 1 ≤ i ≤ e . (2.9)Let P ′ (2 e , w ) = P (2 e , w ) / ∼ . Then we define P ′ (2 e , w ) : = P (2 e , w ) ∪ P ′ (2 e , w ) , (2.10)where the elements of P (2 e , w ) are counted twice. In particular, if w is odd, then P (2 e , w ) isempty and P ′ (2 e , w ) = P ′ (2 e , w ).Let β Γ = Γ ∈ F or Γ ∈ F ∪ F . Then by [50, Prop. (3.7)] and the proofsof [49, Prop. 14 and 15], Ψ Γ ( s , κ ) is in bijection with P ( β Γ e Γ , w Γ ) if Γ ∈ F and κ Γ is non-degenerate or Γ ∈ F ∪ F , and P ′ (2 e , w Γ ) if Γ ∈ F and κ Γ is degenerate . (2.11)Again following the notation of [24] and [4], we denote by V and V ∗ finite-dimensionalsymplectic or orthogonal spaces over F q related as follows: V dimV V ∗ dimV ∗ symplectic 2 n orthogonal 2 n + n + n orthogonal 2 n orthogonal 2 n where η ( V ) = η ( V ∗ ) = η ( V ) = η ( V ∗ ) in the third case. Here η ( V ) = d , we denote by I ( d ) the identity matrix of degree d and by I d theidentity matrix of degree ℓ d . We end the section with the following lemmas. Lemma 2.15.
Let n = md and ε ∈ {± } . Let ι be the natural embedding of GL m ( ε q d ) into SO ε n ( q ) . Then ι (GL m ( ε q d )) * Ω ε n ( q ) = [SO ε n ( q ) , SO ε n ( q )] . roof. First assume that ε =
1. Now we give the structure of the embeddings. First note thatSO + n ( q ) = { A ∈ SL n ( q ) | A tr K n A = K n } , where K n = . . . . Let ξ be a generator of F × q d . Let ι be the embedding of GL m ( q d ) into GL n ( q ) induced by F × q d ֒ → GL d ( q ), ξ ( Λ ξ ),where Λ ξ is the minimal polynomial of ξ over F q . Let A ∈ GL n ( q ), then we take ι ( A ) = diag( A , K n ( A − ) tr K n ). Then ι = ι ◦ ι . Let θ : SO + n ( q ) → F × q / ( F × q ) be the spinor norm (see,for example, [29, §
9] or [34, § Ω + n ( q ) is the kernel of θ . Thus it su ffi ces to show that θ ( ι (GL m ( ε q d ))) = F × q / ( F × q ) .Let V be the orthogonal space with dim( V ) = n with a non-degenerate symmetric bilinearform B and basis ν , . . . , ν n such that B ( ν i , ν j ) = i + j = n + B ( ν i , ν j ) = v ∈ V , let σ v be the reflection along v , i.e. , σ ν ( u ) = u − B ( u , v ) B ( v , v ) v for any u ∈ V .For ζ ∈ F × q and 1 ≤ i ≤ n the matrix corresponding to σ ν i + ζν n − i + with respect to the basis ν , . . . , ν n is I ( i − − ζ I (2 n − i ) − ζ − I ( i − . Now let ξ be a generator of F × q d . Over the field F × q d , ι (diag( ξ, I ( m − )) is conjugate todiag( ξ, ξ q , . . . , ξ q d − , I (2( m − d ) , ξ q − d + , . . . , ξ − q , ξ − ) , which is d − Q i = σ ν i + ξ qi ν n − i + σ ν i + ν n − i + . Let Q ( v ) = B ( v , v ) for v ∈ V be the associated quadratic form.Then d − Q i = Q ( ν i + ξ q i ν n − i + ) Q ( ν i + ν n − i + ) = d − Q i = ξ q i = d ξ qd − q − ∈ F × q . By Remark 3 after [29,Cor. 9.9], the spinor norm can be determined from the Cli ff ord groups, and then is independentof the fields. Then θ ( ι (diag( a , I ( m − ))) = ξ qd − q − ( F × q ) . Note that ξ qd − q − is a generator of F × q , we have θ ( ι (GL m ( ε q d ))) = F × q / ( F × q ) .The proof for ε = − (cid:3) Lemma 2.16.
Let V and V be orthogonal spaces such that dim( V ) is even and dim( V ) = ℓ d .Suppose that C is a subgroup of I ( V ) satisfies that C * Ω ( V ) . Let V = V ⊗ V be theorthogonal space defined as in [34, p. 127]. Then C ⊗ I d * Ω ( V ) .Proof. Let θ : I ( V ) → F × q / ( F × q ) and θ : I ( V ) → F × q / ( F × q ) be the spinor norms. For anon-singular vector v ∈ V , we claim that θ ( σ v ⊗ I d ) = θ ( σ v ) ℓ d . In fact, by [34, Lem. 4.4.13], σ v ⊗ I d = σ v ⊗ w · · · σ v ⊗ w ℓ d , where w , . . . , w ℓ d is a basis of V as in [34, Prop. 2.5.12]. Thus θ ( σ v ⊗ I d ) = θ ( σ v ) ℓ d δ by direct calculation, where δ ∈ F × q is determined by the basis w , . . . , w ℓ d .For g ∈ C , we write g = σ v · · · σ v t . Then g ⊗ I d = ( σ v ⊗ I d ) · · · ( σ v t ⊗ I d ), and then θ ( g ⊗ I d ) = θ ( g ) ℓ d δ t which implies θ ( g ⊗ I d ) = θ ( g ) ℓ d since t is even. From this, C ⊗ I d * Ω ( V ) since ℓ isodd. (cid:3) A basic case of weights for classical groups I ( V ) We first give some more notation and conventions as in [4]. Let e be defined as in § a , and sign ε = ± ℓ a be the exact power of ℓ dividing q e − ε be the sign chosen so that ℓ a divides q e − ε .Let α, γ be non-negative integers, Z α be the cyclic group of order ℓ a + α and E γ be an ex-traspecial ℓ -group of order ℓ γ + . We may assume the exponent of E γ is ℓ by [4, (1B)]. Denoteby Z α E γ the central product of Z α and E γ over Ω ( Z α ) = Z ( E γ ). Let V α,γ be a symplectic ororthogonal space over F q of dimension 2 e ℓ α + γ and η ( V α,γ ) = ε if V α,γ is orthogonal. By [4,(1A)], the group Z α E γ can be embedded into GL ℓ γ ( ε q e ℓ α ) uniquely up to conjugacy in the sensethat Z α is identified with O ℓ ( Z (GL ℓ γ ( ε q e ℓ α ))). We denote by R α,γ the image of Z α E γ under thecomposition Z α E γ ֒ → GL ℓ γ ( ε q e ℓ α ) ֒ → I ( V α,γ ). Then by [4, (1C)], R α,γ is uniquely determinedby Z α E γ up to conjugacy.For an integer m ≥
1, let V m ,α,γ = V α,γ ⊥ · · · ⊥ V α,γ with m terms and let R m ,α,γ = R α,γ ⊗ I ( m ) . For each positive integer c , let A c denote the elementary abelian group of order ℓ c . For asequence of positive integers c = ( c , . . . , c t ) with t >
0, we denote by A c = A c ≀ · · · ≀ A c t and | c | = c + · · · + c t . Then A c can be regarded as an ℓ -subgroup of the symmetric group S ( ℓ | c | ).Let V m ,α,γ, c = V m ,α,γ ⊥ · · · ⊥ V m ,α,γ with ℓ | c | terms. Groups of the form R m ,α,γ, c = R m ,α,γ ≀ A c arecalled the basic subgroups of I ( V m ,α,γ, c ). Then R m ,α,γ, c is determined up to conjugacy in I ( V m ,α,γ, c )and η ( V m ,α,γ, c ) = ε m if V m ,α,γ, c is orthogonal. By [4, (2D)], any ℓ -radical subgroup R of I ( V ) isconjugate to R × R × · · · × R u , where R is a trivial group and R i ( i >
1) is a basic subgroup.Moreover, by the construction in [4], R m ,α,γ, c C I ( V m ,α,γ, c ) ( R m ,α,γ, c ) ≤ I ( V m ,α,γ, c ) (3.1)and N I ( V m ,α,γ, c ) ( R m ,α,γ, c ) (cid:2) I ( V m ,α,γ, c ) . (3.2)By [21, Lem.2.2(ii)], the map Rad ℓ ( I ( V )) → Rad ℓ ( I ( V )) given by R R ∩ I ( V ) is surjec-tive. Since ℓ is odd, we have that Rad ℓ ( I ( V )) = Rad ℓ ( I ( V )). Now assume that V is orthogonaland let Ω ( V ) = [ I ( V ) , I ( V )]. Then | I ( V ) : Ω ( V ) | =
2. Similarly, Rad ℓ ( Ω ( V )) = Rad ℓ ( I ( V )). Lemma 3.1.
Assume that V is orthogonal. Let R be an ℓ -radical subgroup of I ( V ) , thenN I ( V ) ( R ) (cid:2) I ( V ) and N I ( V ) ( R ) * Ω ( V ) .Proof. N I ( V ) ( R ) (cid:2) I ( V ) follows by (3.2). For the second assertion, it su ffi ces to show that θ ( C I ( V ) ( R )) = F × q / ( F × q ) , where θ is the spinor norm. If R = R is trivial, then C I ( V ) ( R ) = I ( V ).Now we assume that R , R . And then it su ffi ces to show that θ ( C I ( V m ,α,γ, c ) ( R m ,α,γ, c )) = F × q / ( F × q ) for any m , α, γ, c . Note that C I ( V m ,α,γ, c ) ( R m ,α,γ, c ) = C I ( V m ,α,γ, c ) ( R m ,α,γ, c ) = C m ,α ⊗ I γ ⊗ I c where C m ,α (cid:27) GL m ( ε q e ℓ α ) by [4, p.12-13]. Here C m ,α is the image of an embedding of GL m ( ε q e ℓ α ) into I ( V m ,α ). Thus the assertion follows by Lemma 2.15 and 2.16. (cid:3) The following lemma follows from Lemma 3.1 immediately.
Lemma 3.2.
Let V be orthogonal and R an ℓ -radical subgroup of I ( V ) . Then N I ( V ) ( R ) / N I ( V ) ( R ) (cid:27) I ( V ) / I ( V ) and N I ( V ) ( R ) / N Ω ( V ) ( R ) (cid:27) I ( V ) / Ω ( V ) . Let R be an ℓ -radical subgroup of I ( V ), then by Lemma 3.1, I ( V ) = I ( V ) N I ( V ) ( R ) and I ( V ) = Ω ( V ) N I ( V ) ( R ). So if two ℓ -radical subgroups of I ( V ) are I ( V )-conjugate, then they are I ( V )-conjugate and Ω ( V )-conjugate. Thus we have:13 orollary 3.3. Assume that V is orthogonal. Then
Rad ℓ ( I ( V )) / ∼ I ( V ) = Rad ℓ ( I ( V )) / ∼ I ( V ) = Rad ℓ ( Ω ( V )) / ∼ Ω ( V ) . Remark 3.4.
By the uniqueness of R m ,α,γ, c proved in [4, (1C)], we know that Aut( I ( V )) actstrivially on Rad ℓ ( I ( V )) / ∼ I ( V ) .Let V be a symplectic or orthogonal space, ˜ G = I ( V ), and R an ℓ -radical subgroup of ˜ G .Then there exists a corresponding decomposition V = V ⊥ V ⊥ · · · ⊥ V t , R = R × R × · · · × R t such that R is the trivial subgroup of I ( V ) and R i is a basic subgroup of I ( V i ) for i >
1. Let σ be an automorphism of ˜ G . Then there is an automorphism σ ′ of ˜ G , which is a composition of σ by some suitable inner automorphism, such that σ ′ stabilizes V i and R i for 0 ≤ i ≤ t . I ( V ) Given Γ ∈ F , let e Γ and β Γ be defined as in § α Γ and m Γ be the following: ℓ α Γ isthe exact power of ℓ dividing d Γ , and m Γ satisfies m Γ e ℓ α Γ = e Γ δ Γ . Recall that there is no directconnection between m Γ and m Γ ( s ).In this section, we let σ : = F p be the field automorphism of G = I ( V ) which sends ( a i j ) to( a pi j ) and let σ ∗ be the automorphism of G ∗ such that σ is dual to σ ∗ as in [55, § σ ∗ isalso the field automorphism which sends ( a i j ) to ( a pi j ).Recall that F ′ is defined to be the subset of F consisting of polynomials whose roots are of ℓ ′ -order. Given Γ ∈ F ′ , we define G Γ , R Γ , C Γ , θ Γ and s Γ as follows: let V Γ denote a symplecticor orthogonal space of dimension 2 e Γ δ Γ over F q and of type ε or ε e Γ Γ according as Γ ∈ F or Γ ∈ F ∪ F if V Γ is orthogonal. Let ˜ G Γ = I ( V Γ ) and G Γ = I ( V Γ ). Thus ˜ G Γ has a primaryelement s ∗ Γ with a unique elementary divisor Γ of multiplicity β Γ e Γ and ˜ G Γ has a basic subgroup R Γ of form R m Γ ,α Γ , by [24, (1.12) and (5.2)]. Let ˜ C Γ = C ˜ G Γ ( R Γ ) and ˜ N Γ = N ˜ G Γ ( R Γ ). Then s ∗ Γ ∈ G Γ ,˜ C Γ ≤ G Γ and ˜ C Γ (cid:27) GL m Γ ( ε q e ℓ α Γ ), so that a Coxeter torus ˜ T Γ of ˜ C Γ has order q m Γ e ℓ α Γ − ε m Γ . Thedual ˜ T ∗ Γ is embedded as a regular subgroup of ˜ C ∗ Γ , and ˜ C ∗ Γ is embedded as a regular subgroup of G Γ ∗ . By [4, p. 22], there exists an element s Γ in ˜ T ∗ Γ such that C ˜ C ∗ Γ ( s Γ ) = ˜ T ∗ Γ and as an element of G Γ ∗ , s Γ and s ∗ Γ are dual to each other in the sense of [4, (3E)]. Here, s Γ has unique elementarydivisor Γ and s Γ is uniquely determined by Γ up to I ( V ∗ Γ )-conjugacy. We denote by b s Γ thecharacter of ˜ T Γ corresponding to s Γ and let ˜ θ Γ = ± R ˜ C Γ ˜ T Γ ( b s Γ ) where the sign is chosen so ˜ θ Γ is anirreducible character of ˜ C Γ . The block ˜ b Γ of ˜ C Γ containing ˜ θ Γ then has defect group R Γ by [22,(4C)] or [9, (3.2)].Let c , γ be a tuple and an integer as in the previous sections, and δ = | c | + γ . Let V Γ ,δ = V Γ ⊥ · · · ⊥ V Γ , where there are ℓ δ terms V Γ on the right-hand side. Then if V Γ is orthogonal, V Γ ,δ has type ε ℓ δ = ε or ε e Γ ℓ δ Γ = ε e Γ Γ according as Γ ∈ F or Γ ∈ F ∪ F . Let ˜ G Γ ,γ, c = I ( V Γ ,δ ), G Γ ,γ, c = I ( V Γ ,δ ), R Γ ,γ, c = R m Γ ,α Γ ,γ, c , ˜ N Γ ,γ, c = N ˜ G Γ ,γ, c ( R Γ ,γ, c ) and ˜ C Γ ,γ, c = C ˜ G Γ ,γ, c ( R Γ ,γ, c ). Then˜ C Γ ,γ, c (cid:27) ˜ C Γ ⊗ I δ . Let ˜ θ Γ ,γ, c = ˜ θ Γ ⊗ I δ , then ˜ θ Γ ,γ, c can be viewed as a canonical character of ˜ C Γ ,γ, c R Γ ,γ, c with R Γ ,γ, c in the kernel and all canonical characters are of this form. Let B Γ ,δ, i = bl ℓ ( ˜ θ Γ ,δ, i ) G Γ ,δ, i .Then B Γ ,δ, i ⊆ E ℓ ( G Γ ,δ, i , x Γ ), where x Γ = s Γ ⊗ I δ by the proof of [4, (4A)].Let R Γ ,δ be the set of all the basic subgroups of the form R Γ ,γ, c with γ + | c | = δ . Label thebasic subgroups in R Γ ,δ as R Γ ,δ, , R Γ ,δ, , . . . and we denote the canonical character associated to R Γ ,δ, i by θ Γ ,δ, i . It is possible that there exists Γ ′ ∈ F ′ such that m Γ ′ = m Γ = : m and α Γ ′ = α Γ = : α .In this case, R Γ ,δ = R Γ ′ ,δ and naturally we may choose the labeling of R Γ ,δ and R Γ ′ ,δ such that R Γ ,δ, i = R Γ ′ ,δ, i for i = , , . . . . By convention, we denote R m ,α,γ, c as R Γ ,δ, i or R Γ ′ ,δ, i depending onwhether the related canonical character of ˜ C m ,α R m ,α = ˜ C m ,α considered is θ Γ or θ Γ ′ .Let C Γ ,δ be the set of characters of ( ˜ N Γ ,δ, i ) ˜ θ Γ ,δ, i lying over ˜ θ Γ ,δ, i and of defect zero as charactersof ( ˜ N Γ ,δ, i ) ˜ θ Γ ,δ, i / R Γ ,δ, i for all i . By Cli ff ord theory, this set is in bijection with the set of characters14f ˜ N Γ ,δ, i lying over ˜ θ Γ ,δ, i and of defect zero as characters of ˜ N Γ ,δ, i / R Γ ,δ, i for all i . We assume C Γ ,δ = { ˜ ψ Γ ,δ, i , j } with ˜ ψ Γ ,δ, i , j a character of ( ˜ N Γ ,δ, i ) ˜ θ Γ ,δ, i . Then | C Γ ,δ | = β Γ e Γ ℓ d by the proof of [4,(4A)].We define σ ∗ Γ to be the unique elementary divisor of σ ∗ ( s Γ ). Obviously m σ ∗ Γ = m Γ , α σ ∗ Γ = α Γ and R Γ ,δ, i = R σ ∗ Γ ,δ, i . By Remark 3.4, we may assume R σ Γ ,δ, i = R Γ ,δ, i up to a composition of σ bya suitable inner automorphism. Then we may assume B σ Γ ,δ, i = B σ ∗− Γ ,δ, i since B Γ ,δ, i ⊆ E ℓ ( G Γ ,δ, i , x Γ ), B σ Γ ,δ, i ⊆ E ℓ ( G Γ ,δ, i , σ ∗− ( x Γ )) and σ ∗− ( x Γ ) = x σ ∗− Γ (see for instance [55, Prop. 7.2]). Since ˜ θ Γ ,δ, i is the canonical character of a root block of B Γ ,δ, i , we have ˜ θ σ Γ ,δ, i = ˜ θ σ ∗− Γ ,δ, i up to a compositionof σ by an inner automorphism. Then we may denote R σ Γ ,δ, i = R σ ∗− Γ ,δ, i , ˜ N σ Γ ,δ, i = ˜ N σ ∗− Γ ,δ, i and˜ C σ Γ ,δ, i = ˜ C σ ∗− Γ ,δ, i although the corresponding terms indexed by Γ and σ ∗− Γ are actually the same.Also (( ˜ N Γ ,δ, i ) ˜ θ Γ ,δ, i ) σ = ( ˜ N Γ ,δ, i ) ˜ θ σ Γ ,δ, i = ( ˜ N σ ∗− Γ ,δ, i ) ˜ θ σ ∗− Γ ,δ, i . We may choose the labeling of C Γ ,δ and C σ ∗− Γ ,δ such that ˜ ψ σ Γ ,δ, i , j = ˜ ψ σ ∗− Γ ,δ, i , j . (3.3) Remark 3.5.
