SCOPES REDUCTION AND MORITA EQUIVALENCE CLASSES OF BLOCKS IN FINITE CLASSICAL GROUPS II

GERHARD HISS AND RADHA KESSAR

Abstract. We extend our work of [23] on unipotent blocks of classical groups to blocks of orthogonal groups containing characters labelled by degenerate symbols. We also show that, for classical groups in the linear prime case, there are only finitely many Morita types of unipotent blocks with a given defect group.

January 15, 2003

1. Introduction

In [23] we began the investigation of the Morita types of unipotent blocks of classical groups (other than the general linear groups). Here we extend this work to to blocks of orthogonal groups containing characters labelled by degenerate symbols (and correct some errors of [23]). In view of Donovan’s conjecture (see [1, Conjecture M]), the ultimate goal is to show that there are only finitely many Morita equivalence classes of unipotent blocks of classical groups with a given defect group.

In [17], Olaf Düvel reduces the proof of Donovan’s conjecture for elementary abelian defect groups of rank two to the corresponding problem for quasi-simple groups. He also obtains a reduction to quasi-simple groups of the weaker problem of bounding the maximum entry of the Cartan matrix of a block in terms of its defect group. This weak form of Donovan’s conjecture is of course motivated by the question of Brauer posed in [5, Problem 22] and his remark in [6].

The following theorem summarizes our results of [23] and of this paper.

Theorem. Let F be an algebraically closed field of characteristic l > 0, and let D be a finite l-group. Then there is a finite set B of F-algebras and a positive integer N, each of B and N depending only on D, such that the following holds:

Let B be a unipotent l-block with defect group isomorphic to D of a group algebra FG where G is one of the classical groups SL_{n}(q), SU_{n}(q), SO_{2m+1}(q), Sp_{2m}(q), or SO_{2m}^{±}(q) (n, m positive integers, q a prime power). Then one of the following cases occurs.

(1) B is Morita equivalent to an algebra in B.

(2) There is a classical group Ḡ of the above list with natural dimension n̄ ≤ N such that B is Morita equivalent to a unipotent l-block B̄ of Ḡ with defect group D.

(3) G = SO_{2m}^{+}(q) or SO_{2m}^{−}(q) for some positive integer m and some prime

power q, and l | q + 1. This theorem is proved at the end of Section 8.

As in [23], the proof of the main finiteness result has two parts. The first one is purely combinatorial, unfortunately rather more complicated than in [23]. The second one, where the Morita equivalence between certain blocks is established, uses a new criterion (Theorem 2.5 below). Our new criterion in fact gives an equivalence of the source algebras of the blocks involved, not only up to an automorphism of the defect group, as in [23, Theorem 8.1], but up to isomorphism of interior algebras. This new criterion, which can of course also be applied to simplify some of the arguments of previous results of the second author in [26, 27], replaces Broué’s criterion [7, Théorème 2.4] and the criterion of Scott and Puig [34, Remark 7.5].

We plan a sequel to this paper, where we intend to treat non-unipotent blocks of classical groups. The idea is to use the result of Bonnafé and Rouquier (cf. Theorem 8.2 below) to reduce to the case of “isolated” blocks. It seems plausible that the isolated blocks can be dealt with in a way similar to the unipotent blocks (even though much more complicated).

We give a brief summary of the contents of the individual chapters of our paper. Chapter 2 contains the new criterion for two blocks to have equivalent source algebras. Chapter 3 recalls some basic material on unipotent blocks and the theorem of Cabanes and Enguehard to descend to commutator subgroups. In Chapter 4 we treat the case of linear primes. These results are not an immediate consequence of [22], since there the bounds on the decomposition numbers are given in terms of the natural dimension of the classical groups. In general, there are classical groups of arbitrarily large dimension containing a unipotent block with a given defect group. Chapter 5 sets up the combinatorial results used in the reduction theorem for orthogonal groups of even dimension, contained in Chapter 6. In Chapter 7 we summarize the consequences of the reduction theorem to the unitary prime case of these groups. In Chapter 8 we treat the general linear and unitary groups for all blocks, not just the unipotent ones. Finally, Chapter 9 collects some remarks and clarifications on [23], the first part of this work. The ordering of the chapters follows their decreasing generality.

2. A criterion for Puig equivalence

We give a sufficient criterion for two blocks to have equivalent source algebras.

2.1. Notation. Let l a prime and let (K, O, F) be an l-modular system. By this we understand a complete discrete valuation ring O with field of fractions K of characteristic 0 and residue class field F of characteristic l. We assume in addition that F is algebraically closed.

Let G and Ḡ be finite groups and let B and B̄ be blocks of OG and OḠ, respectively. Following Broué [8, 6.C], we say that B and B̄ are Puig equivalent, if there are compatible isomorphisms between the defect groups and the source algebras of the two blocks.

From now on, till the end of this section, we assume that (K, O, F) is a splitting lmodular system for G, i.e., that K is a splitting field for G and all of its subgroups. We denote by Irr(G) the set of irreducible K-characters of G. If b is a central idempotent of OG, we write Irr(G, b) for the subset of Irr(G) arising from the constituents of K ⊗_{O} OGb. Similarly, we write IBr(G) for the set of irreducible Brauer characters of G with respect to (K, O, F), and IBr(G, b) for those elements of IBr(G) arising from representations of FGb.

2.2. Algebras and idempotent decompositions. Let A denote either a unitary O-algebra which is O-free of finite rank or a finite dimensional unitary F-algebra.

By an idempotent decomposition of A we will mean a set of primitive idempotents of A that are pairwise orthogonal and whose sum equals the identity of A. We denote by m(A) the cardinality of an idempotent decomposition of A.

Suppose now that A is a G-algebra. Then A^{G} will denote the set of elements of A fixed by G. The algebra A is called an interior G-algebra if there is a group homomorphism from G into the group of invertible elements of A. If A is an interior G-algebra, then A is a G-algebra via the conjugation action (by the elements of the image of G under the homomorphism defining the interior G-algebra structure on A). If H is a subgroup of G, then the inclusion of H into G gives OG the structure of an interior H-algebra. In what follows we will assume this structure without comment.

2.3. The Brauer homomorphism. Suppose A as above is an H-algebra for a finite group H. Let D be a subgroup of H. The Brauer homomorphism on A with respect to D, denoted by Br_{D}^{A}, is the canonical surjection

Br_{D}^{A} : A^{D} → A^{D}/(∑_{Q} Tr_{Q}^{D}(A^{Q}) + J(O)A^{D}),

where Q runs over all proper subgroups of D, Tr_{Q}^{D} is the relative trace map and J(O) is the maximal ideal of O (see [36, §11]). We denote the image A^{D}/(∑_{Q} Tr_{Q}^{D}(A^{Q})+ J(O)A^{D}) of Br_{D}^{A} by \overline{A}(D). Then \overline{A}(D) is non-zero only if D is an l-group. Since ∑_{Q} Tr_{Q}^{D}(A^{Q})+J(O)A^{D} is an N_{G}(D)-invariant ideal of A^{D}, \overline{A}(D) inherits naturally a structure of an N_{H}(D)-algebra, and Br_{D}^{A} respects this structure.

Let u be an idempotent of A^{H}. For any subgroup Q of D, uA^{Q}u = (uAu)^{Q} and ker(Br_{D}^{uAu}) = ker(Br_{D}^{A}) ∩ (uAu)^{D}. Thus, the map

ι : \overline{uAu}(D) → \overline{A}(D)

given by ι(Br_{D}^{uAu}(a)) = Br_{D}^{A}(a) is a well defined and injective homomorphism of N_{H}(D)-algebras. Also, ι(\overline{uAu}(D)) = Br_{D}^{A}(u)\overline{A}(D)Br_{D}^{A}(u).

By the D-multiplicity of H on A we will mean the number of elements i in an idempotent decomposition of A^{H} for which Br_{D}^{A}(i) is non-zero. Note that this makes sense since A^{H} ⊂ A^{D} and that this number is independent of the choice of the idempotent decomposition. We will denote the D-multiplicity of H on A by m_{H,D}(A).

2.4. Group algebras and the Brauer homomorphism. Let D be an l-subgroup of G. By [36, Proposition 37.5], there is an isomorphism of N_{G}(D)-algebras α : \overline{OG}(D) → FC_{G}(D) such that the composition α ∘ Br_{D}^{OG} is the surjective map from (OG)^{D} to FC_{G}(D) which sends an element of OC_{G}(D) to its canonical image in FC_{G}(D) and maps to zero any D-conjugacy class sum involving elements of G outside C_{G}(D). Henceforth we shall identify Br_{D}^{OG} with α ∘ Br_{D}^{OG} and put α ∘ Br_{D}^{OG} =: Br_{D}.

Let H be a subgroup of G containing D and u an idempotent of (OG)^{H}. Letting A = OG, and considering the map ι as in Subsection 2.3, we get that for any a ∈ (uOGu)^{D}, Br_{D}(a) = α ∘ ι ∘ Br_{D}^{uOGu}(a). Since α is an isomorphism and ι is injective it follows that for any element a of (uOGu)^{D}, Br_{D}(a) = 0 if and only if Br_{D}^{uOGu}(a) = 0. Thus, in particular, we have that m_{H,D}(uOGu) is the number of elements i in a primitive idempotent decomposition of (uOGu)^{H} for which Br_{D}(i) is non-zero.

From now on till the end of the section, we fix a subgroup H of G. If u is an idempotent of (OG)^{H}, τ is an irreducible K-character of H and χ is an irreducible K-character of G, we let r(τ, χ, u) be the number of composition factors of the KG-module KGu ⊗_{KH} W_{τ} isomorphic to V_{χ}, where V_{χ} is a KG-module affording χ and W_{τ} is a KH-module affording τ.

Let us now fix block idempotents b of OG, and c of OH, respectively. We will assume that b and c have a common defect group D ⊂ H. For what follows, note that the algebra cOGbc is stable under conjugation by H. Theorem 2.5. Let u be an idempotent of (cOGbc)^{H}. Then for any χ ∈ Irr(G, b),

∑_{τ∈Irr(H,c)} r(τ, χ, u) ≥ m_{H,D}(uOGu),

and for any τ ∈ Irr(H, c),

∑_{χ∈Irr(G,b)} r(τ, χ, u) ≥ m_{H,D}(uOGu).

Further, if |Irr(H, c)| = |Irr(G, b)| and if

∑_{χ∈Irr(G,b),τ∈Irr(H,c)} r(τ, χ, u) ≤ m_{H,D}(uOGu)|Irr(H, c)|,

then for any primitive idempotent i of (uOGu)^{H}, Br_{D}^{uOGu}(i) is non-zero and the map OHc → iOGi given by x ↦ xi is an isomorphism of interior H-algebras. In particular, if the above holds, then OGb and OHc are Puig equivalent.

Proof. For τ ∈ Irr(H, c), let f_{τ} denote the central idempotent of KH corresponding to the character τ. Similarly for χ in Irr(G, b) let e_{χ} denote the central idempotent of KG corresponding to the character χ. Then, if τ ∈ Irr(H, c), χ ∈ Irr(G, b) and i is an idempotent of (cOGbc)^{H}, we have

KGe_{χ}if_{τ} ≅ KGe_{χ}i ⊗_{KH} KHf_{τ}
(1)  ≅ KGe_{χ}i ⊗_{KH} τ(1)W_{τ} ,

as left KG modules. Here W_{τ} denotes a KH-module affording τ. Thus, the character of the left KG-module KGe_{χ}if_{τ} is r(τ, χ, i)τ(1)χ. In particular, r(τ, χ, i) is non-zero if and only if e_{χ}if_{τ} is non-zero.

If i and j are orthogonal idempotents of (cOGbc)^{H}, then KGe_{χ}(i + j)f_{τ} = KGe_{χ}if_{τ} ⊕ KGe_{χ}jf_{τ} and it follows from the above that

(2)  r(τ, χ, i + j) = r(τ, χ, i) + r(τ, χ, j).

Now let i be an idempotent of (ucOGbcu)^{H} = (uOGu)^{H} such that Br_{D}^{uOGu}(i) is non-zero. There is a primitive idempotent t in (uOGu)^{D} such that it = t and Br_{D}^{uOGu}(t) is non-zero, whence, as explained in Subsection 2.4, Br_{D}(t) is non-zero. Also, t is primitive in (OGb)^{D} (if t = t′ + t′′ with t′ and t′′ orthogonal idempotents in (OGb)^{D}, then ut′u = u(tt′t)u = t′ and similarly, ut′′u = t′′). In other words, t is a source idempotent of the block b, so that in particular the algebras tOGt and OGb are Morita equivalent (see [36, Proposition 38.2]). Thus, by [36, Theorem 9.9], if OGbtOGb is the ideal generated by t in OGb, then OGbtOGb = OGb. But since it = i, this clearly means that OGbiOGb = OGb. Hence KGbiKGb = KGb, so that by [36, Theorem 9.9], iKGi and KGb are Morita equivalent. In particular, iKGi and KGb have the same number of isomorphism classes of irreducible modules. On the other hand,

KGb ≅ ∏_{χ∈Irr(G,b)} KGe_{χ},

whence

iKGi ≅ ∏_{χ∈Irr(G,b)} iKGe_{χ}i.

So, for all χ ∈ Irr(G, b), e_{χ}i ≠ 0. Since ic = i, and c = ∑_{τ∈Irr(H,c)} f_{τ}, this means that for each χ ∈ Irr(G, b), there exists some τ ∈ Irr(H, c) such that e_{χ}if_{τ} is non-zero, which by the remark following (1) is equivalent to saying that for some τ ∈ Irr(H, c), r(τ, χ, i) is non-zero. Hence for each χ ∈ Irr(G, b), we have

(3)  ∑_{τ∈Irr(H,c)} r(τ, χ, i) ≥ 1.

The first assertion of the theorem now follows from (2) and (3).

Again, let i be an idempotent of (ucOGbcu)^{H} = (uOGu)^{H} such that Br_{D}^{uOGu}(i) ≠ 0. Then, as is explained in Subsection 2.4, Br_{D}(i) ≠ 0. Since i is fixed by H, the map from OHc to iOGi sending an element x of OHc to the element xi of iOGi is an H-algebra homomorphism and this homomorphism is unitary since ci = c(ui) = c((bc)u)i = bcui = i. Note that D stabilizes an O-basis of iOGi, since iOGi is a D-invariant direct summand of OG. Also, Z(D) commutes with Br_{D}(i), hence left multiplication by elements of Z(D) gives Br_{D}(i(OG)^{D}i) = Br_{D}(i)FC_{G}(D)Br_{D}(i) the structure of a left FZ(D)-module. Further, Br_{D}(i)FC_{G}(D)Br_{D}(i) is clearly a direct summand of FC_{G}(D) as left FZ(D)-modules. In particular, Br_{D}(i(OG)^{D}i) is projective as left FZ(D)-module. Hence, by [33, Proposition 3.8] the above map OHc → iOGi is a split injection of O-modules.

Let τ ∈ Irr(H, c). Then f_{τ} i is non-zero (since the induced map from KHc to iKGi is injective). On the other hand, bi = i, and b = ∑_{χ∈Irr(G,b)} e_{χ}. Thus e_{χ}if_{τ} is non-zero for some χ ∈ Irr(G, b). By the remark following (1), this means that for any τ ∈ Irr(H, c), there exists some χ ∈ Irr(G, b) such that r(τ, χ, i) is non-zero. In particular,

(4)  ∑_{χ∈Irr(G,b)} r(τ, χ, i) ≥ 1

for all τ ∈ Irr(H, c). The second statement of the theorem now follows from (2) and (4).

Suppose from now till the end of the proof that |Irr(G, b)| = |Irr(H, c)| and that ∑_{χ,τ} r(τ, χ, u) ≤ m_{H,D}(uOGu)|Irr(H, c)|.

Let I be a primitive idempotent decomposition of (uOGu)^{H}, and let S be the subset of I consisting of those elements which are not in the kernel of Br_{D}^{uOGu}. Then by (2) and (4), letting χ range over Irr(G, b) and letting τ range over Irr(H, c),

∑_{χ,τ} r(τ, χ, u) = ∑_{j∈S} ∑_{τ,χ} r(τ, χ, j) + ∑_{j∈I\S} ∑_{τ,χ} r(τ, χ, j)
= ∑_{j∈S} ∑_{τ} ∑_{χ} r(τ, χ, j) + ∑_{j∈I\S} ∑_{τ,χ} r(τ, χ, j)
(5)  ≥ m_{H,D}(uOGu)|Irr(H, c)| + ∑_{j∈I\S} ∑_{τ,χ} r(τ, χ, j).

The above implies that r(τ, χ, j) = 0 for any j ∈ I \ S, χ ∈ Irr(G, b), and τ ∈ Irr(H, c). By the remark following (1), this means that e_{χ}jf_{τ} = 0 for any j ∈ I \ S, χ ∈ Irr(G, b), and τ ∈ Irr(H, c). But

j = bjc = ( ∑_{χ∈Irr(G,b)} e_{χ})j( ∑_{τ∈Irr(H,c)} f_{τ}),

hence j = 0 and I = S as required.

In fact, it follows from (4) and (5) that for any τ ∈ Irr(H, c), and i ∈ I,

∑_{χ∈Irr(G,b)} r(τ, χ, i) = 1.

This means that for any τ ∈ Irr(H, c) and any i ∈ I, there is a unique χ ∈ Irr(G, b) for which r(τ, χ, i) = 1 and r(τ, ψ, i) = 0 for all ψ ∈ Irr(G, b) different from χ. Similarly, rewriting (5) by interchanging the order of the summations over χ and τ, and using (3), we get

∑_{τ∈Irr(H,c)} r(τ, χ, i) = 1

for any χ ∈ Irr(G, b) and any i ∈ I, which is to say that for any χ ∈ Irr(G, b) and any i ∈ I, there is a unique τ ∈ Irr(G, b) for which r(τ, χ, i) = 1 and r(ϕ, χ, i) = 0 for all ϕ ∈ Irr(H, c) different from τ.

Now, fix i ∈ I. The above shows that there is a bijection between Irr(H, c) and Irr(G, b) which sends an element τ of Irr(H, c) to the unique element χ of Irr(G, b) for which r(τ, χ, i) is non-zero (and hence r(τ, χ, i) = 1). So, by (1) it follows that for any χ ∈ Irr(G, b) the multiplicity of χ as an irreducible character of the KG module KGi is τ(1), where τ is the element of Irr(H, c) corresponding to χ via the bijection described above.

