ON ROUQUIER BLOCKS FOR FINITE CLASSICAL GROUPS AT LINEAR PRIMES

MICHAEL LIVESEY

Abstract. H. Miyachi and W. Turner have independently proved that Broué’s Abelian Defect Group Conjecture holds for certain unipotent blocks of the finite general linear group, the so-called Rouquier blocks[18] and[19, Section 2, Theorem 1]. This together with A. Marcus[17, Theorem 4.3(b)] and J. Chuang and R. Rouquier[7, Theorem 7.18] proves that the conjecture holds for all blocks of such groups. We prove that other finite classical groups also possess unipotent Rouquier blocks at linear primes.

Contents

1. Introduction and Notation  3
2. Some Combinatorics  5
2.1. Partitions and Symbols  5
2.2. Hooks and Cores  6
2.3. Alternative Description  7
2.4. Littlewood-Richardson Coefficients  7
3. Weyl Groups  8
4. The Brauer Homomorphism  10
5. The Finite Classical Groups  10
5.1. Forms  10
5.2. Description of Groups  11
5.3. Clifford Groups  14
5.4. Levi Subgroups  15
6. Representation Theory of the Finite Classical Groups  16
6.1. Semisimple elements  16
6.2. Characters of Finite Classical groups  19
7. Harish-Chandra Induction  20
7.1. Harish-Chandra Induction  20
7.2. Characters under Harish-Chandra Induction  21
7.3. Characters with Equal Dimension  23
8. Unipotent Blocks of Finite Classical groups  23
8.1. Blocks  24
8.2. Defect Groups and Dual Defect Groups  24
8.3. Characters in Unipotent Blocks  24
8.4. Unipotent Blocks of Sp_{2n}(q) and SO_{2n}^{±}(q)  25
9. Main Theorem  25
9.1. Non-Degenerate Case  27
9.2. Degenerate Case  34

9.3. Sp_{2n}(q) and SO_{2n}^{±}(q)  37
10. Application to Broué’s Conjecture  38
References  41

1. Introduction and Notation

Let p be a prime and consider the p-modular system (K, O, k) such that K contains enough roots of unity for the groups being considered in this paper.

Let G be a finite group and b a block idempotent of OG with defect group P. Recall that to each subgroup H of G containing N_{G}(P) there exists a unique block OHc associated to b through the Brauer homomorphism.

We state Broué’s Abelian Defect Group Conjecture[14, Chapter 6.3.3].

Conjecture 1.0.1 (Broué). Let G be a finite group and P an abelian p-subgroup. Let b be a block idempotent of OG with defect group P and Brauer correspondent c in N_{G}(P). Then OGb and ON_{G}(P)c are derived equivalent.

The conjecture is known to hold for symmetric groups. Recall that to each block B of OS_{n} there is an associated non-negative integer w called the weight of B. The defect group P of B is abelian if and only if w < p, in which case P ≅ (C_{p})^{w}. The proof consists of three steps:

(1) For each w with (0 ≤ w < p) there exists an n ≥ 1 and a block B of OS_{n} which is Morita equivalent to the principal block of O(S_{p} ≀ S_{w})[6, Section 3, Theorem 2].

(2) The principal block of O(S_{p} ≀ S_{w}) is derived equivalent to the Brauer correspondent of B in N_{S_{n}}(P)[17, Theorem 4.3(b)].

(3) Any two blocks B, B′ of OS_{n}, OS_{n′} respectively of the same weight have Morita equivalent Brauer correspondents.

(4) Any two blocks B, B′ of OS_{n}, OS_{n′} respectively of the same weight are derived equivalent[7, Theorem 7.2].

These methods have been adapted for unipotent blocks of finite general linear groups[19, Section 2, Theorem 1], [7, Theorem 7.18]. We investigate (1), (2) and (3) for unipotent blocks of other finite classical groups.

Let q be a prime power (we allow q to be even only in the case of the unitary group) and G = G_{m}(q) be a group of the form U_{n}(q), Sp_{2n}(q), CSp_{2n}(q), SO_{2n+1}(q), SO_{2n}^{+}(q), SO_{2n}^{−}(q), CSO_{2n}^{+}(q) or CSO_{2n}^{−}(q). We adopt the notation that elements of U_{n}(q) have entries in 𝔽_{q} so q is a square, say q = q_{0}^{2}. For a positive integer d we let J_{d} be the d × d matrix with entries in 𝔽_{q} with 1s along the antidiagonal and 0s elsewhere. To each of the above groups types we associate the 2-fold extension GL_{d}(q).2 = ⟨GL_{d}(q), s⟩ of GL_{d}(q) described as follows:

(i)

s^{2} = \begin{cases}−1 & for Sp_{2n}(q) or CSp_{2n}(q) \\ 1 & otherwise \end{cases}

(ii) and for all A ∈ GL_{d}(q)

sAs^{−1} = \begin{cases}J_{d}A^{−q_{0}t}J_{d} & for U_{n}(q) \\ −J_{d}A^{−t}J_{d} & for Sp_{2n}(q) or CSp_{2n}(q) \\ J_{d}A^{−t}J_{d} & otherwise \end{cases}

We have a natural homomorphism GL_{d}(q).2 → {±1} with kernel GL_{d}(q). This extends to a map GL_{d}(q).2 ≀ S_{w} → {±1} and we use M to denote the kernel of this map.

We now recall some facts about the unipotent blocks of OG, more details can be found in section 8. To each unipotent block B of OG there is an associated non-negative integer w called the weight of B. As with the symmetric group the defect group of B is abelian if and only if w < p. In the case of SO_{2n}^{+}(q) or CSO_{2n}^{+}(q) we have the notion of B being degenerate. For a fixed prime power q we have the notion of p being a linear or unitary prime with respect to q.