We can assume (3.3) because ˜ ψ Γ ,δ, i , j is invariant under the action of σ if σ ∗− Γ = Γ .We prove this as follows. First note that σ ∗− Γ = Γ if and only if ˜ θ Γ ,δ, i is invariant under theaction of σ . Let R Γ ,δ, i = R m Γ ,α Γ ,γ, c with c = ( c , . . . , c t ). We also abbreviate R = R Γ ,δ, i , ˜ N = ˜ N Γ ,δ, i ,˜ C = ˜ C Γ ,δ, i and ˜ θ = ˜ θ Γ ,δ, i . By [4, (2E)], ˜ N / R = ˜ N m Γ ,α Γ ,γ / R m Γ ,α Γ ,γ × Q ti = GL c i ( ℓ ) and then we mayassume that | c | = i.e. , R = R m Γ ,α Γ ,γ . Let ˜ N = { g ∈ ˜ N | [ g , Z ( R )] = } , then by the remark after[4, (3I)], ˜ N ≤ ˜ N ˜ θ and Also, ˜ N = CRL , where L ≤ ˜ N satisfies [ L , C ] = L ∩ C = Z ( L ) = Z ( C )and L / Z ( L ) R (cid:27) Sp γ ( ℓ ). Then there is exactly one character ϑ of ˜ N which lies over ˜ θ and of ℓ -defect zero when viewed as a character of ˜ N / R and we may write ϑ = ˜ θ × ζ , where ζ isthe Steinberg character of Sp γ ( ℓ ). Hence ϑ is invariant under the action of σ . On the otherhand, we have ˜ N ˜ θ / ˜ N (cid:27) ˜ N Γ / ˜ C Γ . From this we may assume further that γ = δ = R = R Γ , ˜ N = ˜ N Γ , ˜ C = ˜ C Γ and ˜ θ = ˜ θ Γ . Now ˜ C (cid:27) GL m Γ ( ε q e ℓ α Γ ). By [24, (1.14)],˜ N = h ˜ C , D i , where D is generated by one or two element and every element of D acts on ˜ C asa field or graph automorphism. Then ˜ θ extends to h ˜ N , σ i by a result of Bonnaf´e [5, Thm. 4.3.1and Lem. 4.3.2] (see [21, Prop. 4.17] for details). Thus σ acts trivially on Irr( ˜ N | ˜ θ ).Now let V be a symplectic or even-dimensional orthogonal space and let ˜ B be an ℓ -blockof ˜ G = I ( V ) with defect group D and root block b such that V = [ V , D ] and b G ⊆ E ℓ ( G , s ) forsome semisimple ℓ ′ -element s ∈ G ∗ . Let s ∗ be a dual of s in G in the sense of [4, (3E)]. Then m Γ ( s ∗ ) = w Γ β Γ e Γ for some positive integer w Γ . Similar with [38, p.145] for groups of type A ,now we define i W ℓ ( ˜ B ) to be the set of elements K = K Γ , where K Γ : S δ C Γ ,δ → { ℓ -cores } suchthat P δ, i , j ℓ δ | K Γ ( ψ Γ ,δ, i , j ) | = w Γ . Here, an ℓ -core means ℓ -core of some partition.A bijection between W ℓ ( ˜ B ) and i W ℓ ( ˜ B ) has been constructed implicitly in [4, (4E)] andcan be described as follows. Let ( R , ˜ ϕ ) be an ℓ -weight of ˜ G . Set ˜ C = C ˜ G ( R ) and ˜ N = N ˜ G ( R ).Thus there exists an ℓ -block ˜ b of ˜ CR with R a defect group such that ˜ ϕ = Ind ˜ N ˜ N ( θ ) ˜ ψ where ˜ θ isthe canonical character of ˜ b and ˜ ψ is a character of ˜ N ( ˜ θ ) lying over ˜ θ and of ℓ -defect zero as acharacter of ˜ N ( ˜ θ ) / R .We may suppose that Z ( D ) ≤ Z ( R ) ≤ R ≤ D so that V = [ V , R ]. Assume we havethe following decomposition ˜ θ + = Q Γ ,δ, i ˜ θ t Γ ,δ, i Γ ,δ, i , R + = Q Γ ,δ, i R t Γ ,δ, i Γ ,δ, i . Note that θ Γ determines asemisimple ℓ ′ -element of G Γ with a unique elementary divisor Γ of multiplicity β Γ e Γ . Thus m Γ ( s ∗ ) = Q δ, i t Γ ,δ, i β Γ e Γ ℓ δ for each Γ .Now we have ˜ N ( ˜ θ ) = Y Γ ,δ, i ˜ N Γ ,δ, i ( ˜ θ Γ ,δ, i ) ≀ S ( t Γ ,δ, i ) , ˜ ψ = Y Γ ,δ, i ˜ ψ Γ ,δ, i ψ Γ ,δ, i a character of ˜ N Γ ,δ, i ( ˜ θ Γ ,δ, i ) ≀ S ( t Γ ,δ, i ) covering ˜ θ t Γ ,δ, i Γ ,δ, i and of ℓ -defect zero as a character of (cid:16) ˜ N Γ ,δ, i ( ˜ θ Γ ,δ, i ) ≀ S ( t Γ ,δ, i ) (cid:17) / R t Γ ,δ, i Γ ,δ, i . By Cli ff ord theory, ˜ ψ Γ ,δ, i is of the formInd ˜ N Γ ,δ, i ( θ Γ ,δ, i ) ≀ S ( t Γ ,δ, i )˜ N Γ ,δ, i ( θ Γ ,δ, i ) ≀ Q j S ( t Γ ,δ, i , j ) Y j ˜ ψ t Γ ,δ, i , j Γ ,δ, i , j · Y j φ κ Γ ,δ, i , j (3.4)where t Γ ,δ, i = P j t Γ ,δ, i , j , Q j ˜ ψ t Γ ,δ, i , j Γ ,δ, i , j is an extension of Q j ˜ ψ t Γ ,δ, i , j Γ ,δ, i , j from ˜ N Γ ,δ, i ( ˜ θ Γ ,δ, i ) t Γ ,δ, i to ˜ N Γ ,δ, i ( ˜ θ Γ ,δ, i ) ≀ Q j S ( t Γ ,δ, i , j ), κ Γ ,δ, i , j ⊢ t Γ ,δ, i , j is an e Γ -core and φ κ Γ ,δ, i , j a character of S ( t Γ ,δ, i , j ) corresponding to κ Γ ,δ, i , j . Now, define K Γ : S δ C Γ ,δ → { ℓ -cores } , ψ Γ ,δ, i , j κ Γ ,δ, i , j . Then K = Q Γ K Γ is the labelingof the ℓ -weight ( R , ˜ ϕ ).We can define the action of σ ∗ on K by ( σ ∗ K ) σ ∗ Γ = K Γ . Lemma 3.6.
With the notation above, if ( R , ˜ ϕ ) is a ˜ B-weight with label K, ( R , ˜ ϕ ) σ is a ˜ B σ -weightwith label σ ∗− K.Proof.
Let K ′ be the label of ( R , ˜ ϕ ) σ . First note that R σ = R , ˜ C σ = ˜ C , ˜ N σ = ˜ N , and σ stabilizesevery ˜ C Γ ,δ, i up to conjugacy.By the argument above, we may denote R σ Γ ,δ, i = R σ ∗− Γ ,δ, i , ˜ N σ Γ ,δ, i = ˜ N σ ∗− Γ ,δ, i and ˜ C σ Γ ,δ, i = ˜ C σ ∗− Γ ,δ, i although the corresponding terms indexed by Γ and σ ∗− Γ are actually the same. Todetermine K ′ , we note that ˜ ψ σ = Q Γ ,δ, i ˜ ψ σ Γ ,δ, i . By (3.4), ˜ ψ σ Γ ,δ, i isInd ˜ N Γ ,δ, i (˜ θ Γ ,δ, i ) σ ≀ S ( t Γ ,δ, i )˜ N Γ ,δ, i (˜ θ Γ ,δ, i ) σ ≀ Q j S ( t Γ ,δ, i , j ) Y j ˜ ψ t Γ ,δ, i , j Γ ,δ, i , j σ · Y j φ κ Γ ,δ, i , j . Here, we note that σ acts trivially on S ( t Γ ,δ, i ) and S ( t Γ ,δ, i , j ) by Remark 3.4. Since ˜ θ σ Γ ,δ, i = ˜ θ σ ∗− Γ ,δ, i ,we have ˜ N Γ ,δ, i ( ˜ θ Γ ,δ, i ) σ = ˜ N σ ∗− Γ ,δ, i ( ˜ θ σ ∗− Γ ,δ, i ). We can fix the way to extend Q j ˜ ψ t Γ ,δ, i , j Γ ,δ, i , j as in [30,Lem. 25.5] , then we have that Y j ˜ ψ t Γ ,δ, i , j Γ ,δ, i , j σ = Y j (cid:16) ˜ ψ σ Γ ,δ, i , j (cid:17) t Γ ,δ, i , j . Since ˜ ψ σ Γ ,δ, i , j = ψ σ ∗− Γ ,δ, i , j by (3.3), ˜ ψ σ Γ ,δ, i isInd ˜ N σ ∗− Γ ,δ, i (˜ θ σ ∗− Γ ,δ, i ) ≀ S ( t Γ ,δ, i )˜ N σ ∗− Γ ,δ, i (˜ θ σ ∗− Γ ,δ, i ) ≀ Q j S ( t Γ ,δ, i , j ) Y j ˜ ψ t Γ ,δ, i , j σ ∗− Γ ,δ, i , j · Y j φ κ Γ ,δ, i , j . Then K ′ σ ∗− Γ = K Γ which is just K ′ = σ ∗− K . (cid:3) Thus, by a similar proof as in [2, (1A)], there is a canonical bijection between P ( β Γ e Γ , w Γ )(defined as in (2.7)) and K Γ for every Γ ∈ F by [50, Prop.(3.7)]. Let P ( ˜ B ) = Q Γ P ( β Γ e Γ , w Γ ).Then by the argument above, we have a bijection between P ( ˜ B ) and W ℓ ( ˜ B ). We also define σ ∗ µ = Q Γ ( σ ∗ µ ) Γ with ( σ ∗ µ ) σ ∗ Γ = µ Γ . By Lemma 3.6, we have Corollary 3.7.
With the notation above, if ( R , ˜ ϕ ) is a ˜ B-weight with label µ ∈ P ( ˜ B ) , then ( R , ˜ ϕ ) σ is a ˜ B σ -weight with label σ ∗− µ . Now we consider the action of diagonal automorphisms on the weights of ˜ B . The followingtwo lemmas will be useful. 16 emma 3.8. Let H be an arbitrary finite group, L , K E H and M = L ∩ K such that | H / K | = and H / L is cyclic. Suppose that ϕ ∈ Irr( L ) such that θ = Res LM ϕ ∈ Irr( M ) . Assume further θ isH-invariant. Then | H : H ϕ | ≤ . Moreover, exactly one of the following statements hold.(i) | H : H ϕ | = and Res HK χ is irreducible for every χ ∈ Irr( H θ ) .(ii) | H : H ϕ | = and Res HK χ is not irreducible for every χ ∈ Irr( H θ ) .Proof. Let χ ∈ Irr( H | ϕ ). First by Cli ff ord theory, we may write χ = Ind HH ϕ ψ for some ψ ∈ Irr( H ϕ | ϕ ) and then Mackey formula implies that Res HK χ = Ind KK ϕ (Res H ϕ K ϕ ψ ). Note that ψ isan extension of θ . So Res H ϕ K ϕ ψ is irreducible. Thus Ind KK ϕ (Res H ϕ K ϕ ψ ) is a sum of | H : H ϕ | irreducibleconstituents. Hence | H : H ϕ | ≤ | H / K | = (cid:3) Lemma 3.9.
Let H be an arbitrary finite group and M E H such that H = C H ( M ) M. Supposethat θ ∈ Irr( M ) and η ∈ Irr( C H ( M )) such that η (1) = and Irr( Z ( M ) | θ ) = Irr( Z ( M ) | η ) . Thenthere exists a unique extension ϕ of θ to H with Irr( C H ( M ) | ϕ ) = { η } . In particular, ϕ ( c ) , forevery c ∈ C H ( M ) .Proof. In fact, this follows from [53, 2.1]. But for convenience, we still give the details here. Let D : M → GL θ (1) ( C ) be a C -representation of M a ff ording θ . Then define D ′ : H → GL θ (1) ( C ), cg η ( c ) D ( g ) for c ∈ C H ( M ) and g ∈ M . It is easy to check that D ′ is well-defined and is a C -representation of H . Let ϕ be the character a ff orded by D ′ . Then ϕ is the unique extensionof θ to H with Irr( C H ( M ) | ϕ ) = { η } . Let c ∈ C H ( M ), then ϕ ( c ) = η ( c ) θ (1). Now η is a linearcharacter, then η ( c ) , c ∈ C H ( M ). Thus ϕ ( c ) , c ∈ C H ( M ). (cid:3) Following the notation of [24], we denote by J ( V ) the group of all conformal endomor-phisms of V when dim( V ) is even. Then J ( V ) = CSp( V ) or CO( V ) according as V is sym-plectic or orthogonal. We also let J ( V ) = CSp( V ) or CSO( V ) according as V is symplecticor orthogonal. Then | J ( V ) / I ( V ) Z ( J ( V )) | =
2. Let J ( V ) = h I ( V ) , τ i where τ ∈ J ( V ). Ob-viously, J ( V ) = h I ( V ) , τ i . Then for a basic subgroup R Γ ,δ, i , up to a composition of τ by somesuitable inner automorphism, which is denoted by τ ′ , we have that N J ( V Γ ,δ ) ( R Γ ,δ, i ) = h ˜ N Γ ,δ, i , τ ′ i , C J ( V Γ ,δ ) ( R Γ ,δ, i ) = h ˜ C Γ ,δ, i , τ ′ i and τ ′ commutes with R Γ ,δ, i ˜ C Γ ,δ, i by [24, § Lemma 3.10.
Keep the hypothesis and setup above.(i) If Γ , x + , then every element of C Γ ,δ is invariant under τ ′ .(ii) If Γ = x + , then no element of C Γ ,δ is invariant under τ ′ .Proof. Similar with the argument of Remark 3.5, we may assume that δ =
0. Then R Γ ,δ, i = R Γ ,˜ N Γ ,δ, i = ˜ N Γ , ˜ C Γ ,δ, i = ˜ C Γ and ˜ θ Γ ,δ, i = ˜ θ Γ . In this way C Γ ,δ is the set of extensions of ˜ θ Γ to ˜ N Γ .Recall that ˜ θ Γ = ± R ˜ C Γ ˜ T Γ ( b s Γ ). We abbreviate R = R Γ , ˜ N = ˜ N Γ , ˜ C = ˜ C Γ , ˜ θ = ˜ θ Γ , ˜ N ′ = N J ( V Γ ) ( R Γ )and ˜ C ′ = C J ( V Γ ) ( R Γ ). Also, there is an extension ˜ θ ′ of ˜ θ to ˜ C ′ . Claim that ˜ θ ′ ( τ ′ ) ,
0. Inorder to do this, we choose a canonical ˜ θ ′ . First note that C ˜ C ′ ( ˜ C ) = h Z ( ˜ C ) , τ ′ i is abelian. Let { η } = Irr( Z ( ˜ C ) | ˜ θ ) and take η ′ to be an extension of η to C ˜ C ′ ( ˜ C ). By Lemma 3.9, there is aunqiue extension ˜ θ ′ of ˜ θ to ˜ C ′ with { η ′ } = Irr( C ˜ C ′ ( ˜ C ) | ˜ θ ′ ). In particular, ˜ θ ′ ( τ ′ ) ,
0, as claimed.Hence for g ∈ ˜ N , ( ˜ θ ′ ) g = ˜ θ ′ if and only if ˜ θ ′ ( g τ ′ ) = ˜ θ ′ ( τ ′ ) by [51, Cor. 1.22]. Also by [24, (5A)],[ τ ′ , ˜ N ] ≤ Z ( ˜ C ). So ˜ θ ′ ( g τ ′ ) = ˜ θ ([ g , τ ′ ]) ˜ θ ′ ( τ ′ ) and thus ( ˜ θ ′ ) g = ˜ θ ′ if and only if [ g , τ ′ ] ∈ ker( ˜ θ ).Now we calculate the stabilizer of ˜ θ ′ in ˜ N ′ .If Γ = x −
1, then ˜ θ ′ is the trivial character. Thus ˜ N ′ ˜ θ ′ = ˜ N ′ . If Γ ∈ F ∪ F , then by [24,(6A)(2)], we also have ˜ N ′ ˜ θ ′ = ˜ N ′ . If Γ = x +
1, then by [24, (6A)(3)], we have ˜ N ′ ˜ θ ′ , ˜ N ′ . Hencethe assertion holds by Lemma 3.8. (cid:3) Γ = x +
1, we recall that | C Γ ,δ | = e ℓ δ . Thus by Lemma 3.10, we may rewrite C x + ,δ = { ˜ ψ x + ,δ, i , j | ≤ i ≤ e , ≤ j ≤ ℓ δ } such that ˜ ψ τ ′ x + ,δ, i , j = ˜ ψ x + ,δ, e + i , j for every 1 ≤ i ≤ e . By Lemma3.10 again, we have the following result by a similar argument with Lemma 3.6 and 3.7. Proposition 3.11.
With the notation of Corollary 3.7, we let ( R , ˜ ϕ ) be a ˜ B-weight with label µ ∈ P ( ˜ B ) and write µ = Q Γ µ Γ , where µ Γ = ( µ (1) Γ , . . . , µ ( β Γ e Γ ) Γ ) . Then the image of ( R , ˜ ϕ ) under thenon-trivial action of J ( V ) / Z ( J ( V )) I ( V ) is a ˜ B-weight with label µ † ∈ P ( ˜ B ) , where µ † = Q Γ µ † Γ with µ † Γ = ( µ † Γ (1) , . . . , µ † Γ ( β Γ e Γ ) ) such that µ † Γ = µ Γ if Γ , x + and µ † x + i ) = µ ( e + i ) x + , µ † x + e + i ) = µ ( i ) x + for every ≤ i ≤ e. B In this section, we let G = SO n + ( q ) and G = SO n + ( F q ) (a connected reductive algebraicgroup). As usual, we always denote by F p the field automorphism which sends ( a i j ) to ( a pi j ) andwe write E = h F p i . Let F : = F fp be the standard Frobenius endomorphism over G . We write G F for the group of fixed points, then G = G F . As before, we denote by G ∗ = Sp n ( F q ) the dualof G and G ∗ = G ∗ F = Sp n ( q ). ℓ -Brauer characters of SO n + ( q ) and Ω n + ( q ) Let s ∈ G ∗ F be a semisimple element and let Ψ ( s ) = Q Γ Ψ Γ ( s ) be defined as in (2.4). Thenthe unipotent characters of C G ∗ ( s ) F are in bijection with Ψ ( s ). For µ ∈ Ψ ( s ), we denote ψ µ the unipotent character of C G ∗ ( s ) F corresponding to µ . Now we define i Irr( G ) to be the set of G ∗ -conjugacy classes of pairs ( s , µ ) where s is a semisimple element of G ∗ F and µ ∈ Ψ ( s ).Here two pairs ( s , µ ) and ( s , µ ) are said to be G ∗ -conjugate if there exists g ∈ G ∗ such that s = g s and µ = µ . The irreducible characters of G have been classified by Lusztig [39].By the Jordan decomposition of characters of G , there is a bijection between E ( C G ∗ F ( s ) ,
1) and E ( G F , s ) for every semisimple element s of G ∗ . For µ ∈ Ψ ( s ), we denote χ s ,µ the character in E ( G F , s ) corresponding to ψ µ . So i Irr( G ) is a labeling set of the characters of G .In this section, we assume that σ = F p is the field automorphism, then σ ∗ is also the fieldautomorphism F p of G ∗ . For µ ∈ Ψ ( s ), we define σµ = Q Γ ( σµ ) Γ , with ( σµ ) σ ∗ Γ = µ Γ . Proposition 4.1.