In particular,

rank_{O}(iOGi) = dim_{K}(iKGi)
= dim_{K}End_{KG}(KGi)
= ∑_{τ∈Irr(H,c)} τ^{2}(1)
= dim_{K}(KHc)
= rank_{O}(OHc).

Since we have already shown that the map x → xi from OHc to iOGi is a split injection of O-modules, the above proves that in fact the map is an isomorphism as required.

Finally, let g be a source idempotent of OHc, that is, g is a primitive idempotent of (OHc)^{D} such that Br_{D}^{OHc}(g) is non-zero. Since the isomorphism between OHc and iOGi displayed above is an interior H-algebra isomorphism, it is in particular an isomorphism of interior D-algebras. So, gi is a primitive idempotent of (iOGi)^{D} such that Br_{D}^{iOGi}(gi) ≠ 0. By [36, Proposition 4.12], gi is primitive in (OGb)^{D}, and, as explained in Subsection 2.4 Br_{D}^{OGb}(gi) ≠ 0. In other words, gi is a source idempotent of b. Now, restricting the above map to gOHg provides the required isomorphism of source algebras. □

With D as above, let π : FC_{G}(D) → F(C_{G}(D)/Z(D)) denote the canonical surjection. This is a map of N_{G}(D)/D-algebras. For the sake of notational convenience let us denote the composition π ∘ Br_{D} by Ψ, again a N_{G}(D)/D-algebra homomorphism. Thus if u ∈ (OG)^{H}, then Ψ(u) is stable under N_{G}(H). Finally, let us denote by X and X̄ the groups C_{G}(D) and C_{G}(D)/Z(D), respectively, and by Y and Ȳ the groups C_{H}(D) and C_{H}(D)/Z(D), respectively. We then have the following. Proposition 2.6. Let G, H, b, c, u and D be as in the previous theorem. The image of (uOGu)^{H} under Ψ equals [Ψ(u)FX̄Ψ(u)]^{N_{H}(D)}. In particular, m_{H,D}(uOGu) = m([Ψ(u)FX̄Ψ(u)]^{N_{H}(D)}) (recall that the last quantity is the cardinality of an idempotent decomposition of [Ψ(u)FX̄Ψ(u)]^{N_{H}(D)}).

Proof. By the Mackey formula (see [36, Corollary 11.10]), Br_{D} gives a surjection from Tr_{D}^{H}((uOGu)^{D}) to Tr_{{1}}^{N_{H}(D)/D}(Br_{D}(u)FXBr_{D}(u)). Clearly the image of

Tr_{{1}}^{N_{H}(D)/D}(Br_{D}(u)FXBr_{D}(u)) under π equals Tr_{{1}}^{N_{H}(D)/D}(Ψ(u)FX̄Ψ(u)). On the other hand c ∈ Tr_{D}^{H}((OH)^{D}), since D is a defect group of OHc. Now the fact that H stabilizes u implies that u = uc ∈ Tr_{D}^{H}((OG)^{D}). Thus, Tr_{D}^{H}((uOGu)^{D}) = (uOGu)^{H}. Since Br_{D} and π are unitary algebra homomorphism it follows that Ψ((uOGu)^{D}) = Tr_{{1}}^{N_{H}(D)/D}(Ψ(u)FX̄Ψ(u)). The latter is an ideal in [Ψ(u)FX̄Ψ(u)]^{N_{H}(D)} containing Ψ(u), giving the first assertion The second assertion follows from the first by the idempotent lifting lemma and the fact that ker(π) ⊂ J(FX). □

The above proposition allows us to calculate m_{H,D}(uOGu) in certain situations. Recall that by Brauer’s first main theorem, b determines a set, Λ(b), of ordinary irreducible defect 0 characters of KX̄; the elements of Λ(b) are transitively permuted by N_{G}(D) (see [31, 5.8.13]).The set Λ(b) is characterized as follows: For θ ∈ Λ(b), if b_{θ} is the block (of defect zero) of FX̄ corresponding to the character θ, then

Ψ(b) = ∑_{θ∈Λ(b)} b_{θ}.

Let Λ(c) be defined similarly. For θ in Irr(X̄), η ∈ Irr(Ȳ), let r(η, θ, Ψ(u)) stand for the multiplicity of θ as a character of the KX̄-module KX̄{ \atop Ψ(u)}⊗_{KȲ} W_{η}, where { \atop Ψ(u)} is an idempotent of (OX̄)Ȳ lifting Ψ(u) and where W_{η} is a KȲ -module affording the character η. Proposition 2.7. Suppose that θ ∈ Λ(b) and α ∈ N_{H}(D, θ) := {h ∈ N_{H}(D) | ^{h}θ = θ} are such that ⟨Y, α⟩ and N_{H}(D, θ) have the same image in the group of algebra automorphisms of FX̄b_{θ}. If α stabilizes every element of Λ(c), then

m_{H,D}(uOGu) ≥ ∑_{η∈Λ(c)} r(η, θ, Ψ(u)).

Proof. Since b_{θ} is a central idempotent of FX̄ and N_{H}(D, θ) = N_{H}(D, b_{θ}), the map from [Ψ(u)FX̄Ψ(u)b_{θ}]^{N_{H}(D,θ)} to [Ψ(u)FX̄Ψ(u)]^{N_{H}(D)} given by sending an element x to Tr_{N_{H}(D,θ)}^{N_{H}(D)}(x) is an injective algebra homomorphism (since for r, s ∈ N_{H}(D), x, y ∈ [Ψ(u)FX̄Ψ(u)b_{θ}]^{N_{H}(D,θ)}, rxr^{−1}.sys^{−1} = 0, unless rs^{−1} ∈ N_{H}(D, θ)).

It follows from the previous proposition that

(6)  m_{H,D}(uOGu) ≥ m([Ψ(u)FX̄Ψ(u)b_{θ}]^{N_{H}(D,θ)}).

On the other hand, by hypothesis,

(7)  [Ψ(u)FX̄Ψ(u)b_{θ}]^{N_{H}(D,θ)} = [Ψ(u)FX̄Ψ(u)b_{θ}]^{⟨Y,α⟩}.

Note that since Ψ(u) = Ψ(uc) = Ψ(u)Ψ(c) and Ψ(c) is the sum of the blocks c_{η} of FȲ corresponding to elements η of Λ(c), there is an inclusion of algebras

∏_{η∈Λ(c)} c_{η}Ψ(u)FX̄b_{θ}Ψ(u)c_{η} ⊆ Ψ(u)FX̄b_{θ}Ψ(u).

Since α ∈ N_{H}(D, θ) stabilizes Ψ(u), b_{θ}, and, by hypothesis, c_{η} for all η ∈ Λ(c), we have the inclusion of algebras

∏_{η∈Λ(c)}[c_{η}Ψ(u)FX̄b_{θ}Ψ(u)c_{η}]^{⟨Y,α⟩} ⊆ [Ψ(u)FX̄b_{θ}Ψ(u)]^{⟨Y,α⟩}

which yields in particular that

(8)  m([Ψ(u)FX̄b_{θ}Ψ(u)]^{⟨Y,α⟩}) ≥ ∑_{η∈Λ(c)} m([c_{η}Ψ(u)FX̄b_{θ}Ψ(u)c_{η}]^{⟨Y,α⟩}).

Suppose that η ∈ Λ(c) is such that b_{θ}Ψ(u)c_{η} is non-zero. Then, since FȲ c_{η} and Ψ(u)FX̄b_{θ}Ψ(u) (and hence c_{η}Ψ(u)FX̄b_{θ}Ψ(u)c_{η}) are simple algebras, the map FȲ c_{η} → c_{η}Ψ(u)b_{θ}FX̄Ψ(u)c_{η} given by x → xb_{θ}Ψ(u) is an injective unitary algebra homomorphism (note that elements of FȲ stabilize Ψ(u)). Identifying FȲ c_{η} with its image in c_{η}Ψ(u)FX̄b_{θ}Ψ(u)c_{η} under the above homomorphism, by the double centralizer property, there is an algebra isomorphism

c_{η}Ψ(u)FX̄b_{θ}Ψ(u)c_{η} ≅ FȲ c_{η} ⊗_{F} [c_{η}Ψ(u)FX̄b_{θ}Ψ(u)c_{η}]Ȳ .

On the other hand, by the same type of arguments as used in the proof of Theorem 2.5, we can show that the F-dimension of c_{η}Ψ(u)FX̄b_{θ}Ψ(u)c_{η} equals η(1)^{2}r^{2}(η, θ, Ψ(u)) (the change from K to F is not significant in this situation as all blocks considered are of defect zero). This, along with the above isomorphism gives that [c_{η}Ψ(u)FX̄b_{θ}Ψ(u)c_{η}]Ȳ is a matrix algebra and that

(9)  m([c_{η}Ψ(u)FX̄b_{θ}Ψ(u)c_{η}]Ȳ) = r(η, θ, Ψ(u)).

Also, note that since b_{θ}Ψ(u)c_{η} is non-zero precisely when r(η, θ, Ψ(u)) is a positive integer, the above holds for all η ∈ Λ(c).

Since Ȳ is normalized by α, α acts on [c_{η}Ψ(u)FX̄b_{θ}Ψ(u)c_{η}]Ȳ. Further, α induces an l′-automorphism of [c_{η}Ψ(u)FX̄b_{θ}Ψ(u)c_{η}]Ȳ since N_{G}(D, θ)/DX is an l′-group ([31, 5.8.13]). Hence, by the Noether-Skolem theorem, there is a semi-simple, invertible element t in the matrix algebra [c_{η}Ψ(u)FX̄b_{θ}Ψ(u)c_{η}]Ȳ such that

[c_{η}Ψ(u)FX̄b_{θ}Ψ(u)c_{η}]^{⟨Y,α⟩} ≅ [c_{η}Ψ(u)FX̄b_{θ}Ψ(u)c_{η}]^{⟨Ȳ ,t⟩}.

Since t is diagonalizable, [c_{η}Ψ(u)FX̄b_{θ}Ψ(u)c_{η}]^{⟨Ȳ ,t⟩} is clearly isomorphic to a product of matrix algebras, so that in particular

(10)  m([c_{η}Ψ(u)FX̄b_{θ}Ψ(u)c_{η}]^{⟨Y,α⟩}) = m([c_{η}Ψ(u)FX̄b_{θ}Ψ(u)c_{η}]Ȳ).

The desired results now follow from the equations (6)–(10). □

3. Unipotent blocks of classical groups

As in the first paper [23] of this series we are concerned with unipotent blocks of classical groups. Let us recall the usual setup of Deligne-Lusztig theory. For a prime power q, let 𝔽̄_{q} denote an algebraic closure of the finite field 𝔽_{q}, with q elements. Let G be a connected reductive algebraic group over 𝔽̄_{q}, and let F be a Frobenius morphism of G (relative to some 𝔽_{q}-structure of G). We then put G := G^{F}.

Let l be a prime not dividing q, and let (K, O, F) be a splitting l-modular system for G. For further notation concerning groups of Lie type and their unipotent lblocks we refer the reader to [23, Sections 5,6].

3.1. A theorem of Cabanes and Enguehard. To make use of the full power of Deligne-Lusztig theory, it is convenient to assume that G has connected center. Since we are ultimately interested in quasi-simple groups, we have to descend to commutator subgroups. For this purpose the following general result will be useful.

Let θ : G_{sc} → [G, G] denote a simply-connected covering, defined over 𝔽_{q}, of the commutator group [G, G]. The theorem below, due to Cabanes and Enguehard gives a bijection, preserving defect groups, between the unipotent l-blocks of G and those of [G, G]^{F}. Theorem 3.2. (Cabanes, Enguehard [12]) Let K be a subgroup of G containing θ(G_{sc}^{F}). Then K contains the commutator subgroup of G. Let l be a prime which does not divide any of the numbers q, |G/K| or |Z(G_{sc})^{F} |. Then restriction of characters from G to K induces an injection from E_{l}(G, (1)) into Irr(K).

In particular, every block B of OG contained in E_{l}(G, (1)) covers a unique block b of OK. Moreover, B and b are isomorphic and have a common defect group.

Proof. The first assertion follows from the fact that θ(G_{sc}^{F}) is a normal subgroup of G with abelian factor group (see, e.g. [28, (7.4.2)]), the second one is a reformulation of [12, Lemma 2(iii)]. Let B be a unipotent l-block of G covering a block b of OK. Then b is uniquely determined by B. If ρ ∈ Irr(K, b), then there is an irreducible character of B restricting to ρ. Therefore, restriction of ordinary irreducible characters from B to b is a bijection. The result now follows from [12, Propostion 6]. □

The correspondence of blocks described in the previous theorem is compatible with Harish-Chandra restriction (for subgroups K = K^{F}, where K is a closed connected F -stable subgroup of G). This is a general property of Harish-Chandra restriction, a special case of which was stated as Lemma 5.7 in [23]. However, the proof of the reduction theorem [23, Theorem 7.10] uses a more general version of this lemma, valid for Lusztig induction from d-split Levi subgroups (see e.g., the last paragraph of [23, Page 406]). For this reason we now state this more general result, which was already needed in [23] and which will again be used later on. Lemma 3.3. Let K be a closed connected F -stable subgroup of G containing [G, G]. Moreover, let L be an F -stable Levi subgroup of G, which is a Levi complement in the parabolic subgroup P of G. Then P∩K is a parabolic subgroup of K, and L∩K is an F -stable Levi complement of P ∩ K containing [L, L]. Moreover,

Res_{K}^{G}(R_{L⊂P}^{G}(ζ)) = R_{L∩K⊂P∩K}^{K}(Res_{L∩K}^{L}(ζ))

for every ζ ∈ E_{l}(L, (1)).

Proof. The first two statements are clear. For the last, see [2, Corollary 2.1.3] and the subsequent remarks. □

3.4. Notations for unipotent blocks. We shall continue with the notation of [23, Section 6]. Thus let G be a connected reductive algebraic group with connected center such that G = G^{F} = G_{n}(q) is a classical group as in [23, Subsection 4.1]. (The group G can be taken to be one of the classical groups described in [23, 4.2.3–4.2.5].)

Let d denote the order of q modulo l. If G is a unitary group, we let e := e(G, l) denote the order of −q modulo l, whereas e := e(G, l) is the order of q^{2} modulo l in the other cases. We say that l is unitary for G, if l = 2 or if l is odd and e = d/2. Otherwise, l is called linear for G (see [23, 6.1]).

Let B be a unipotent l-block of G. By work of Fong and Srinivasan (see [18]), one can associate a non-negative integer w := w(B) to B, called the weight of B (see [23, Subsection 6.2]). By work of Cabanes and Enguehard, one can associate a d-cuspidal pair (K, ψ) to B, uniquely determined by B up to G-conjugacy (see [13, Theorem 4.4]). The partition or symbol γ labelling the unipotent character ψ of K is called the unipotent label of B.

The set of unipotent K-characters in B will be denoted by Irr_{u}(G, B).

3.5. Unipotent blocks for linear primes. Suppose that l is a linear prime for G. In this case the d-cuspidal pair associated to B can be chosen in a particular way which simplifies some of the arguments below. If G = GU_{n}(q), then e is even (see [23, Subsection 6.1]) an we put e′ = e/2. In the other cases we let e′ = e. Put w = w(B). Using the notation of [23, Subsection 4.4], the (1-split) Levi subgroup M_{e′w} is isomorphic to GL_{e′w}(𝔽̄_{q})×G_{n−2e′w}, where G_{n−2e′w} is a group of the same type as G, with natural dimension n − 2e′w. The F -stable direct factor GL_{e′w}(𝔽̄_{q}) of M_{e′w} contains a Φ_{d}-torus S with |S^{F} | = ϕ_{d}(q)^{w}.

Using an argument similar to the one in the last paragraph of the proof of [21, Theorem 8.3], one sees that C_{G}(S) ≤ M_{e′w}. Moreoever, the image of the direct factor G_{n−2e′w} of M_{e′w} in G (under the embedding of [23, Subsection 4.4]) centralizes S. It follows that K := C_{G}(S) ≅ T × G_{n−2e′w}, where T is an F -stable maximal torus of GL_{e′w}(𝔽̄_{q}) containing S with T^{F} ≅ (q^{e} − 1)^{w}.

Let K be as above and let L be a maximal d-split Levi subgroup of G containing K. We claim that L = C_{G}(S_{0}) for some Φ_{d}-torus contained in S. Indeed, there is some Φ_{d}-torus S′ of G with L = C_{G}(S′). Since S ≤ K ≤ L, we have S′ ≤ Z(L) ≤ K. Hence S′ ≤ Z(K) = T × Z(G_{n−2e′w}). Now Z(G) ≤ Z(K) and hence S′Z(G) = S_{0} × Z(G) for some Φ_{d}-torus S_{0} of GL_{e′w}(𝔽̄_{q}) contained in T and hence in S. Since C_{G}(S_{0}) = C_{G}(S′Z(G)) = L, this proves the above claim. Since L is maximal, this easily implies that there is some 1 ≤ r ≤ w such that L ≅ GL_{r}(𝔽̄_{q}) × G_{n−2e′r} and L = L^{F} ≅ GL_{r}(q^{e}) × G_{n−2e′r}(q).

4. Donovan’s conjecture for linear primes and unipotent blocks

In this section we prove Donovan’s conjecture for unipotent blocks of classical groups in the linear prime case. We begin by showing that the decompositon numbers in a unipotent block are bounded by a function in the defect of the block and the decomposition numbers occuring in unipotent characters. The results of Gruber and Hiss [22] then give bounds for the decompositon numbers in terms of the defect. Finally, we use the fact that the unipotent characters of classical groups are rational valued.

4.1. Bounds on decomposition numbers. Here we derive bounds on decomposition numbers in unipotent blocks in terms of decomposition numbers of unipotent characters. Lemma 4.2. Let G, l, d, e, B and w be as in Subsection 3.4 and let (K, ψ) be a d-cuspidal pair in G giving rise to B.

Let L be a d-split Levi subgroup of G containing K, and let ρ be an irreducible constituent of R_{K}^{L}(ψ). Write

R_{L}^{G}(ρ) = ∑_{χ∈Irr_{u}(G)} a_{χ}χ

with a_{χ} ∈ ℤ. Then a_{χ} = 0 if χ does not lie in B, and |a_{χ}| ≤ (2e)^{w}w! for χ ∈ Irr_{u}(B).