Now let p be a linear prime with respect to q and d the multiplicative order of q mod p.

We now state our main theorem.

Theorem 1.0.2. (cf. Theorem 9.1.1, Theorem 9.2.1, Corollary 9.3.1, Lemma 9.3.2) Let p be a linear prime with respect to q. For CSp_{2n}(q) and CSO_{2n}^{±}(q) let’s denote by Z_{p} the p-part of the the multiplicative group 𝔽_{q}^{×}. For all other groups Z_{p} will be the trivial group. For all w (0 ≤ w < p) there exists some m = m(w) and some unipotent block B of OG_{m}(q) of weight w such that:

(1) B is Morita equivalent to the principal block of O((GL_{d}.2≀S_{w})×Z_{p}) if G = U_{n}(q), Sp_{2n}(q), CSp_{2n}(q) or SO_{2n+1}(q) or if G = SO_{2n}^{±}(q) or CSO_{2n}^{±}(q) and B is nondegenerate.

(2) B is Morita equivalent to the principal block of O(M × Z_{p}) if G = SO_{2n}^{+}(q) or CSO_{2n}^{+}(q) and B is degenerate.

We will then prove that the Brauer correspondent of B in N_{G}(P) is derived equivalent to the appropriate block in the above theorem and thus we obtain:

Corollary 1.0.3. (cf. Corollary 10.0.3) The block B of OG satisfies Broué’s Abelian Defect Group Conjecture.

Finally we will prove that if B and B′ are unipotent blocks of OG_{m}(q) and OG_{m′}(q) respectively with the same weight and either both non-degenerate or both degenerate then they have Morita equivalent Brauer correspondents (see Corollary 10.0.6). We will then have an analogue for parts (1), (2) and (3) in the proof of the conjecture for the symmetric groups for unipotent blocks at linear primes of our finite classical groups.

We should note that the Morita equivalences we construct in 9.1.1, 9.2.1 and 9.3.1 are splendid in the sense of[14, Chapter 9.2.5] and that they ultimately gives rise to splendid derived equivalences between B and its Brauer correspondent in N_{G}(P).

Sections 2,3 and 4 give some basic background information before our finite classical groups are introduced. In section 2 we describe the combinatorial objects partitions and symbols. They will later be used to label the characters and blocks of our finite classical groups. In section 3 we will take a brief look at the Weyl groups of type A_{n}, B_{n} and D_{n} including branching rules which will ultimately be used to describe what happens to characters of our finite classical groups under Harish-Chandra induction. Section 4 consists of a brief overview of the Brauer homomorphism and the Brauer correspondence.

Section 5 is where we first introduce the finite classical groups that are the subject of this paper. We do this in two ways. First as the group of fixed points of some algebraic group under some Frobenius endomorphism and secondly as the group of linear maps that preserve or scale some bilinear form. We then go on to describe certain Levi subgroups of the finite classical groups. In section 6 we set up a labeling for the characters of the groups in question. Section 7 describes Harish-Chandra induction from the Levi subgroups described in section 5 and we look at what effect this has on characters. Unipotent blocks are introduced in section 8 and we describe the defect groups as well as exactly what characters are in such a block.

Our main theorems 9.1.1 and 9.2.1 are proved in section 9 and we then go on to look at how the main theorems are applied to Broué’s Abelian Defect Group Conjecture. In section 10 we go on to prove the analogues of steps (2) and (3) in the proof of Broué’s Abelian Defect Group Conjecture for the symmetric groups.

2. Some Combinatorics

2.1. Partitions and Symbols. For our description of partitions and symbols we follow G. Hiss and R. Kessar[12, Section 2.2].

A partition λ of a non-negative integer n is a finite ordered set (α_{1}, α_{2}, …, α_{t}) of positive integers with α_{1} ≥ α_{2} ≥ ⋯ ≥ α_{t} and α_{1} + α_{2} + ⋯ + α_{t} = n. We write λ ⊢ n or |λ| = n. A beta-set of a partition λ = (α_{1}, α_{2}, …, α_{t}) is a finite set of non-negative integers {β_{1}, β_{2}, …, β_{s}} where β_{1} < β_{2} < ⋯ < β_{s} and α_{i} = β_{s−i+1} − s + i for 1 ≤ i ≤ s and α_{i} is considered to be zero for i > t. Now if d is a non-negative integer then the d-shift of a beta-set {β_{1}, β_{2}, …, β_{s}} is {0, 1, …, d − 1} ∪ {β_{1} + d, β_{2} + d, …, β_{s} + d}. Two beta-sets are said to be equivalent if one is the d-shift of the other for some nonnegative integer d and two beta-sets give rise to the same partition if and only if they are equivalent.

A symbol is an unordered pair of beta-sets {X, Y }. A symbol {X, Y } is said to be degenerate if X = Y. Two symbols are said to be equivalent if one of them can be obtained from the other by simultaneous d-shifts on both parts of the symbol for some non-negative integer d.

The defect of a symbol {X, Y } is the quantity:

||X| − |Y ||

The rank is:

∑_{x∈X} + ∑_{y∈Y} −⌊(\frac{|X|+|Y |−1}{2})^{2}⌋

Both defect and rank are constant on equivalence classes of symbols.

2.2. Hooks and Cores. The Young diagram for a partition λ = (α_{1}, α_{2}, …, α_{t}) is an arrangement of boxes with α_{i} boxes in the ith row. For example the partition (5, 5, 3, 2) has Young diagram:

A hook of a partition is any box in its Young diagram together with everything directly below it and directly to the right of it in the diagram. The length of a hook is equal to the number of boxes in the hook. If a hook is of length e it is said to be an e-hook. The partition corresponding to the diagram obtained by deleting a hook and moving everything below and to the right of the hook one box up and one box left is said to be obtained by removing this hook. For example:

• • • • ; • ; • ⟶

shows a 6-hook and how to delete it.