With the above definitions, we have χ σ s ,µ = χ σ ∗− ( s ) ,σ ∗− µ in the sense that thepair ( s , µ ) in the subscript means in fact a G ∗ -conjugacy class.Proof. This follows by [17, Thm. 3.1] (or [20, Prop. 1.3.1(iv)]) and the fact that the unipotentcharacters of symplectic groups with odd defining characteristic and general linear and unitarygroups are invariant under the automorphism groups (see, for example, [41, Thm. 2.5]). (cid:3)
Let s be a semisimple ℓ ′ -element and let C ( s ) = Q Γ C Γ ( s ) as (2.5). We define i Bl ℓ ( G ) tobe the set of G ∗ -conjugacy classes of pairs ( s , κ ) where s is a semisimple ℓ ′ -element of G ∗ and κ ∈ C ( s ). Then by [24, (10B)], there is a bijection ( s , κ ) B ( s , κ ) from i Bl ℓ ( G ) to Bl ℓ ( G ). Alsoby [24, (11A)], Irr( B ( s , κ )) ∩ E ( G , ℓ ′ ) = { χ s ,µ | µ ∈ Ψ ( s , κ ) } , where Ψ ( s , κ ) is as defined in (2.6).We define the action of σ on i Bl ℓ ( G ) similarly as the action on i Irr( G ). The following resultcan be deduced directly from Proposition 4.1. Proposition 4.2. B ( s , κ ) σ = B ( σ ∗− ( s ) , σ ∗− κ ) . We define i IBr ℓ ( G ) : = { ( s , µ ) ∈ i Irr( G ) | s is of ℓ ′ -order } . Then for ( t , µ ) ∈ i IBr ℓ ( G ), χ t ,µ liesin the ℓ -block B ( s , κ ) if and only if t is G ∗ -conjugate to s and µ ∈ Ψ ( s , κ ).18 roposition 4.3. Assume that ℓ is linear. There is a bijection ( s , µ ) φ s ,µ from i IBr ℓ ( G ) to IBr ℓ ( G ) such that φ σ s ,µ = φ σ ∗− ( s ) ,σ ∗− µ .Proof. By Theorem 2.14, E ( G F , ℓ ′ ) is a basic set of IBr ℓ ( G ). By [27], the decomposition matrixwith respect to E ( G F , ℓ ′ ) is unitriangular since ℓ is linear. Then by [17, Lem. 7.5], there is a σ -equivariant bijection from E ( G F , ℓ ′ ) to IBr ℓ ( G ) which preserves blocks. Thus the assertionfollows from Proposition 4.1. (cid:3) Remark 4.4.
In fact, by the construction in [17, Lem. 7.5], the proof of Proposition 4.3 gives abijection D : E ( G F , ℓ ′ ) → IBr ℓ ( G ) such that φ s ,µ = D ( χ s ,µ ) for all ( s , µ ) ∈ i IBr ℓ ( G ). In addition,there is a partial order relation ≤ on IBr ℓ ( G ), such that χ ◦ s ,µ = φ s ,µ + P ϕ ∈ IBr ℓ ( G ) ,ϕ (cid:12) ϕ d ϕ ϕ with d ϕ ∈ Z .Note that Z ( G ∗ ) = h z i , where z = − I (2 n ) . For a semisimple element s ∈ G ∗ , we write − s : = zs = − I (2 n ) · s . For Γ ∈ F , let ξ be a root of Γ . We define z . Γ to be the unique polynomialin F such that − ξ is a root of z . Γ . For µ ∈ Ψ ( s ), we define − µ = ( − µ ) z . Γ , with ( − µ ) z . Γ = µ Γ .Let ˆ z ∈ E ( G F , z ) be the character corresponding under Jordan decomposition to 1 G ∈ E ( G F , z is the (unique) non-trivial linear character of G (the definition of ˆ z also follows from [19,Prop. 13.30]). Then by [20, Prop. 1.3.1(ii)], we have the following result. Proposition 4.5. ˆ z χ s ,µ = χ − s , − µ in the sense that the pair ( s , µ ) in the subscript means in fact aG ∗ -conjugacy class. Since ℓ is odd and z has order 2, the character ˆ z in Proposition 4.5 can be regarded as a linear ℓ -Brauer character of G . Proposition 4.6.
With the notation of Proposition 4.3 and 4.5, if ℓ is linear, then ˆ z φ s ,µ = φ − s , − µ in the sense that the pair ( s , µ ) in the subscript means in fact a G ∗ -conjugacy class.Proof. Here, we use [21, Lem. 2.4]. By its proof, ˆ z induces an automorphism of the associatedgroup algebra. Then it permutes the irreducible ordinary and ℓ -Brauer characters in the wayindicated. Thus the assertion follows from Remark 4.4, Proposition 4.3 and 4.5. (cid:3) SO n + ( q ) Now we let V be an odd-dimensional orthogonal space, ˜ G = I ( V ) and G = I ( V ). Then˜ G = Z ( ˜ G ) × G . Define i W ℓ ( G ) = ( s , κ, K ) G ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s is a semisimple ℓ ′ -element of G ∗ , κ ∈ C ( s ) , K = K Γ , K Γ : S δ C Γ ,δ → { ℓ -cores } s.t. P δ, i , j ℓ δ | K Γ ( ψ Γ ,δ, i , j ) | = w Γ , m Γ ( s ) /β Γ = | κ Γ | + e Γ w Γ . Here, ( s , κ, K ) G ∗ means a G ∗ -conjugacy class of ( s , κ, K ).A bijection between W ℓ ( G ) and i W ℓ ( G ) has been constructed implicitly in the proof of [4,(4G)] and can be described as follows. Let ( R , ϕ ) be an ℓ -weight of G . Then ( R , ˜ ϕ ) is an ℓ -weightof ˜ G , where ˜ ϕ = Z ( ˜ G ) × ϕ . Set ˜ C = C ˜ G ( ˜ R ) and ˜ N = N ˜ G ( ˜ R ). Thus there exists an ℓ -block ˜ b of ˜ C ˜ R with ˜ R a defect group such that ˜ ϕ = Ind ˜ N ˜ N ( θ ) ˜ ψ where ˜ θ is the canonical character of ˜ b and ˜ ψ is acharacter of ˜ N ( ˜ θ ) lying over ˜ θ and of ℓ -defect zero as a character of ˜ N ( ˜ θ ) / ˜ R .Let V = C V ( R ) and V + = [ V , R ]. Then V = V ⊥ V + and V + is an even-dimensionalorthogonal space. Suppose that dim( V ) = n +
1. In addition, let ˜ G = I ( V ), G = I ( V ),˜ G + = I ( V + ) and G + = I ( V + ). Then R = R × R + , ˜ b = ˜ b × ˜ b + , ˜ θ = ˜ θ × ˜ θ + , where R = h V i ≤ ˜ G , R + ≤ ˜ G + , ˜ b , ˜ b + are ℓ -blocks of ˜ G , C ˜ G + ( R + ) respectively, and ˜ θ ∈ Irr(˜ b ), ˜ θ + ∈ Irr(˜ b + ).19irst, we let ˜ C = ˜ N = ˜ G , C + = C I ( V + ) ( R + ), and ˜ N + = N I ( V + ) ( R + ). Then ˜ ϕ = ˜ ψ = ˜ θ acharacter of ˜ G of ℓ -defect zero. Let ˜ θ = h− V i × θ , where θ is a character ( ℓ -defect zero) of G (cid:27) SO n + ( q ). So it is of the form χ s ,κ where s is a semisimple ℓ ′ -element of G ∗ (cid:27) Sp n ( q )and κ ∈ Ψ ( s ) such that κ Γ is an e Γ -core which a ff ords the second component of the triple( s , κ, K ).Secondly, assume we have the following decomposition ˜ θ + = Q Γ ,δ, i ˜ θ t Γ ,δ, i Γ ,δ, i , R + = Q Γ ,δ, i R t Γ ,δ, i Γ ,δ, i .θ Γ determines a semisimple ℓ ′ -element of G Γ with a unique elementary divisor Γ of multiplic-ity β Γ e Γ . Let s + be a semisimple element of G + such that s + has divisors Γ with multiplicity Q δ, i t Γ ,δ, i β Γ e Γ ℓ δ . Then s = s × s + is the first component of the triple ( s , κ, K ). We can viewthe block ˜ b as a block of C ˜ G ( R ). Thus ( R , ϕ ) belongs to an ℓ -block of G with label ( s , κ ). Inparticular, κ ∈ C ( s ).Finally, the correspondence ( R , Ind ˜ N ˜ N ( θ ) ˜ ψ )) ( R + , Ind ˜ N + ˜ N + ( θ + ) ˜ ψ + )) is a bijection from { ( R , Ind ˜ N ˜ N ( θ ) ˜ ψ )) | ˜ ψ ∈ Irr ( ˜ N ( θ ) | ˜ θ ) } to { ( R + , Ind ˜ N + ˜ N + ( θ + ) ˜ ψ + )) | ˜ ψ + ∈ Irr ( ˜ N + ( θ + ) | ˜ θ + ) } . Then the third component K = Q Γ K Γ of the triple ( s , κ, K ) is given as in the statement preceding Lemma 3.6.Let ( R , ϕ ) be the ℓ -weight of G with label ( s , κ , K ). Then by the proof of [4, (4G)], ( R , ϕ )is a B ( s , κ )-weight if and only if s and s are G ∗ -conjugate and κ = κ . Proposition 4.7.
Let ( R , ϕ ) be the ℓ -weight of G with label ( s , κ, K ) G ∗ . Then ( R , ϕ ) σ is the ℓ -weight of G with label ( σ ∗− ( s ) , σ ∗− κ, σ ∗− K ) G ∗ .Proof. We want to find which triple corresponds to ( R , ϕ ) σ . Assume it be ( s ′ , κ ′ , K ′ ). First, R σ = R , ˜ C σ = ˜ N σ = ˜ C = ˜ N , ˜ C σ + = ˜ C + and ˜ N σ + = ˜ N + by Remark 3.4.. Now, ˜ ϕ σ = ˜ ϕ σ × ˜ ϕ σ + . ϕ = Res ˜ C C ( ˜ ϕ ) is of the form χ s ,κ by construction. By Proposition 4.1, χ σ s ,κ = χ σ ∗− ( s ) ,σ ∗− κ .Then we have κ ′ = σ ∗− κ .Secondly, σ stabilizes every ˜ C Γ ,δ, i . Now ˜ θ σ Γ ,δ, i = ˜ θ σ ∗− Γ ,δ, i corresponds to a semisimple elementwith a unique elementary divisor Γ of multiplicity β Γ e Γ ℓ δ . Up to conjugacy, we have s ′ = σ ( s ).Finally, by the argument above, we may denote R σ Γ ,δ, i = R σ Γ ,δ, i , ˜ N σ Γ ,δ, i = ˜ N σ Γ ,δ, i and ˜ C σ Γ ,δ, i = ˜ C σ Γ ,δ, i although the corresponding terms indexed by Γ and σ ∗− Γ are actually the same. Then K ′ = σ ∗− K follows by Lemma 3.6. (cid:3) Let i W ′ ℓ ( G ) be the set of G ∗ -conjugacy classes of triples ( s , κ, µ ) such that s is a semisimple ℓ ′ -element of G ∗ , κ ∈ C ( s ), µ ∈ Q Γ P ( β Γ e Γ , w Γ ). Then by [50, Prop.(3.7)], i W ′ ℓ ( G ) is also alabeling set for W ℓ ( G ). Now by Corollary 3.7 and Proposition 4.7, we have Corollary 4.8.
Let ( R , ϕ ) be an ℓ -weight of G corresponding to ( s , κ, µ ) ∈ i W ′ ℓ ( G ) , then ( R , ϕ ) σ corresponds to ( σ ∗− ( s ) , σ ∗− κ, σ ∗− µ ) . Theorem 4.9.
Let G = SO n + ( q ) , where q = p f is a power of an odd prime p, ℓ , p is an oddprime and n ≥ . Assume that ℓ is linear. Then the blockwise bijection between IBr ℓ ( G ) and W ℓ ( G ) given in [4] is equivariant under the action of field automorphisms.Proof. By Theorem 2.14, the set E ( G , ℓ ′ ) is a basic set of IBr ℓ ( G ) and by [27], the corresponding ℓ -decomposition matrix of G is lower unitriangular since ℓ is linear. Then there is a canonicalbijection Ξ from i IBr ℓ ( G ) to i W ′ ℓ ( G ). By the construction of Ξ there, Ξ is E -equivariant. Thusthe assertion follows by Proposition 4.3 and Corollary 4.8. (cid:3) Ω n + ( q ) Recall that S = Ω n + ( q ) and z = − I n ∈ Z (Sp n ( q )). We may identify ˆ z with 1 Z ( ˜ G ) × ˆ z and regard ˆ z as a linear character of ˜ G . We may assume R Γ ,δ, i = R z . Γ ,δ, i , N Γ ,δ, i = N z . Γ ,δ, i , and20 Γ ,δ, i = C z . Γ ,δ, i . We also may regard ˆ z as a non-trivial linear character of C Γ ,δ, i by Lemma 3.1.Then by [19, Prop. 12.6], ˆ z ˜ θ Γ = ± ˆ zR ˜ C Γ ˜ T Γ ( b s Γ ) = ± R ˜ C Γ ˜ T Γ ( d − s Γ ), and then ˆ z ˜ θ Γ = ˜ θ z . Γ . Thus ˆ z θ Γ ,δ, i = θ z . Γ ,δ, i .So we may choose the labeling of C Γ ,δ and C σ ∗− Γ ,δ such thatˆ z ψ Γ ,δ, i , j = ψ z . Γ ,δ, i , j . (4.1)We define − K = Q Γ ( − K ) z . Γ with ( − K ) z . Γ = K Γ . Since N G ( R ) / N S ( R ) (cid:27) G / S , we may regardˆ z as a linear character of N G ( R ) whose kernel is N S ( R ). Proposition 4.10.
Let ( R , ϕ ) be the ℓ -weight of G with label ( s , κ, K ) G ∗ . Then ( R , ˆ z ϕ ) is the ℓ -weight of G with label ( − s , − κ, − K ) G ∗ .Proof. We want to find which triple corresponds to ( R , ˆ z ϕ ). Assume it be ( s ′ , κ ′ , K ′ ). Now˜ ϕ = Z ( ˜ G ) × ϕ , so ˆ z ˜ ϕ = Z ( ˜ G ) × ˆ z ϕ . First, ˆ z ˜ ϕ = ˆ z ˜ ϕ × ˆ z ˜ ϕ + . ϕ = Res ˜ G G ˜ ϕ is of the form χ s ,κ byconstruction. By Proposition 4.5, ˆ z χ s ,κ = χ − s , − κ . Then we have κ ′ = − κ .Secondly, we have ˆ z ˜ θ Γ ,δ, i = ˜ θ z . Γ ,δ, i as above. Note that ˆ z ˜ θ Γ ,δ, i corresponds to a semisimpleelement with a unique elementary divisor Γ of multiplicity β Γ e Γ ℓ δ and ˜ θ z . Γ ,δ, i corresponds to asemisimple element with a unique elementary divisor z . Γ of multiplicity β Γ e Γ ℓ δ . Up to conju-gacy, we have s ′ = − s .Finally, by the conventions above, we may assume R Γ ,δ, i = R z . Γ ,δ, i , ˜ N Γ ,δ, i = ˜ N z . Γ ,δ, i , and ˜ C Γ ,δ, i = ˜ C z . Γ ,δ, i . To determine K ′ , we note that ˆ z ˜ ψ + = Q Γ ,δ, i ˆ z ˜ ψ Γ ,δ, i . By (3.4), ˆ z ˜ ψ Γ ,δ, i isˆ z Ind ˜ N Γ ,δ, i ( θ Γ ,δ, i ) ≀ S ( t Γ ,δ, i )˜ N Γ ,δ, i ( θ Γ ,δ, i ) ≀ Q j S ( t Γ ,δ, i , j ) Y j ˜ ψ t Γ ,δ, i , j Γ ,δ, i , j · Y j φ κ Γ ,δ, i , j = Ind ˜ N z . Γ ,δ, i ( θ z . Γ ,δ, i ) ≀ S ( t Γ ,δ, i )˜ N z . Γ ,δ, i ( θ z . Γ ,δ, i ) ≀ Q j S ( t Γ ,δ, i , j ) ˆ z Y j ˜ ψ t Γ ,δ, i , j Γ ,δ, i , j · Y j φ κ Γ ,δ, i , j . Since ˆ z ˜ θ Γ ,δ, i = ˜ θ z . Γ ,δ, i , we have ˜ N Γ ,δ, i ( ˜ θ Γ ,δ, i ) = ˜ N z . Γ ,δ, i ( ˜ θ z . Γ ,δ, i ). We can fix the way to extend Q j ˜ ψ t Γ ,δ, i , j Γ ,δ, i , j as in [30, Lem. 25.5], then we have that ˆ z (cid:18)Q j ˜ ψ t Γ ,δ, i , j Γ ,δ, i , j (cid:19) = Q j (cid:16) ˆ z ˜ ψ Γ ,δ, i , j (cid:17) t Γ ,δ, i , j . Since ˆ z ˜ ψ Γ ,δ, i , j = ˜ ψ z . Γ ,δ, i , j by (4.1), ˆ z ˜ ψ Γ ,δ, i would be Ind ˜ N z . Γ , ˜ δ, i ( θ z . Γ ,δ, i ) ≀ S ( t Γ ,δ, i )˜ N z . Γ ,δ, i (˜ θ z . Γ ,δ, i ) ≀ Q j S ( t Γ ,δ, i , j ) Y j ˜ ψ t Γ ,δ, i , j z . Γ ,δ, i , j · Y j φ κ Γ ,δ, i , j . Then K ′ z . Γ = K Γ which is just K ′ = z . K . Thus we complete the proof. (cid:3) Corollary 4.11.
Let ( R , ϕ ) be an ℓ -weight of G corresponding to ( s , κ, µ ) ∈ i W ′ ℓ ( G ) , then ( R , ˆ z ϕ ) corresponds to ( − s , − κ, − µ ) . Let W ℓ ( G ) be a complete set of representatives of all G -conjugacy classes of ℓ -weights of G . We may assume that for ( R , ϕ ) , ( R , ϕ ) ∈ W ℓ ( G ), R and R are G -conjugate if and only if R = R .Let S = Ω n + ( q ), then Rad ℓ ( G ) = Rad ℓ ( S ). Now define a equivalence relation on W ℓ ( G )such that for ( R , ϕ ) , ( R , ϕ ) ∈ W ℓ ( G ), ( R , ϕ ) ∼ ( R , ϕ ) if and only if R = R and ϕ = ϕ η for some η ∈ Irr( N G ( R ) / N S ( R )). Then by [21, Lem. 2.4] and Corollary 3.3, the set { ( R , ψ ) } , where ( R , ϕ ) runs through a complete set of representatives of the equivalence classesof W ℓ ( G ) / ∼ and ψ runs through Irr( N S ( R ) | ϕ ), is a complete set of representatives of all S -conjugacy classes of ℓ -weights of S . 21et ( R , ϕ ) be an ℓ -weight of G , ( R , ψ ) an ℓ -weight of S such that ϕ ∈ Irr( N G ( R ) | ψ ). Let b = bl ℓ ( ϕ ), b = bl ℓ ( ψ ) and B = b G and B = b S . By [35, Lem. 2.3], if b covers b , then B covers B .Let B be an ℓ -block of S . Denote by B the union of the ℓ -blocks of S which are G -conjugate to B and B the union of the ℓ -blocks of G which cover B . Then • if ( R , ϕ ) is an ℓ -weight of G belonging to B , then for every ψ ∈ Irr( N S ( R ) | ϕ ), ( R , ψ ) is an ℓ -weight of S belonging to B , and • if ( R , ψ ) is an ℓ -weight of S belonging to B , then there exists ϕ ∈ Irr( N G ( R ) | ψ ) such that( R , ϕ ) is an ℓ -weight of G belonging to B . Proposition 4.12.