Proof. The set of irreducible constituents of R_{K}^{G}(ψ) coincides with Irr_{u}(B) by [13, Theorem 4.4]. The fact that a_{χ} = 0 if χ ∉ Irr_{u}(B) now follows from the transitivity of d-Lusztig induction (see [9, Theorem 3.11]).

Let W_{G}(K, ψ) := N_{G}(K, ψ)/K denote the relative Weyl group of (K, ψ). Then W_{G}(K, ψ) ≅ C_{e} ≀ S_{w} if G is a unitary group. Otherwise W_{G}(K, ψ) is isomorphic to a subgroup of index at most 2 in C_{2e} ≀ S_{w}. (For proofs of these facts see [9, pp. 45–52].)

It follows from [9, Fundamental Theorem 3.2] that the multiplicities of the irreducible constituents of R_{L}^{G}(ρ) are (up to signs) the same as the multiplicities of the irreducible constituents of Ind_{W_{L}(K,ψ)}^{W_{G}(K,ψ)}(ϑ), where ϑ is a suitable irreducible character of W_{L}(K, ψ). Hence these multiplicities are at most equal to (2e)^{w}w!. □

Lemma 4.3. Let G, l, d, e, B and w be as in Subsection 3.4. Suppose that l is linear for G and that w > 0. Let (K, ψ) be a d-cuspidal pair in G giving rise to B.

Let L be a maximal d-split Levi subgroup of G containing K. Then (K, ψ) is a d-cuspidal pair in L. The structure of L = L^{F} is given as L ≅ GL_{r}(q^{e}) × M, where M = M^{F} is a classical group of the same type as G and where r ≤ w.

Let b be the unipotent block of L arising from (K, ψ). Then b ≅ b_{0} ⊗ b_{1}, where b_{0} is the principal block of GL_{r}(q^{e}), and where b_{1} is a unipotent block of M of weight w(b_{1}) < w.

Proof. The fact that (K, ψ) is a d-cuspidal pair in L follows from the remarks in [13, Subsection 3.1]. The structure of L is derived in Subsection 3.5.

Write b = b_{0} ⊗ b_{1} with unipotent blocks of GL_{r}(q^{e}) and of M, respectively. Certainly, b_{0} is the principal block of GL_{r}(q^{e}) since l | q^{e} − 1. The d-cuspidal pair of M giving rise to b_{1} equals (K ∩ M, Res_{K∩M}(ψ)). If w′ = w(b_{1}) and e′ is defined as in Subsection 3.5, then n − 2e′r − 2e′w′ = n − 2e′w. Since r ≥ 1, this implies that w′ < w. □

If G is any finite group, and l is a prime, we write γ_{G,l′} for the characteristic function on the l′-elements of G, i.e., γ_{G,l′}(g) = 1 if g is l-regular, and γ_{G,l′}(g) = 0, otherwise. Then γ_{G,l′} is a class function on G. Moreoever, for a prime p ≠ l, we have γ_{G,l′}(g) = γ_{G,l′}(g_{p′}) for all g ∈ G (where g_{p′} denotes the p′-part of g).

We shall view the l-modular Brauer characters as class functions defined on all of G by extending them by 0 to the l-singular elements of G. For χ ∈ Irr(G, B) we may thus write

γ_{G,l′}χ = ∑_{φ∈IBr(G,B)} d_{χ,φ}φ.

For a prime l, an integer e dividing l − 1 and a non-negative integer w we write B̃_{u}(l, e, w) for the set of pairs (G, B) where G is a classical group as in Subsection 3.4 such that e(G, l) = e and l is linear for G, and B is a unipotent block of G of weight w(B) ≤ w. Next, we put

k̃_{u}(e, w) = max{|Irr_{u}(G, B)| | (G, B) ∈ B̃_{u}(l, e, w)}

and

m̃_{u}(l, e, w) := max{d_{χ,φ} | χ ∈ Irr_{u}(G, B), φ ∈ IBr(G, B) for some (G, B) ∈ B̃_{u}(l, e, w)}.

Notice that k̃_{u}(e, w) only depends on e and not on l, and that k̃_{u}(e, w) and m̃_{u}(l, e, w) are non-decreasing as functions in w (for fixed l and e). Proposition 4.4. Let G, l, d, e, B and w be as in Subsection 3.4 and suppose that l is linear for G. Put k := k̃_{u}(e, w) and M := m̃_{u}(l, e, w).

Then the decomposition numbers in B are bounded above by

f(l, e, w) := k^{w(k+3)}M^{w(k+1)}(2e)^{w(w+1)/2} ∏_{i=1}^{w} (i!)^{4} .

Proof. The proof is by induction on w. Note that f(l, e, 0) = 1, so the assertion is true for w = 0. Suppose then that w > 0. We choose a d-cuspidal pair (K, ψ) giving rise to B. Let D be a Sylow l-subgroup of C_{G}^{0}([K, K])^{F}. Then D is a defect group of B by [13, Theorem 4.4(ii)].

Let ξ be an ordinary irreducible character of B. Then there is an element t of D, a linear character t̂ of C_{G}(t) of l-power order, and a unipotent character λ of C_{G}(t) such that ξ = ε_{G}ε_{C_{G}(t)}R_{C_{G}(t)}^{G}(t̂λ) (see [13, Theorem 4.4] and [23, Subsections 6.2, 6.3, and Lemma 7.7]).

If t ∈ Z(G), then ξ = t̂λ and thus ξ has the same restriction to the l-regular classes as the unipotent character λ of G. Certainly m̃_{u}(l, e, w) = M ≤ f(l, e, w).

We may thus assume that t does not lie in the center of G. By [13, Propositon 2.5], there is a proper d-split Levi subgroup L of G containing C := C_{G}(t). Let us take a maximal such Levi subgroup L. By [13, Theorem 4.4(iii)], we have [K, K] ⊆ C ⊆ L. It follows from [13, Proposition 1.7], that there is an element g ∈ G centralizing [K, K] such that K ≤ gLg^{−1}. Replacing t and L by their g-conjugates, we may assume that K ⊆ L and C ⊆ L.

Again by [13, Theorem 4.4], there is a unipotent l-block b of L arising from the d-cuspidal pair (K, ψ). We claim that the decomposition numbers in b are bounded above by w!f(l, e, w − 1). Indeed, by Lemma 4.3, L ≅ GL_{r}(q^{e}) × G_{n−2e′r}(q), and b is the tensor product of the principal block b_{0} of GL_{r}(q^{e}) and of an l-block b_{1} of weight smaller than w of G_{n−2e′r}(q) (here, e′ has the same meaning as in Subsection 3.5). Since r ≤ w, the claim follows from [22, Corollary 8.10] and the induction hypothesis.

Put μ := R_{C}^{L}(t̂λ). Then ±μ is an ordinary irreducible character in b. Write γ_{L,l′}μ = ∑_{ϑ∈IBr(L,b)} z_{ϑ}ϑ. Then, by the above,

(11)  |z_{ϑ}| ≤ w!f(l, e, w − 1),

for all ϑ ∈ IBr(L, b).

Let ϑ ∈ IBr(L, b). It follows from the main result of [20] that we may write ϑ = ∑_{ρ∈Irr_{u}(L,b)} y_{ρ,ϑ}γ_{L,l′}ρ for some y_{ρ,ϑ} ∈ ℤ. The decomposition numbers of unipotent characters in b_{0} are bounded above by w! and those in b_{1} by m̃_{u}(l, e, w−1). The decomposition matrix of b_{0} is upper triangular. Hence the entries in the inverse of the matrix of unipotent decomposition numbers of b_{0} are at most w!^{2} in absolute value. By crudely applying Hadamard’s inequality (see, e.g., [15, Proposition 2.2.4]) to the matrix of unipotent decomposition numbers of b_{1} we find that the absolute values of the entries in its inverse matrix are bounded above by k^{k}m̃_{u}(l, e, w − 1)^{k}. Hence

(12)  |y_{ρ,ϑ}| ≤ w!^{2}k^{k}m̃_{u}(l, e, w − 1)^{k},

for all ρ ∈ Irr_{u}(L, b).

Let ρ ∈ Irr_{u}(L, b). Then R_{L}^{G}(ρ) = ∑_{χ∈Irr_{u}(G)} a_{χ,ρ}χ, with a_{χ,ρ} ∈ ℤ. By Lemma 4.2 we have

(13)  |a_{χ,ρ}| ≤ (2e)^{w}w!.

Moreover, a_{χ,ρ} = 0 if χ ∉ B. Put ε := ε_{G}ε_{C_{G}(t)}. By [16, Proposition 12.6], Lusztig induction commutes with the restriction to l′-elements, in the sense that it respects multiplication by γ_{G,l′}. We thus have

εγ_{G,l′}ξ = R_{L}^{G}(γ_{L,l′}μ)
= ∑_{ϑ∈IBr(L,b)} z_{ϑ}R_{L}^{G}(ϑ)
= ∑_{ρ∈Irr_{u}(L,b)} ∑_{ϑ∈IBr(L,b)} z_{ϑ}y_{ρ,ϑ}R_{L}^{G}(γ_{L,l′}ρ)

= ∑_{χ∈Irr_{u}(G,B)} ( ∑_{ρ∈Irr_{u}(L,b)} ∑_{ϑ∈IBr(L,b)} z_{ϑ}y_{ρ,ϑ}a_{χ,ρ}) γ_{G,l′}χ.

Now write

γ_{G,l′}χ = ∑_{φ∈IBr(G,B)} d_{χ,φ}φ.

Since χ is unipotent, we have

(14)  d_{χ,φ} ≤ m̃_{u}(l, e, w).

Then the multiplicity d_{ξ,φ} of φ ∈ IBr(G, B) in γ_{G,l′}ξ is given by

d_{ξ,φ} = |∑_{ρ} ∑_{ϑ} ∑_{χ} z_{ϑ}y_{ρ,ϑ}a_{χ,ρ}d_{χ,φ}| .

Each of the sums above has at most k summands. Thus, using Equations (11), (12), (13), (14), and induction, we find that

d_{χ,ξ} ≤ k^{3} [w!f(l, e, w − 1)] · [w!^{2}k^{k}m̃_{u}(l, e, w − 1)^{k}] · [(2e)^{w}w!] · [m̃_{u}(l, e, w)]
≤ k^{3} [w!k^{(w−1)(k+3)}M^{(w−1)(k+1)}(2e)^{w(w−1)/2} ∏_{i=1}^{w−1} (i!)^{4}] ·
[w!^{2}k^{k}M^{k}] · [(2e)^{w}w!] · M
= k^{w(k+3)}M^{w(k+1)}(2e)^{w(w+1)/2} ∏_{i=1}^{w} (i!)^{4} .
□

If the defect groups of B are abelian, we obtain a much better bound, without the assumption that l is linear. Proposition 4.5. Let G, l, d, e, B and w be as in Subsection 3.4 and suppose that B has an abelian defect group. Let m_{u}(B) denote the largest decomposition number occurring in some unipotent character of B.

Then the decomposition numbers of B are bounded above by

(2e)^{w}w!m_{u}(B)|Irr_{u}(G, B)|.

Proof. Let (K, ψ) be a d-cuspidal pair giving rise to B. Since B has an abelian defect group, a Sylow l-subgroup D of Z^{0}(K)^{F} is a defect group of B (see [11, (2.3)]). In particular, K ⊆ C_{G}(t) for all t ∈ D. By assumption, the prime l is (Z(K)/Z(G), F, d) adapted (see [11, Définiton 1.14]) and thus C_{G}(t) is a d-split Levi subgroup of G for all t ∈ D (see [11, Proposition 1.15]).

Let ξ be an ordinary irreducible character of B. Then, as in the proof of Proposition 4.4, ξ = ε_{G}ε_{C_{G}(t)}R_{C_{G}(t)}^{G}(t̂λ), for some t ∈ D, a linear character t̂ of C_{G}(t) and a unipotent character λ of C_{G}(t). Moreover, by [13, Theorem 4.4(iii)], there is a d-cuspidal pair (K_{t}, ψ_{t}) in L := C_{G}(t) such that [K, K] = [K_{t}, K_{t}], and the restrictions of ψ and ψ_{t} to [K, K]^{F} agree. Since K ≤ L, the pair (K, ψ) is d-cuspidal in L, by the remarks in [13, Subsection 3.1]. This implies, by the statement in [13, Definition 3.4] that the two d-cuspidal pairs (K, ψ) and (K_{t}, ψ_{t}) are conjugate in L. We may thus assume without loss of generality that they are equal. In particular, λ is a constituent of R_{K}^{L}(ψ).

Let us write ε_{G}ε_{L}R_{L}^{G}(λ) = ∑_{χ∈Irr_{u}(G,B)} a_{χ}χ. Since Lusztig induction commutes with the restriction to l′-elements, we find

γ_{G,l′}ξ = ε_{G}ε_{L}γ_{G,l′}R_{L}^{G}(t̂λ) = ε_{G}ε_{L}R_{L}^{G}(γ_{L,l′} t̂λ)
= ε_{G}ε_{L}R_{L}^{G}(γ_{L,l′}λ) = ε_{G}ε_{L}γ_{G,l′}R_{L}^{G}(λ)
= ∑_{χ∈Irr_{u}(G,B)} a_{χ}γ_{G,l′}χ

= ∑_{χ∈Irr_{u}(G,B)} a_{χ} ( ∑_{φ∈IBr(G,B)} d_{χ,φ}φ)

= ∑_{φ∈IBr(G,B)} ( ∑_{χ∈Irr_{u}(G,B)} a_{χ}d_{χ,φ}) φ.

Since |a_{χ}| ≤ (2e)^{w}w! by Lemma 4.2, and since d_{χ,φ} ≤ m_{u}(B) by definition, the result follows. □

4.6. Linear primes and unipotent blocks. Here we present the main result of this section. This considerably generalizes [23, Theorem 9.4], where we fixed the dimension of the classical groups considered. In this and the next subsection we let l be a prime and F an algebraically closed field of characteristic l. Note that, given a finite group G, there is always a splitting l-modular system (K, O, F) for G. Theorem 4.7. Let D be a finite l-group. Then, up to Morita equivalence, there are only finitely many l-blocks with defect group D occurring as unipotent blocks in the group algebras FG for groups G in the set

(15) {G_{n}(q) | G_{n}(q) as in Subsection 3.4, l ∤ q and l linear for G_{n}(q)}.

Proof. Let B be a unipotent block of FG, with defect group isomorphic to D, for a group G = G_{n}(q) as in (15). Let w denote the weight of B and consider a unipotent character χ in B. By Harish-Chandra theory, χ corresponds to an ordered or unordered pair (μ_{1}, μ_{2}) of partitions, such that |μ_{1}| + |μ_{2}| = n. Since B has e-weight w, and since l is linear for G, each of μ_{1}, μ_{2}, has e-weight at most w. By [22, Theorem 8.11], every modular constituent of χ occurs with multiplicity at most M_{w}^{2}, where M_{w} is the largest decomposition number of a unipotent character lying in a block of weight at most w of some group GL_{n′}(q′), where n′ ≤ n and q′ ∈ {q, q^{2}}. Theorem 9.3 of [23] implies that M_{w} is bounded independently of n′ and q′, hence m̃_{u}(l, e, w) is finite. By Proposition 4.4 the decomposition numbers in B are bounded from above by some number depending only on l and w.

Since l > 2, the unipotent characters of B form a basic set by [20]. The unipotent characters in a classical group are rational valued (see the proof of [23, Theorem 9.4]). This implies by [23, Propositions 9.1, 9.2] that there are only finitely many Morita equivalence classes among the unipotent blocks with defect group D occurring in groups algebras of the groups of (15). □

4.8. Descent to commutator subgroups. We close this section by an investigation of commutator subgroups. This is necessary for the intended applications to Donovan’s conjecture. Since the unitary groups will be dealt with in more detail in the final section, we restrict ourselves to the case of classical groups of types B, C, and D here.

Let G_{n}(q) denote a classical group as in [23, Subsection 4.1], but exclude Case (1), i.e., assume that G_{n}(q) ≠ GU_{n}(q). We denote the commutator subgroup of G_{n}(q) by K_{n}(q), and call an l-block of K_{n}(q) unipotent, if it is covered by a unipotent l-block of G_{n}(q). Theorem 4.9. Let D be a finite l-group. Then, up to Morita equivalence, there are only finitely many l-blocks with defect group D occurring as unipotent blocks in the group algebras FG for groups G in the class

(16)  {K_{n}(q) | K_{n}(q) as above, l ∤ q and l linear for G_{n}(q)}.

Proof. There is nothing to prove if D is the trivial group, hence we assume |D| > 1 in the following. Let G = K_{n}(q) for a group K_{n}(q) as in (16), and put G̃ := G_{n}(q). Let B be a unipotent block of FG with defect group isomorphic to D, and let B̃ be a unipotent block of G̃ covering B.

There is a simply connected covering θ : G_{sc} → [G, G], defined over 𝔽_{q}, of the commutator group [G, G] such that G = K_{n}(q) = θ(G_{sc}^{F}). We may thus apply Theorem 3.2. If l ∤ q − 1, then B and B̃ are Morita equivalent and the latter has defect group D. Suppose then that l | q − 1 and let G_{1} denote the subgroup of G̃ containing G such that G_{1}/G is the Sylow l-subgroup of G̃/G. Then, by Theorem 3.2, B̃ covers a unique block B_{1} of G_{1}, and B̃ and B_{1} are Morita equivalent. Moreover, B_{1} covers B.

Let Z denote the Sylow l-subgroup of Z(G̃). Then G_{1} = Z × G, and thus the defect group of B_{1} is isomorphic to Z × D. Since D is non-trivial, the weight of B̃ is larger than 1, and so |Z| | |D| by the description of the defect groups (see [13, Theorem 4.4(ii)]). In particular, |Z × D| ≤ |D|^{2}. By Theorem 4.7 the Morita equivalence class of B_{1} lies in a finite set depending only on the order of Z × D. This implies that there is an upper bound, depending only on |D|, for the Cartan invariants of B_{1}.