If β is a corresponding beta-set then an e-hook corresponds to a pair of non-negative integers (x, y) such that x ∈ β and y ∉ β with x − y = e. Removing this hook corresponds to replacing x with y in β. Similarly if β is a beta-set and (x, y) a pair of non-negative integers with x ∈ β, y ∉ β and y − x = e then the beta-set obtained obtained by replacing x with y is said to obtained by adding an e-hook to β. The corresponding transformation to the partition is also called adding an e-hook.

For the rest of the section let’s fix a positive integer e. Beta-sets can be displayed on an e-abacus as follows: Take an abacus with e columns (runners) labeled from left to right by 0, 1, …, e − 1 and with the rows labeled from the top by 0, 1, 2, . . .. From now on this will be known as an e-abacus. Now given some beta-set β we can represent it on the abacus by putting a bead on the jth runner and ith row if and only if ei + j ∈ β. Removing/adding an e-hook corresponds to moving a bead up/down a runner one place into an unoccupied position. From this description it is clear that removing e-hooks until there are no more e-hooks to remove gives a well-defined beta-set called the e-core of β. The corresponding partition is called the e-core of the partition corresponding to β. Again this is well-defined, in other words equivalent beta-sets have equivalent cores. If λ is a partition and λ′ its e-core then the e-weight of λ is defined to be \frac{|λ|−|λ′|}{e}. In other words the weight is defined to be the number of e-hooks removed to obtain the e-core.

Given a beta-set β we can construct a beta-set β^{j} by letting i ∈ β_{j} if and only if ei+j ∈ β. These beta-sets are called the e-quotients of β. Then a beta-set is completely defined by the β^{j}s (0 ≤ j ≤ e − 1). The e-quotients are written as [β^{0}, β^{1}, …, β^{e−1}]. If λ is the partition corresponding to β then we say [λ^{0}, λ^{1}, …, λ^{e−1}] are the e-quotients of λ where λ^{i} is the partition corresponding to β^{i}. This is uniquely determined by λ up to a cyclic permutation of the λ^{i}s.

Given a symbol {X, Y } we again have a notion of hook. An e-hook is a pair (x, y) of non-negative integers with x ∈ X but y ∉ X (or X replaced with Y) with x − y = e. Removing this e-hook means replacing x with y in X (or Y). There is also a completely analogous notion of adding hooks.

Now consider the 2e-abacus. For integers i and j with 0 ≤ i and 0 ≤ j ≤ e − 1 put a bead on the ith row and jth runner if and only if ei + j ∈ X and on the ith row and (e + j)th runner if and only if ei + j ∈ Y. This is called the 2e-linear diagram of {X, Y }.

This diagram shows that removing e-hooks until no more can be removed is a welldefined process producing what is called the e-core of {X, Y }. Also equivalent symbols have equivalent e-cores. If when we remove all e-hooks from a symbol we get a degenerate symbol and this process involved removing a positive number of e-hooks we say that the e-core is 2 copies of this degenerate symbol. However, if this process involved removing no e-hooks then we say the e-core is just 1 copy of this degenerate symbol. From now on we will use symbol to mean an equivalence class of symbols.

2.3. Alternative Description. When we are looking at characters of our finite classical groups we will need the following alternative description of partitions and symbols as given in[12, Section 5.2].

Given a partition we display it on a 2-abacus making sure the 0th runner has more beads than the 1th. Say it has s more beads. Then we relabel the partition (s, μ, ν) where [μ, ν] are the 2-quotient partitions with respect to the abacus representation given. Given a symbol {X, Y } with |X| > |Y | we relabel it (s, μ, ν) where s = |X| − |Y |, μ is the partition corresponding to X and ν is the partition corresponding to Y. If we have a symbol {X, Y } with |X| = |Y | we relabel it (0, μ, ν) where μ is the partition corresponding to X and ν is the partition corresponding to Y. Note that in this final case (0, μ, ν) and (0, ν, μ) correspond to the same symbol.

2.4. Littlewood-Richardson Coefficients. Let μ, ν and λ be partitions such that |μ| = m, |ν| = n and |λ| = (n + m). The Littlewood Richardson coefficient g_{μ,ν}^{λ} is described as follows[13, Definition 16.1]: Let μ = (α_{1}, α_{2}, …, α_{s}), ν = (β_{1}, β_{2}, …, β_{t}) and λ = (γ_{1}, γ_{2}, …, γ_{u}). If s > u or α_{i} > γ_{i} for some i with 0 ≤ i ≤ s then g_{μ,ν}^{λ} is zero. Otherwise lie the Young diagram of μ on top of that of λ so that the top left boxes coincide. Now g_{μ,ν}^{λ} is the number of ways the complement of the Young diagram of μ inside that of λ can be filled with positive integers according to the following rules.

(1) There must appear β_{i} i’s.

(2) The integers must be non-decreasing from left to right along rows and strictly increasing down columns.

(3) Reading each row from right to left starting with the top row and continuing downwards gives a sequence of integers such that at no point does the number of (i + 1)s exceed the number of is for all positive integers i.

3. Weyl Groups

For the branching rules of the following Weyl groups we follow G. Hiss and K. Kessar[12, Section 3]. We will have repeated use of the Littlewood-Richardson coefficients 2.4.

A_{n}: The Weyl group of type A_{n}, the symmetric group S_{n+1} on (n + 1) letters, has

presentation:

S_{n+1} = ⟨s_{1}, …, s_{n}|s_{i}^{2} = 1, (s_{i}s_{i+1})^{3} = 1, (s_{i}s_{j})^{2} = 1 |i − j| > 1⟩ The ordinary characters of S_{n} are labeled by partitions of n. If α ⊢ n we use

χ^{α} to denote the corresponding character of S_{n}.