Let q be a power of an odd prime and ℓ an odd prime. Assume that ℓ is linear.Then the Alperin weight conjecture 1.1 holds for every ℓ -block of the group S = Ω n + ( q ) .Proof. The proof here is analogous to [21, Thm. 1.2]. Let Θ be the canonical blockwise bi-jection between IBr ℓ ( G ) and W ℓ ( G ). For φ ∈ IBr ℓ ( G ), let ( R , ϕ ) = Θ ( φ ). By Proposition 4.6and Corollary 4.11, ˆ z φ = φ if and only if ˆ z ϕ = ϕ . Thus the assertion follows by the argumentabove. (cid:3) By [28, § S ) (cid:27) G ⋊ E , where E = h F p i . Theorem 4.13.
Let S = Ω n + ( q ) , where q = p f is a power of an odd prime p, ℓ , p is an oddprime and n ≥ . Assume further that f is odd and ℓ is linear. Then there exists a blockwise Aut( S ) -equivariant bijection between IBr ℓ ( S ) and W ℓ ( S ) .Proof. It is analogous to the proof of [21, Prop. 5.19]. By Theorem 4.9 and the proof of Prop-sition 4.12, it su ffi ce to show that for any φ ∈ IBr ℓ ( G ) and any ( R , ϕ ) ∈ W ℓ ( G ), E acts triviallyon IBr ℓ ( S | φ ) and N G ⋊ E ( R ) acts trivially on Irr( N S ( R ) | ϕ ). Let m be the length of an orbit of E on IBr ℓ ( S | φ ) and m the length of an orbit of N G ⋊ E ( R ) on Irr( N S ( R ) | ϕ ). Then m , m ≤ m | f and m | f . Now f is odd, so m = m =
1. This completes the proof. (cid:3)
Note that X = Spin n + ( q ) is the universal ℓ ′ -covering group of the simple group S =Ω n + ( q ) (with n ≥ q odd) unless when n = q = § Proof of Theorem 1.2.
By assumption, Aut( S ) / S is cyclic for S = X / Z ( X ) = Ω n + ( q ). Thus byCorollary 2.13, it su ffi ces to show that there exists a blockwise Aut( X ) B -equivariant bijectionbetween IBr ℓ ( B ) and W ℓ ( B ).By Lemma 2.5, it su ffi ces to show that there exists a blockwise Aut( S )-equivariant bijectionbetween IBr ℓ ( S ) and W ℓ ( S ), which follows by Theorem 4.13. (cid:3) Proposition 4.14.
Let G = SO n + ( q ) and S = Ω n + ( q ) with n ≥ and q odd, and ℓ ∤ q an oddprime. Suppose B is a unipotent ℓ -block of G. Then • B covers a unique ℓ -block b of S , • Res GS : IBr ℓ ( B ) → IBr ℓ ( b ) is bijective, and • W ℓ ( B ) → W ℓ ( b ) , ( R , ϕ ) ( R , Res N G ( R ) N S ( R ) ϕ ) is a bijection. roof. For ℓ -weights, this follows by Proposition 4.10. Now we prove that Res GS : IBr ℓ ( B ) → IBr ℓ ( b ) is bijective. First, ˆ z ⊗ B is also an ℓ -block of G covering b (for the definition of ˆ z ⊗ B ,see Page 5). Also, ˆ z ⊗ B , B since ˆ z ⊗ B ⊆ E ℓ ( G , z ). Thus there are two ℓ -blocks covering b . Soby Lemma 2.6, Res GS : IBr ℓ ( B ) → IBr ℓ ( b ) is bijective. (cid:3) Lemma 4.15.
Let G = SO n + ( q ) with n ≥ and q odd, and ℓ ∤ q an odd prime. Suppose B isa unipotent ℓ -block of G. Let φ ∈ IBr ℓ ( B ) and ( R , ϕ ) ∈ W ℓ ( B ) . Then • φ extends to G ⋊ E, and • ϕ extend to ( G ⋊ E ) R .Proof. By Theorem 2.14, the set of unipotent characters of B form a basic set of B . By [41,Thm. 2.5], every unipotent character of G is E -invariant. So every ℓ -Brauer character of B is E -invariant. On the other hand we have ϕ is ( G ⋊ E ) R -invariant by Corollary 4.8. Thus thisassertion holds since E is cyclic. (cid:3) Proof of Theorem 1.5 for type B n . Let b be a unipotent ℓ -block of X . Then b dominates an ℓ -block ¯ b of S = Ω n + ( q ). Thus there exists a unique unipotent ℓ -block ¯ B of G = SO n + ( q ) whichcovers ¯ b . By Proposition 4.14, Res GS : IBr ℓ ( ¯ B ) → IBr ℓ (¯ b ) is bijective and W ℓ ( ¯ B ) → W ℓ (¯ b ),( ¯ R , ¯ ϕ ) ( ¯ R , Res N G ( ¯ R ) N S ( ¯ R ) ¯ ϕ ) is a bijection. Thus by Theorem 4.9, there exists a blockwise Aut( S ) ¯ b -equivariant bijection between IBr ℓ (¯ b ) and W ℓ (¯ b ). This gives a blockwise Aut( X ) b -equivariantbijection between IBr ℓ ( b ) and W ℓ ( b ) by Lemma 2.5. Then by Lemma 2.12, it su ffi ce to showthe condition (iii) in Definition 2.10.For every Q ∈ Rad ℓ ( X ) and every φ ∈ IBr ℓ ( b | Q ), we let A : = A ( φ, Q ) = G ⋊ E . Note thatall irreducible character of b have Z ( X ) in their kernel. By Theorem 2.14, Irr( b ) ∩ E ( X , ℓ ′ ) is abasic set of b , so all irreducible ℓ -Brauer characters of b have Z ( X ) in their kernel. By Lemma2.3 (iii), all weight characters of b also have Z ( X ) in their kernel. Thus conditions (1)-(3) ofDefinition 2.10 (iii) hold by Proposition 4.14 and Lemma 4.15. For Definition 2.10 (iii)(4), theproof of [21, Lem. 7.2] applies here. Then the (iBAW) condition holds for b , which completesthe proof. (cid:3) C In this section, we denote by G = Sp n ( F q ), ˜ G = CSp n ( F q ) with q odd and n ≥
2. Let F p : ˜ G → ˜ G be the raising of matrix entries to the p -th power, F = F fp , for some f such that q = p f . Let G = G F = Sp n ( q ), ˜ G = ˜ G F = CSp n ( q ). Then | ˜ G / GZ ( ˜ G ) | =
2. We denote by V theunderlying space of ˜ G and G . First note that ˜ G ∗ is the corresponding special Cli ff ord group (then ˜ G ∗ = ( ˜ G ∗ ) F is a specialCli ff ord group over F q ) and G ∗ = SO n + ( F q ). For the definition of the Cli ff ord groups oforthogonal spaces, see [29, §
9] or [34, § ff ord groups are calledeven Cli ff ord groups in [29]. Thus there is a natural epimorphism π : ˜ G ∗ → G ∗ (see also [24,(2.3)]). Clearly, π ( ˜ G ∗ ) = G ∗ = G ∗ F = SO n + ( q ). For a semisimple element s of ˜ G ∗ , we write¯ s = π ( s ). Note that m x − ( ¯ s ) is odd and m x − ( ¯ s ) is even. In particular, m x − ( ¯ s ) ,
0. Let i Irr( ˜ G )be the set of ˜ G ∗ -conjugacy classes of pairs ( s , µ ), where s is a semisimple ℓ ′ -element of G ∗ and µ ∈ Ψ ( ¯ s ) (where Ψ ( ¯ s ) is defined as (2.4)). Here, ( s , µ ) and ( s , µ ) are conjugate if and only if23 and s are ˜ G ∗ -conjugate and µ = µ . With the parametrization of pairs involving semisimpleelements and unipotent characters, the irreducible characters of ˜ G were constructed by Lusztig[39]; by Jordan decomposition of characters, there is a bijection from i Irr( ˜ G ) to Irr( ˜ G ) (see also[24, (4.5)]). We write ˜ χ s ,µ for the character of ˜ G corresponding to ( s , µ ).In this section, we always assume that σ = F p is the field automorphism and E = h F p i asabove. Then σ ∗ is also a field automorphism. Note that σ ∗ commutes with π . Lemma 5.1.
Let ( s , µ ) ∈ i Irr( ˜ G ) . Then ˜ χ σ s ,µ = ˜ χ σ ∗− ( s ) ,σ ∗− µ .Proof. Similar as in Proposition 4.1, this follows from [17, Thm. 3.1] and the fact that everyunipotent character of groups of type A , A , B , D and D is invariant under field automor-phisms (see [41, Thm. 2.5]). (cid:3) We will make use of the following result.
Theorem 5.2.
Let ˜ χ ∈ Irr( ˜ G ) and ∆ = Irr( G | ˜ χ ) . Then E ∆ acts trivially on ∆ .Proof. This is [18, Thm. 3.1] (or [55, Thm. 16.2]). (cid:3) If χ ∈ Irr( G | ˜ χ s ,µ ), then we say χ corresponds to the pair ( ¯ s , µ ). Lemma 5.3.
Let ( s , µ ) ∈ i Irr( ˜ G ) , ¯ s = π ( s ) and ˜ χ = ˜ χ s ,µ .(i) If − is not an eigenvalue of ¯ s, then Res ˜ GG ˜ χ is irreducible.(ii) If − is an eigenvalue of ¯ s, then • if µ x + is degenerate, then Res ˜ GG ˜ χ is irreducible, and • if µ x + is non-degenerate, then Res ˜ GG ˜ χ is a sum of two irreducible constituents.Proof. First note that G ֒ → ˜ G is a regular embedding. Let ψ µ be the unipotent character of C ˜ G ( s ) F corresponding to µ . Then we may regard ψ µ as a unipotent character of C ◦ G ( ¯ s ) F since E ( C ˜ G ( s ) F , = E ( C ◦ G ( ¯ s ) F , | Irr( G | ˜ χ ) | = | Irr( C ˜ G ( s ) F | ψ µ ) | .Let V ∗ be the underlying space of G ∗ and let ¯ s = Q Γ ¯ s ( Γ ) and V ∗ = P Γ V ∗ Γ ( ¯ s ) be the primarydecomposition. In particular, we abbreviate V ∗ : = V ∗ x − ( ¯ s ), V ∗− : = V ∗ x + ( ¯ s ) and V ∗ Γ : = V ∗ Γ ( ¯ s ).Here, V ∗ is of odd-dimension and V ∗− is of even-dimension. Then C G ( ¯ s ) F = (SO( V ∗ ⊥ V ∗− ) ∩ (GO( V ∗ ) × GO( V ∗− ))) × Q Γ C GO( V ∗ Γ ) ( ¯ s ( Γ )), C ◦ G ( ¯ s ) F = SO( V ∗ ) × SO( V ∗− ) × Q Γ C GO( V ∗ Γ ) ( ¯ s ( Γ )) and C GO( V ∗ Γ ) ( ¯ s ( Γ )) ≤ SO( V ∗ Γ ). We also write ψ µ = Q Γ ψ µ Γ where ψ x − : = ψ and ψ x + : = ψ − are theunipotent characters of SO( V ∗ ) and SO( V ∗− ) corresponding to µ x − and µ x + respectively and ψ µ Γ is the unipotent character of C SO( V ∗ Γ ) ( ¯ s Γ ) corresponding to µ Γ for Γ ∈ F ∪ F . If − s , then C G ( ¯ s ) is connected and then Res ˜ GG ˜ χ is irreducible.Now let − s . Then | C G ( ¯ s ) F / C ◦ G ( ¯ s ) F | = ˜ GG ˜ χ is irreducible ifand only if ψ µ is not C G ( ¯ s ) F / C ◦ G ( ¯ s ) F -invariant. Let g = g × g − × Q Γ g Γ with g ∈ GO( V ∗ ), g − ∈ GO( V ∗− ) and g Γ ∈ C GO( V ∗ Γ ) ( ¯ s ( Γ )) such that g and g − are of determinant −
1. Then ψ µ = ψ g × ψ g − − × Q Γ ψ g Γ µ Γ = ψ × ψ g − − × Q Γ ψ µ Γ by [41, Thm. 2.5] and then ψ g µ = ψ µ if and onlyif κ x + is non-degenerate by [41, Thm. 2.5] again. Thus the assertion holds. (cid:3) Remark 5.4.
In Lemma 5.3 (ii), if µ x + is degenerate, then by Jordan decomposition, Res ˜ GG ˜ χ s ,µ = Res ˜ GG ˜ χ s ,µ ′ , where µ ′ is defined as in § G | Res ˜ GG ˜ χ s ,µ ) ∩ E ( ˜ G , s ) = { ˜ χ s ,µ , ˜ χ s ,µ ′ } .24 .2 The blocks of symplectic groups Recall that we let ℓ be an odd prime with ℓ ∤ q and e the multiplicative order of q modulo ℓ . The labeling of ℓ -blocks of ˜ G F and G F (using e -Jordan-cuspidal pairs) described in [16]and [32] can be stated as following. Theorem 5.5.
Let H ∈ { ˜ G , G } and e = e ( q , ℓ ) as defined in Equation (2.1).(i) For any e -Jordan-cuspidal pair ( L , ζ ) of H such that ζ ∈ E ( L F , ℓ ′ ) , there exists a unique ℓ -block b H F ( L , ζ ) of H F such that all irreducible constituents of R HL ( ζ ) lie in b H F ( L , ζ ) .(ii) Moreover, the map Ξ : ( L , ζ ) b H F ( L , ζ ) is a bijection from the set of H F -conjugacyclasses of e -Jordan-cuspidal pairs ( L , ζ ) of H such that ζ ∈ E ( L F , ℓ ′ ) to the ℓ -blocks of H F . Now we give the relationship between the e -cuspidal pairs of ˜ G and the e -cuspidal pairsof G . Proposition 5.6. (i) Let ( ˜ L , ˜ ζ ) be an e -cuspidal pair of ˜ G and B an ℓ -block of G F coveredby ˜ B = b ˜ G F ( ˜ L , ˜ ζ ) , then B = b G F ( L , ζ ) , where L = ˜ L ∩ G and ζ is an irreducible constituentof Res ˜ L F L F ˜ ζ .(ii) Let ( L , ζ ) be an e -cuspidal pair of G and ˜ B an ℓ -block of ˜ G F which covers B = b G F ( L , ζ ) ,then ˜ B = b ˜ G F ( ˜ L , ˜ ζ ) for some e -cuspidal pair ( ˜ L , ˜ ζ ) satisfying that L = ˜ L ∩ G and ζ is anirreducible constituent of Res ˜ L F L F ˜ ζ .Proof. This follows by [32, Lem. 3.7 and 3.8] (see [21, Prop. 4.5] for details). (cid:3)
Note that we have ˜ L = Z ( ˜ G ) L in Proposition 5.6. In fact, the F -stable Levi subgroups of ˜ G and G have been classified in [24, (3A) and (3B)]. Lemma 5.7.
Let ˜ L be an F-stable Levi subgroup of ˜ G , ˜ ζ ∈ Irr( ˜ L F ) and L = ˜ L ∩ G . Let ∆ : = Irr( L F | ˜ ζ ) , then N G F ( L ) ∆ acts trivially on ∆ .Proof. Let L = L F and ˜ L = ˜ L F . Follow [24, (3A) and (3B)], we may assume that there is anorthogonal decomposition V = V ⊥ V + of V , where V + = t P i = V i such that • L = L × L + , where L = Sp( V ), L + = t Q i = L i such that L i ≤ Sp( V i ) isomorphic to somegeneral linear or unitary group for 1 ≤ i ≤ t . • ˜ L = h τ, L i , where τ satisfies ˜ G = h G , τ i and [ τ, L + ] =
1. Moreover, τ = τ × τ + such that τ ∈ CSp( V ) and τ + ∈ CSp( V + ).Thus, | ˜ L / LZ ( ˜ L ) | = N G ( L ) = N × N + , with N = L and N + ≤ Sp( V + ). So | ∆ | ≤
2. If | ∆ | =
1, then the assertion is obvious. Now we may assume that | ∆ | = ∆ = { ζ, ζ ′ } , then ζ and ζ ′ are ˜ L -conjugate. We write ζ = ζ × ζ + and ζ ′ = ζ ′ × ζ ′ + with ζ , ζ ′ ∈ Irr( L ) and ζ + , ζ ′ + ∈ Irr( L + ). Since [ τ, L + ] =
1, we know ζ + = ζ ′ + . Hence ζ , ζ ′ . For n ∈ N G ( L ) ∆ , we let n = n × n + where n ∈ N and n + ∈ N + . If ζ n = ζ ′ , then ζ n = ζ ′ and this isimpossible since n ∈ L . So ζ n , ζ ′ , which implies that N G ( L ) ∆ acts trivially on ∆ . (cid:3) emark 5.8. Let ˜ L an F -stable Levi subgroup of ˜ G , and L = ˜ L ∩ G . Then by (4) of [24, (3B)]˜ L F / L F (cid:27) ˜ G F / G F and then ˜ G F = G F N ˜ G F ( L ). So the ˜ G F -conjugacy classes of e -split Levisubgroups of G are just the G F -conjugacy classes of e -split Levi subgroups of G .We denote by e L a complete set of representatives of the ˜ G F -conjugacy classes of e -Jordan-cuspidal pairs of ˜ G such that ˜ ζ ∈ E ( ˜ L F , ℓ ′ ). We may assume that for ( ˜ L , ˜ ζ ), ( ˜ L ′ , ˜ ζ ′ ) ∈ e L ,if ˜ L and ˜ L ′ are ˜ G F -conjugacy, then ˜ L = ˜ L ′ . Now we define an equivalence relation on e L :( ˜ L , ˜ ζ ) ∼ ( ˜ L ′ , ˜ ζ ′ ) if and only if ˜ L = ˜ L ′ and Res ˜ L F L F ˜ ζ = Res ˜ L F L F ˜ ζ ′ where L = ˜ L ∩ G . Then byProposition 5.6, Lemma 5.7 and Cli ff ord theory, { ( ˜ L ∩ G , ζ ) } is a complete set of representativesof G F -conjugacy classes of e -Jordan-cuspidal pairs of G such that ζ ∈ E (( ˜ L ∩ G ) F , ℓ ′ ), where( ˜ L , ˜ ζ ) runs through a complete set of representatives of the equivalence classes of e L / ∼ and ζ runs through Irr(( ˜ L ∩ G ) F | ˜ ζ ).Now we recall the classification of ℓ -blocks of ˜ G F in [24, § i Bl ℓ ( ˜ G ) be the set of ˜ G ∗ -conjugacy classes of pairs ( s , κ ) where s is a semisimple ℓ ′ -element of ˜ G ∗ and κ ∈ C ( ¯ s ) (where C ( ¯ s ) is defined as in (2.5)). Here, ( s , κ ) and ( s , κ ) are ˜ G ∗ -conjugate if and only if s and s are ˜ G ∗ -conjugate and κ = κ . Then [24, (11E)] gives a bijection ( s , κ ) ˜ B ( s , κ ) from i Bl ℓ ( ˜ G )to Bl ℓ ( ˜ G ).For ( s , κ ) ∈ i Bl ℓ ( ˜ G ), [24, (13B)] also gave a criterion for when an irreducible character of ˜ G lies in the ℓ -block ˜ B = ˜ B ( s , κ ). In particular, the irreducible characters of Irr( ˜ B ) ∩ E ( ˜ G , ℓ ′ ) areof the form ˜ χ s ,µ with µ ∈ Ψ ( ¯ s , κ ) (where Ψ ( ¯ s , κ ) is defined as in (2.6)). In addition, by (2.11), Ψ Γ ( ¯ s , κ ) is in bijection with P ( β Γ e Γ , w Γ ) if Γ , x + Γ = x + κ x + is non-degenerate andin bijection with P ′ (2 e , w x + ) if Γ = x + κ x + is degenerate. Here, the sets P ( β Γ e Γ , w Γ ) and P ′ (2 e , w x + ) are defined as in (2.7) and (2.10) respectively.Fix ( s , κ ) ∈ i Bl ℓ ( ˜ G ). Now we give an e -Jordan-cuspidal pair of ˜ G F corresponding to ˜ B ( s , κ ).First, we define an e -split Levi subgroup L of G . Let ( s , µ ) ∈ i Irr( ˜ G ) such that χ s ,µ ∈ ˜ B ( s , κ ).Recall that we have integers w Γ = e − Γ ( m Γ ( s ) − | κ Γ | ) if Γ ∈ F ∪ F and w Γ is the number of e Γ -hooks (or e Γ -cohooks) removed from µ Γ to get κ Γ if Γ ∈ F (see § w Γ do not depend on the choice of µ and are determined by ( s , κ ). Let ˜ D be a defect group ofthe ℓ -block ˜ B ( s , κ ) and D = ˜ D ∩ G . Then by [24, § V = V ⊥ V ⊥ · · · ⊥ V t and D = D × D × · · ·× D t such that D = h V i and for i > D i = R m i ,α i , ,β i for integers m i , α i , β i . Here, R m i ,α i , ,β i = R m i ,α i ,γ, c with γ = c = ( β i ) definedas in § D = D Q Γ ( R m Γ ,α Γ , ,β ) t Γ ,β , where t Γ ,β are the coe ffi cientsoccuring in the ℓ -adic expansion P β t Γ ,β ℓ β of w Γ for Γ ∈ F . Thus V = C V ( D ) and V + = [ D , V ],where V + = V ⊥ · · · ⊥ V t .Let L be an F -stable Levi subgroup of G (described as in [24, (3A)]) such that L = L F = L × L + with L = Sp( V ), L + = Q Γ w Γ Q i = L Γ , i ≤ Sp( V + ) and L Γ , i (cid:27) GL e Γ δ Γ e ( ǫ q e ) for 1 ≤ i ≤ w Γ ,where ǫ = ℓ is linear and ǫ = − ℓ is unitary. Let V = V + P Γ w Γ P i = V Γ , i be the correspondingorthogonal decomposition of V . Obviously, V + = P Γ w Γ P i = V Γ , i . From this we obtain an e -splitLevi subgroup L of G . Then ˜ L = L Z ( ˜ G ) is an e -split Levi subgroup of ˜ G and the structure of L ∗ and ˜ L ∗ are described in [24, (3A) and (3B)]. In fact, L ֒ → ˜ L is a regular embedding. Clearly,¯ s ∈ L ∗ : = L ∗ F and s ∈ ˜ L ∗ : = ( ˜ L ∗ ) F up to conjugacy. Also, ˜ L = h L , τ i · L + is the central productof h L , τ i and L + , where τ is as in the proof of Lemma 5.7. Write ¯ s = ¯ s × ¯ s + , with ¯ s ∈ L ∗ and¯ s + ∈ L ∗ + .Let V ∗ be the underlying space of G ∗ and V ∗ = V ∗ + P Γ w Γ P i = V ∗ Γ , i , L ∗ = L ∗ × L ∗ + with L ∗ + = Q Γ w Γ Q i = L ∗ Γ , i be the corresponding orthogonal decompositions of V ∗ and L ∗ as in [24, (3A)]. Denote26 ∗ + = P Γ w Γ P i = V ∗ Γ , i . In addition, we have the primary decompositions V ∗ = P Γ V ∗ Γ ( ¯ s ) and ¯ s = Q Γ ¯ s ( Γ ) of V ∗ and ¯ s . Thus C ◦ L ∗ ( ¯ s ) F = SO( V ∗ x − ( ¯ s )) × SO( V ∗ x + ( ¯ s )) × Q Γ ∈F ∪F GL m Γ (¯ s ) ( ε Γ q δ Γ ) × C L ∗ + ( s + ), where C L ∗ + ( s + ) = Q Γ ∈F w Γ Q i = GL (( ǫ q e ) e Γ δ Γ / e ). Let φ κ = Q Γ φ Γ × C L ∗ + ( s + ) be the unipotentcharacter of C ◦ L ∗ ( ¯ s ) F , where φ Γ is the unipotent character of SO( V ∗ Γ ( ¯ s )) corresponding to κ Γ if Γ ∈ F and the unipotent character of GL m Γ (¯ s ) ( ε Γ q δ Γ ) corresponding to κ Γ if Γ ∈ F ∪ F . Then φ κ is an e -cuspidal unipotent character of C ◦ L ∗ ( ¯ s ) F . Now note that E ( C ˜ L ∗ ( s ) F , = E ( C ◦ L ∗ ( ¯ s ) F , φ κ as an e -cuspidal unipotent character of C ˜ L ∗ ( s ) F . Let ˜ ζ be thecharacter of ˜ L F corresponding under the Jordan decomposition to φ κ ∈ E ( C ˜ L ∗ ( s ) F , L , ˜ ζ ) is an e -Jordan-cuspidal pair of ˜ G . Lemma 5.9.