Every irreducible character of B is the restriction of an irreducible character of B_{1}. In particular, the Cartan invariants of B are bounded. Also, the irreducible Brauer character of B are rational valued (since the same is true for those of B_{1}). By [23, Propositons 9.1, 9.2], there are only finitely many possibilities for the Morita type of B. □

5. Combinatorics

Here we collect the combinatorial results on symbols needed later on in the statement and proof of the reduction theorem.

5.1. Basic assumptions. Throughout this section let e, w, i, k denote integers with e > 1, w ≥ 0, 1 ≤ i ≤ e − 1, and k > 0. We will use freely the language and notation for symbols as given in [23, Section 2] except for the following change. Since e is fixed in this section, if Λ is a 2e-abacus diagram, we will use Λ^{i_{g}} to denote Λ_{e}^{i_{g}}, and w(Λ) to denote w_{e}(Λ). Also, by a diagram we will always mean a 2e-abacus diagram. Lemma 5.2. Suppose that that Λ and Ψ are diagrams such that Ψ can be obtained from Λ by moving t beads on string i_{g} one position to the left. If 0 < t < |Λ^{i_{g}}| − |Λ^{(i−1)_{g}}|, then w(Ψ^{i_{g}}) + w(Ψ^{(i−1)_{g}}) ≥ |Λ^{i_{g}}| − |Λ^{(i−1)_{g}}| − 1.

Proof. Let S be the set of row indices s such that there is a bead on row s and string i_{g} of Λ but there is no bead on row s and string (i − 1)_{g} of Λ. Then |S| ≥ |Λ^{i_{g}}| − |Λ^{(i−1)_{g}}| and there is a subset T of S of cardinality t such that if for each s in T the bead on row s of string i_{g} of Λ is moved one position to the left, one obtains Ψ.

Let s_{1} = min(T), s_{2} = max(T), s_{3} = min(S \ T), and s_{4} = max(S \ T). Note that the assumptions on t imply that T and S \ T are non-empty and hence that these integers are all defined.

If s_{1} > s_{4}, then s_{2} = max(S) and s_{3} = min(S). Since there is no bead on row s_{3} and string (i − 1)_{g} of Ψ and there is a bead on row s_{2} and string (i − 1)_{g} of Ψ, we get w(Ψ^{(i−1)_{g}}) ≥ s_{2} − s_{3} = |S| − 1 ≥ |Λ^{i_{g}}| − |Λ^{(i−1)_{g}}| − 1.

Similarly, if s_{3} > s_{2}, then s_{1} = min(S) and s_{4} = max(S). There is no bead on row s_{1} and string i_{g} of Ψ but there is a bead on row s_{4} and string i_{g} of Ψ, hence w(Ψ^{i_{g}}) ≥ s_{4} − s_{1} = |S| − 1 ≥ |Λ^{i_{g}}| − |Λ^{(i−1)_{g}}| − 1. So we may assume that both both s_{1} and s_{3} are less than or equal to each of s_{2} and s_{4}. We have w(Ψ^{(i−1)_{g}}) + w(Ψ^{i_{g}}) ≥ (s_{2} − s_{3}) + (s_{4} − s_{1}) ≥ |S| − 1 ≥ |Λ^{i_{g}}| − |Λ^{(i−1)_{g}}| − 1. This completes the proof. □

5.3. Diagram operations. Let Δ be a diagram. Denote by c(Δ) the core of Δ. Let Δ′ denote the diagram obtained from Δ by swapping all pairs of strings j_{g} and j_{b}, 0 ≤ j ≤ e − 1. Let Δ_{1} be the diagram obtained from Δ by swapping strings i_{g} and (i − 1)_{g} and let Δ^{∗} be the diagram obtained from Δ by swapping the pair of strings i_{g} and (i−1)_{g}, and the pair of strings i_{b} and (i−1)_{b}. Note that Δ ↦ Δ_{1} and Δ ↦ Δ^{∗} are weight preserving bijections of diagrams. Also note that c(Δ′) = c(Δ)′ and c(Δ_{1}) = c(Δ)_{1}.

In addition, when e = 2, we define the diagram Δ̃ as follows. There is a bead on row t and string 1_{g} of Δ̃ if and only if there is a bead on row t and string 0_{g} of Δ; there is a bead on row t and string 1_{b} of Δ̃ if and only if there is a bead on row t and string 0_{b} of Δ; there is a bead on row 0 and string 0_{g} of Δ̃ and for all t > 0, there is a bead on row t and string 0_{g} of Δ̃ if and only if there is a bead on row t − 1 and string 1_{b} of Δ. Similarly, there is a bead on row 0 and string 0_{b} of Δ̃ and for all t > 0, there is a bead on row t and string 0_{b} of Δ̃ if and only if there is a bead on row t − 1 and string 1_{g} of Δ. (Note that if Δ is a 2e-unitary diagram of a symbol δ, then Δ̃ is the 2e-unitary diagram of the 1-shift of δ.)

5.4. Symbols. Symbols can be represented on an abacus with 2e strings either through 2e-linear diagrams or 2e-unitary diagrams. We fix a choice of either 2elinear diagrams or 2e-unitary diagrams. So, whenever we speak of a diagram for a symbol (or an equivalence class of symbols) we will consistently either mean a 2e-linear or a 2e-unitary diagram. Similarly, when we speak of the weight or core of a symbol we will mean its e-linear core and e-linear weight if we are in the linear case and e-unitary core and e-unitary weight if we are in the unitary case. If λ and δ are equivalence classes of symbols, let S_{λ,δ} denote the set of sequences

λ = δ(0), δ(1), ⋯, δ(k) = δ

of classes of symbols such that for each j, 1 ≤ j ≤ k, a representative of δ(j) can be obtained from a representative of δ(j −1) by the removal of a 1-hook. For S ∈ S_{λ,δ}, we denote by d(S) the number of integers, j, 1 ≤ j ≤ k − 1 for which the j-th term of S is a degenerate symbol.

5.5. The symbol γ. From now on, till the end of this section let γ be an equivalence class of symbols of weight 0. Let k = š_{e}(γ) if we are in the linear case and k = ŝ_{e}(γ) if we are in the unitary case. Assume that k > 0. Choose a diagram Γ for γ with the following properties: There are k more beads on string i_{g} of Γ than there are on string (i − 1)_{g} of Γ and there are enough beads on Γ so that for any equivalence class of symbols λ whose weight is at most w and whose core is γ, there is a diagram Λ for λ with core Γ.

5.6. The symbol γ_{1}. Let γ_{1} be the class of symbols corresponding to Γ_{1} (see 5.3). Since there are exactly k more beads on string i_{g} of Γ than on string (i − 1)_{g} and since all the beads of Γ are as high as possible on their strings, we can also obtain Γ_{1} from Γ by moving the bottom k beads on string i_{g} one position to the left. Note that rank(γ_{1}) = rank(γ) − k and that if either γ or γ_{1} is degenerate, the other is not.

5.7. The symbol γ̄. If neither γ nor γ_{1} is degenerate, we put γ̄ := γ_{1}. In this case γ̄ is non-degenerate and rank(γ̄) = rank(γ) − k.

If γ is degenerate and does not contain {{1}, {1}}, define γ̄ to be the symbol corresponding to the diagram Γ^{∗} (if Γ is a diagram for {{1}, {1}}, then Γ^{∗} is not the diagram of a symbol). Note that since γ is degenerate, there are also k more beads on string i_{b} of Γ_{1} than on string (i − 1)_{b} and so Γ^{∗} can be obtained from Γ_{1} by moving the bottom k beads on string i_{b} to string (i − 1)_{b}. Note that γ̄ is degenerate and rank(γ̄) = rank(γ) − 2k.

If γ_{1} is degenerate, e = 2 and we are in the situation where symbols are represented by 2e-unitary diagrams, we define γ̄ as the class of symbols corresponding to the diagram (Γ̃_{1})_{1}. Note that since we are in the case of 2e-unitary diagrams, Γ̃_{1} represents γ_{1}. Also, Γ̃_{1} has k − 1 more beads on string 1_{g} than on 0_{g} and k − 1 more beads on string 1_{b} than on string 0_{b}. In this case γ̄ is non-degenerate and rank(γ̄) = rank(γ) − 2k + 1.

5.8. The correspondence λ ↦ λ_{1}. For each class of symbols λ of weight at most w and core γ choose once and for all a diagram Λ representing λ such that Γ is the core of Λ. Note that if γ is non-degenerate, or if λ is degenerate, there is a unique choice for Λ, but if γ is degenerate and Λ is non-degenerate, there are two choices for Λ and we fix one.

If λ is a class of symbols of weight w and core γ and if Λ is the chosen diagram for λ, define λ_{1} to be the class of symbols corresponding to the diagram Λ_{1}. Then, λ ↦ λ_{1} is a map from the set of classes of symbols of weight w and core γ to the set of classes of symbols of weight w and core γ_{1}. Lemma 5.9. Let w̃ be an integer with 0 ≤ w̃ ≤ w. Suppose that both γ and γ_{1} are non-degenerate and that k ≥ w̃ + 2. Let λ and θ be classes of symbols of weight w̃ and core γ, and suppose that δ is a class of symbols of weight w̃ and core γ_{1}.

Then |S_{λ,λ_{1}}| = k!, |S_{λ,δ}| = 0 if δ ≠ λ_{1}, and |S_{θ,λ_{1}}| = 0 if θ ≠ λ. Further, d(S) = 0 for any S ∈ S_{λ,λ_{1}}. The map λ ↦ λ_{1} is a bijection between classes of symbols of weight w̃ and core γ and classes of symbols of weight w̃ and core γ_{1}.

Proof. Let Λ be the chosen diagram for λ and suppose that |S_{λ,δ}| is non-zero. Then there is a sequence of abacus diagrams

Λ = Δ(0), Δ(1), ⋯, Δ(k)

such that for each j, 1 ≤ j ≤ k, Δ(j) is obtained from Δ(j − 1) by moving a bead one position to the left, and such that δ is the class of symbols defined by Δ(k). Let us consider such a sequence. The core of Δ(k) represents the class γ_{1}. On the other hand, the number of beads on Δ(k) is equal to the number of beads on the diagram Γ_{1}. Hence, either the core of Δ(k) is Γ_{1} or it is (Γ_{1})′ (recall that (Γ_{1})′ is the swap of Γ_{1}, see 5.3).

We claim that the core of Δ(k) is Γ_{1}. If not, we would have the following equalities.

(a) |Δ(k)^{i_{g}}| = |Γ_{1}^{i_{b}}| = |Γ^{i_{b}}| = |Λ^{i_{b}}|.

(b) |Δ(k)^{i_{b}}| = |Γ_{1}^{i_{g}}| = |Γ^{(i−1)_{g}}| = |Λ^{(i−1)_{g}}|.
(c) |Δ(k)^{(i−1)_{g}}| = |Γ_{1}^{(i−1)_{b}}| = |Γ^{(i−1)_{b}}| = |Λ^{(i−1)_{b}}|.
(d) |Δ(k)^{(i−1)_{b}}| = |Γ_{1}^{(i−1)_{g}}| = |Γ^{i_{g}}| = |Λ^{i_{g}}|.
(e) |Δ(k)^{j_{g}}| = |Γ_{1}^{j_{b}}| = |Γ^{j_{b}}| = |Λ^{j_{b}}| for all j different from i and i − 1.
(f) |Δ(k)^{j_{b}}| = |Γ_{1}^{j_{g}}| = |Γ^{j_{g}}| = |Λ^{j_{g}}| for all j different from i and i − 1.

Let l_{1} = |Λ^{i_{g}}| − |Λ^{i_{b}}| and l_{2} = |Λ^{i_{b}}| − |Λ^{(i−1)_{g}}|. If l_{1} ≥ 0, then by (a) in order to get from Λ to Δ(k), at least l_{1} beads on string i_{g} of Λ must be moved one position to the left. In particular, l_{1} ≤ k. Similarly, if l_{2} ≥ 0, then by (b) at least l_{2} beads must be moved from string i_{b} of Λ to get from Λ to Δ(k), hence l_{2} ≤ k. But l_{1} + l_{2} = k. Thus 0 ≤ l_{1}, l_{2} ≤ k, and the only strings from which beads are removed in going from Λ to Δ(k) are the strings i_{g} and i_{b}. In particular, by (a)–(f) we have |Γ^{j_{g}}| = |Δ(k)^{j_{b}}| = |Λ^{j_{b}}| = |Γ^{j_{b}}| for all j different from i and i − 1, and |Γ^{(i−1)_{g}}|+|Γ^{i_{g}}| = |Δ(k)^{(i−1)_{b}}|+|Δ(k)^{i_{b}}| = |Λ^{(i−1)_{b}}|+|Λ^{i_{b}}| = |Γ^{(i−1)_{b}}|+|Γ^{i_{b}}|. Since neither Γ nor Γ_{1} is degenerate, we must have 0 < l_{1}, l_{2} < k. On the other hand, since Δ(k) is obtained from Λ by moving l_{1} beads on string i_{g} and l_{2} beads on string i_{b} one position to the left, there exists a sequence of diagrams

Λ = Π(0), Π(1), ⋯, Π(k) = Δ(k)

such that for each j, 1 ≤ j ≤ l_{1}, Π(j) is obtained from Π(j − 1) by moving a bead on string i_{g} one position to the left, and for l_{1} + 1 ≤ j ≤ k, Π(j) is obtained from Π(j − 1) by moving a bead on string i_{b} one position to the left. By Lemma 5.2 we have w(Π(l_{1})^{i_{g}}) + w(Π(l_{1})^{(i−1)_{g}}) ≥ k − 1. Since in going from Π(l_{1}) to Δ(k) the j_{g} strings are not disturbed, w(Δ(k)) ≥ w(Π(l_{1})^{i_{g}}) + w(Π(l_{1})^{(i−1)_{g}}). Hence w̃ = w(Δ(k)) ≥ k − 1, a contradiction. This proves the claim.

Thus the only way to get from Λ to Δ(k) is by successively moving k beads from string i_{g} of Λ one position to the left. Since k ≥ w̃, there is no row index s such that there is a bead on row s of string (i − 1)_{g} and there is no bead on row s and string i_{g} of Λ. Thus, there are exactly k beads on string i_{g} of Λ such that there is no bead on the corresponding position of string (i − 1)_{g} of Λ, and it is exactly these k beads that will be moved. In other words, Δ(k) is Λ_{1} and δ = λ_{1}. (The reasoning in this paragraph follows exactly the original argument of Joanna Scopes in the proof of [35, Lemma 2.1].)

By reversing the argument after (a) through (f) and the argument in Lemma 5.2 in the sense of moving beads to the right rather than to the left, we see that if we start with Λ_{1} and successively move k beads to the right in such a way that we arrive at a diagram with core representing γ, then this diagram has to be Λ. It follows that if θ is a class of symbols of weight w̃ and core λ, then |S_{θ,λ_{1}}| = 0 unless θ = λ.

The above also shows that |S_{λ,λ_{1}}| is at most equal to the number of sequences of diagrams of the above type linking Λ to Λ_{1} and that this latter number is k!. Conversely, since any sequence of diagrams linking Λ to Λ_{1} only involves moving beads from string i_{g} to (i − 1)_{g}, it is clear that any two such sequences represent distinct sequences of classes of symbols linking λ and λ_{1}. Thus |S_{λ,λ_{1}}| = k!.

Next, we claim that d(S) = 0 for any S in S_{λ,λ_{1}}. If not, there is a sequence Λ = Δ(0), ⋯, Δ(k) = Λ_{1} obtained from Λ by successively moving a bead from string i_{g} to string (i − 1)_{g} such that for some j, 0 ≤ j ≤ k, Δ(j) is a degenerate diagram. Both γ and γ_{1} are non-degenerate, hence 0 < j < k. By Lemma 5.2, we have w((Δ(j))^{i_{g}}) + ((Δ(j))^{(i−1)_{g}}) ≥ k − 1. On the other hand, the degeneracy of Δ(j) along with the fact that the b-strings are not disturbed at any stage implies that

w̃ ≥ w(Λ^{i_{b}}) + w(Λ^{(i−1)_{b}}) = w((Δ(j))^{i_{g}}) + w((Δ(j))^{(i−1)_{g}}),

a contradiction.

The fact that λ ↦ λ_{1} is a bijection between classes of symbols of weight w̃ and core γ and classes of symbols of weight w̃ and core γ_{1} is obvious from the assumption that γ and γ_{1} are both non-degenerate. □

5.10. Examples. Here we give a couple of examples for the failure of Lemma 5.9 if some of its hypotheses are not satisfied.

(1) Consider the following 4-unitary (e = 2) diagrams Γ and Λ.

[figure: Γ; 0_{g}; 1_{g}; 0_{b}; 1_{g}; 0; 1; 2; Λ; 0_{g}; 1_{g}; 0_{b}; 1_{g}; 0; 1; 2]

Thus Γ is a 2-unitary core, w(Λ) = 1 and c(Λ) = Γ. Also, k = ŝ_{e}(γ) = 2, where γ is the class of symbols represented by Γ.

The next four pictures give the diagram Λ_{1} and three diagrams Δ(i), i = 1, 2, 3.

[figure: Λ_{1}; 0_{g}; 1_{g}; 0_{b}; 1_{g}; 0; 1; 2; Δ(1); 0_{g}; 1_{g}; 0_{b}; 1_{g}; 0; 1; 2]

[figure: Δ(2); 0_{g}; 1_{g}; 0_{b}; 1_{g}; 0; 1; 2; Δ(3); 0_{g}; 1_{g}; 0_{b}; 1_{g}; 0; 1; 2]

Denoting by λ_{1} the class of symbols represented by Λ_{1}, and by δ(i) the classes of symbols represented by Δ(i), i = 1, 2, 3, we find S_{λ,λ_{1}} = {(λ, δ(i), λ_{1}) | i = 1, 2, 3}, the elements of which correspond to the sequences of diagrams (Λ, Δ(i), Λ_{1}), i = 1, 2, and (Λ, Δ(3), Λ_{1}′). It follows that |S_{λ,λ_{1}}| = 3 > 2 = k!. Note also that d(S) = 2 for the second of the above sequences. This example shows that it does not suffice to assume that k ≥ w̃ + 1 in Lemma 5.9 (or even only k ≥ w̃, as we did in [23, Lemma 7.4]).