We are concerned with inducing characters from S_{n−k} × S_{k}, generated by:

{s_{1}, …, s_{n−k−1}, s_{n−k+1}, …, s_{n−1}}

to S_{n}.

If α ⊢ (n−k), β ⊢ k and γ ⊢ n then the multiplicity of χ^{γ} in Ind_{S_{n−k}×S_{k}}^{S_{n}}(χ^{α}⊗χ^{β}) is g_{α,β}^{γ}.

B_{n}: The Weyl group of type B_{n}, denoted W_{n}, has presentation:

W_{n} = ⟨s_{1}, …, s_{n}|s_{i}^{2} = 1, (s_{1}s_{2})^{4} = 1, (s_{i}s_{i+1})^{3} = 1 i > 1, (s_{i}s_{j})^{2} = 1 |i − j| > 1⟩

The ordinary characters of W_{n} are labeled by bi-partitions of n, a bi-partition of n is an ordered pair of partitions (α^{0}, α^{1}) such that |α^{0}|+|α^{1}| = n and we write (α^{0}, α^{1}) ⊢ n. If (α^{0}, α^{1}) ⊢ n we use χ^{α} to denote the corresponding character of W_{n}.

We are concerned with inducing characters from W_{n−k} × S_{k}, generated by:

{s_{1}, .., s_{n−k}, s_{n−k+2}, …, s_{n}}

to W_{n}.

If (α^{0}, α^{1}) ⊢ (n − k), γ ⊢ k and (β^{0}, β^{1}) ⊢ n then set j = |β^{0}| − |α^{0}|. The multiplicity of χ^{β} in Ind_{W_{n−k}×S_{k}}^{W_{n}}(χ^{α} ⊗ χ^{γ}) is zero if j < 0 or j > k. Otherwise it is:

∑_{δ^{0}⊢j} ∑_{δ^{1}⊢k−j} g_{α^{0},δ^{0}}^{β^{0}}g_{α^{1},δ^{1}}^{β^{1}}g_{δ^{0},δ^{1}}^{γ}

D_{n}: The Weyl group of type D_{n}, denoted W̃_{n}, has presentation:

W̃_{n} = ⟨s_{1}, …, s_{n}|s_{i}^{2} = 1, (s_{1}s_{2})^{2} = 1, (s_{1}s_{3})^{3} = 1, (s_{i}s_{i+1})^{3} = 1 i > 1, (s_{i}s_{j})^{2} = 1 |i − j| > 1 {i, j} ≠ {1, 3}⟩

Note W̃_{n} can be viewed as a subgroup of W_{n} of index 2 with generators {s_{1}s_{2}s_{1}, s_{2}, …, s_{n}}.

If (α^{0}, α^{1}) ⊢ n with α^{0} ≠ α^{1} then the character χ^{(α^{0},α^{1})} of W_{n} restricts to a single character of W̃_{n} and χ^{(α^{1},α^{0})} restricts to the same character. We denote this character χ^{{α^{0},α^{1}}} and describe it as non-degenerate. However, if α^{0} = α^{1} then χ^{(α^{0},α^{0})} restricts to the sum of 2 distinct characters of W̃_{n}. We denote the 2 characters χ^{{α^{0},α^{0}}} and χ^{′{α^{0},α^{0}}} and describe them as degenerate. Now conjugation by any element g ∈ W_{n}\W̃_{n} induces an automorphism on W̃_{n} that swaps χ^{{α^{0},α^{0}}} and χ^{′{α^{0},α^{0}}}.

We are concerned with inducing characters from W̃_{n−k} × S_{k}, generated by:

{s_{1}, …, s_{n−k}, s_{n−k+2}, …, s_{n}}

to W̃_{n}.

Given the information above we can deduce the multiplicity of an irreducible character of W̃_{n} in that of an irreducible character of W̃_{n−k} × S_{k} induced up to W̃_{n}. We consider three cases. When the characters of W̃_{n−k} and W̃_{n} are both non-degenerate, when the character of W̃_{n−k} is non-degenerate but that of W̃_{n} is degenerate and when the character of W̃_{n−k} is degenerate but that of W̃_{n} is non-degenerate. We consider the following commutative diagram to obtain our results:

W̃_{n−k} × S_{k} ⟶^{Ind} W̃_{n}
↓_{Ind}  ↓_{Ind}
W_{n−k} × S_{k} ⟶^{Ind} W_{n}

(1) Suppose (α^{0}, α^{1}) ⊢ (n−k) with α^{0} ≠ α^{1} and (β^{0}, β^{1}) ⊢ n with β^{0} ≠ β^{1}. Additionally let δ ⊢ k. Then the multiplicity of χ^{{β^{0},β^{1}}} in IndW̃_{n} ; W̃_{n−k}×S_{k}(χ^{{α^{0},α^{1}}}⊗

χ^{δ}) is equal to the multiplicity of χ^{(β^{0},β^{1})} plus the multiplicity of χ^{(β^{1},β^{0})} in Ind_{W_{n−k}×S_{k}}^{W_{n}}(χ^{(α^{0},α^{1})} ⊗ χ^{δ}).

(2) Suppose that (α, β) ⊢ (n − k) with α ≠ β and (α, α) ⊢ n. Additionally let δ ⊢ k. Then the multiplicity of χ^{{α,α}} in IndW̃_{n} ; W̃_{n−k}×S_{k}(χ^{{α,β}} ⊗ χ^{δ}) is equal

to the multiplicity of χ^{(α,α)} in Ind_{W_{n−k}×S_{k}}^{W_{n}}(χ^{(α,β)} ⊗ χ^{δ}). The multiplicity of χ^{′{α,α}} is exactly the same.