With the notation above, the ℓ -block b ˜ G F ( ˜ L , ˜ ζ ) of ˜ G F corresponding to the e -Jordan-cuspidal pair ( ˜ L , ˜ ζ ) is ˜ B ( s , κ ) .Proof. We prove that there is one irreducible constituent of R ˜ G ˜ L ( ˜ ζ ) lying in ˜ B ( s , κ ). In fact, thisis essentially contained in [24, § Q and ˜ Q be the F -fixed point of some F -stable Levisubgroups (say, Q and ˜ Q ) defined in [24, p. 178], centralizer of a certain ℓ -element in Z ( D ) in G and ˜ G respectively. Then Q (cid:27) Q × Q + and ˜ Q = h Q , τ i Q + with Q = L and L + ≤ Q + .Also, we let ˜ b be the ℓ -block of ˜ Q defined in [24, p. 179]. Now ˜ L ≤ ˜ Q , so R ˜ G ˜ L ( ˜ ζ ) = R ˜ G ˜ Q ( R ˜ Q ˜ L ( ˜ ζ )).In addition C ˜ Q ∗ ( s ) ≤ ˜ L ∗ , then R ˜ Q ˜ L ( ˜ ζ ) lies in E ( ˜ Q , s ) and then by [24, (13A)], lies in ˜ b . Thuswe conclude from the proof of [24, (13B)] that there exists one irreducible constituent of R ˜ G ˜ L ( ˜ ζ )lying in ˜ B ( s , κ ). (cid:3) Now keep the hypotheses and setup above and we wish to investigate how many ℓ -blocks of G are covered by ˜ B ( s , κ ). This number is equal to the cardinality of the set Irr( L | ˜ ζ ) by Remark5.8.Let e be the identity element of the cli ff ord algebra over V ∗ . Then Z (( ˜ G ∗ ) F ) = { k e | k ∈ F × q } .For z ∈ Z (( ˜ G ∗ ) F ), we denote by ˆ z the corresponding linear character (by [19, Prop. 13.30]) of˜ G as before. Moreover, we may regard ˆ z as a linear character of ˜ L / L since ˜ L / L (cid:27) ˜ G / G . Fromthis, ( ˜ L , ˆ z ˜ ζ ) is also an e -Jordan-cuspidal pair of ˜ G . Also, ˆ z ˜ ζ ∈ E ( ˜ L F , ℓ ′ ) if and only if z is of ℓ ′ -order. Conversely, if ˜ B ′ is an ℓ -block of ˜ G such that ˜ B ′ and b ˜ G F ( ˜ L , ˜ ζ ) cover the same ℓ -blocksof G , then ˜ B ′ = b ˜ G F ( ˜ L , ˆ z ˜ ζ ) for some z ∈ O ℓ ′ ( Z ( ˜ G ∗ )).The relations between conjugacy classes of ˜ G ∗ and G ∗ are given in [39, § C be the conjugacy class of G ∗ containing ¯ s and C = π − ( ¯ C ). If − s ,then C is the union of | Z ( ˜ G ∗ ) | conjugacy classes of ˜ G ∗ and each class contains a unique elementof { zs | z ∈ Z ( ˜ G ∗ ) } . If − s , then C is the union of | Z ( ˜ G ∗ ) | conjugacyclasses of ˜ G ∗ and each class contains exactly two elements z and z of { zs | z ∈ Z ( ˜ G ∗ ) } suchthat z = − z .If − s , then ˜ B ( zs , κ ) are distinct ℓ -blocks of ˜ G for z ∈ O ℓ ′ ( Z ( ˜ G ∗ ))and they cover the same ℓ -block of G . In addition, b ˜ G F ( ˜ L , ˆ z ˜ ζ ) = ˜ B ( zs , κ ) = ˆ z ⊗ ˜ B ( s , κ ). Nowsuppose − s . Then ˜ B ( zs , κ ) and ˜ B ( zs , κ ′ ) (where κ ′ is defined as in page11) are ℓ -blocks of ˜ G , where z runs through a complete set of representatives of h− e i -cosets in O ℓ ′ ( Z ( ˜ G ∗ )), and they cover the same ℓ -blocks of G . If w x + , κ x + is non-degenerate, then˜ B ( zs , κ ) = ˜ B ( zs , κ ′ ) by Remark 5.4 and then b ˜ G F ( ˜ L , ˆ z ˜ ζ ) = ˜ B ( zs , κ ) = ˆ z ⊗ ˜ B ( s , κ ) for z ∈ O ℓ ′ ( Z ( ˜ G ∗ )).Let w x + = κ x + be degenerate. Fix z ∈ O ℓ ′ ( Z ( ˜ G ∗ )), then ˜ B ( zs , κ ) and ˜ B ( zs , κ ′ ) are distinct ℓ -blocks of ˜ G . In addition, if b ˜ G F ( ˜ L , ˆ z ˜ ζ ) = ˜ B ( zs , κ ) = ˆ z ⊗ ˜ B ( s , κ ) for z ∈ O ℓ ′ ( Z ( ˜ G ∗ )), then b ˜ G F ( ˜ L , d − e s ˜ ζ ) = ˜ B ( − e zs , κ ) = d − e s ⊗ ˜ B ( s , κ ). 27ow L ֒ → ˜ L is a regular embedding, so by Jordan decomposition, we have | Irr( L F | ˜ ζ ) | = | Irr( C L ∗ ( ¯ s ) F | φ κ ) | . Thus if − s , then | Irr( L F | ˜ ζ ) | =
1, Now supposethat − s . Then by Cli ff ord theory, Res ˜ L F L F ˜ ζ is not irreducible if and onlyif φ κ is C L ∗ ( ¯ s ) F -invariant. Note that C L ∗ ( ¯ s ) F = SO( V ∗ x − ( ¯ s ) ⊥ V ∗ x + ( ¯ s )) ∩ (GO( V ∗ x − ( ¯ s )) × GO( V ∗ x + ( ¯ s ))) × Q Γ ∈F ∪F GL m Γ (¯ s ) ( ε q δ Γ ) × C L ∗ + ( s + ). Here, V ∗ x − has odd-dimension and V ∗ x + haseven-dimension. In this way, by a similar proof as in Lemma 5.3, Res ˜ L F L F ˜ ζ is irreducible if andonly if κ x + is degenerate.By the argument above, we have the following result. Theorem 5.10.
Let ( s , κ ) ∈ i Bl ℓ ( ˜ G ) , ¯ s = π ( s ) , ˜ B = ˜ B ( s , κ ) and B an ℓ -block of G covered by ˜ B.(i) If − is not an eigenvalue of ¯ s, then B is the unique ℓ -block of G covered by ˜ B and there are |O ℓ ′ ( F × q ) | ℓ -blocks of ˜ G covering B. In addition, the ℓ -blocks covering B of ˜ G are ˜ B ( zs , κ ) ,where z runs through O ℓ ′ ( Z ( ˜ G ∗ )) .(ii) If − is an eigenvalue of ¯ s, w x + = and κ x + is degenerate, then B is the unique ℓ -blockof G covered by ˜ B and there are |O ℓ ′ ( F × q ) | ℓ -blocks of ˜ G covering B. In addition, the ℓ -blocks covering B of ˜ G are ˜ B ( zs , κ ) and ˜ B ( zs , κ ′ ) , where z runs through a complete set ofrepresentatives of h− e i -cosets in O ℓ ′ ( Z ( ˜ G ∗ )) .(iii) If w x + , or κ x + is non-degenerate, then there are |O ℓ ′ ( F × q ) | ℓ -blocks of ˜ G covering Band they are ˜ B ( zs , κ ) , where z runs through a complete set of representatives of h− e i -cosetsin O ℓ ′ ( Z ( ˜ G ∗ )) . Moreover, • if w x + , and κ x + is degenerate, then B is the unique ℓ -block of G covered by ˜ Band • if κ x + is non-degenerate, then there are two ℓ -blocks of G covered by ˜ B. Now let i Bl (1) ℓ ( G ) be the set of G ∗ -conjugacy classes of pairs ( s , κ ), where s ∈ G ∗ is asemisimple ℓ ′ -element and κ ∈ C ( s ) such that − s or κ x + is degenerate.Here, we identify ( s , κ ) with ( s , κ ′ ). Let i Bl (2) ℓ ( G ) be the set of G ∗ -conjugacy classes of pairs( s , κ ), where s ∈ G ∗ is a semisimple ℓ ′ -element and κ ∈ C ( s ) is such that w x + ( s ) , κ x + is non-degenerate. Also ( s , κ ) and ( s , κ ) are G ∗ -conjugate means s and s are G ∗ -conjugateand κ = κ . Then i Bl ℓ ( G ) : = i Bl (1) ℓ ( G ) ∪ i Bl (2) ℓ ( G ), where the elements of i Bl (2) ℓ ( G ) are countedtwice, is a labeling set for Bl ℓ ( G ) by Theorem 5.10. If ( s , κ ) ∈ i Bl (1) ℓ ( G ), we denote by B ( s , κ ) the ℓ -blocks of G corresponding to ( s , κ ). If ( s , κ ) ∈ i Bl (2) ℓ ( G ), then B (1) ( s , κ ) and B ( − ( s , κ ) denotethe two ℓ -blocks of G corresponding to ( s , κ ). Aut( G ) on the Brauer characters and weights Now let i IBr (1) ℓ ( G ) be the set of G ∗ -conjugacy classes of pairs ( s , µ ), where s ∈ G ∗ is asemisimple ℓ ′ -element and µ ∈ Ψ ( s ) such that either − s or − s and µ x + is degenerate. Here, we identify ( s , µ ) with ( s , µ ) in i IBr (1) ℓ ( G ), whichmeans degenerate symbols are not counted twice in any case. Let i IBr (2) ℓ ( G ) be the set of G ∗ -conjugacy classes of pairs ( s , µ ), where s ∈ G ∗ is a semisimple ℓ ′ -element and µ ∈ Ψ ( s ) such that − s and µ x + is non-degenerate. Then i IBr ℓ ( G ) : = i IBr (1) ℓ ( G ) ∪ i IBr (2) ℓ ( G ),where the elements of i IBr (2) ℓ ( G ) are counted twice, is a labeling set for Irr( G ) ∩ E ( G , ℓ ′ ) byLemma 5.3. If ( s , µ ) ∈ i IBr (1) ℓ ( G ), we denote by χ s ,µ the character of G corresponding to ( s , µ ).If ( s , µ ) ∈ i IBr (2) ℓ ( G ), then χ (1) s ,µ and χ ( − s ,µ denote the two characters of G corresponding to ( s , µ ).28urthermore, if ( s , κ ) ∈ i Bl (1) ℓ ( G ) and B = B ( s , κ ), then Irr( B ) ∩ E ( G , s ) = { χ s ,µ | µ ∈ Ψ ( s , κ ) , ( s , µ ) ∈ i IBr (1) ℓ ( G ) } ∪ { χ ( ± s ,µ | µ ∈ Ψ ( s , κ ) , ( s , µ ) ∈ i IBr (2) ℓ ( G ) } . If ( s , κ ) ∈ i Bl (2) ℓ ( G ), B (1) = B (1) ( s , κ ) and B ( − = B ( − ( s , κ ), then Irr( B (1) ∪ B ( − ) ∩ E ( G , s ) = { χ ( ± s ,µ | µ ∈ Ψ ( s , κ ) } .We may assume that Irr( B ( i ) ) ∩ E ( G , s ) = { χ ( i ) s ,µ | µ ∈ Ψ ( s , κ ) } for i = ±
1. Note that, if κ x + isnon-degenerate, then µ x + is also non-degenerate, and then we always have ( s , µ ) ∈ i IBr (2) ℓ ( G ). Remark 5.11.
Let g ∈ ˜ G \ GZ ( ˜ G ). Then g induces the non-trivial diagonal automorphism on G . By Lemma 5.1 and 5.3, we have χ σ s ,µ = χ σ ∗− ( s ) ,σ ∗− µ and χ gs ,µ = χ s ,µ if ( s , µ ) ∈ i IBr (1) ℓ ( G ). ByTheorem 5.2, we may assume that ( χ ( i ) s ,µ ) σ = χ ( i ) σ ∗− ( s ) ,σ ∗− µ and ( χ ( i ) s ,µ ) g = χ ( − i ) s ,µ for ( s , µ ) ∈ i IBr (2) ℓ ( G )and i = ± s , κ ) ∈ i Bl ℓ ( G ) and an ℓ -block B of G corresponding to ( s , κ ), we define P ( B ) : = Q Γ P ( β Γ e Γ , w Γ ), where the sets P ( β Γ e Γ , w Γ ) are defined as in (2.7). Proposition 5.12.
With the preceding notation, P ( B ) is a labeling set for Irr( B ) ∩ E ( G , ℓ ′ ) .Proof. Let ˜ B be an ℓ -block of ˜ G covering B . If ( s , κ ) ∈ i Bl (2) ℓ ( G ), then every character ofIrr( ˜ B ) ∩ E ( ˜ G , ℓ ′ ) is parametrized by an element of i IBr (2) ℓ ( G ). Thus the map Res ˜ GG : Irr( ˜ B ) ∩E ( ˜ G , ℓ ′ ) → Irr( B ) ∩ E ( G , ℓ ′ ) is bijective. So Ψ ( s , κ ) is a labeling set for Irr( B ) ∩ E ( G , ℓ ′ ). Fromthis there is a canonical bijection between Ψ ( s , κ ) and P ( B ) by (2.11). So P ( B ) is a labeling setfor Irr( B ) ∩ E ( G , ℓ ′ ).Now we assume that ( s , κ ) ∈ i Bl (1) ℓ ( G ). If − s , then every characterof Irr( ˜ B ) ∩ E ( ˜ G , ℓ ′ ) is parametrized by an element of i IBr (1) ℓ ( G ). From this we obtain the mapIrr( ˜ B ) ∩ E ( ˜ G , ℓ ′ ) → Irr( B ) ∩ E ( G , ℓ ′ ), which sends ˜ χ to the unique element of Irr( B ) ∩ Irr( G | ˜ χ ).So Ψ ( s , κ ) is a labeling set for Irr( B ) ∩ E ( G , ℓ ′ ). Thus P ( B ) is a labeling set for Irr( B ) ∩ E ( G , ℓ ′ )as in the last paragraph.Now we assume that − s . Let µ ∈ Ψ ( s , κ ). If ( s , µ ) ∈ i IBr (1) ℓ ( G ), then µ x + is degenerate and if ( s , µ ) ∈ i IBr (2) ℓ ( G ), then µ x + is non-degenerate. By the proof of [49,Prop. 15 (2)], if µ x + is degenerate, then it corresponds to an element of P (2 e , w x + ) (definedas in (2.8)) and if µ x + is non-degenerate, then µ x + and its copy correspond to the two elementof P (2 e , w x + ) \ P (2 e , w x + ) which are equivalent in the sense of (2.9). Thus we have a naturalbijection between P ( B ) and Irr( B ) ∩ E ( G , ℓ ′ ). (cid:3) By the proof of Proposition 5.12 and Remark 5.11, we have the following result immedi-ately.
Proposition 5.13.