(2) The next example shows why the assumption e > 1 in Lemma 5.9 is necessary. Let k, w be integers with w > 0 and k ≥ w odd. Let X = {0, 2, 4, …, k − 1} and Y = {1, 3, 5, …, k}, and let γ denote the equivalence class of symbols containing {X, Y }. Then γ is its own 1-unitary core, has defect 0 and rank (k + 1)^{2}/4.

According to our conventions in 5.8, we choose a 2-unitary (e = 1) diagram Γ for γ which has at least w beads on each string. For example, if w is even, we can take the following diagram (of the symbol {[0, w −1]∪(X +w), [0, w −1]∪(Y +w)} equivalent to {X, Y }):

[figure: 0_{g}; 0_{b}; .; .; .; .; .; .; .; .; .; 0; 1; w − 1; 0; 1; w − 1; w; w + k]

(a) Let λ be the class of symbols of 1-unitary weight w and 1-unitary core γ represented by the diagram Λ obtained by shifting every bead on the 0_{b}-string of Γ one position down. Here is a picture of Λ:

[figure: 0_{g}; 0_{b}; .; .; .; .; .; .; .; .; .; 0; 1; w; 0; 1; w − 1; w; w + k]

Now consider the classes of symbols λ_{1} and δ of weight w and core γ_{1}, represented by the following two diagrams.

[figure: 0_{g}; 0_{b}; .; .; .; .; .; .; .; .; .; 0; 1; 2; w; w + 1; 0; 1; w − 1; w; w + k − 1; 0_{g}; 0_{b}; .; .; .; .; .; .; .; .; .; 0; 1; 2; w; 0; 1; k − 1; k; k + 1; w + k]

It is easy to see that |S_{λ,λ_{1}}| ≠ 0 ≠ |S_{λ,δ}|, contrary to the assertion in Lemma 5.9. Note that this is true for any odd k ≥ w.

(b) Now suppose that w is even and let λ be the class of symbols of 1-unitary weight w and 1-unitary core γ obtained by shifting the last w beads on the 0_{g}-string of Γ one position down. It is easy to see that S_{λ,λ_{1}} does contain sequences S with d(S) ≠ 0, again contrary to the assertions in Lemma 5.9.

5.11. The correspondences λ ↦ λ_{1}^{∗} and λ ↦ λ^{∗}. Suppose that λ is a class of symbols of weight at most w and core γ, and that Λ is the chosen diagram for λ. Define λ_{1}^{∗} to be the class of symbols corresponding to the diagram which is obtained from Λ by swapping the strings i_{b} and (i−1)_{b} and define λ^{∗} to be the class of symbols corresponding to the diagram which is obtained from Λ_{1} by swapping the strings i_{b} and (i − 1)_{b}. Clearly λ_{1}^{∗} and λ^{∗} have the same weight as λ and λ_{1}. Lemma 5.12. Let λ be a class of symbols of weight at most w and core γ. If γ is degenerate then λ_{1}^{∗} has core γ_{1} and λ_{1} = λ_{1}^{∗} if and only if λ is degenerate. If γ_{1} is degenerate, then λ^{∗} has core γ, and λ^{∗} = λ if and only if λ_{1} is degenerate.

Proof. Suppose first that γ is degenerate. Let Θ be the diagram obtained form Λ by swapping strings i_{b} and (i − 1)_{b}. Then Θ is a diagram for λ_{1}^{∗}. Note that Θ′ = (Λ′)_{1}. From 5.3 we get that

c(Θ)′ = c(Θ′) = c((Λ′)_{1}) = c(Λ′)_{1} = (Γ′)_{1}.

Since γ is degenerate, Γ′ = Γ, and hence c(Θ) = (Γ_{1})′. This proves that γ_{1} is the core of λ_{1}^{∗}.

Now, since

|Θ^{i_{g}}| = |Λ^{i_{g}}| ≠ |Λ^{(i−1)_{g}}| = |Λ_{1}^{i_{g}}|,

Θ is not the same as Λ_{1}. Hence, λ_{1}^{∗} = λ_{1} if and only if Θ′ = Λ_{1}. But Θ′ = (Λ′)_{1}. Hence λ_{1}^{∗} = λ_{1} if and only if (Λ′)_{1} = Λ_{1} thus if and only if Λ′ = Λ.

This settles the lemma in the situation where γ is degenerate. The statements for the case that γ_{1} is degenerate follow in an entirely analogous way. □

Lemma 5.13. Let w̃ be an integer with 0 ≤ w̃ ≤ w. Suppose that γ is degenerate and that k ≥ w̃. Let λ and θ be classes of symbols of weight w̃ and core γ, and suppose that δ is a class of symbols of weight w̃ and core γ_{1}. Then |S_{λ,λ_{1}}| = |S_{λ,λ_{1}^{∗}}| = k!, |S_{λ,δ}| = 0 if δ ∉ {λ_{1}, λ_{1}^{∗}}, and |S_{θ,λ_{1}}| = |S_{θ,λ_{1}^{∗}}| = 0 if θ ≠ λ. Further, d(S) = 0 for any element S of S_{λ,λ_{1}} or S_{λ,λ_{1}^{∗}}.

Proof. Suppose that |S_{λ,δ}| is non-zero. Then, as in the proof of Lemma 5.9, there is a sequence of abacus diagrams

Λ = Δ(0), Δ(1), ⋯, Δ(k)

such that for each j, 1 ≤ j ≤ k, Δ(j) is obtained from Δ(j − 1) by moving a bead one position to the left, and such that the core of Δ(k) is Γ_{1} or (Γ_{1})′. By the usual Scopes argument, if the core of Δ(k) is Γ_{1}, then Δ(k) = Λ_{1} and there are k! sequences of diagrams linking Λ to Λ_{1}; each such sequence of diagrams gives a sequence of classes of symbols linking λ to λ_{1} and no two sequences of such diagrams give the same sequence of equivalence classes of symbols.

Similarly, if the core of Δ(k) is (Γ_{1})′, then Δ(k) is the diagram obtained from Λ by swapping strings i_{b} and (i − 1)_{b}. (Note that a problem as discussed in the proof of Lemma 5.9, namely that beads are moved from i_{g} to (i − 1)_{g} as well as from i_{b} to (i − 1)_{b}, in order to get from Λ to Δ(k), can not happen in this situation since γ is degenerate.) In other words, Δ(k) = ((Λ′)_{1})′ and δ = λ_{1}^{∗}. Again, there are k! diagrams linking Λ to ((Λ′)_{1})′; each such sequence of diagrams gives a sequence of classes of symbols linking λ to λ_{1}^{∗} and no two sequences of such diagrams give the same sequence of equivalence classes of symbols. In particular, |S_{λ,δ}| = 0 if δ ∉ {λ_{1}, λ_{1}^{∗}}.

By reversing the above argument, it also follows that |S_{θ,λ_{1}}| = |S_{θ,λ_{1}^{∗}}| = 0 if θ ≠ λ. If λ is non-degenerate, then λ_{1} ≠ λ_{1}^{∗} by Lemma 5.12, and it follows that |S_{λ,λ_{1}}| = |S_{λ,λ_{1}^{∗}}| = k!. If λ (and hence Λ is degenerate), it is easy to see that there is a bijection between the set of sequences of diagrams linking Λ to Λ_{1} and the set of sequences of diagrams linking Λ to ((Λ′)_{1})′ such that a pair of corresponding sequences yields the same sequence of equivalence classes of symbols. Thus again, |S_{λ,λ_{1}}| = k!. Since γ is degenerate, and any sequence linking λ to λ_{1} or to λ_{1}^{∗} comes from a sequence of diagrams (starting with Λ) in which beads are only moved either in the g-strings or only in the b-strings, the last assertion of the lemma follows. □

Lemma 5.14. Let w̃ be an integer with 0 ≤ w̃ ≤ w. Suppose that γ_{1} is degenerate and that k ≥ w̃. Let λ and θ be classes of symbols of weight w̃ and core γ, and suppose that δ is a class of symbols of weight w̃ and core γ_{1}. Then |S_{λ,λ_{1}}| = |S_{λ^{∗},λ_{1}}| = k!, |S_{λ,δ}| = |S_{λ^{∗},δ}| = 0 if δ ≠ λ_{1}, and |S_{θ,λ_{1}}| = 0 if θ ∉ {λ, λ^{∗}}. Further, d(S) = 0 for any element S of S_{λ,λ_{1}} or S_{λ^{∗},λ_{1}}.

Proof. This is analogous to that of Lemma 5.13. □

5.15. The correspondence λ ↦ λ̄. Let λ be a class of symbols of weight at most w and core γ. If neither γ nor γ_{1} is degenerate, let λ̄ denote the class of symbols corresponding to the diagram Λ_{1}. If γ is degenerate, define λ̄ to be the class of symbols corresponding to the diagram Λ^{∗}. If γ_{1} is degenerate, e = 2 and we are in the unitary situation, define λ̄ to be the class of symbols corresponding to the diagram (Λ̃_{1})_{1} (see 5.3 and 5.7). Lemma 5.16. Let w̃ be an integer with 0 ≤ w̃ ≤ w. Suppose that γ is degenerate and that k ≥ w̃. Let λ be a class of symbols of weight w̃ and core γ, δ a class of symbols of weight w̃ and core γ_{1}, and ζ a class of symbols of weight w̃ and core γ̄. Then |S_{λ_{1},λ̄}| = |S_{λ_{1}^{∗},λ̄}| = k!, |S_{λ_{1},ζ}| = |S_{λ_{1}^{∗},ζ}| = 0 if ζ ≠ λ̄, and |S_{δ,λ̄}| = 0 if δ ∉ {λ_{1}, λ_{1}^{∗}}. Further, d(S) = 0 for any element S of S_{λ_{1},λ̄} or S_{λ_{1}^{∗},λ̄}.

The map λ ↦ λ̄ is a bijection between classes of symbols of weight w̃ and core γ and classes of symbols of weight w̃ and core γ̄, preserving degeneracy of symbols.

Proof. Omitted. □

In the following two lemmas, we put Δ̄ := (Δ̃_{1})_{1} for a diagram Δ (see 5.7). Also, λ̄^{∗} is defined as in 5.11, except that the underlying core is γ̄ instead of γ. Lemma 5.17. Let w̃ be an integer with 0 ≤ w̃ ≤ w. Suppose that γ_{1} is degenerate, e = 2 and that we are in the unitary situation.

If k > 1, then λ ↦ λ̄ is a bijection between classes of symbols of weight w̃ and core γ and classes of symbols of weight w̃ and core γ̄. This bijection has the property that \overline{λ^{∗}} = λ̄^{∗}. Further, if λ is a class of symbols of weight w̃ and core γ, λ̄^{∗} = λ̄ if and only if λ_{1} is degenerate.

Proof. Let λ be a class of symbols of weight w and core γ. Clearly, λ̄ has weight w̃ and core γ̄ whence the above map is well defined. We show first that it is onto. Let ζ be a class of symbols of weight w̃ and core γ̄ and let Θ be a diagram for ζ with core Γ̄. Let Θ(1) be the diagram obtained from Θ by swapping strings 0_{g} and 1_{g}, Θ(2) the diagram such that { \atop Θ(2)} = Θ(1). So Θ(2) has weight w̃ and core Γ_{1}. Let Θ(3) be the diagram obtained by swapping strings 0_{g} and 1_{g} of Θ(2). Then Θ(3) has weight w̃ and core Γ. Let λ be the class of symbols represented by Θ(3). Since γ is non-degenerate, Θ(3) is the unique diagram for λ with core Γ, from which it is immediate that Θ = Λ̄.

Next, suppose that λ and δ are classes of symols of weight w̃ and core γ. Let Λ and Δ be the chosen diagrams for λ and δ respectively. By our hypothesis on k, we have

|Λ̄^{0_{g}}| = |Γ^{1_{g}}| ≠ |Γ^{0_{g}}| + 1 = |Δ̄^{0_{b}}|.

So Δ̄ is not the swap of Λ̄. It follows that λ̄ = δ̄ if and only if Λ̄ = Δ̄ or, in other words, if and only if λ = δ. Thus λ ↦ λ̄ is a bijection as claimed.

Let λ be a class of symbols of weight w̃ and core γ and let Λ be the chosen diagram for λ. Now (Λ^{∗})′ is a diagram for λ^{∗} and Λ^{∗}′ has core Γ (see 5.12). Since γ is non-degenerate, this means that (Λ^{∗})′ is the chosen diagram for λ^{∗}. Put Δ := (Λ^{∗})′. It can be checked that Δ̄′ = Λ̄^{∗} from which it follows that \overline{λ^{∗}} = λ̄^{∗}. □

Lemma 5.18. Let the assumptions be as in Lemma 5.17 and suppose that k ≥ w̃+1. Let λ be a class of symbols of weight w̃ and core γ, δ a class of symbols of weight w̃ and core γ_{1}, and ζ a class of symbols of weight w̃ and core γ̄. Then |S_{λ_{1},λ̄}| = |S_{λ_{1},λ̄^{∗}}| = (k − 1)!. Moreoever, |S_{λ_{1},ζ}| = 0 if ζ ∉ {λ̄, λ̄^{∗}} and |S_{δ,λ̄}| = |S_{δ,λ̄^{∗}}| = 0 if δ ≠ λ_{1}. Further, d(S) = 0 for any element S of S_{λ_{1},λ̄} or S_{λ_{1},λ̄^{∗}}.

Proof. Omitted. □

6. Morita equivalences

In this section we fix an odd prime l and a prime power q not divisible by l.

6.1. Introductory remarks. In our previous paper, [23], we showed that the Donovan and Puig conjectures hold for the unipotent l-blocks of the families of finite unitary groups {GU_{n}(q) | n ≥ 1}, the odd dimensional orthogonal groups {SO_{n}(q) | n odd} and the conformal symplectic groups {CSp_{n}(q) | n even} when l is a unitary prime for the family concerned. (For the definiton of unitary prime see Subsection 3.4.)

However, we were unable to settle the case of the even dimensional special orthogonal groups; the difficulty being a degeneracy phenomenon in the combinatorics describing the character theory of these groups (see [23, Theorem 8.3 (iii)]). In the present paper, we circumvent these problems and prove that the Donovan and Puig conjectures hold also for the unipotent l-blocks in the familes of the even dimensional orthogonal groups {CSO_{n}^{±}(q) | n even} (for odd q) and {SO_{n}^{±}(q) | n even} (for even q), when l is unitary and does not divide q + 1.

Since the publication of [23] we have discovered that the proof of the reduction theorem (Theorem 7.9) therein does not apply to all the situations covered by its hypothesis. Specifically, if we are in the symplectic or orthogonal case, we require that the equivalence class of symbols γ being considered have defect different from 0 and 2, and not merely that γ is an equivalence class of non-degenerate symbols. Since all relevant symbols for the odd dimensional orthogonal groups and the symplectic groups have odd defect, the stronger condition is always satisfied for these groups, and the reduction theorem, as well as all consequences thereof remain valid as stated for these groups. For the even dimensional orthogonal groups however, this extra restriction implies that the phrase “non-degenerate cores” should be replaced by “cores of defect at least 4” in [23, Theorem 8.3(iii)]. Note that this extra hypothesis effectively rules out the case e = 1, i.e., l | q + 1. In what follows, we prove an anlogue of the reduction theorem which places no restriction on the defect of cores, but where we assume that e > 1.

6.2. The groups. For a positive even integer n we let G_{n}(q) denote one of the following groups.

(1) The conformal special orthogonal group CSO_{n}^{+}(q) of an n-dimensional vector space over 𝔽_{q} with a quadratic form of maximal Witt index, where q is odd.

(2) The conformal special orthogonal group CSO_{n}^{−}(q) of an n-dimensional vector space over 𝔽_{q} with a quadratic form of Witt index n/2 − 1, where q is odd.

(3) The special orthogonal group SO_{n}^{+}(q) of a quadratic space as in (1), where q is even.

(4) The special orthogonal group SO_{n}^{−}(q) of a quadratic space as in (2), where q

is even. If n is clear from the context, we simply write G for G_{n}(q). Let us recall some notation introduced in [23].

If Y is any finite-dimensional vector space and t a semisimple invertible linear transformation on Y, we put Y_{t} for the set of fixed points of t on Y, and we let [t, Y] = {ty − y | y ∈ Y }. Then Y = Y_{t} ⊕ [t, Y]. Moreoever, if Y carries a nondegenerate t-invariant bilinear form, then Y_{t} is the orthogonal complement of [t, Y].

Recall from [23, Section 4,4.2.5–6] that G = G^{F}, where G is a special (conformal) orthogonal group of a non-degenerate quadratic form on an 𝔽̄_{q}-vecor space V of dimension n, and where F is a Frobenius endomorphism of G. We have G = J(V)^{∘}, if q is odd, and G = I(V)^{∘}, if q is even. To simplify notation we shall write J_{0}(V) := J(V)^{∘} and I_{0}(V) := I(V)^{∘}. (Note that the latter notation differs from the one introduced in [23, 4.2.1] in case q is even.) There is a suitable 𝔽_{q}-form V of V such that J_{0}(V) = J_{0}(V)^{F} and I_{0}(V) = I_{0}(V)^{F}. Here, I_{0}(V) = I(V)∩SL(V) if q is odd, and, if q is even, I_{0}(V) denotes the kernel of the Dickson invariant, i.e., the set of isometries t of V with dim_{𝔽_{q}}([t, V]) even.

The form on V is defined with respect to a particular basis e_{1}, …, e_{m}, e′_{m}, …, e′_{1}. For an integer k with 0 ≤ k ≤ m we let V_{n−2k} denote the 𝔽̄_{q}-subspace of V spanned by e_{k+1}, …, e_{m}, e′_{m}, …, e′_{k+1} (see [23, 4.2.7]). If Y = V_{n−2k} for some k, we write Y for the 𝔽_{q}-span of the basis elements e_{k+1}, …, e_{m}, e′_{m}, …, e′_{k+1}, so that Y is a particular 𝔽_{q}-form of Y.

6.3. The reduction theorem. Let G = G_{n}(q) be a group as in Subsection 6.2 and let (K, O, F) be a splitting l-modular system for G such that F is algebraically closed. Let d and e denote the orders of q and of q^{2} modulo l, respectively. Then l is linear for G if e = d, and unitary, if e = d/2 (it is easy to see that l is linear for G if and only if d is odd).