(3) Suppose that (α, α) ⊢ (n − k) and (α, β) ⊢ n with α ≠ β. Additionally let δ ⊢ k. Then the multiplicity of χ^{{α,β}} in IndW̃_{n} ; W̃_{n−k}×S_{k}(χ^{{α,α}} ⊗χ^{δ}) is equal to the multiplicity of χ^{(α,β)} in Ind_{W_{n−k}×S_{k}}^{W_{n}}(χ^{(α,α)} ⊗ χ^{δ}). The same statement

is true with χ^{{α,α}} replaced with χ^{′{α,α}}.

4. The Brauer Homomorphism

Let G be a finite group and M an OG-module. The Brauer homomorphism Br_{P}^{G}(M) of M with respect to some p-subgroup P of G is the natural surjection:

Br_{P}^{G} : M^{P} → M^{P} /(∑_{Q<P} T r_{Q}^{P}(M^{Q}) + J M^{P})

If M is indecomposable with trivial source then Br_{P}^{G}(M^{P}) ≠ 0 if and only if M has a vertex containing P.

We consider the specific case where M = OG with the action of G given by conjugation. In this case Br_{P}^{G}(M^{P}) ≅ kC_{G}(P) as kN_{G}(P)-modules with Br_{P}^{G} given by:

Br_{P}^{G}(∑_{g∈G} α_{g}g) = ∑_{g∈C_{G}(P)} \overline{α_{g}}g

Now if H is a subgroup of G containing N_{G}(P) and b and c are block idempotents of OG and OH respectively such that the corresponding blocks both have defect group P, then OHc is described as the Brauer correspondent of OGb in H if Br_{P}^{G}(b) = Br_{P}^{H}(c). When H = N_{G}(P) we describe OHc simply as the Brauer correspondent.

5. The Finite Classical Groups

Let q be a power of a prime. We describe each of the finite classical groups G_{m}(q) in question in two ways. First we view G_{m}(q) as the fixed points of some algebraic group G_{m}(q) under a Frobenius morphism F and secondly as the group of linear maps that preserve or scale some bilinear form over 𝔽_{q}. For both these descriptions we follow G. Hiss and K. Kessar[12, Section 4].

5.1. Forms. Let W be an m-dimensional vector space over 𝔽_{q}. We describe the following bilinear forms on W. We require q to be odd for the symplectic and orthogonal forms only.

Unitary: To define a unitary form we require q to be a square, say q = q_{0}^{2}. We want a non-degenerate sesquilinear form on W. Up to isomorphism there is one such form:

< u, v >= u_{1}v_{m}^{q_{0}} + ⋯ + u_{m}v_{1}^{q_{0}}

Symplectic: We want a non-degenerate anti-symmetric bilinear form on W. In this case m must be even. Say m = 2n. Up to isomorphism there is one such form:

< u, v >= (u_{1}v_{2n} + ⋯ + u_{n}v_{n+1}) − (u_{n+1}v_{n} + ⋯ + u_{2n}v_{1})

Orthogonal: We want a non-degenerate symmetric bilinear form on W. There

are two non-isomorphic such forms:

< u, v >= u_{1}v_{m} + u_{2}v_{m−1} + ⋯ + u_{m−1}v_{2} + u_{m}v_{1}

and

< u, v >= u_{1}v_{1} + δu_{m}v_{m} + (u_{2}v_{m−1} + u_{3}v_{m−2} + ⋯ + u_{m−2}v_{3} + u_{m−1}v_{2})

where −δ is a non-square in 𝔽_{q}.

These two forms are known as type 1 and type -1 respectively.

Throughout this section if W is a vector space over K with some bilinear form g then we introduce the sets[10, Section 1]:

I(W, g) = {x ∈ GL(W)|g(xu, xv) = g(u, v), ∀u, v ∈ W }
I_{0}(W, g) = I(W, g) ∩ SL(W)

When dim(W) is even, say 2n, then we define:

J(W, g) = {x ∈ GL(W)|∃λ_{x} ∈ K, g(xu, xv) = λ_{x}g(u, v), ∀u, v ∈ W }
J_{0}(W, g) = {x ∈ J(W, g)| det(x) = λ_{x}^{n}}

5.2. Description of Groups. First we fix an algebraic closure \overline{𝔽_{q}} of 𝔽_{q} and introduce the automorphism:

GL_{n}(\overline{𝔽_{q}}) → GL_{n}(\overline{𝔽_{q}})
M ↦ M^{[q]}

Where M^{[q]} is obtained from the matrix M by raising all its entries to the power q. In all cases except where stated F will be the above automorphism.

When q is odd we fix a non-square δ ∈ 𝔽_{q}. We will use J_{n} to denote the n × n matrix

with 1s along the antidiagonal and 0s elsewhere, J′_{2n} the matrix ( \begin{matrix}0 & J_{n} \\ −J_{n} & 0 \end{matrix} ) and J′′_{2n}

the matrix ( \begin{matrix}0 & 0 & 0 & J_{n−1} \\ 0 & 1 & 0 & 0 \\ 0 & 0 & δ & 0 \\ J_{n−1} & 0 & 0 & 0 \end{matrix} ).

Let V be an m-dimensional vector space over \overline{𝔽_{q}} with basis \{e_{1}, …, e_{m}\} and V the 𝔽_{q}-vector space given by the 𝔽_{q}-span of {e_{1}, …, e_{m}}. f will be a bilinear form on V but we can also consider it as a bilinear form f on V in a natural way.

We allow q to be even in our description of GL_{n}(q) and U_{n}(q) but assume q is odd for all the other groups. This convention will hold throughout this paper.