Let B be an ℓ -block of G, µ ∈ P ( B ) and χ be the irreducible character in Irr( B ) ∩ E ( G , ℓ ′ ) corresponding to µ . Let g be an element of ˜ G which induces a non-trivialdiagonal automorphism and σ be a field automorphism. Then(i) χ σ is a character of B σ corresponding to σ ∗− µ ,(ii) χ g is a character of B g corresponding to µ † , where µ † is defined as in Proposition 3.11. Recall that V is a 2 n -dimensional symplectic space over F q with n ≥ G = I ( V ) = I ( V ) = Sp( V ) throughout this section. Let B be an ℓ -block of G covered by ˜ B = ˜ B ( s , κ )and define i W ℓ ( B ) to be the set of K = Q Γ K Γ where K Γ : S δ C Γ ,δ → { ℓ -cores } such that P δ, i , j ℓ δ | K Γ ( ψ Γ ,δ, i , j ) | = w Γ .A bijection between W ℓ ( B ) and i W ℓ ( B ) has been constructed implicitly in the proof of [4,(4F)] and can be described as follows. Let D be a defect group of B , V = C V ( D ) and V + = [ V , D ]29o that V = V ⊥ V + as above. Let ( D , b ) be a maximal Brauer pair of G containing (1 , B ), and ϑ be the canonical character of b . Then D = D × D + , b = b × b + and ϑ = ϑ × ϑ + , where D = h V i ≤ Sp( V ), D + ≤ Sp( V + ), b , b + are ℓ -blocks of Sp( V ) and C Sp( V + ) ( D + ) respectively,and ϑ ∈ Irr( b ), ϑ + ∈ Irr( b + ).Let ( R , ϕ ) be a B -weight, C = C G ( R ) and N = N G ( R ). Then there is an ℓ -block b of CR with defect group R and canonical character θ such that b G = B and ϕ = Ind NN θ ψ for some ψ ∈ Irr ( N θ | θ ). We may suppose Z ( D ) ≤ Z ( R ) ≤ R ≤ D . Thus V = C V ( R ) and V + = [ V , R ], sothat R = R × R + , C = C × C + , N = N × N + , where R = D , R + ≤ Sp( V + ), C = N = Sp( V ), C + = C Sp( V + ) ( R + ) and N + = N Sp( V + ) ( R + ). Let b = b × b + and θ = θ × θ + be the correspondingdecompositions. Then θ = ϑ and b Sp( V + ) + = b Sp( V + ) + . Note that N θ = N × N θ + . If ψ ∈ Irr ( N θ | θ ),then ψ = ϑ × ψ + , where ψ + ∈ Irr ( N θ + | θ + ). The map ( R , Ind NN θ ψ ) ( R + , Ind N + ( N + ) θ + ψ + ) is abijection from { ( R , Ind NN θ ψ ) | ψ ∈ Irr ( N θ | θ ) } to { ( R + , Ind N + ( N + ) θ + ψ + ) | ψ + ∈ Irr ( N θ + | θ + ) } . Thenthe bijection between W ℓ ( B ) and i W ℓ ( B ) can be given as in § i W ℓ ( B ) and P ( B ) : = Q Γ P ( β Γ e Γ , w Γ ). So P ( B )is also a labeling set for W ℓ ( B ) by [4, (4F)]. Proof of Theorem 1.3.
Let B be an ℓ -block of G = Sp n ( q ). Then P ( B ) is a labeling set for bothIrr( B ) ∩ E ( G , ℓ ′ ) and W ℓ ( B ) by the above argument. So | Irr( B ) ∩ E ( G , ℓ ′ ) | = |W ℓ ( B ) | . Thus theassertion follows by Theorem 2.14. (cid:3) By Corollary 3.7 and Proposition 3.11, we have the following result.
Proposition 5.14.
Let B be an ℓ -block of G, ( R , ϕ ) be a B-weight corresponding to µ ∈ P ( B ) Let g be an element of ˜ G which induces a non-trivial diagonal automorphism and σ be a fieldautomorphism of G. Then(i) ( R , ϕ ) σ is a B σ -weight corresponding to σ ∗− µ ,(ii) ( R , ϕ ) g is a B g -weight corresponding to µ † , which is defined as in Proposition 3.11. Note that X = G = Sp n ( q ) is the universal ℓ ′ -covering group of the simple group S = PSp n ( q ) by [28, § § G ⋊ E equal Aut( G ).Recall that E = h F p i .By Proposition 5.13 and 5.14, we have Theorem 5.15.
Let G = Sp n ( q ) , B an ℓ -block of G where q = p f is a power of an odd primep, n ≥ and ℓ ∤ q is an odd prime. Then there is an Aut( G ) B -equivariant bijection between Irr( B ) ∩ E ( G , ℓ ′ ) and W ℓ ( B ) . Theorem 5.16.
Keep the hypothesis and setup of Theorem 5.15. Assume further that ℓ is linear.Then there exists an Aut( G ) B -equivariant bijection between IBr ℓ ( B ) and W ℓ ( B ) .Proof. By Theorem 2.14, Irr( B ) ∩ E ( G , ℓ ′ ) is a basic set for B . Since ℓ is linear, the decompo-sition matrix is unitriangular by [27]. Hence there is an ( ˜ G ⋊ E ) B -equivariant bijection betweenIrr( B ) ∩ E ( G , ℓ ′ ) and IBr ℓ ( B ) by [17, Lem. 7.5]. Thus the assertion follows from Theorem5.15. (cid:3) Now we prove the main result of this paper for simple groups of type C n . Proof of Theorem 1.4.
With the assumption that f is odd, we know Aut( S ) / S is cyclic. Thenby Corollary 2.13, it su ffi ces to show that there exists an Aut( G ) B -equivariant bijection betweenIBr ℓ ( B ) and W ℓ ( B ) for every ℓ -block B of G . Then the assertion follows by Theorem 5.16. (cid:3) .4 The unipotent blocks We first summarize the description for the unipotent ℓ -blocks of symplectic groups above.Let G = Sp n ( q ), with n ≥ q odd. Then the unipotent ℓ -blocks of G are parametrisedby C (1). Let w = w x − . We also write B ( κ, w ) for B (1 , κ ) to emphasize w . Then P (2 e , w ) is alabeling set for the unipotent characters of B ( κ, w ).Let B = B ( κ, w ) be a unipotent ℓ -block of G . Then by the above argument, there is a bijectionbetween W ℓ ( B ) and P ( B ) = P (2 e , w ). By Proposition 5.14, we have Lemma 5.17.
Let B be a unipotent ℓ -block of G = Sp n ( q ) with q, ℓ as above. Then everyB-weight (in the sense of G-conjugacy class) is invariant under the action of ˜ G ⋊ E. Remark 5.18.
Now we give the relationship between ℓ -weights of ˜ G and ℓ -weights of G in theproof above. Since | ˜ G / GZ ( ˜ G ) | =
2, by the same argument as in [21, § § • ˜ R ˜ R ∩ G gives a bijection from Rad ℓ ( ˜ G ) to Rad ℓ ( G ) with inverse given by R R ×O ℓ ( Z ( ˜ G )). • Let ˜ R = R O ℓ ( Z ( ˜ G )) for R ∈ Rad ℓ ( G ). If ( ˜ R , ˜ ϕ ) is an ℓ -weight of ˜ G , then ( R , ϕ ) is an ℓ -weight of G for every ϕ ∈ Irr( N G ( R ) | ˜ ϕ ). Conversely, if ( R , ϕ ) is an ℓ -weight of G , thenthere exists ˜ ϕ ∈ Irr( N ˜ G ( ˜ R ) | ϕ ) such that ( ˜ R , ˜ ϕ ) is an ℓ -weight of ˜ G . • Let R ∈ Rad ℓ ( G ), ˜ R = R O ℓ ( Z ( ˜ G )), ( ˜ R , ˜ ϕ ) an ℓ -weight of ˜ G and ϕ ∈ Irr( N G ( R ) | ˜ ϕ ).Suppose that ˜ B is an ℓ -block of ˜ G and B is an ℓ -block of G . If ( ˜ R , ˜ ϕ ) is an ˜ B -weight and( R , ϕ ) is an B -weight, then ˜ B covers B . Lemma 5.19.
Let R ∈ Rad ℓ ( G ) , ˜ R = R O ℓ ( Z ( ˜ G )) and ( R , ϕ ) an ℓ -weight of G. Then |{ ˜ ϕ ∈ Irr( N ˜ G ( ˜ R ) | ϕ ) | ( ˜ R , ˜ ϕ ) is an ℓ -weight of ˜ G }| ≤ |O ℓ ′ ( ˜ G / G ) | .Proof. Let ˜ ϕ ∈ Irr( N ˜ G ( ˜ R ) | ϕ ) such that ( ˜ R , ˜ ϕ ) is an ℓ -weight of ˜ G . Since N ˜ G ( ˜ R ) / N G ( R ) ≤ ˜ G / G is cyclic, by Cli ff ord theory we have that Irr( N ˜ G ( ˜ R ) | ϕ ) = { ˜ ϕ τ | τ ∈ Irr( N ˜ G ( ˜ R ) / N G ( R )) } . If( ˜ R , ˜ ϕ τ ) is an ℓ -weight of ˜ G , then ˜ R is contained in the kernel of ϕ τ , and then O ℓ ( Z ( ˜ G )) iscontained in the kernel of τ . Thus τ ∈ Irr( N ˜ G ( ˜ R ) / O ℓ ( Z ( ˜ G )) N G ( R )). Now N ˜ G ( ˜ R ) / O ℓ ( Z ( ˜ G )) N G ( R )is an ℓ ′ -subgroup of ˜ G / G , thus the assertion holds. (cid:3) Proof of Theorem 1.5 for type C n . Let B = B ( κ, w ) be a unipotent ℓ -block of X = G = Sp n ( q ).Then by the argument above, P (2 e , w ) is a labeling set of W ℓ ( B ). By Lemma 5.17, everyelement of W ℓ ( B ) is ˜ G ⋊ E -invariant. On the other hand, Irr( B ) ∩ E ( G ,
1) is a basic set of B by Theorem 2.14. By [41, Thm. 2.5], every unipotent character of G is ˜ G ⋊ E -invariant. Thus P (2 e , w ) is a labeling set of IBr ℓ ( B ) and every element of IBr ℓ ( B ) is ˜ G ⋊ E -invariant. Hencethere exists an Aut( G )-equivariant bijection between IBr ℓ ( B ) and W ℓ ( B ). Then it su ffi ces toshow condition (iii) of Definition 2.10. Note that the number of ℓ -blocks of ˜ G covering B is |O ℓ ′ ( F × q ) | and there is a unique unipotent ℓ -block ˜ B of ˜ G covering B by Theorem 5.10. Then˜ B z : = ˆ z ⊗ ˜ B for z ∈ O ℓ ′ ( Z ( ˜ G ∗ )) are (all) the ℓ -blocks of ˜ G covering B . Claim : Every φ ∈ IBr ℓ ( B ) extends to ˜ G ⋊ E .Let ˜ φ ∈ IBr ℓ ( ˜ B ) be an extension of φ . Note that every element of IBr ℓ ( ˜ B ) is ˜ G ⋊ E -invariantsince Irr( ˜ B ) ∩ E ( ˜ G ,
1) is a basic set of ˜ B by Theorem 2.14 and every unipotent character of ˜ G is E -invariant. Thus ˜ φ extends to ˜ G ⋊ E and so does φ . Claim : If ( R , ϕ ) is a B -weight, then ϕ extends to ( ˜ G ⋊ E ) R .By Lemma 5.17, ϕ is ( ˜ G ⋊ E ) R -invariant. For z ∈ O ℓ ′ ( Z ( ˜ G ∗ )), there exists ˜ ϕ z ∈ Irr( N ˜ G ( ˜ R ) | ϕ )such that ( ˜ R , ˜ ϕ z ) is a ˜ B z -weight by Corollary 2.8. Now the number of ℓ -blocks of ˜ G which31over B is |O ℓ ′ ( F × q ) | , by Lemma 5.19, ˜ ϕ z is unique. Let ˜ ϕ = ˜ ϕ , that is, ˜ ϕ ∈ Irr( N ˜ G ( ˜ R ) | ϕ ) and( ˜ R , ˜ ϕ ) is a ˜ B -weight. Now ˜ B is D -invariant, so we have ˜ ϕ x ∈ Irr( N ˜ G ( ˜ R ) | ϕ ) and ( ˜ R , ˜ ϕ x ) is a˜ B -weight for all x ∈ ( ˜ G ⋊ E ) R . Thus ˜ ϕ x = ˜ ϕ and then ˜ ϕ is ( ˜ G ⋊ E ) R -invariant. Note that E acts trivially on Rad ℓ ( ˜ G ) / ∼ G . Thus there exists g ∈ G such that g − σ stabilizes R , and then( ˜ G ⋊ E ) R = N ˜ G ( R ) h g − σ i , which implies ( ˜ G ⋊ E ) R / N ˜ G ( ˜ R ) is cyclic. Hence ˜ ϕ extends to ( ˜ G ⋊ E ) R and so does ϕ .The remaining process is similar to the case of type B . For a set IBr ℓ ( B | Q ) as in Lemma2.12 (for definition, see the proof of [52, Lemma 2.10]) and φ ∈ IBr ℓ ( B | Q ), we let A : = A ( φ, Q ) = ( ˜ G ⋊ E ) / Z ( ˜ G ). By a similar argument as in the proof of [21, Prop. 7.1], conditions(1)-(3) of Definition 2.10 (iii) hold. For Definition 2.10 (iii)(4), the proof of [21, Lem. 7.2]applies here. (cid:3) D Let V be a 2 n -dimensional orthogonal space over F q with n ≥ G = I ( V ) = GO( V ) = GO ǫ n ( q ) and G = I ( V ) = SO( V ) = SO ǫ n ( q ) with ǫ ∈ {±} . As before, F p denotes the fieldautomorphism which sends ( a i j ) to ( a pi j ) and we write E = h F p i .In this section, we write J = CSO ǫ n ( q ) for the special conformal orthogonal groups and˜ G = GO ǫ n ( q ) for the general orthogonal groups for convention (of description of ℓ -weight ofspecial orthogonal groups in [4]), which is not the same as the notation in Appendix B.The blocks of SO ǫ n ( q ) can be obtained from CSO ǫ n ( q ) as we did for Sp n ( q ) in § ℓ -blocks, which are classified byCabanes and Enguehard [15], which is easier to describe. So we do not use the results ofAppendix B in this section. Let ℓ be an odd prime with ℓ ∤ q and e and e defined as before.As in § ℓ -blocks of G = SO ǫ n ( q ) are parametrised by the G -conjugacy classes of e -cuspidal pairs ( L , λ ). Here L satisfies that L = L F = SO δ n − we ) ( q ) × T we ,with either T e = GL ( q e ) if e is odd, or T e = GU ( q e ) if e is even, and δ = ǫ if e is odd or w is even, and δ = − ǫ else, and λ is an e -cuspidal unipotent character of L (for the structure of e -cuspidal pairs, see [11]). Following the notation of [44, § B = B ( L , λ ) for thecorresponding (unipotent) ℓ -block.Write λ = λ × T we , where λ is an e -cuspidal unipotent character of SO δ n − we ) ( q ). Let κ be the symbol corresponding to λ . Then by [11, § κ is an e -core. Moreover, the unipotentcharacters in the block B ( L , λ ) are then the members of the e -Harish-Chandra series above( L , λ ), and then the ones parametrised by the symbols of rank n and having e -core κ . Thus wealso write B ( κ, w ) for B ( L , λ ).If B = B ( κ, w ), then we let P ( B ) : = P (2 e , w ) if κ is non-degenerate, and P ( B ) : = P ′ (2 e , w )if κ is degenerate. Thus by (2.11), P ( B ) is a labeling set for Irr( B ) ∩ E ( G ,
1) . Also, B is a defectzero ℓ -block if and only if w =
0. So we always assume that w > B ( κ, w ) is then obtained as a Sylow ℓ -subgroup of C G ([ L , L ]), which isisomorphic to a Sylow ℓ -subgroup of GL ew ( q ) (if e is odd) or GU ew ( q ) (if e is even) by [45, § J : = CO ǫ n ( q ) and S = P Ω ǫ n ( q ). Then S is simple. By [28, § A : = ˜ J ⋊ E on S equal Aut( S ) except when n = ǫ = + . Recall that E = h F p i isthe group generated by the field automorphism F p which sends ( a i j ) to ( a pi j ). Here we start witha lemma following from [41, Thm. 2.5] immediately. Lemma 6.1. (i) Every unipotent character of ˜ G is ˜ A-invariant. ii) Any element g ∈ ˜ A fixes every unipotent character of G except when ǫ = + , the action of gon G can be induced by some element of ˜ G \ G and the unipotent character is labelled bya degenerate symbol (or an element of P (2 e , w ) ). Furthermore, such g interchanges thetwo unipotent characters in all pairs labeled by the same degenerate symbol. Lemma 6.2.