We use the notation and assumptions of Section 5. The integer e now has the specific meaning introduced above, and γ and w are the unipotent label and weight of a unipotent l-block of G, respectively. As in 5.5, we let k = š_{e}(γ) if l is linear for G, and k = ŝ_{e}(γ), otherwise. In order to use the results of Section 5, we assume that e > 1 for the rest of this section. We also assume that γ is not equivalent to the symbol {{1}, {1}}, that k > 0, and that e = 2 and k > 1 whenever γ_{1} is degenerate (note that l is a unitary prime for G if e = 2).

Let k̃ = 0 if both γ and γ_{1} are non-degenerate, k̃ = k if γ is degenerate and let k̃ = k − 1 if γ_{1} is degenerate.

Let G_{1} be the group G_{n−2k}(q) and let L_{k} be the standard Levi subgroup of G isomorphic to G_{n−2k}(q) × GL_{1}(q)^{k} as described in [23, Section 4.4]. Let Ḡ be G_{n−2(k+k̃)}(q) and L_{k+k̃} the corresponding Levi subgroup of G. Note that L_{k+k̃} is naturally a Levi subgroup of L_{k}. If χ is a character of G_{1} (or Ḡ), we will denote the inflation of χ to L_{k} (or to L_{k+k̃}) also by χ. Note that since e > 1, any block of G_{1} (or of Ḡ) may be regarded as a block of L_{k} (or of L_{k+k̃}), and we shall do this without comment. Recall from 5.7 that if γ and γ_{1} are non-degenerate, then γ̄ = γ_{1}. Theorem 6.4. Suppose that w > 0 and that B, B_{1} and B̄ are unipotent blocks of G, G_{1} and Ḡ, respectively, each of weight w, and with unipotent labels γ, γ_{1} and γ̄, respectively, and suppose that rank(γ̄) ≥ 2. Also, suppose that if γ and γ_{1} are non-degenerate, then k ≥ w + 2, if γ is degenerate then k ≥ w, and if γ_{1} is degenerate, then e = 2 and k ≥ w + 1.

(i) Suppose that γ and γ_{1} are non-degenerate. Then there is a bijection χ ↦ χ̄ between Irr(G, B) and Irr(Ḡ, B̄) such that R_{L_{k}}^{G}(χ̄)_{B} = (k!)χ for all χ̄ ∈ Irr(Ḡ, B̄). (Here, the subscript B on a character indicates its restriction to the block B).

(ii) Suppose that γ or γ_{1} is degenerate. Then there is an involution ι permuting Irr(G, B) and Irr(Ḡ, B̄), and a bijection χ ↦ χ̄ between Irr(G, B) and Irr(Ḡ, B̄) commuting with ι, such that the following holds. If χ and χ̄ denote the sums of the characters in corresponding ι-orbits on Irr(G, B) and Irr(Ḡ, B̄), repectively, then R_{L_{k}}^{G}(R_{L_{k+k̃}}^{L_{k}}(χ̄)_{B_{1}})_{B} = 2(k!)(k̃!)χ.

Proof. The proof is very similar to that of the reduction theorem [23, Theorem 7.9]. First we construct suitable d-cuspidal pairs for B, B_{1} and B̄ as follows. Let V_{1} := V_{n−2k} and V̄ := V_{n−2(k+k̃)} be the subspaces of V introduced in Subsection 6.2, so that G_{1} = J_{0}(V_{1}) or I_{0}(V_{1}) and Ḡ = J_{0}(V̄) or I_{0}(V̄). Since B̄ is a block of weight w of Ḡ, there is an orthogonal decomposition V̄ = V′ ⊕ V̄′′ of V̄, is defined over 𝔽_{q}, such that dim𝔽̄_{q}(V′) equals 2ew and I_{0}(V′) has a Sylow ϕ_{d}-torus S with |S^{F} | = ϕ_{d}^{w}(q). Put K̄ := CḠ(S). Then V′ is the fixed point space of [K̄, K̄]. There is a unipotent character ψ̄ of K̄^{F} whose restriction to I_{0}(V̄′′)^{F} has label γ̄ and (K̄, ψ̄) is a unipotent d-cuspidal pair for B̄. Let V′′_{1} and V′′ be the orthogonal complements to V′ in V_{1} and V, respectively. Put K_{1} = C_{G_{1}}(S) and K = C_{G}(S). Then clearly, V′ is the fixed point space of [K_{1}, K_{1}] and of [K, K]. Let ψ_{1} be a character of K_{1}^{F} whose restriction to I_{0}(V′′_{1})^{F} is a unipotent character with label γ_{1} and let ψ be a character of K^{F} whose restriction to I_{0}(V′′) is a unipotent character with label γ. Then (K_{1}, ψ_{1}) is a d-cuspidal pair for B_{1} and (K, ψ) is a d-cuspidal pair for B.

Since rank(γ̄) ≥ 2, we can argue just as in [23, Section 7.6 and Lemma 7.7] to conclude that there is an integer k′ with k′ ≥ k + k̃ such that S ⊆ G_{n−2k′} and such that any Sylow l-subgroup D of G_{n−2k′}(q) is a defect group of B, B_{1} and B̄.

Let T denote the set of G^{∗}-conjugacy classes of F -stable l-elements of G^{∗} for which the corresponding Lusztig series of G contains a character in the block B. Here G^{∗} denotes the dual group of G. Let T_{1} and T̄ have analogous meanings for the pairs (G_{1}, B_{1}) and (Ḡ, B̄), respectively. Then, as in the proof of [23, Theorem 7.9], each of T , T_{1} and T̄ is in bijection with Ḡ-conjugacy classes of l-elements intersecting D non-trivially. If the class of an element t of D corresponds to the classes t̃^{∗} of G̃^{∗}, where G̃^{∗} is one of G^{∗}, G_{1}^{∗} or Ḡ^{∗}, then C_{G}(t) is a Levi subgroup of G in duality with CG̃^{∗}(t̃^{∗}).

Let t ∈ D correspond to each of t^{∗} ∈ G^{∗}, t_{1}^{∗} ∈ G_{1}^{∗} and t̄^{∗} ∈ Ḡ^{∗}. Note that [t, V] = [t, V_{1}] = [t, V̄] ⊆ V′. Put w(t) := dim𝔽̄_{q}[t, V]/2e. Then w(t) ≤ w. Let C = C_{I_{0}([t,V])}(t) and C = C^{F}. Then [C_{G}(t), C_{G}(t)] ⊆ I_{0}(V_{t}) × C ⊂ C_{G}(t), and it follows from Lemma 3.3 that a unipotent character of C_{G}(t) is completely determined by its restriction to I_{0}(V_{t}) × C. Using this lemma once more and [13, Theorem 4.4], it follows that a character χ_{t^{∗},ρ} of G lies in B if and only if ρ restricts to a unipotent character ρ′′ × ρ′ of I_{0}(V_{t}) × C such that the label of ρ′′ has e-linear core γ (and hence e-linear weight w(t)) if l is a linear prime for G, and has eunitary core γ (and hence e-unitary weight w(t)) if l is a unitary prime for G. Similar characterizations hold for the elements of E(G_{1}, t_{1}^{∗}) in B_{1} and the elements of E(Ḡ, t̄^{∗}) in B̄.

Now let χ = χ_{t^{∗},ρ} be a character of B such that ρ restricts to the character ρ′′ × ρ′ of I_{0}(V_{t}) × C, and suppose that ρ′′ has label λ. Note that λ has weight w(t) and core γ. Let λ_{1} and λ̄ be the symbols defined in Subsections 5.8 and 5.15, respectively. Then λ_{1} and λ̄ have weight w(t) and cores γ_{1} and γ̄, respectively.

Suppose first that both γ and γ_{1} are non-degenerate. Then λ and λ_{1} = λ̄ are both non-degenerate, hence there is a unique unipotent character ρ̄′′ of I_{0}((V_{1})_{t}) with label λ_{1}. Define χ̄ to be the character χ_{t_{1}^{∗},ρ̄} in B_{1} such that ρ̄ restricts to the character ρ̄′′ × ρ′ of I_{0}((V_{1})_{t}) × C. By Lemma 5.9, applied with w̃ = w(t), χ ↦ χ̄ provides a bijection between the set of characters of E(G, t^{∗}) in B and the set of characters of E(G_{1}, t_{1}^{∗}) in B_{1}.

If γ is degenerate, then by Lemma 5.16, λ is degenerate if and only if λ̄ is degenerate. So, if λ is non-degenerate, there is a unique unipotent character ρ̄′′ of I_{0}(V̄_{t}) with label λ̄ and we define χ̄ to be the character χ_{t̄^{∗},ρ̄} in B̄ such that ρ̄ restricts to the character ρ̄′′ × ρ′ of I_{0}(V̄_{t}) × C. Also, set χ^{ι} := χ and χ̄^{ι} := χ̄ in this case. If λ is degenerate, let (ρ′′)^{ι} be the other character of I_{0}(V_{t}) labelled by λ, let ρ^{ι} = (ρ′′)^{ι} × ρ′ and set χ^{ι} := χ_{t^{∗},ρ^{ι}}. Also, if λ is degenerate, there are two characters ρ̄′′ and (ρ̄′′)^{ι} of I_{0}(V̄_{t}) labelled by λ̄; set ρ̄ to be the character of CḠ(t) which restricts to ρ̄′′ × ρ′ and ρ̄^{ι} to be the character of CḠ(t) which restricts to (ρ̄′′)^{ι} × ρ′. Set χ̄ = χ_{t̄^{∗},ρ̄} and χ̄^{ι} := χ_{t̄^{∗},ρ̄^{ι}}. Note that ι acts as an involution on both E(G, t^{∗}) and E(Ḡ, t^{∗}). Moreover, the map χ ↦ χ̄ is a bijection between these two sets commuting with ι.

If γ_{1} is degenerate, then λ and λ̄ are both non-degenerate. Let ρ̄′′ be the unique character of I_{0}(V̄_{t}) with label λ̄, set ρ̄ = ρ̄′′ ×ρ′ and set χ̄ = χ_{t̄^{∗},ρ̄}. Now Lemma 5.17 provides a bijection between the set of characters of E(G, t^{∗}) in B and the set of characters of E(Ḡ, t̄^{∗}) in B̄.

Letting t run over the Ḡ-conjugacy classes of l-elements intersecting D, in all three cases we get a bijection χ ↦ χ̄ between the irreducible characters of G in B and those of Ḡ in B̄. We now prove the required properties of the bijection χ ↦ χ̄. For the rest of the section, let t ∈ D and let χ = χ_{t^{∗},ρ} be an irreducible character of G in B as above.

Let us first assume that both γ and γ_{1} are non-degenerate. Let τ be an irreducible character of G in B such that τ is a constituent of the Harish-Chandra induced character R_{L_{k}}^{G}(χ̄). Since Harish-Chandra induction preserves Lusztig series, we may assume that τ ∈ E(G, t^{∗}). Suppose that τ = χ_{t^{∗},η} and suppose that η restricts to the character η′′ × η′ of I_{0}(V_{t}) × C. Then, arguing as in the proof of [23, Theorem 7.9], we have that η′ = ρ′ and the calculation of the multiplicity of τ as a constituent of R_{L_{k}}^{G}(χ̄) reduces to the calculation of the number

⟨R_{I_{0}(V̄_{t})×GL_{1}(q)^{k}}^{I_{0}(V_{t})}(ρ̄′′), η′′⟩_{I_{0}(V_{t})} .

The Iwahori-Hecke algebras corresponding to the Harish-Chandra series of ρ̄′′ and of η′′ are either both of type B or of type D, and the calculation of the above inner products reduces to a calculation in the corresponding Weyl groups. Let θ be the equivalence class of symbols of weight w(t) and e-core γ labelling η′′. From the branching rules (see [23, Section 5.5]), if the Iwahori-Hecke algebras are of type B, then the above number is equal to |S_{θ,λ_{1}}| (for the definition of the set S_{θ,λ_{1}} see Subsection 5.4). If the Iwahori-Hecke algebras are of type D, then the above number is ∑_{S∈S_{θ,λ_{1}}} 2^{d(S)}. But now it follows from Lemma 5.9 applied with w̃ = w(t) that in either case, the above number is k! if θ = λ and is zero if θ ≠ λ. This completes the proof of the reduction theorem in the case that both γ and γ_{1} are non-degenerate.

Now suppose that γ is degenerate. We have already defined the permutation χ ↦ χ^{ι} and have shown that it commutes with the bijection χ ↦ χ̄. Let χ be as above. Let ρ′′_{1} be the unique character of I_{0}((V_{1})_{t}) labelled by λ_{1}, set ρ_{1} = ρ′′_{1} × ρ′ and set χ_{1} = χ_{t_{1}^{∗},ρ_{1}}. Also, let ρ′′_{1}^{∗} be the unique character of I_{0}((V_{1})_{t}) labelled by λ_{1}^{∗}, where λ_{1}^{∗} is defined in Subsection 5.11, set ρ_{1}^{∗} = ρ′′_{1}^{∗}×ρ′ and set χ_{1}^{∗} = χ_{t_{1}^{∗},ρ_{1}^{∗}}. Then by Lemma 5.12, χ_{1} and χ_{1}^{∗} are characters of G_{1} in B_{1} and χ_{1} = χ_{1}^{∗} if and only if λ is degenerate.

Let τ be an irreducible character of G_{1} in B_{1} such that τ is a constituent of the Harish-Chandra induced character R_{L_{2k}}^{L_{k}}(χ̄). Then, again, we may assume that τ ∈ E(G_{1}, t_{1}^{∗}). Suppose that τ = χ_{t_{1}^{∗},η}, and let θ be the class of symbols to which η corresponds as in the case when γ and γ_{1} were both non-degenerate. Then, arguing as above, we get that the multiplicity of τ as a constituent of R_{L_{2k}}^{L_{k}}(χ̄) is either |S_{θ,λ̄}| or ∑_{S∈S_{θ,λ̄}} 2^{d(S)}. By Lemma 5.16, it follows that θ is either λ_{1} or λ_{1}^{∗} and

thus that R_{L_{2k}}^{L_{k}}(χ̄)_{B_{1}} = k!(χ_{1} +χ_{1}^{∗}) if λ is non-degenerate, and R_{L_{2k}}^{L_{k}}(χ̄)_{B_{1}} = (k!)χ_{1} if λ is degenerate. Similarly, if λ is degenerate, R_{L_{2k}}^{L_{k}}(\overline{χ^{ι}})_{B_{1}} = (k!)χ_{1}. By invoking Lemma 5.13, one sees in exactly the same way as above that if λ is non-degenerate, then R_{L_{k}}^{G}(χ_{1}) = R_{L_{k}}^{G}(χ_{1}^{∗}) = (k!)χ and that if λ is degenerate, then R_{L_{k}}^{G}(χ_{1}) = k!(χ + χ^{ι}). This proves statement (ii) of the theorem in case γ is degenerate.

Finally, let us assume that γ_{1} is degenerate. Let ρ′′_{1} be a character of I_{0}((V_{1})_{t}) labelled by λ_{1}. Note that if λ_{1} is degenerate, there are two choices for ρ′′_{1}. Set ρ_{1} = ρ′′_{1} × ρ′ and set χ_{1} = χ_{t_{1}^{∗},ρ_{1}}. If λ_{1} is degenerate, let (ρ′′_{1})^{ι} be the other character of I_{0}((V_{1})_{t}) labelled by λ_{1}, set ρ_{1}^{ι} = (ρ′′_{1})^{ι} × ρ′ and χ_{1}^{ι} = χ_{t_{1}^{∗},ρ_{1}^{ι}}. Again, let ρ^{∗} be the character of I_{0}(V_{t}) labelled by λ^{∗} where λ^{∗} is defined in Subsection 5.11. Note that by Lemma 5.12, χ^{ι} := χ_{t^{∗},ρ^{∗}} is a character of G in B, χ = χ^{ι} if and only if λ_{1} is degenerate, and χ ↦ χ^{ι} is a permutation of the set of irreducible characters of G in B. We can define the permutation τ ↦ τ^{ι} on the characters of Ḡ in B̄ analogously.Note that \overline{χ^{ι}} = χ̄^{ι} by Lemma 5.17.

It follows from Lemma 5.18 that if λ_{1} is degenerate, then R_{L_{2k−1}}^{L_{k}}(χ̄)_{B_{1}} = (k − 1)!(χ_{1} + χ_{1}^{ι}) and if λ_{1} is non-degenerate, then R_{L_{2k−1}}^{L_{k}}(χ̄^{ι})_{B_{1}} = R_{L_{2k−1}}^{L_{k}}(χ̄)_{B_{1}} = (k −1)!χ_{1}. Again, by Lemma 5.14, we get that if λ_{1} is degenerate then R_{L_{k}}^{G}(χ_{1}^{ι})_{B} = R_{L_{k}}^{G}(χ_{1})_{B} = (k!)χ and if λ_{1} is non-degenerate, then R_{L_{k}}^{G}(χ_{1})_{B} = k!(χ + χ^{ι}). □

6.5. Source algebra equivalence. In order to show that the blocks B and B̄ of the reduction theorem have equivalent source algebras, we shall invoke Theorem 2.5. However, it is easier to apply this theorem to the orthogonal groups instead of to the conformal groups. For this reason we introduce the following notation. If G is one of the groups introduced in 6.2, we put G′ := [G, G] (and G′ := G′^{F}), where G is the (conformal) orthogonal group over 𝔽̄_{q} with G = G^{F}. Thus if V is the natural module for G, its 𝔽_{q}-form V introduced in Subsection 6.2 is the natural module for G′ and we have G′ = I_{0}(V)^{F}. Similarly, we define L′_{k} := L_{k} ∩ G′ etc. Theorem 6.6. Under the hypotheses of Theorem 6.4, the blocks B and B̄ are Puig equivalent.

Proof. By Theorem 3.2 there is a unique block B′ of G′ covered by B, and restriction yields an isomorphism between the corresponding block algebras. Similarly, there is a unique block B′_{1} of G′_{1} covered by B_{1} and a unique block B̄′ of Ḡ′ covered by B̄. Since

L′_{k} ≅ G′_{1} × GL_{1}(q)^{k}

and l does not divide q − 1, it follows that the block B′_{1} can naturally be considered as a block of L′_{k} etc. Then, by Lemma 3.3, it follows that the reduction theorem holds on replacing G by G′, L_{k} by L′_{k}, B by B′ etc.