GL_{n}(q): Let G = GL_{n}(\overline{𝔽_{q}}). Then G = GL_{n}(q) = G^{F}.

GL_{n}(q) has order q^{\frac{n(n−1)}{2}} ∏_{i=1}^{n}(q^{i} − 1).

U_{n}(q): Here we require q to be a square, say q = q_{0}^{2}. We set G = GL_{n}(\overline{𝔽_{q}}) and set F to be the automorphism:

F : M ↦ J_{n}M^{−t[q_{0}]}J_{n}

Then G = U_{n}(q) = G^{F}. Alternatively if m = n and

f(u, v) = u_{1}v_{n}^{q_{0}} + ⋯ + u_{n}v_{1}^{q_{0}}

then U_{n}(q) = I(V, f) = I(V, f). U_{n}(q) has order q0^{\frac{n(n−1)}{2}} ∏_{i=1}^{n}(q_{0}^{i} − (−1)^{i})[20, Section 2.6].

Sp_{2n}(q), CSp_{2n}(q):

Sp_{2n}(\overline{𝔽_{q}}) = \{x ∈ GL_{2n}(\overline{𝔽_{q}})|x^{t}J′_{2n}x = J′_{2n}\}
CSp_{2n}(\overline{𝔽_{q}}) = \{x ∈ GL_{2n}(\overline{𝔽_{q}})|x^{t}J′_{2n}x = λ_{x}J′_{2n}, λ_{x} ∈ \overline{𝔽_{q}}\}

For G = Sp_{2n}(\overline{𝔽_{q}}), CSp_{2n}(\overline{𝔽_{q}}) we have G = G^{F} = Sp_{2n}(q), CSp_{2n}(q) respectively. Alternatively if m = 2n and

f(u, v) = (u_{1}v_{2n} + ⋯ + u_{n}v_{n+1}) − (u_{n+1}v_{n} + ⋯ + u_{2n}v_{1}) then

Sp_{2n}(q) = I(V, f)  Sp_{2n}(\overline{𝔽_{q}}) = I(V, f)
CSp_{2n}(q) = J(V, f)  CSp_{2n}(\overline{𝔽_{q}}) = J(V, f)

CSp_{2n}(q) has order q^{n^{2}}(q − 1) ∏_{i=1}^{n}(q^{2i} − 1)[20, Section 2.6]. O_{2n+1}(q), SO_{2n+1}(q), CO_{2n+1}(q), CSO_{2n+1}(q):

O_{2n+1}(\overline{𝔽_{q}}) = \{x ∈ GL_{2n+1}(\overline{𝔽_{q}})|x^{t}J_{2n+1}x = J_{2n+1}\}
SO_{2n+1}(\overline{𝔽_{q}}) = O_{2n+1}(\overline{𝔽_{q}}) ∩ SL_{2n+1}(\overline{𝔽_{q}})

For G = O_{2n+1}(\overline{𝔽_{q}}), SO_{2n+1}(\overline{𝔽_{q}}) we have G = G^{F} = O_{2n+1}(q), SO_{2n+1}(q) respectively. Alternatively if m = 2n + 1 and

f(u, v) = u_{1}v_{2n+1} + ⋯ + u_{2n+1}v_{1}

then

O_{2n+1}(q) = I(V, f)  O_{2n+1}(\overline{𝔽_{q}}) = I(V, f)
SO_{2n+1}(q) = I_{0}(V, f)  SO_{2n+1}(\overline{𝔽_{q}}) = I_{0}(V, f)

SO_{2n+1}(q) has order q^{n^{2}} ∏_{i=1}^{n}(q^{2i} − 1)[20, Section 2.6]. We note that when dim(V) is odd although there are two non-isomorphic nondegenerate orthogonal forms on V they produce isomorphic groups.

O_{2n}^{+}(q), SO_{2n}^{+}(q), CO_{2n}^{+}(q), CSO_{2n}^{+}(q):

O_{2n}^{+}(\overline{𝔽_{q}}) = \{x ∈ GL_{2n}(\overline{𝔽_{q}})|x^{t}J_{2n}x = J_{2n}\}
SO_{2n}^{+}(\overline{𝔽_{q}}) = O_{2n}^{+}(\overline{𝔽_{q}}) ∩ SL_{2n}(\overline{𝔽_{q}})

For G = O_{2n}^{+}(\overline{𝔽_{q}}), SO_{2n}^{+}(\overline{𝔽_{q}}) we have G = G^{F} = O_{2n}^{+}(q), SO_{2n}^{+}(q) respectively.

CO_{2n}^{+}(\overline{𝔽_{q}}) = \{x ∈ GL_{2n}(\overline{𝔽_{q}})|x^{t}J_{2n}x = λ_{x}J_{2n}, λ_{x} ∈ \overline{𝔽_{q}}\}
CSO_{2n}^{+}(\overline{𝔽_{q}}) = \{x ∈ CO_{2n}^{+}(\overline{𝔽_{q}})|λ_{x}^{n} = det(x)\}

For G = CO_{2n}^{+}(\overline{𝔽_{q}}), CSO_{2n}^{+}(\overline{𝔽_{q}}) we have G = G^{F} = CO_{2n}^{+}(q), CSO_{2n}^{+}(q) respectively. Alternatively if m = 2n and

f(u, v) = u_{1}v_{2n} + ⋯ + u_{2n}v_{1}

then

O_{2n}^{+}(q) = I(V, f)  O_{2n}^{+}(\overline{𝔽_{q}}) = I(V, f)
SO_{2n}^{+}(q) = I_{0}(V, f)  SO_{2n}^{+}(\overline{𝔽_{q}}) = I_{0}(V, f)
CO_{2n}^{+}(q) = J(V, f)  CO_{2n}^{+}(\overline{𝔽_{q}}) = J(V, f)
CSO_{2n}^{+}(q) = J_{0}(V, f)  CSO_{2n}^{+}(\overline{𝔽_{q}}) = J_{0}(V, f)

SO_{2n}^{+}(q) has order q^{n(n−1)}(q^{n} − 1) ∏_{i=1}^{n−1}(q^{2i} − 1)[20, Section 2.6]. CSO_{2n}^{+}(q) has order q^{n(n−1)}(q^{n} − 1)(q − 1) ∏_{i=1}^{n−1}(q^{2i} − 1)[20, Section 2.6].