Let φ ∈ IBr ℓ ( B ) where B is a unipotent ℓ -block of G.(i) If ǫ = − , then φ extends to ˜ A.(ii) If ǫ = + and ℓ is linear, then φ extends to ˜ A φ .Proof. Let B = B ( κ, w ). If w =
0, then B is a defect zero ℓ -block, and then the assertionfollows from [41, Thm. 2.4]. Now we assume that w >
0. For every z ∈ O ℓ ′ ( Z ( J ∗ )) (notethat J ∗ is the special Cli ff ord group), we let ˆ z be the corresponding linear character of J (cf.[19, Prop. 13.30]) for z ∈ O ℓ ′ ( Z ( J ∗ )). Then by [19, Prop. 13.30], ˆ z ⊗ E ( J , = E ( J , z ). Thusthere exists an ℓ -block B z of J covering B such that Irr( B z ) ⊆ E ℓ ( J , z ) (see also Theorem B.2).Moreover, B z is unique. In this way, there are |O ℓ ′ ( F × q ) | ℓ -blocks of J covering B . We denote B = B . Then B z = ˆ z ⊗ B . Since z is of ℓ ′ -order, we may regard ˆ z as a linear ℓ -Brauer characterof J . Thus IBr ℓ ( B z ) = { ˆ z φ | φ ∈ IBr ℓ ( B ) } by [21, Lem. 2.4].If κ is non-degenerate, then there are two unipotent ℓ -blocks ˜ B (1) , ˜ B (2) of ˜ G covering B . Let ˜B ( i ) be the unique unipotent ℓ -block of ˜ J covering ˜ B ( i ) for i = ,
2. Since J / G (cid:27) ˜ J / ˜ G , we mayregard ˆ z as a linear character (or linear ℓ -Brauer character) of ˜ J for z ∈ O ℓ ′ ( Z ( J ∗ )). Note that ˜ G acts trivially on J / G , and then ˆ z is ˜ G -invariant. Thus B z is ˜ J -invariant for every z ∈ O ℓ ′ ( Z ( J ∗ )).Let ˜B ( i ) z = ˆ z ⊗ ˜B ( i ) for z ∈ O ℓ ′ ( Z ( J ∗ )). Then ˜B ( i ) z covers B z . From this, ˜B ( i ) z for i = , z ∈ O ℓ ′ ( Z ( J ∗ )) are distinct ℓ -blocks of ˜ J . In particular, there are |O ℓ ′ ( F × q ) | ℓ -blocks of ˜ J covering B ( i ) for i = ,
2. Now every character of Irr( B ) ∩ E ( G ,
1) is ˜ A -invariant by Lemma 6.1. ByTheorem 2.14, Irr( B ) ∩ E ( G ,
1) is a basic set of B . So every irreducible ℓ -Brauer character of B is ˜ A -invariant. Then there is an extension ˜ φ ∈ IBr ℓ ( ˜ B (1) ) of φ to ˜ G . By Lemma 6.1 again, ˜ B (1) is˜ A -invariant. So ˜ φ is ˜ A -invariant by Lemma 2.6. Then there exists an extension ˜ φ ′ ∈ IBr ℓ ( ˜B (1) )of ˜ φ . Note that the number of extensions of ˜ φ to ˜ J is at most |O ℓ ′ ( F × q ) | . By Lemma 2.6 again, ˜ φ ′ is ˜ A -invariant and then extends to ˜ A since ˜ A / ˜ J is cyclic.If ǫ = − , then every κ is non-degenerate, and then this assertion holds.Now we let ǫ = + . Then ℓ is linear. By [27], with a suitable order, the decomposition matrixof B with respect to the basic set Irr( B ) ∩ E ( G ,
1) is unitriangular. By [41, Thm. 2.4], every χ ∈ Irr( B ) ∩ E ( G ,
1) extends to ˜ A χ . Then by Lemma 2.9, every φ ∈ IBr ℓ ( B ) extends to ˜ A φ . Thiscompletes the proof. (cid:3) Now let B = B ( κ, w ) be a unipotent ℓ -block of G and D a defect group of B . We state theresults for B -weights which follows from the proof of [4, (4H)]. Let V = C V ( D ) and V + = [ V , D ]so that V = V ⊥ V + . Then dim( V + ) = ew . Let ( D , b ) be a maximal Brauer pair of G containing(1 , B ), and ϑ be the canonical character of b . Let ˜ G = GO( V ), G = SO( V ), ˜ G + = GO( V + )and G + = SO( V + ). Then D = D × D + , b = b × b + and ϑ = ϑ × ϑ + , where D = h V i ≤ G , D + ≤ G + , b , b + are ℓ -blocks of G and C G + ( D + ) respectively, and ϑ ∈ Irr( b ), ϑ + ∈ Irr( b + ).Let ( R , ϕ ) be a B -weight, ˜ C = C ˜ G ( R ), C = C G ( R ), ˜ N = N ˜ G ( R ) and N = N G ( R ). Thenthere is an ℓ -block b of CR with defect group R and canonical character θ such that b G = B and ϕ = Ind NN θ ψ for some ψ ∈ Irr ( N θ | θ ). We may suppose Z ( D ) ≤ Z ( R ) ≤ R ≤ D . Thus V = C V ( R ) and V + = [ V , R ], so that R = R × R + , C = G × C + , ˜ C = ˜ G × C + , N = h τ, G × N + i and ˜ N = ˜ G × ˜ N + , where R = D , R + ≤ G + , C + = C G + ( R + ), N + = N G + ( R + ), ˜ C + = C ˜ G + ( R + ),˜ N + = N ˜ G + ( R + ) and τ = τ × τ + with τ ∈ ˜ G , τ + ∈ ˜ G + of determinant −
1. Then ˜ N = h τ , N i . Let33 = b × b + and θ = θ × θ + be the corresponding decompositions. Then θ = ϑ and b ˜ G + + = b ˜ G + + and we suppose that ( R , b ) ≤ ( D , b ). Case . ϑ σ = ϑ for some σ ∈ ˜ G of determinant − i.e. , κ is non-degenerate. There aretwo irreducible characters ϑ ′ and ϑ ′′ of ˜ G covering ϑ . Let ϑ ′ = ϑ ′ × ϑ + , ϑ ′′ = ϑ ′′ × ϑ + , and b ′ , b ′′ be the ℓ -blocks of C ˜ G ( D ) containing ϑ ′ , ϑ ′′ respectively. Then b ′ ˜ G and b ′′ ˜ G are two ℓ -blocksof ˜ G . Let ˜ B = b ′ ˜ G . Case . V = ϑ σ , ϑ for any σ ∈ ˜ G of determinant − i.e. , κ is degenerate. If V =
0, then it is the case in Lemma A.1. Now assume that V , ϑ σ , ϑ for any σ ∈ ˜ G of determinant −
1. Also, we may assume that θ = ϑ × θ + for some character θ + of C + . Then˜ N θ = G × ˜ N θ + and N θ = G × N θ + . Thus each character ˜ ψ ∈ Irr( ˜ N θ | θ ) and each ψ ∈ Irr( N θ | θ )decompose as ˜ ψ = ϑ × ˜ ψ + and ψ = ϑ × ψ + for some ˜ ψ ∈ Irr( ˜ N θ + | θ + ) and ψ ∈ Irr( N θ + | θ + ).Then Res ˜ N θ N θ ˜ ψ is irreducible if and only if Res ˜ N θ + N θ + ˜ ψ + is irreducible (which is the case in LemmaA.1). Let ϑ ′ = ϑ + ϑ τ and b ′ be the ℓ -block of C ˜ G ( D ) D containing ϑ ′ = ϑ ′ × ϑ + . Then ˜ B = b ′ ˜ G is the unique ℓ -block of ˜ G covering B .For both cases, we define i W ℓ ( ˜ B ) to be the set of K : S δ C x − ,δ → { ℓ -cores } such that P δ, i , j ℓ δ | K ( ψ x − ,δ, i , j ) | = w . Note that N ϑ ′ = ˜ G × N ϑ + . If ψ ∈ Irr ( N ϑ ′ | ϑ ′ ), then ψ = ϑ ′ × ψ + , where ψ + ∈ Irr ( N ϑ + | ϑ + ).The map ( R , Ind NN ϑ ′ ψ ) ( R + , Ind N + ( N + ) ϑ + ψ + ) is a bijection from { ( R , Ind NN ϑ ′ ψ ) | ψ ∈ Irr ( N ϑ ′ | ϑ ′ ) } to { ( R + , Ind N + ( N + ) ϑ + ψ + ) | ψ + ∈ Irr ( N ϑ + | ϑ + ) } . Then the bijection between W ℓ ( ˜ B ) and i W ℓ ( ˜ B ) canbe given as in § i W ℓ ( ˜ B )and P (2 e , w ), so P (2 e , w ) is also a labeling set of W ℓ ( ˜ B ) (see also Appendix A.2.1).Similar with Lemma 5.17 (using Corollary 3.7 and Proposition 3.11), we have Lemma 6.3.
Every ˜ B-weight (in the sense of ˜ G-conjugacy class) is invariant under the actionof ˜ A = CO ǫ n ( q ) ⋊ E. Now we give ℓ -weights of G by the argument above and Lemma A.1. If κ is non-degenerate,then ( R , ˜ ϕ ) ( R , Res N ˜ G ( R ) N G ( R ) ˜ ϕ ) is a bijection from W ℓ ( ˜ B ) to W ℓ ( B ). If κ is degenerate and w isodd, then ( R , ˜ ϕ ) ( R , Res N ˜ G ( R ) N G ( R ) ˜ ϕ ) is also a bijection from W ℓ ( ˜ B ) to W ℓ ( B ). Now let κ bedegenerate and let w be even. If ( R , ϕ ) ∈ W ( ˜ B ) corresponds to some element in P (2 e , w ) \P (2 e , w ), then Res N ˜ G ( R ) N G ( R ) ˜ ϕ is irreducible. If ( R , ϕ ) ∈ W ( ˜ B ) corresponds to some element in P (2 e , w ), then Res N ˜ G ( R ) N G ( R ) ˜ ϕ is a sum of two irreducible constituents (for the construction of thesetwo irreducible constituents, see the proof of Lemma A.1). In addition, by the argument above, P ( B ) is a labeling set for W ℓ ( B ) . Corollary 6.4.
Suppose that g ∈ ˜ A and let ( R , ϕ ) be a B-weight. Let ˜ ϕ ∈ Irr( N ˜ G ( R ) | ϕ ) suchthat ( R , ˜ ϕ ) is a ˜ B-weight. Then the G-conjugacy class of ( R , ϕ ) is invariant under the action ofg except when ǫ = + , the action of g on G can be induced by some element of ˜ G \ G and ( R , ˜ ϕ ) corresponds to an element of P (2 e , w ) . Furthermore, when ( R , ˜ ϕ ) corresponds to some elementof P (2 e , w ) , g interchanges the two G-conjugacy classes of ( R , ϕ ) and ( R , ϕ ) , where ϕ and ϕ are the irreducible constituents of Res N ˜ G ( R ) N G ( R ) ˜ ϕ .Proof. By Lemma 6.3, it su ffi ce to show that the G -conjugacy class of ( R , ϕ ) is invariant underthe action of CSO ǫ n ( q ) ⋊ E if κ is degenerate, w is even and ( R , ˜ ϕ ) corresponds to some elementof P (2 e , w ). It is similar to the proof of Lemma 6.3. In fact, by the remark after [4, (4A)],for every δ , the restriction of every weight character in C x − ,δ to the subgroup of N x − ,δ, i withdeterminant 1 is irreducible and then is invariant under the action of CSO ǫ n ( q ) ⋊ E by Lemma3.10 (i). So we conclude from Lemma A.1 a similar result to Proposition 3.6 and 3.11 (i), hencethe proof of Lemma 5.17 also applies here, and finally that this assertion holds. (cid:3) emma 6.5. Let ( R , ϕ ) be a B-weight, where B is a unipotent ℓ -block of G. Then ϕ extends to ˜ A R ,ϕ .Proof. The proof here is similar to the proof of
Claim in the proof of Theorem 1.5 for type C ,using the argument in the proof of Lemma 6.2. (cid:3) Lemma 6.6.
Let T = [ G , G ] = Ω ǫ n ( q ) , B a unipotent ℓ -block of G and b an ℓ -block of T coveredby B. Then(i) Res GT : IBr ℓ ( B ) → IBr ℓ ( b ) is a bijection, and(ii) ( R , ϕ ) ( R , Res N G ( R ) N T ( R ) ϕ ) is a bijection from W ℓ ( B ) to W ℓ ( b ) .Proof. (ii) is similar with Proposition 4.10. For (i), note that there exists another ℓ -block of G covering b . Indeed, let 1 , z ∈ Z ( G ∗ ) = Z ( G ) (we have G ∗ = G here) and ˆ z be the correspondinglinear character (cf. [19, Prop. 13.30]), then ˆ z ⊗ B is an ℓ -block of G covering b and is containedin E ℓ ( G , z ). So ˆ z ⊗ B , B . Thus (i) holds by Lemma 2.6. (cid:3) Note that X = Spin ǫ n ( q ) is the universal ℓ ′ -covering group of the simple group S = P Ω ǫ n ( q )by [28, § Proof of Theorem 1.5 (ii) and (iii).
Let b be a unipotent ℓ -block of X = Spin ǫ n ( q ). Then b domi-nates an ℓ -block b of T = Ω ǫ n ( q ). Thus there exists a unique unipotent ℓ -block B of G = SO ǫ n ( q )covering b . By Lemma 6.1 and Proposition 6.4, there is an ˜ A -equivariant bijection between W ℓ ( B ) and Irr( B ) ∩ E ℓ ( G , ǫ = − , then every character of Irr( B ) ∩ E ℓ ( G ,
1) is ˜ A -invariant.If ǫ = + , then by assumption, ℓ is linear and thus by [17, Lem 7.5], there is an ˜ A -equivariantbijection between Irr( B ) ∩ E ℓ ( G ,
1) and IBr ℓ ( B ) (since the corresponding decomposition matrixis unitriangular by [27]). In both cases, there exist ˜ A -equivariant bijections between W ℓ ( B )and IBr ℓ ( B ). From this, by Lemma 6.6, we get an ˜ A -equivariant bijection between W ℓ ( b ) andIBr ℓ ( b ). Then by Lemma 2.5, there exists an Aut( X )-equivariant bijection between W ℓ ( b ) andIBr ℓ ( b ). By Lemma 2.12, we only need to verify Definition 2.10 (iii) now.The remaining process is similar with the case of type B since all irreducible characterof b have Z ( X ) in their kernel. Let ¯ b be the ℓ -block of S = P Ω ǫ n ( q ) dominated by b . Fora set IBr ℓ ( b | Q ) as in Lemma 2.12 (for definition, see the proof of [52, Lemma 2.10]) and φ ∈ IBr ℓ ( b | Q ), we denote by φ ′ ∈ IBr ℓ ( b ) and ¯ φ ∈ IBr ℓ (¯ b ) the ℓ -Brauer characters associatedwith φ . Let A : = A ( φ, Q ) = ˜ A φ ′ / Z ( G ), where ˜ A = ˜ J ⋊ E as above. Then by Lemma 6.2 and6.5, conditions (1)-(3) of Definition 2.10 (iii) hold. For Definition 2.10 (iii)(4), the proof of [21,Lem. 7.2] applies here. (cid:3) A Appendix A: Remarks on [4]
In line 35 of [4, p. 33], after “Then w ∈ X and so ξ ( w ) = ξ x ( w )”, we can only get ξ i ( h ) ξ j ( h ) = ξ ′ i ( h ) ξ ′ j ( h ) but can not conclude ξ i ( h ) = ξ ′ i ( h ), nor does the claim X S m = K in line 29 of [4, p. 33]follow. For this reason, we give a new proof for [4, (4C)].We will completely follow the notation in [4] and all references in this proof are to this paperthroughout Appendix A. 35 .1 For (4C) A.1.1
Now we give some conventions for orthogonal cases first. Let Γ ∈ F and C Γ , d = { ϕ Γ , d , i , j | ≤ i ≤ e , ≤ j ≤ r d } be the set as page 32.Let ( R , θ ) be a pair of type Γ and R = R , ,γ, c ( c = ( c , . . . , c l )) a basic subgroup such that γ + c + · · · + c l = d . Let V be the underlying (orthogonal) space of R . Then | Irr ( N ( θ ) | θ ) | = e ( r − l . Let N ( θ ) = N ( θ ) ∩ SO( V ). Then | N ( θ ) : N ( θ ) | = ( N ( θ ) | θ ) to N ( θ ) is irreducible by the remark of (4A). For ϕ , ϕ ∈ N ( θ ), wewrite ϕ ∼ ϕ if ϕ | N ( θ ) = ϕ | N ( θ ) . Then if ϕ ∼ ϕ and ϕ , ϕ , then ϕ ( n ) = − ϕ ( n ) for any n ∈ N ( θ ) with determinant − . (A.1)We may assume that ϕ Γ , d , i , j ∼ ϕ Γ , d , e + i , j for all 1 ≤ i ≤ e , ≤ j ≤ r d . (A.2)Now we keep the notation and assumption preceding (4B). Let B be the block in (4B). Thenthe proof of (4B) gives a bijection between B -weights and the assignments a d ≥ C Γ , d → { r -cores } , ϕ Γ , d , i , j κ Γ , d , i , j (A.3)such that P d ≥ r d β Γ e Γ P i = r d P j = | κ Γ , d , i , j | = w Γ .(4C) should be as follows. Lemma A.1.
With the notation and hypothesis preceding (4C), let G = O ( V ) be an orthogonalgroup, G = SO( V ) , and R a radical subgroup of G such that [ V , R ] = V. Let ( R , b ) a Brauerpair of G labeled by ( R , s , − ) and θ the canonical character of b. Then | N ( θ ) : N ( θ ) | = β Γ .Moreover, if we write N ( θ ) = N ( θ ) ∩ G , then the restriction ψ | N ( θ ) of each ψ ∈ Irr ( N ( θ ) , θ ) toN ( θ ) is irreducible unless when Γ ∈ F , w Γ is even and the associated assignment (A.3) of ψ satisfies that κ Γ , d , i , j = κ Γ , d , e + i , j for all ≤ i ≤ e , ≤ j ≤ r d . A.1.2 ProofLemma A.2.
Let M be arbitrary finite group, M , M E M with M M = M and L = M ∩ M .Let ξ ∈ Irr( M ) such that ϕ = ξ L is irreducible. Let ψ ∈ Irr( M ( ξ ) | ξ ) and η = ψ | M ( ξ ) ∩ M . Then η is irreducible and the following statements hold.(i) If M ( η ) = M ( ξ ) ∩ M , then the restriction of Ind MM ( ξ ) ( ψ ) to M is irreducible.(ii) If | M ( η ) : M ( ξ ) ∩ M | = , then the restriction of Ind MM ( ξ ) ( ψ ) to M is a sum of twoirreducible characters.(iii) If M ( ϕ ) = M ( ξ ) ∩ M , then the restriction defines a bijection from Irr( M | ξ ) onto Irr( M | ϕ ) . roof. By [31, Cor. (4.2)], the restriction defines a bijection from Irr( M ( ξ ) | ξ ) onto Irr( M ( ξ ) ∩ M | ϕ ). So η is irreducible.(i) By Mackey formula, (Ind MM ( ξ ) ( ψ )) | M = Ind M M ( ξ ) ∩ M ( ψ | M ( ξ ) ∩ M ) = Ind M M ( η ) ( η ). Then (i)follows by Cli ff ord theory.(ii) By Gallagher’s theorem, Ind M ( η ) M ( ξ ) ∩ M ( η ) = ˜ η + ˜ η ′ , where ˜ η and ˜ η ′ are two extensions of η to M ( η ) (and ˜ η , ˜ η ′ ). Thus Ind M M ( ξ ) ∩ M ( η ) = Ind M M ( η ) ( ˜ η ) + Ind M M ( η ) ( ˜ η ′ ) and both Ind M M ( η ) ( ˜ η ) andInd M M ( η ) ( ˜ η ′ ) are irreducible (and Ind M M ( η ) ( ˜ η ) , Ind M M ( η ) ( ˜ η ′ )) by Cli ff ord theory. Then (ii) followsby Mackey formula.(iii) follows from (i) immediately. (cid:3) Proof of Lemma A.1.