Let b, b_{1}, b̄ be the central primitive idempotents of OG′, OL′_{k} and OL′_{k+k̃}, respectively, corresponding to the blocks B′, B′_{1} and B̄′. We let

u := \frac{1}{|U|} ∑_{v∈U} v ∈ OG′,

where U is the unipotent radical of Q, the standard parabolic subgroup of G with Levi complement L′_{k}. Also, we let

u_{1} := \frac{1}{|U_{1}|} ∑_{v∈U_{1}} v ∈ OL′_{k},

where U_{1} is the unipotent radical of Q_{1}, the standard parabolic subgroup of L′_{k} with Levi complement L′_{k+k̃}.

Note that by Theorem 6.4, we have (17)

∑ r(τ, χ, bub_{1}u_{1}b̄) = \begin{cases}|Irr(L′_{k}, b̄)|k!, & if γ and γ_{1} are non-degenerate, \\ 2|Irr(L′_{2(k+k̃)}, b̄)|(k!)^{2}(k̃!)^{2}, & if γ or γ_{1} is degenerate, \end{cases}

where χ in the above sum ranges over Irr(G′, b) and τ ranges over Irr(L′_{k+k̃}, b̄). We shall apply Theorem 2.5 with G replaced by G′, H replaced by L′_{k+k̃}, c by b̄ and u by bub_{1}u_{1}b̄.

In order to proceed, we choose a particular defect group D of the blocks involved as follows. With the notation introduced at the beginning of the proof of Theorem 6.4, recall that S is a Sylow ϕ_{d}-torus of I_{0}(V′) with |S^{F} | = ϕ_{d}^{w}(q). Put T := C_{I_{0}(V′)}(S). Then T is a maximal F -stable torus of I_{0}(V′) with |T^{F} | = (q^{e} ± 1)^{w}. Let D_{0} be a Sylow l-subgroup of T := T^{F} and let D be Sylow l-subgroup of I_{0}(V′)^{F} ≅ SO_{2ew}^{−}(q) containing D_{0}. Then D is a defect group of the blocks B′, B′_{1} and B̄′. Since I_{0}(V′)^{F} is a subgroup of index 2 of I(V′)^{F}, N_{I_{0}(V′)^{F}} (D) has index 2 in N_{I(V′)^{F}} (D). Let us fix an element τ of N_{I(V′)^{F}} (D)\N_{I_{0}(V′)^{F}} (D). Also, let us fix an element ρ of I(V̄′′)^{F} \ I_{0}(V̄′′)^{F}. (Note that dim𝔽̄_{q}(V̄′′) ≥ 4 since rank(γ̄) ≥ 2.) Lemma 6.7. (i) C_{G′}(D) = C_{T}(D) × I_{0}(V′′).

(ii) N_{G′}(D) is generated by N_{I_{0}(V′)}(D) × I_{0}(V′′) and ρτ.

Proof. It is clear that an element of C_{G′}(D) fixes the orthogonal decomposition V = V′ ⊕ V′′. Since l is odd, the centralizers in G′ of l-elements are connected and thus C_{G′}(D) = C_{I_{0}(V′)}(D) × I_{0}(V′′). Now D_{0} certainly contains a regular element of T. Hence T = C_{I_{0}(V′)}(D_{0}) ≥ C_{I_{0}(V′)}(D), and the first result follows. Part (ii) is obvious. □

Taking F -fixed points in Lemma 6.7 gives

C_{G′}(D) = C_{T} (D) × I_{0}(V′′),
C_{L′_{k}}(D) = C_{T} (D) × I_{0}(V′′_{1}) × GL_{1}(q)^{k},

and

C_{L′_{k+k̃}}(D) = C_{T} (D) × I_{0}(V̄′′) × GL_{1}(q)^{k+k̃}.

Moreover, I_{0}(V′′_{1}) × GL_{1}(q)^{k} is a standard Levi subgroup of I_{0}(V′′) and I_{0}(V̄′′) × GL_{1}(q)^{k+k̃} is a standard Levi subgroup of I_{0}(V′′_{1}) × GL_{1}(q)^{k}. Since D is a Sylow l-subgroup of I_{0}(V′), and T contains C_{I_{0}(V′)}(D), putting Z = C_{T} (D)/Z(D), we have that Z is a an abelian l′-group and the above gives canonical isomorphisms

C_{G′}(D)/Z(D) ≅ Z × I_{0}(V′′),
C_{L′_{k}}(D)/Z(D) ≅ Z × I_{0}(V′′_{1}) × GL_{1}(q)^{k},

and

C_{L′_{k+k̃}}(D)/Z(D) ≅ Z × I_{0}(V̄′′) × GL_{1}(q)^{k+k̃}.

Henceforth we will use these isomorphisms without comment. To simplify notation, we denote a chosen unipotent character of I_{0}(V′′) labelled by γ with the same symbol γ. Similarly, γ_{1} denotes the character of I_{0}(V′′_{1}) × GL_{1}(q)^{k} which is the product of the unipotent character of I_{0}(V′′_{1}) labelled by γ_{1} with the trivial character of GL_{1}(q)^{k} and γ̄ the unipotent character of I_{0}(V̄′′) × GL_{1}(q)^{k+k̃} which is the product of the unipotent character of I_{0}(V̄′′) labelled by γ̄ with the trivial character of GL_{1}(q)^{k+k̃}. Also, whenever γ is degenerate, let γ^{ι} denote the other character labelled by γ. We will also use γ etc to denote the inflation of γ to C_{G′}(D)/Z(D); b_{γ} will be used to denote the defect 0 block of FI_{0}(V′′) containing γ as well as the defect 0 block of F(C_{G′}(D)/Z(D)) containing γ. The terms γ_{1}^{ι}, γ̄^{ι}, b_{γ^{ι}}, b_{γ_{1}} etc will all have the obvious meanings.

As in Proposition 2.7 we denote by Λ(b) the set of irreducible characters of defect zero of C_{G′}(D)/Z(D) connected with b through the Brauer homomorphism. It follows from [13, Lemma 4.5] that if γ and γ_{1} are both non-degenerate, then Λ(b) = {γ}, and Λ(b̄) = {γ̄} (recall that b̄ = b_{1} and γ̄ = γ_{1} in this case). If γ is degenerate, Λ(b) = {γ, γ^{ι}}, Λ(b_{1}) = {γ_{1}} and Λ(b̄) = {γ̄, γ̄^{ι}}. If γ_{1} is degenerate, then Λ(b) = {γ}, Λ(b_{1}) = {γ_{1}, γ_{1}^{ι}} and Λ(b̄) = {γ̄}. Also, it is clear from the descriptions given above that there is a natural isomorphism

F(C_{G′}(D)/Z(D))b_{γ} ≅ FI_{0}(V′′)b_{γ}.

In case both γ and γ_{1} are non-degenerate, N_{G′}(D, γ) = N_{G′}(D), so by Part (iii) of Lemma 6.7 and the above isomorphism it follows that N_{I_{0}(V′)}(D) acts trivially on F(C_{G′}(D)/Z(D))b_{γ} and the image of N_{L′_{k+k̃}}(D, γ) in the automorphism group of F(C_{G′}(D)/Z(D))b_{γ} is generated by ρτ and C_{L′_{k+k̃}}(D). Since Λ(b_{1}) contains a single element, the hypotheses of Proposition 2.7 apply in this case.

In case γ is degenerate, by Part (iii) of Lemma 6.7, we have N_{G′}(D, γ) = N_{I_{0}(V′)}(D) × I_{0}(V′′), from which it is clear that the image of N_{L′_{k+k̃}}(D) in the automorphism group of (FC_{G′}(D)/Z(D))b_{γ} is generated by C_{L′_{k+k̃}}(D, γ). So, clearly the hypotheses of Proposition 2.7 apply in this case as well. An argument similar to the first case shows that the hypotheses of Proposition 2.7 apply in case γ_{1} is degenerate.

On the other hand, note that by the equations following the proof of Lemma 6.7, the intersection of C_{G′}(D) with U is the unipotent radical of the standard parabolic subgroup of I_{0}(V′′) containing I_{0}(V′′_{1}) × GL_{1}(q)^{k} as Levi complement. Thus, denoting by Ψ the homomorphism introduced before Proposition 2.6 (with G = G′) one sees that Ψ(u) = v, where v is the central idempotent corresponding to the trivial representation of this unipotent group. (Here we are considering I_{0}(V′′) as a subgroup of C_{G}(D)/Z(D) via the isomorphism given above.) Similarly Ψ(u_{1}) = v_{1}, where v_{1} is the central idempotent corresponding to the trivial representation of the unipotent radical of the standard parabolic subgroup of I_{0}(V′′_{1}) containing I_{0}(V̄′′) × GL_{1}(q)^{k+k̃} as Levi complement.

In case both γ and γ_{1} are non-degenerate, Ψ(b) = b_{γ} and Ψ(b_{1}) = Ψ(b̄) = b_{γ_{1}}. Thus, r(γ_{1}, γ, Ψ(bub_{1}u_{1}b̄)) is the multiplicity of γ in

R_{I_{0}(V′′_{1})×GL_{1}(q)^{k}}^{I_{0}(V′′)}(b_{γ_{1}})_{b_{γ}} .

But the latter number is |S_{γ,γ_{1}}| which by Lemma 5.9 equals k!. Now (17), Proposition 2.7 and Theorem 2.5 imply that b and b̄ are Puig equivalent.

In case γ is degenerate, Ψ(b) = b_{γ} + b_{γ^{ι}}, Ψ(b_{1}) = b_{γ_{1}} and Ψ(b̄) = bγ̄ + b_{}_{γ̄^{ι}}. So, r(γ̄, γ, Ψ(bub_{1}u_{1}b̄)) is the multiplicity of γ in

R_{I_{0}(V′′_{1})×GL_{1}(q)^{k}}^{I_{0}(V′′)}(R_{I_{0}(V̄′′)×GL_{1}(q)^{2k}}^{I_{0}(V′′_{1})×GL_{1}(q)^{k}}(bγ̄)_{b_{γ_{1}}} )_{b_{γ}}

and again by Lemma 5.13 and Lemma 5.16, the above number equals (k!)^{2}. Similarly, we get that r(γ̄^{ι}, γ, Ψ(bub_{1}u_{1}b̄)) is (k!)^{2}. Again, (17), Proposition 2.7 and Theorem 2.5 imply that b and b̄ are Puig equivalent.

The proof for the case that γ_{1} is degenerate is similar to the above two cases, except that since in this case, Ψ(b_{1}) = b_{γ_{1}} + b_{γ_{1}}^{ι}, r(γ̄, γ, Ψ(bub_{1}u_{1}b̄)) is equal to the sum of the multiplicities of γ in

R_{I_{0}(V′′_{1})×GL_{1}(q)^{k}}^{I_{0}(V′′)}(R_{I_{0}(V̄′′)×GL_{1}(q)^{2k−1}}^{I_{0}(V′′_{1})×GL_{1}(q)^{k}}(bγ̄)_{b_{γ_{1}}} )_{b_{γ}}

and in

R_{I_{0}(V′′_{1})×GL_{1}(q)^{k}}^{I_{0}(V′′)}(R_{I_{0}(V̄′′)×GL_{1}(q)^{2k−1}}^{I_{0}(V′′_{1})×GL_{1}(q)^{k}}(bγ̄)_{b_{γ_{1}}^{ι}})_{b_{γ}} .
□

7. Finiteness results

As a consequence of the results of the previous section, we get the following finiteness theorem. Again, we let l be a prime and choose an l-modular system (K, O, F) with F algebraically closed. Theorem 7.1. Let q be a prime power not divisible by l, and let w be a positive integer. Let {G_{n}(q) | n = 2, 4, …} be one of the families in 6.2 and suppose that l is a unitary prime for this family. Also, assume that l does not divide q^{2} − 1. Then the number of Puig (Morita) equivalence classes of unipotent l-blocks of weight w occurring in the group algebras OG_{n}(q), n = 2, 4, …, is finite. There are at most

[\frac{l − 1}{2}(w + 1) + 2] ∏_{i=1}^{(l−3)/2} [(\frac{l − 1}{2}i + i^{2})(w + 1)^{2} + (\frac{l − 1}{2} + 3i)(w + 1) + 2]

(18)  + [(\frac{l − 3}{2})(w + 1) + 2] ∏_{i=1}^{(l−5)/2} [i(w + 1) + 1]

such blocks. Every unipotent block is Puig (Morita) equivalent to a block of OG_{n}(q) for some n where n is less than or equal to

\frac{1}{2}(l − 1)^{2} [(l − 2)(w + 1) + 1]^{2} + (l − 1)(w + 2).

Proof. Let G = G_{n}(q) for some n. Choose a finite extension (K̃, Õ, F) of (K, O, F) such that K̃ is a splitting field for all subgroups of G. Let B be a unipotent block of FG with label γ. If the hypothesis of Theorem 6.4 is satisfied, we may apply Theorem 6.6 to obtain an equivalence of source algebras of B and B̄ over Õ. Thus by [36, Proposition 38.8], there is an equivalence of the source algebras of B and B̄ over O.

Suppose there does not exist a γ̄ such that the hypotheses of Theorem 6.4 are satisfied.

If γ and γ_{1} are non-degenerate (so that γ̄ = γ_{1}) and rank(γ̄) = 1, then γ̄ = {{1}, {0}} and γ = {{2}, {0}}, or {{1, 2}, {0, 1}} or γ̄ = {{0, 1}, {}} and γ = {{0, 2}, {}} or {{0, 1, 3}, {1}} (the latter only occurs for e = 2). If γ is degenerate then γ is not equivalent to {{1}, {1}} by assumption, and hence rank(γ̄) ≥ 2. If γ_{1} is degenerate, e = 2 and rank(γ̄) = 1, then γ̄ = {{1}, {0}} and γ = {{1, 2}, {0, 3}. In any of these cases, ŝ_{e}(γ) ≤ 2 < w + 2 and rank(γ) ≤ 4.

Suppose then that either ŝ_{e}(γ) = 0 or that ŝ_{e}(γ) > 0 and there exists a γ̄ of rank at least 2, but the hypothesis of Theorem 6.4 is not satisfied for any such γ̄.

If e > 2, this can happen only if γ has the property given in [23, Proposition 8.2(iii)] (with w replaced by w + 2). In other words, if Γ is a 2e-unitary abacus diagram for some element of γ, then for all i, 1 ≤ i ≤ e − 1 and for any a ∈ {g, b} such that ŝ_{e}^{i_{a}}(Γ) > 0, either ŝ_{e}^{i_{a}}(Γ) < w + 2 or the diagram obtained by swapping strings i_{a} and (i − 1)_{a} is degenerate. Then the result follows from Theorem 6.6 and [23, Proposition 8.2(iii)]. If e = 2, then ŝ_{e}(γ) < w + 2 and the result follows from Theorem 6.6 and [23, Proposition 8.2(ii)]. □

Example 7.2. To illustrate the difficulties in case e = 1, let us consider the simplest non-trivial example, the case w = 1. Let k be an odd positive integer and let γ denote the equivalence class of symbols introduced in Example 5.10. We choose a 2-unitary diagram Γ for γ which has exactly one bead on string 0_{b}. A picture of Γ as well as of Γ̄ (where this time Γ̄ is the diagram associated to Γ as in [23, Subsection 7.5, Case 2.2.2]) is given below.

[figure: Γ; .; .; .; 0; 0; 1; k; k + 1; Γ̄; .; .; .; 0; 0; 1; k − 1; k]

Let l be an odd prime and let q be a prime power with l | q + 1. Then γ determines a unipotent 2-cuspidal character ψ of H := CSO_{n−2}^{+}(q) with n − 2 = (k + 1)^{2}/2. In turn, ψ determines a unipotent l-block B of G := CSO_{n}^{−}(q) of weight 1 and 1-unitary core γ. The unipotent characters of B are labelled by symbols λ_{1} and λ_{2}. We obtain diagrams Λ_{1} and Λ_{2} for these two symbols by adding a 1-unitary hook to Γ in two ways.

[figure: Λ_{1}; .; .; .; 0; 0; 1; k; k + 2; Λ_{2}; .; .; .; 1; 0; 1; k; k + 1]

The symbols λ̄_{1} and λ̄_{2} (notation as in the proof of [23, Theorem 7.10]) can be represented by the following diagrams Λ̄_{1} and Λ̄_{2}.

[figure: Λ̄_{1}; .; .; .; 0; 0; 1; k − 1; k + 1; Λ̄_{2}; .; .; .; 0; 0; 1; k − 1; k]

Of course, they can also be represented by the swaps Λ̄′_{1} and Λ̄′_{2} (see the diagram operations introduced in 5.3) of the above diagrams.

[figure: Λ̄′_{1}; .; .; .; 0; 1; 0; k − 1; k + 1; Λ̄′_{2}; .; .; .; 0; 2; 0; k − 1; k]

Let χ_{1} and χ_{2} denote the two unipotent characters of G with labels λ_{1} and λ_{2}, respectively. Similarely, χ̄_{1} and χ̄_{2} denote the two unipotent characters of the Levi subgroup L_{k} of G with labels λ̄_{1} and λ̄_{2}, respectively. Then, with the notation of Theorem 6.4 we find in this case that

R_{L_{k}}^{G}(χ̄_{1})_{B} = (k!)χ_{1} + f(k)χ_{2},

and

R_{L_{k}}^{G}(χ̄_{2})_{B} = f(k)χ_{1} + (k!)χ_{2}

for some function f with f(k) > 0. More precisely, f(k) is the number of “paths” of length k, from the diagram Λ̄′_{2} to the diagram Λ_{1}, where a single step consists in the addition of a 1-hook. This is also equal to the number of such paths from Λ̄′_{1} to Λ_{2}. For example, f(3) = 2 and f(5) = 16.

8. Groups of type A

In this section we let l denote a prime and F an algebraically closed field of characteristic l.

Let ℭ be a class of finite groups. We say that Donovan’s conjecture (for l or over F) holds for ℭ, if for all finite l-groups D, the number of Morita equivalence classes of blocks with defect groups isomorphic to D occuring in group algebras FG for groups G in ℭ, is finite. If the number of Cartan matrices of such blocks is finite, we say that the weak Donovan conjecture holds for ℭ.

8.1. The theorem of Bonnafé and Rouquier. Bonnafé and Rouquier have recently proved a conjecture of Broué. In this section we discuss some consequences of this result for groups of type A. For a prime power q and a positive integer n, we shall write GL_{n}(−q) for the unitary group GU_{n}(q) in the following.