O_{2n}^{−}(q), SO_{2n}^{−}(q), CO_{2n}^{−}(q), CSO_{2n}^{−}(q):

O_{2n}^{−}(\overline{𝔽_{q}}) = \{x ∈ GL_{2n}(\overline{𝔽_{q}})|x^{t}J′′_{2n}x = J′′_{2n}\}
SO_{2n}^{−}(\overline{𝔽_{q}}) = O_{2n}^{−}(\overline{𝔽_{q}}) ∩ SL_{2n}(\overline{𝔽_{q}})

For G = O_{2n}^{−}(\overline{𝔽_{q}}), SO_{2n}^{−}(\overline{𝔽_{q}}) we have G = G^{F} = O_{2n}^{−}(q), SO_{2n}^{−}(q) respectively.

CO_{2n}^{−}(\overline{𝔽_{q}}) = \{x ∈ GL_{2n}(\overline{𝔽_{q}})|x^{t}J′′_{2n}x = λ_{x}J′′_{2n}, λ_{x} ∈ \overline{𝔽_{q}}\}
CSO_{2n}^{−}(\overline{𝔽_{q}}) = \{x ∈ CO_{2n}^{−}(\overline{𝔽_{q}})|λ_{x}^{n} = det(x)\}

For G = CO_{2n}^{−}(\overline{𝔽_{q}}), CSO_{2n}^{−}(\overline{𝔽_{q}}) we have G = G^{F} = CO_{2n}^{−}(q), CSO_{2n}^{−}(q) respectively.

Alternatively if m = 2n and

f(u, v) = u_{1}v_{2n} + ⋯ + u_{n}v_{n} + δu_{n+1}v_{n+1} + ⋯ + u_{2n}v_{1}

then

O_{2n}^{−}(q) = I(V, f)  O_{2n}^{−}(\overline{𝔽_{q}}) = I(V, f)
SO_{2n}^{−}(q) = I_{0}(V, f)  SO_{2n}^{−}(\overline{𝔽_{q}}) = I_{0}(V, f)
CO_{2n}^{−}(q) = J(V, f)  CO_{2n}^{−}(\overline{𝔽_{q}}) = J(V, f)
CSO_{2n}^{−}(q) = J_{0}(V, f)  CSO_{2n}^{−}(\overline{𝔽_{q}}) = J_{0}(V, f)

SO_{2n}^{−}(q) has order q^{n(n−1)}(q^{n} + 1) ∏_{i=1}^{n−1}(q^{2i} − 1)[20, Section 2.6]. CSO_{2n}^{−}(q) has order q^{n(n−1)}(q^{n} + 1)(q − 1) ∏_{i=1}^{n−1}(q^{2i} − 1)[20, Section 2.6].

5.3. Clifford Groups. For our description of Clifford groups we follow P. Fong and B. Srinivasan[10, Section 2]. Let V be a vector space over K of finite dimension greater than 1 endowed with a non-degenerate quadratic form Q. Let <, > be the corresponding symmetric bilinear form. The Clifford algebra C(V) is the K-algebra generated by V subject to all the linear relations in V as well as the additional condition:

v^{2} = Q(v)1 for all v ∈ V

C(V) has a ℤ_{2}-grading given by demanding all the non-zero v ∈ V are odd. We use C_{+}(V) to denote the even part. Now define the Clifford group Cl(V) and special Clifford group Cl_{+}(V) as follows:

Cl(V) ={x ∈ C(V)^{×}|xV x^{−1} = V }
Cl_{+}(V) ={x ∈ C_{+}(V)^{×}|xV x^{−1} = V }

Now if x ∈ Cl(V) and v ∈ V then:

Q(xvx^{−1}) = (xvx^{−1})^{2} = xv^{2}x^{−1} = xQ(v)x^{−1} = Q(v)

Therefore conjugation by an element in Cl(V) preserves <, > on V. This gives us the map:

Cl(V) → O(V)

where O(V) is the group of linear maps on V that preserve <, >.

When we restrict to Cl_{+}(V) we get the short exact sequence:

0 ⟶ K^{×} ⟶ Cl_{+}(V) ⟶^{π} SO(V) ⟶ 0

Let q be the power of an odd prime. We use Cl_{2n+1}(q) (respectively Cl_{2n}^{+}(q), Cl_{2n}^{−}(q)) to denote the special Clifford groups defined over 𝔽_{q} with respect to the bilinear form given by the Gram matrix J_{2n+1} (respectively J_{2n}, J′′_{2n}).

Similarly we can define Cl_{2n+1}(\overline{𝔽_{q}}), Cl_{2n}^{+}(\overline{𝔽_{q}}) and Cl_{2n}^{−}(\overline{𝔽_{q}}) over \overline{𝔽_{q}}.