Note that the first and second paragraph of (4C) also apply here. Keepthe notation in (4C) (and in the first and second paragraph of its proof). First, we have | N ( θ ) : N ( θ ) | = β Γ and then we may assume that Γ ∈ F . Also, we suppose that u = d = d .For ψ ∈ Irr ( N ( θ ) , θ ), we write ψ = Ind HX S m ( ˜ ξχ ) (as in the second paragraph of the proof of(4C)), where χ is some character of X S m trivial on X . Let ξ = ξ | X and let ξ = ⊠ dk = ξ k , where ξ k ∈ Irr( T ). By the remark of (4A), ξ k | T is irreducible for every 1 ≤ k ≤ d and hence ξ | T d isirreducible. So ξ is irreducible. Let K be the stabilizer of ξ in H and let ˜ ξ = ˜ ξ | X S m . Then ˜ ξ is an extension of ξ to X S m . By Cli ff ord theory, each irreducible character of X S m covering ξ has the form ˜ ξ χ , where χ is an irreducible character of X S m trivial on X . Then by LemmaA.2 (iii), ψ | N ( θ ) is irreducible if K = X S m .Now we assume that K , X S m . Let x ∈ K such that x < X S m = H ( ξ ) first. Then d > x ∈ S ( d ). Write ξ ′ : = ξ x = ⊠ dk = ξ ′ k , where ξ ′ k ∈ Irr( T ). Then ξ ′ , ξ and ξ ′ | X = ξ | X = ξ . Thus both ξ ′ k and ξ k are extensions of the irreducible character ξ k | T to T for every 1 ≤ k ≤ d ( i.e. , ξ ′ k ∼ ξ k ). Let 1 ≤ i , j ≤ n with i , j and h ∈ T with determinant − w = diag { w , . . . , w d } such that w i = h = w j and w k = k , i , j . Then w ∈ X andso ξ ( w ) = ξ ′ ( w ) and then ξ i ( h ) ξ j ( h ) = ξ ′ i ( h ) ξ ′ j ( h ). Thus if ξ ′ i , ξ i , then ξ ′ j , ξ j by (A.1). Henceeither ξ ′ k = ξ k for all 1 ≤ k ≤ d or ξ ′ k ∼ ξ k and ξ ′ k , ξ k for all 1 ≤ k ≤ d holds. On the other hand,if this holds, it is easy to check that K , X S m . Also, by the argument above, | K : X S m | ≤ ξ = ⊠ d , i , j ϕ t Γ , d , i , j Γ , d , i , j . By the argument above, K , X S m , if and only if t Γ , d , i , j = t Γ , d , e + i , j for all 1 ≤ i ≤ e , ≤ j ≤ r d . (A.4)This occurs only when w Γ is even.Now we assume that K , X S m and thus | K : X S m | = ζ = ˜ ξχ and ζ = ˜ ξ χ , then ζ = ζ | X S m . Then X S m ≤ H ( ζ ) ≤ K . Then by (i) and (ii) of Lemma A.2, ψ | N ( θ ) is irreducible if H ( ζ ) = X S m , and ψ | N ( θ ) is a sum of two irreducible characters if H ( ζ ) = K .Now let x ∈ K \ X S m and without loss of generality we assume that x ∈ S ( d ). Now we maywrite ξ = Q si = ( ξ t i i × ξ ′ i t i ) with ξ i ∼ ξ ′ i and ξ i , ξ ′ i for all 1 ≤ i ≤ s . Then ξ x = Q si = ( ξ ′ i t i × ξ it i ).Note that the values of the extension of ξ t i i to T ≀ S ( t i ) on S ( t i ) only depend on ξ i (1) (see, forexample, [30, Lem. 25.5]), i.e. , there exists extension η i (resp. η ′ i ) of ξ t i i (resp. ξ ′ t i i ) to T t i ⋊ S ( t i )such that η i | S ( t i ) = η ′ i | S ( t i ) . So we may assume that ˜ ξ x | S m = ˜ ξ | S m . On the other hand, ˜ ξ | X = ξ , then˜ ξ x | X = ˜ ξ | X , and then ˜ ξ x | X S m = ˜ ξ | X S m . Thus ˜ ξ x = ˜ ξ . Hence H ( ζ ) = K ⇔ ˜ ξ χ = ( ˜ ξ χ ) x = ˜ ξ χ x ⇔ χ = χ x . So H ( ζ ) = K if and only if κ Γ , d , i , j = κ Γ , d , e + i , j for all 1 ≤ i ≤ e , ≤ j ≤ r d . This completes theproof. (cid:3) A.2 (4E) and (4H)
The remark of (4E) and (4H) used (4C). Thus we give some remarks here.37 .2.1 The remark of (4E)
By Lemma A.1, the remark of (4E) should be stated as follows.With the assumption of (4E), we have a bijection between B -weights and Q Γ T Γ , where T Γ is the set of β Γ e Γ -tuples ( κ , κ , . . . , κ β Γ e Γ ) of partitions k i such that β Γ e Γ P i = | κ i | = w Γ (by the proof of(4E).Let G = O ( V ), G = SO( V ), ( R , ϕ ) a B -weight of G , and θ an irreducible character of C = C G ( R ) covered by ϕ . Then | N ( θ ) : N ( θ ) | = m X ± = m X ± ,
0. Moreover,for each ψ ∈ Irr ( N ( θ ) | θ ), the restriction ψ | N ( θ ) ∩ G is irreducible unless when m X ± , w X ± ’sare even and the element κ = Q Γ κ Γ (with κ Γ = ( κ Γ , κ Γ , . . . , κ Γ β Γ e Γ )) in Q Γ T Γ corresponding to ψ satisfies that κ Γ i = κ Γ e + i for every Γ ∈ F and every 1 ≤ i ≤ e . A.2.2 (4H)
For Γ ∈ F , we recall that the integer f Γ is defined to be the number of β Γ e Γ -tuples ( κ , . . . , κ β Γ e Γ )of partitions such that β Γ e Γ P i = | κ i | = w Γ .If Γ ∈ F and w Γ is even, we define f ′ Γ to be the number of e -tuples ( κ , . . . , κ e ) of partitionssuch that e P i = | κ i | = w Γ .The conclusion of (4E) should be as follows.(1) The number of B -weights is Q Γ f Γ if one of the following statements holds:(a) m X ± ( s + ) = ϑ σ = ϑ for some σ ∈ O ( V ) of determinant − m X ± ( s + ) , V = ϑ σ , ϑ for any σ ∈ O ( V ) of determinant − w X − or w X + is odd, then the number of B -weights is Q Γ f Γ .(b) If both w X − and w X + are even, then the number of B -weights is f X − f X + + f ′ X − f ′ X + Q Γ < F f Γ . B Appendix B: The blocks of special orthogonal groups ineven-dimension
Let G = SO ǫ n ( q ) with ǫ ∈ {±} , q odd and n ≥
4. Now we give a classification for ℓ -blocks of G for an odd prime ℓ ∤ q , which is completely analogous with the case of Sp n ( q ) in § V be the underlying space of G and ˜ G = CSO( V ). Then ˜ G ∗ is the special Cli ff ord group over V ∗ and G ∗ = G . Let π : ˜ G ∗ → G ∗ be the natural epimorphism. As usual, we let G = SO ǫ n ( F q ) and˜ G = CSO ǫ n ( F q ) for the corresponding algebraic groups and F the Frobenius endomorphism.Note that the notation of special conformal orthogonal groups and general orthogonal groupsare not the same with those in §
6, since the relations of ℓ -blocks of SO ǫ n ( q ) and CSO ǫ n ( q ) issimilar with the relations of ℓ -blocks of Sp n ( q ) and CSp n ( q ). So we use the notation which isanalogous to the one in § G have been classified by Lusztig [39]. For a semisimpleelement s of ˜ G ∗ , we write ¯ s = π ( s ). Note that both m x − ( ¯ s ) and m x + ( ¯ s ) are even. Let i Irr( ˜ G )38e the set of ˜ G ∗ -conjugacy classes of pairs ( s , µ ), where s is a semisimple ℓ ′ -element of ˜ G ∗ and µ ∈ Ψ ( ¯ s ) (where Ψ ( ¯ s ) is defined as (2.4)). Here, ( s , µ ) and ( s ′ , µ ′ ) are conjugate if and only if s and s ′ are ˜ G ∗ -conjugate and µ = µ ′ . By Jordan decomposition of characters, there is a bijectionfrom i Irr( ˜ G ) onto Irr( ˜ G ) (see also [24, (4.5)]). We write ˜ χ s ,µ for the character of ˜ G correspondingto ( s , µ ).If χ ∈ Irr( G | ˜ χ s ,µ ), then we say χ corresponds to the pair ( ¯ s , µ ). Furthermore, we have thefollowing result about the characters of G , which is similar to Lemma 5.3. Lemma B.1.
Let ( s , µ ) ∈ i Irr( ˜ G ) , ¯ s = π ( s ) and ˜ χ = ˜ χ s ,µ .(i) If or − is not an eigenvalue of ¯ s, then Res ˜ GG ˜ χ is irreducible.(ii) If both and − are eigenvalues of ¯ s, then • if µ x − or µ x + is degenerate, then Res ˜ GG ˜ χ is irreducible, and • if both µ x − and µ x + are non-degenerate, then Res ˜ GG ˜ χ is a sum of two irreducibleconstituents. Let I = GO ǫ n ( q ). We recall the action of I on Irr( G ) which was given in [24, (4D)]. Let( s , µ ) ∈ i Irr( ˜ G ), ˜ χ = ˜ χ s ,µ and g ∈ I of determinant −
1. Then ˜ χ g corresponds to the pair( g ∗ sg ∗− , µ ∗ ) (the operator ∗ is defined as in [24, §
2, p. 132]). More precisely, we have(a) ˜ χ g = ˜ χ if • s , − s , and µ x − is non-degenerate, or • both 1 and − s and µ x − is non-degenerate, and(b) ˜ χ g , ˜ χ and Res ˜ GG ( ˜ χ g ) = Res ˜ GG ( ˜ χ ) if • s , − s , and µ x + is non-degenerate, or • both 1 and − s and µ x − is degenerate and µ x + is non-degenerate,and(c) ˜ χ g , ˜ χ and Res ˜ GG ( ˜ χ g ) , Res ˜ GG ( ˜ χ ) if • both 1 and − s , or • s , − s , and µ x − is degenerate, or • s , − s , and µ x + is degenerate, or • both 1 and − s and both µ x − and µ x + are degenerate.Now let χ ∈ Irr( G | ˜ χ ). By Lemma B.1, if we are in case (b) or (c), then χ = Res ˜ GG ˜ χ . Moreover, χ is I -invariant in case (b) and χ is not I -invariant in case (c). If 1 is an eigenvalue of ¯ s and − s or both 1 and − s and µ x − is non-degenerateand µ x + is degenerate, then we also have χ = Res ˜ GG ˜ χ and thus χ is I -invariant. If both 1 and − s and both µ x − and µ x + are non-degenerate, then Res ˜ GG ˜ χ is a sum of twoirreducible constituents, and then ˜ χ is I -invariant but we do not know whether χ is I -invariantor not in this case now.Now we recall the classification of ℓ -blocks of ˜ G F given in [24, § i Bl ℓ ( ˜ G ) be the setof ˜ G ∗ -conjugacy classes of pairs ( s , κ ) where s is a semisimple ℓ ′ -element of ˜ G ∗ and κ ∈ C ( ¯ s ),39here ¯ s = π ( s ) and C ( ¯ s ) is defined as (2.5). Here, ( s , κ ) and ( s ′ , κ ′ ) are ˜ G ∗ -conjugate if and onlyif s and s ′ are ˜ G ∗ -conjugate and κ = κ ′ . Also note that both m x − ( ¯ s ) and m x + ( ¯ s ) are even and | κ | = ,
2, or 4. By [24, (11E)], there is a bijection ( s , κ ) ˜ B ( s , κ ) from i Bl ℓ ( ˜ G ) to Bl ℓ ( ˜ G ).For ( s , κ ) ∈ i Bl ℓ ( ˜ G ), [24, (13B)] also gave a criterion that when an irreducible character of˜ G lies in the ℓ -block ˜ B = ˜ B ( ¯ s , κ ). In particular, the irreducible characters of Irr( ˜ B ) ∩ E ( ˜ G , ℓ ′ )are of form ˜ χ s ,µ with µ ∈ Ψ ( ¯ s , κ ) (where Ψ ( ¯ s , κ ) is defined as in (2.6)). In addition, by (2.11), Ψ Γ ( ¯ s , κ ) is in bijection with P ( β Γ e Γ , w Γ ) if Γ ∈ F ∪ F or Γ ∈ F and κ Γ is non-degenerateand in bijection with P ′ (2 e , w Γ ) if Γ ∈ F and κ Γ is degenerate. Here, the sets P ( β Γ e Γ , w Γ ) and P ′ (2 e , w Γ ) are defined as in (2.7) and (2.10) respectively.Let e be the identity element of the Cli ff ord algebra over V ∗ . With the similar argument with § e -Jordan-cuspidal pair for an ℓ -block ˜ B ( s , κ ) of ˜ G = CSO ǫ n ( q ),which is completely analogous with the case of ˜ G = CSp n ( q ) and then we have the followingresult which is completely analogous with Theorem 5.10. Theorem B.2.
Let ( s , κ ) ∈ i Bl ℓ ( ˜ G ) , ¯ s = π ( s ) , ˜ B = ˜ B ( s , κ ) and B an ℓ -block of G covered by ˜ B.(i) If or − is not eigenvalue of ¯ s, then B is the unique ℓ -block of G covered by ˜ B andthere are |O ℓ ′ ( F × q ) | ℓ -blocks of ˜ G covering B. In addition, the ℓ -blocks covering B of ˜ G are ˜ B ( zs , κ ) , where z runs through O ℓ ′ ( Z ( ˜ G ∗ )) .(ii) If both and − are eigenvalues of ¯ s, and there exists Γ ∈ F such that w Γ = and κ Γ isdegenerate, then B is the unique ℓ -block of G covered by ˜ B and there are |O ℓ ′ ( F × q ) | ℓ -blocksof ˜ G covering B. In addition, the ℓ -blocks covering B of ˜ G are ˜ B ( zs , κ ) and ˜ B ( zs , κ ′ ) , wherez runs through a complete set of representatives of h− e i -cosets in O ℓ ′ ( Z ( ˜ G ∗ )) .(iii) Suppose that both and − are eigenvalues of ¯ s and w Γ , if Γ ∈ F and κ Γ is degenerate.Then there are |O ℓ ′ ( F × q ) | ℓ -blocks of ˜ G covering B and they are ˜ B ( zs , κ ) , where z runsthrough a complete set of representatives of h− e i -cosets in O ℓ ′ ( Z ( ˜ G ∗ )) . Moreover, • if κ x − or κ x + is degenerate, then B is the unique ℓ -block of G covered by ˜ B, and • if both κ x − and κ x + are non-degenerate, then there are two ℓ -blocks of G covered by ˜ B. Now let i Bl (1) ℓ ( G ) be the set of G -conjugacy classes of pairs ( s , κ ), where s ∈ G ∗ is asemisimple ℓ ′ -element and κ ∈ C ( s ) such that either (1) 1 or − s or(2) κ x − or κ x + is degenerate. Here, we identify ( s , κ ) with ( s , κ ′ ). Let i Bl (2) ℓ ( G ) be the set of G ∗ -conjugacy classes of pairs ( s , κ ), where s ∈ G ∗ is a semisimple ℓ ′ -element and κ ∈ C ( s )such that both 1 and − s and both κ x − and κ x + are non-degenerate. Then i Bl ℓ ( G ) : = i Bl (1) ℓ ( G ) ∪ i Bl (2) ℓ ( G ), where the elements of i Bl (2) ℓ ( G ) counting twice, is a labelingset for Bl ℓ ( G ) by Theorem B.2.Let i IBr (1) ℓ ( G ) be the set of G ∗ -conjugacy classes of pairs ( s , µ ), where s ∈ G ∗ is a semisimple ℓ ′ -element and µ ∈ Ψ ( s ) such that either 1 or − s or both 1 and − s and µ x − or µ x + is degenerate. Let i IBr (2) ℓ ( G ) be the set of G ∗ -conjugacyclasses of pairs ( s , µ ), where s ∈ G ∗ is a semisimple ℓ ′ -element and µ ∈ Ψ ( s ) such that both1 and − s and both µ x − and µ x + are non-degenerate. Then i IBr ℓ ( G ) : = i IBr (1) ℓ ( G ) ∪ i IBr (2) ℓ ( G ), where the elements of i IBr (2) ℓ ( G ) counting twice, is a labeling set forIrr( G ) ∩ E ( G , ℓ ′ ) by Lemma B.1. If ( s , µ ) ∈ i IBr (1) ℓ ( G ), we denote by χ s ,µ the character of G corresponding to ( s , µ ). If ( s , µ ) ∈ i IBr (2) ℓ ( G ), then χ (1) s ,µ and χ ( − s ,µ denote the two characters of G corresponding to ( s , µ ). 40urthermore, if ( s , κ ) ∈ i Bl (1) ℓ ( G ) and B = B ( s , κ ), then Irr( B ) ∩ E ( G , s ) = { χ s ,µ | µ ∈ Ψ ( s , κ ) , ( s , µ ) ∈ i IBr (1) ℓ ( G ) } ∪ { χ ( ± s ,µ | µ ∈ Ψ ( s , κ ) , ( s , µ ) ∈ i IBr (2) ℓ ( G ) } . If ( s , κ ) ∈ i Bl (2) ℓ ( G ), B (1) = B (1) ( s , κ ) and B ( − = B ( − ( s , κ ), then Irr( B (1) ∪ B ( − ) ∩ E ( G , s ) = { χ ( ± s ,µ | µ ∈ Ψ ( s , κ ) } . Wemay assume that Irr( B ( i ) ) ∩ E ( G , s ) = { χ ( i ) s ,µ | µ ∈ Ψ ( s , κ ) } for i = ±
1. Note that, if both 1 and − s , and κ x − and κ x + are non-degenerate, then both µ x − and µ x + are alsonon-degenerate, and then we always have ( s , µ ) ∈ i IBr (2) ℓ ( G ).Now we give a labeling set for Irr( B ) ∩ E ( G , ℓ ′ ). First, we define a set P (2 e , w , w ) : = P (2 e , w ) × P (2 e , w ) for integers e ≥ w , w ≥
0, where P (2 e , w ) and P (2 e , w ) are definedas in (2.7). Now we define P (2 e , w , w ) : = P (2 e , w ) × P (2 e , w ), where P (2 e , w ) and P (2 e , w ) are defined as in (2.8). First we define an equivalent relation on the set P (2 e , w , w ).For µ ( k ) = µ ( k , × µ ( k , , where µ ( k , = ( µ ( k , , . . . , µ ( k , e ) ∈ P (2 e , w ), µ ( k , = ( µ ( k , , . . . , µ ( k , e ) ∈P (2 e , w ) and k = ,
2, we let µ (1) ∼ µ (2) if µ (1 , ∼ µ (2 , and µ (1 , ∼ µ (2 , in the sense of (2.9).Then we define P ′ (2 e , w , w ) = ( P (2 e , w , w ) \ P (2 e , w , w ))) / ∼ . Let P ′ (2 e , w , w ) : = P ′ (2 e , w , w ) ∪ P (2 e , w , w ), where the elements of P (2 e , w , w ) are counted twice.Define P ( B ) : = (a) Q Γ P ( β Γ e Γ , w Γ ) if one of the following holds, • w x − = w x + =
0, or • κ x − or κ x + is non-degenerate,(b) P ′ (2 e , w Γ ) × Q Γ , Γ P ( β Γ e Γ , w Γ ) if both κ x − and κ x + are degenerate and there exists a unique Γ ∈ F such that w Γ is odd, where P ′ (2 e , w Γ ) is defined as in (2.10),(c) P ′ (2 e , w x − , w x + ) × Q Γ < F P ( e Γ , w Γ ), if one of the following holds, • both κ x − and κ x + are degenerate and both w x − and w x + are odd, or • w x − or w x + is non-zero, both κ x − and κ x + are degenerate and both w x − and w x + areeven.Similar with Proposition 5.12, we have Proposition B.3.
With the preceding notation, P ( B ) is a labeling set for Irr( B ) ∩ E ( G , ℓ ′ ) . Let f Γ and f ′ Γ be defined as in Appendix A.2.2. Then f Γ = |P ( β Γ e Γ , w Γ ) | and f ′ Γ = |P ( β Γ e Γ , w Γ ) | .We end the appendix by giving the number of irreducible ℓ -Brauer characters in an ℓ -block of G = SO ± n ( q ), which follows by Proposition B.3 immediately. Theorem B.4.
Let B be an ℓ -block corresponding to ( s , κ ) ∈ i Bl ℓ ( G ) and l ( B ) = | IBr ℓ ( B ) | .(i) l ( B ) = Q Γ f Γ if one of the following statements holds. • w x − = w x + = . • κ x − or κ x + is non-degenerate.(ii) Suppose that w x − or w x + is non-zero and both κ x − and κ x + are degenerate. • If either w x − or w x + is odd, then l ( B ) = Q Γ f Γ . • If both w x − and w x + are even, then l ( B ) = f x − f x + + f ′ x − f ′ x + Q Γ < F f Γ . G = SO ± n ( q ). Proposition B.5.
Let B be an ℓ -block corresponding to ( s , κ ) ∈ i Bl ℓ ( G ) . If • or − is not an eigenvalue of s, or • κ x − or κ x + is degenerate,then the Alperin weight conjecture 1.1 holds for B, i.e., l ( B ) = |W ℓ ( B ) | .Proof. Let V , ϑ and s be defined as in [4, (4H)]. Then by the proof of [4, (4H)], ϑ ∈ G and˜ ϑ ∈ Irr( ˜ G ) with G = SO( V ) and ˜ G = CSO( V ) such that ˜ ϑ = ˜ χ t ,κ , where t ∈ ˜ G ∗ satisfiesthat s = π ( t ). Then the assertion follows by § A.2.2 and Theorem B.4 and the criterion forwhen ϑ is GO( V )-invariant given in the statements after Lemma B.1. (cid:3) Now we consider the following properties about the action of I = GO ǫ n ( q ) on the charactersof G .( † ) Let ˜ χ ∈ Irr( ˜ G ) and ∆ = Irr( G | ˜ χ ). Then I ∆ acts trivially on ∆ .( ‡ ) Let s be a semisimple element of G , µ ∈ Ψ ( s ) and χ be a character of G correspondingto ( s , µ ). Suppose that both 1 and − s and both µ x − and µ x + arenon-degenerate. Then χ is I -invariant.Then by the proof of Proposition B.5 and the statements after Lemma B.1, we have Theorem B.6. (i) The Alperin weight conjecture 1.1 holds for every ℓ -block of the specialorthogonal group G = SO ǫ n ( q ) with every n ≥ , odd q and ǫ = ± if ( † ) is true for thespecial orthogonal group G = SO ǫ n ( q ) with every n ≥ , odd q and ǫ = ± .(ii) ( † ) holds if and only if ( ‡ ) holds. Acknowledgements
The first author wishes to thank Gunter Malle for fruitful conversations, vital informationand useful indications. The authors thank him for commenting on an earlier version of thispaper.
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