Theorem 8.2. (Bonnafé and Rouquier) Let G = GL_{n}(εq) (ε = ±1), such that l does not divide q and let s be a semisimple l′-element of G^{∗} = G. Let B be an FGblock with defect group D. Assume that the ordinary characters of B are contained in E_{l}(G, (s)).

Then B is Morita equivalent to a unipotent block B′ of a group isomorphic to GL_{n_{1}}((εq)^{m_{1}}) × ⋯ × GL_{n_{k}}((εq)^{m_{k}}) with ∑_{i=1}^{k} n_{i}m_{i} = n. Moreoever, B and B′ have isomorphic defect groups. Thus B′ ≅ B_{1} ⊗ ⋯ ⊗ B_{k}, where B_{i} is a unipotent block of GL_{n_{i}}((εq)^{m_{i}}) with defect group D_{i} such that D ≅ D_{1} × ⋯ × D_{k}.

Proof. By the main result of Bonnafé and Rouquier [4, Théorème 11.8], B is Morita equivalent to a unipotent block B′ of C_{G}(s). The centralizer of every semisimple element of G is of the form given in the theorem. The statement about the defect group follows from a result of Cabanes and Enguehard [14, Remark 3.6]. □

This theorem shows that, in order to prove Donovan’s conjecture for the general linear or unitary groups, it suffices to consider only their unipotent blocks.

8.3. Donovan’s conjecture for the linear groups. Let us begin with an example which combines the results of Jost [25], and [23, Theorem 9.3] with the new results of Bonnafé and Rouquier stated above.

In the following, we denote the set of non-trivial powers of prime numbers by ℙ̂. Theorem 8.4. Donovan’s conjecture holds for ℭ = {GL_{n}(q) | n ∈ ℕ, q ∈ ℙ̂}.

Proof. Let D be a finite non-trivial l-group and let B be an l-block with defect group isomorphic to D of the group algebra FGL_{n}(q) for some GL_{n}(q) ∈ ℭ. If q is a power of l, then B has maximal defect. This implies that GL_{n}(q) is uniquely determined by l and D and the result follows in this case.

We may thus assume that q is not divisible by l. Next, by Theorem 8.2, we may assume that B is unipotent. In this case, the result follows from [23, Theorem 9.3]. □

It is natural to ask for the number of Morita equivalence classes occuring in the above theorem, or, more demanding, for the exact invariants of a block determining its Morita equivalence class. Although it is not too diffcult to conjecture the answer, we can only give a weak result in this direction. We show that under certain, very natural necessary conditions on two prime powers q_{1} and q_{2}, the unipotent blocks of GL_{n}(q_{i}), i = 1, 2, have the same decompositon numbers. Theorem 8.5. Let n be a positive integer. Suppose that q_{1}, q_{2} ∈ ℙ̂ are not divisible by l and have the same residue modulo l. In particular, they have the same order d modulo l. Assume in addition that the l-parts of q_{1}^{d} − 1 and of q_{2}^{d} − 1 are the same.

Then there is a bijection between E_{l}(GL_{n}(q_{1}), (1)) and E_{l}(GL_{n}(q_{2}), (1)) such that corresponding characters have the same l-modular decompositon numbers.

Proof. We match the unipotent characters of G_{i} := GL_{n}(q_{i}), i = 1, 2 via their canonical labelling by partitions of n. The result in this case follows from Geck’s work on factorizations of decomposition maps. A survey of this is given in Section 3 of [19], and Example 3.3 thereof treats the case of Iwahori-Hecke algebras. This gives, by the way, a proof for a somewhat weaker form of [30, Conjecture 6.38].

The G_{i}-conjugacy classes of l-elements are classified, via their rational Jordan canonical form, by multisets of irreducible polynomials over 𝔽_{q_{i}} of degree at most n, whose roots have l-power order. Let f be the degree of such a polynomial. Then f ≤ n and l | (q_{i}^{f} −1)/(q_{i}^{t}−1) for all divisors t ≠ f of f. Hence l | Φ_{f}(q_{i}) (recall that Φ_{f} denotes the f-th cyclotomic polynomial). It follows from [24, Lemma IX.8.1] that f = l^{a}d for some non-negative integer a. Let a be the largest integer such that l^{a}d ≤ n and put f := l^{a}d. Then the field 𝔽_{q_{i}^{f}} contains the roots of all such polynomials. Our assumption, together with [24, Lemma IX.8.1] now show, that q_{1}^{f} −1 and q_{2}^{f} −1 are divisible by the same power of l. Hence there is an isomorphism between the Sylow l-subgroup of 𝔽_{q_{1}^{f}}^{∗} and that of 𝔽_{q_{2}^{f}}^{∗}, compatible with the action of the Galois groups of 𝔽_{q_{i}^{f}} over 𝔽_{q_{i}}.

This implies that there is a bijection between the G_{1}-conjugacy classes of lelements and the G_{2}-conjugacy classes of l-elements such that if t_{1} ∈ G_{2} and t_{2} ∈ G_{2} lie in corresponding classes, then C_{1} := C_{G_{1}}(t_{1}) and C_{2} := C_{G_{2}}(t_{2}) arise from the same generic subgroup of 𝔾𝕃_{n}. In particular, there is a natural bijection λ_{1} ↦ λ_{2} between their unipotent characters. Moreover, R_{C_{1}}^{G_{1}}(λ_{1}) and R_{C_{2}}^{G_{2}}(λ_{2}) have the same decompositon into unipotent characters of G_{1} and G_{2} respectively (if these unipotent characters are matched as above).

There is a linear character t̂_{i} of C_{G_{i}}(t_{i}) such that ±R_{C_{i}}^{G_{i}}(t̂_{i}λ_{i}) is an irreducible character χ_{t_{i},λ_{i}} of G_{i} lying in E(G_{i}, (t_{i})), the sign being independent of i. Also, χ_{t_{i},λ_{i}} and R_{C_{i}}^{G_{i}}(λ_{i}) have the same restriction to the l-regular conjugacy classes of G_{i}. This restriction is thus determined by the coefficients of the unipotent characters occuring in R_{C_{i}}^{G_{i}}(λ_{i}) and the decomposition numbers of unipotent characters of G_{i}. Since, under the above bijection of unipotent characters, these coefficients are the same for each i, the result follows. □

We finally describe the consequences for the special linear groups of our results for the general linear groups. Theorem 8.6. (a) Let D be a finite l-group. Then, up to Morita equivalence, there are only finitely many l-blocks with defect group D occurring as unipotent blocks in the group algebras FG for groups G in ℭ′ := {SL_{n}(q) | n ∈ ℕ, q ∈ ℙ̂}.

(b) The weak Donovan conjecture holds for ℭ′.

Proof. As in the proof of Theorem 8.4 we may assume that q is not divisible by l. To prove the result in this case, we introduce the following notation. If G = SL_{n}(q) for some n ≥ 1 and q not divisible by l, we write G̃ for the corresponding general linear group containing G as a normal subgroup of index q − 1, i.e., G̃ = GL_{n}(q).

Let B be an l-block with defect group isomorphic to D of the group algebra FG for G = SL_{n}(q). Let B̃ be an l-block of G̃ covering B.

Suppose first that B is unipotent. We may choose B̃ to be unipotent as well. If l ∤ q − 1, then B and B̃ are isomorphic by Theorem 3.2. Moreoever, they have the same defect group. By Theorem 8.4, there are only finitley many possible Morita equivalence classes for B. On the other hand, if l | q − 1, then B̃ is the principal block by the results of Fong and Srinivasan. Hence B is the principal block of FG, and there are only finitely many possibilities for G.

Let us now assume that B is not a unipotent block. By a result of Broué and Michel [10], there is a semisimple l′-element s̃ ∈ G̃ such that B̃ is contained in E_{l}(G̃, s̃). Then B is contained in E_{l}(G, s), where s is the image of s̃ under the canonical epimorphism of G̃ = GL_{n}(q) onto P GL_{n}(q), the dual group of G (see, e.g., [3, Proposition 2.3.5]).

Let χ ∈ Irr(B), and let χ̃ ∈ Irr(B̃) such that χ occurs in the restriction of χ̃ to G. By [29, Proposition, p. 162], the multiplicity of χ in the restriction is 1. Moreoever, if χ̃′ is another irreducible character in B̃ whose restriction to G contains χ, then χ̃′ = λχ̃ for some linear character λ of l-power order (see, e.g., [3, (0.7.5) and Corollary 2.3.4]).

Let D̃ be a defect group of B̃ such that D = D̃∩G. Then D̃/D ≅ D̃G/G ≤ G̃/G. Hence |D̃: D| is bounded above by the l-part l^{a} of q − 1. If l does not divide q − 1, then D̃ = D. Suppose that l divides q − 1. By the description of the blocks of GL_{n}(q) given by Dipper and James, B̃ = E_{l}(G̃, (s̃)). Moreoever, a Sylow l-subgroup of CG̃(s̃) is a defect group of B̃. Now CG̃(s̃) ≅ ×_{i=1}^{k}GL_{n_{i}}(q^{m_{i}}) with ∑_{i=1}^{k} n_{i}m_{i} = n. If k, or one of the n_{i} is greater than 1, then l^{2a} divides |D̃|. In this case l^{a} | |D|, and so |D̃| ≤ |D|^{2}. On the other hand, if k = 1 and n_{1} = n, then CG̃(s̃) ≅ GL_{1}(q^{n}) is the Coxeter torus of G̃. Hence D̃ and thus also D are cyclic. Thus if l | q − 1, either D is cyclic or |D̃| ≤ |D|^{2}.

If D is cyclic, there is nothing to prove. So suppose that D is non-cyclic and let M be the largest decomposition number occuring in an l-block with defect group of order at most |D|^{2} of some group in ℭ. Let Φ̃ be the character of a projective indecomposable module of B̃ corresponding to the irreducible Brauer character φ̃. Then, for χ ∈ Irr(G, B), we find

(Φ̃|_{G}, χ) = ∑_{χ̃∈Irr(G̃,B̃),(χ̃|_{G},χ)≠0} d_{χ̃,φ̃}
≤ M |{χ̃ ∈ Irr(G̃, B̃) | (χ̃|_{G}, χ) ≠ 0}|
≤ M l^{a}
≤ M |D|.

The last inequality follows from l^{a} ≤ |D|. Since every projective indecomposable character of B occurs in the restriction of a Φ̃ as above, this implies the desired result. □

8.7. Donovan’s conjecture for the unitary groups. Theorem 8.8. Let D be a finite l-group. Then the following holds.

(a) Up to Morita equivalence, there are only finitely many l-blocks with defect group D occurring in group algebras FG for groups G in

𝔇 := {GU_{n}(q) | n ∈ ℕ, q ∈ ℙ̂, l ∤ q^{2s+1} + 1 for all s ∈ ℕ}.

(b) Up to Morita equivalence, there are only finitely many l-blocks with defect group D occurring as unipotent blocks in group algebras FG for groups G in

𝔇′ := {SU_{n}(q) | n ∈ ℕ, q ∈ ℙ̂, l ∤ q^{2s+1} + 1 for all s ∈ ℕ}.

(c) The weak Donovan conjecture holds for 𝔇′.

Proof. The condition on q just means that l is a linear prime for GU_{n}(q). By Theorem 8.2, B is Morita equivalent to a unipotent block B′ of a group isomorphic to GL_{n_{1}}((−q)^{m_{1}}) × ⋯ × GL_{n_{k}}((−q)^{m_{k}}) with ∑_{i=1}^{k} n_{i}m_{i} = n. Now GL_{n_{i}}((−q)^{m_{i}}) equals GL_{n_{i}}(q^{m_{i}}), if m_{i} is even, whereas for odd m_{i}, it is equal to GU_{n_{i}}(q^{m_{i}}). In the latter case, the order of −q^{m_{i}} modulo l is even, i.e., l is linear for GU_{n_{i}}(q^{m_{i}}).

We conclude that B′ ≅ B_{1}⊗⋯⊗B_{k}, where B_{i} is a unipotent block of GL_{n_{i}}(q^{m_{i}}) whose defect is contained in D, or of GU_{n_{i}}(q^{m_{i}}) with defect group isomorphic to a subgroup of D. Moreover, l is linear for GU_{n_{i}}(q^{m_{i}}). In view of Theorem 8.4, we may thus assume that B is a unipotent block of G = GU_{n}(q) and that l is linear for G. In this case the result follows from Theorem 4.7.

The proof for the special unitary groups is similar to the one for the special linear groups (Theorem 8.6). Note that by assumption, l does not divide q + 1, so that the corresponding part of that proof is not relevant here. □

8.9. The proof of the theorem in the introduction. Let l, F and D be as in the theorem and suppose that D is non-trivial. Let B be a finite set of basic F-algebras, such that each of the unipotent blocks occurring in Theorem 4.9, in Theorem 8.6(a) and in Theorem 8.8(b) is Morita equivalent to an element of B. Suppose also that B contains the basic algebras of the principal l-blocks of the finitely many groups occurring in the theorem, which are of characteristic l and have a Sylow l-subgroup isomorphic to D.

Let G be one of the classical groups listed in the theorem and let B be an l-block of FG with defect group isomorphic to D. Suppose that B is not Morita equivalent to an element of B and that G and l are not as in Case (3) of the theorem.

Let N be an integer which is larger than any of the bounds (on the dimension n of the classical groups involved) given in [23, Theorem 8.3.(i)(ii)] and Theorem 7.1. (Notice that l^{w} ≤ |D| so that N only depends on D.) Suppose also that N is larger than the maximum n such that SU_{n}(q) for some q with l | q + 1 contains a Sylow l-subgroup isomorphic to D. Finally, if l = 2, suppose also that N is larger than the natural dimension of any other classical group with a Sylow 2-subgroup isomorphic to D.

If G = SU_{n}(q) and l ∤ q+1, it follows from [23, Theorem 8.3.(i)] and Theorem 3.2 that Case (2) occurs. If l | q +1, let B̃ be a unipotent l-block of GU_{n}(q) covering B. Then, by the work of Fong and Srinivasan, B̃ is the principal l-block of GU_{n}(q) and thus D is isomorphic to a Sylow l-subgroup of SU_{n}(q). It follows that n ≤ N.

Finally suppose that G is one of the other classical groups. If l > 2, then [23, Theorem 8.3.(ii)], Theorem 7.1 and Theorem 3.2 show that we are in Case (2). If l = 2, then B is the principal block of G (this follows from [12, Theorem 13]) and thus D is a Sylow 2-subgroup of G. This implies that the natural dimension of G is at most equal to N.

9. Remarks and corrections

The following page numbers refer to our paper [23].

Conjecture 1.2 on Page 379 is incorrectly credited to [34]. Puig’s conjecture is not formulated in this reference, but in the unpublished manuscript [32].

The definition of a symbol {X, Y } on Page 382, Lines 19, 20 should include the condition that not both of X and Y are empty.

Replace w̌_{e}(λ) by ŵ_{e}(λ) on Page 386, Line 21.

Delete “conformal” and replace CSO_{n}^{−}(q) by SO_{n}^{−}(q) on Page 390, Line 6. Replace “of” by “to” on Page 390, Line 22. Omit “and by I_{0}(V) the intersection of I(V) with SL(V)” on the same page, Lines −12, −11.

Replace G := I_{0}(V) by G := I_{0}(V) := I(V) ∩ SL(V) on Page 391, Line 11. The embedding of G_{n−2k} into M_{k} on Page 394 is of course F -equivariant. Replace SCO_{n}^{−}(q) by CSO_{n}^{−}(q) on Page 397, Line 14. It is helpful to note that

the unique unipotent character of (C)SO_{2}^{±}(q) is labelled by the class of symbols of rank 1 containing {{1}, {0}} (in the ‘+’-case), respectively {{0, 1}, {}} (in the ‘−’-case). Moreover, there is no symbol of rank 0 and even defect.

On Page 403, Line −7 we write: “Cases 2.1 and 2.2.1 are proved similarly.” This is of course not true under the assumptions of [23, Lemma 7.4], as Example 5.10(1) above shows.

Insert “in” before “Case 1” on Page 404, Line −18. Replace K̄ := CḠ(T) by K̄ := CḠ(S) on the last line of Page 404. The resulting K̄ is of course the same, but the new version makes it evident that K̄ is d-split.

Replace [K, K] ⊆ C_{G}(t), C_{G}(t)] by [K, K] ⊆ [C_{G}(t), C_{G}(t)] on Page 405, Line −12. Replace k!_{χ} by k!χ on Page 406, Line 14. The notation I_{0}(V̄) on Line −16,

Page 406, stands for the identity component I(V̄)^{0}. The reference to Lemma 5.7 on the same page, Line −2, is not sufficient for the purpose of the proof, since it is stated only for Harish-Chandra induction. This reference should be replaced by a reference to Lemma 3.3 of this paper. Replace ψ_{γ} in Equation (10) of the same page by ψ.

On Page 408, Lines −3, −2, we write: “Arguing as in Case 1, one sees immediately that Δ = Λ̄ … ”. In Case 2.2.1 it can also happen that Δ = Λ̄′ (notation of Subsection 5.3), as Example 5.10(1) shows. For the same reason, our statement “This shows immediately that Δ = Λ̄ … ” on Page 409, Line 8, is wrong, as shown by Examples 5.10(2) and 7.2.

Replace w. by w, on Page 412, Line −12.

Replace \frac{l−2}{2} by \frac{l−1}{2} in Equation (20) on Page 416. Equation (21) on the same page appears to contain various misprints. It is, however, made obsolete by Equation (18) of this paper.

Replace “in” by “is” on Page 417, Line 9.

Replace {{0, 1, 2}, {0}} by {{1, 2}, {0, 1}} and rank(γ̄) > 2 by rank(γ̄) ≥ 2 on Page 418, Line 8. (Notice that the extra case γ = {{0, 1, 3}, {1}} in the proof of Theorem 7.1 of this paper only occurs for e = 2.)

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G.H.: Lehrstuhl D für Mathematik, RWTH Aachen, Templergraben 64, 52062 Aachen, Germany

R.K.: Department of Mathematics, Ohio State University, 231, W.18th Avenue, Columbus, OH 43210, USA

  E-mail address: G.H.:  gerhard.hiss@math.rwth-aachen.de
  E-mail address: R.K.:  kessar@math.ohio-state.edu