If V is the \overline{𝔽_{q}}-vector space with basis \{e_{1} … e_{m}\} then we have an 𝔽_{q}-vector space automorphism of V given by:

∑ α_{i}e_{i} ↦ ∑ α_{i}^{q}e_{i}

This extends to an automorphism F of Cl_{2n+1}(\overline{𝔽_{q}}) (respectively Cl_{2n}^{+}(\overline{𝔽_{q}}), Cl_{2n}^{+}(\overline{𝔽_{q}})) (where {e_{1} … e_{m}} is the basis with respect to which the Gram matrices are taken). Then we have:

Cl_{2n+1}(q) = Cl_{2n+1}(\overline{𝔽_{q}})^{F}
Cl_{2n}^{+}(q) = Cl_{2n}^{+}(\overline{𝔽_{q}})^{F}
Cl_{2n}^{−}(q) = Cl_{2n}^{−}(\overline{𝔽_{q}})^{F}

5.4. Levi Subgroups. We will now describe certain subgroups of the finite classical and Clifford groups described above as in[12, Section 4.4].

GL_{n}(q): If we have positive integers (n_{1}, n_{2}, …, n_{t}) with ∑_{i} n_{i} = n then GL_{n_{1}}(q)× ⋯ × GL_{n_{t}}(q) embeds in GL_{n}(q) in the natural way:

GL_{n_{1}}(q) × ⋯ × GL_{n_{t}}(q) ↪ GL_{n}(q)

A_{1} × ⋯ × A_{t} ↦ ( \begin{matrix}A_{1} \\ & . & . & . \\ & & & & A_{t} \end{matrix} )

We will use L_{(n_{1},...,n_{t})}(q) to denote this Levi subgroup.

U_{n}(q), Sp_{2n}(q), CSp_{2n}(q), SO_{2n+1}(q), SO_{2n}^{±}(q), CSO_{2n}^{±}(q): Let G_{m}(q) be one of the above groups and (n_{1}, n_{2}, …, n_{t}) positive integers with 2s < m where s = ∑_{i} n_{i}, then GL_{n_{1}}(q) × ⋯ × GL_{n_{t}}(q) × G_{m−2s}(q) embeds in G_{m}(q) via the map:

GL_{n_{1}}(q) × ⋯ × GL_{n_{t}}(q) × G_{m−2s}(q) ↪ G_{m}(q)

A_{1} × ⋯ × A_{t} × B ↦ ( \begin{matrix}A_{1} \\ & . & . & . \\ & & & & A_{t} \\ & & & & & B \\ & & & & & & A′_{t} \\ & & & & & & & . & . & . \\ & & & & & & & & & & A′_{1} \end{matrix} )

where the A′_{i} are chosen to stay in G_{m}(q). So A′_{i} = J_{n_{i}}A_{i}^{−t[q_{0}]}J_{n_{i}} for the unitary group and A′_{i} = λ_{B}J_{n_{i}}A_{i}^{−t[q]}J_{n_{i}} in all other cases. Note that λ_{B} is only needed for the conformal groups and we set λ_{B} = 1 for all the other groups. We will use L_{m,(n_{1},...,n_{t})}(q) to denote this Levi subgroup of G_{m}(q).

Cl_{2n+1}(q), Cl_{2n}^{±}(q): Let Cl_{m}^{(±)}(q) denote one of the above groups and SO_{m}^{(±)}(q) the corresponding special orthogonal group giving us the exact sequence.

0 ⟶ 𝔽_{q} ⟶ Cl_{m}^{(±)}(q) ⟶ SO_{m}^{(±)}(q) ⟶ 0

Then if we have positive integers (n_{1}, n_{2}, …, n_{t}) with 2s < m where ∑_{i} n_{i} = s then we have L_{m,(n_{1},...,n_{t})}(q) ≅ GL_{n_{1}}(q) × ⋯ × GL_{n_{t}}(q) × SO_{m−2s}^{(±)}(q) is a Levi subgroup of SO_{m}^{(±)}(q) as described above. The pre-image of L_{m,(n_{1},...,n_{t})}(q) in Cl_{m}^{(±)}(q) is isomorphic to GL_{n_{1}}(q) × ⋯ × GL_{n_{t}}(q) × Cl_{m−2s}^{(±)}(q) and we also denote it by L_{m,(n_{1},...,n_{t})}(q).

6. Representation Theory of the Finite Classical Groups

6.1. Semisimple elements. Semisimple elements will be important when we describe the ordinary characters of the finite classical groups. By a semisimple element we mean a matrix whose minimal polynomial has no repeated roots. Equivalently they are the matrices that are diagonalisable over some field extension. If Γ is a polynomial let d_{Γ} denote the degree of Γ. Finally we let Δ be the set of monic irreducible polynomials over 𝔽_{q} with non-zero roots. We follow P. Fong and B. Srinivasen in[9, Section 1] and[10, Section 1] for our description of semisimple elements.

GL_{n}(q): A conjugacy class of semisimple elements in GL_{n}(q) is completely defined by their common characteristic polynomial. If Γ ∈ Δ let m_{Γ}(s) be the multiplicity of Γ in the characteristic polynomial of s. So we have the function

Δ → ℕ_{0}
Γ ↦ m_{Γ}(s)

with ∑_{Γ∈Δ} m_{Γ}(s).d_{Γ} = n. Conversely any such function uniquely defines a semisimple conjugacy class in GL_{n}(q).

U_{n}(q): A conjugacy class of semisimple elements in U_{n}(q) is also completely defined by their common characteristic polynomial. However, if s ∈ U_{n}(q) and ω is a root of the characteristic polynomial of s then ω^{−q_{0}} is also a root occurring with the same multiplicity. With that in mind we define the following involution on Δ:

Γ =(X^{m} + a_{m−1}X^{m−1} + ⋯ + a_{1}X + a_{0}) ↦
\overline{Γ} =a_{0}^{−q_{0}}(a_{0}^{q_{0}}X^{m} + a_{1}^{q_{0}}X^{m−1} + ⋯ + a_{m−1}^{q_{0}}X + 1)

So \overline{Γ} is the unique monic polynomial with roots those of Γ raised to the power −q_{0}.

Λ_{1} =\{Γ|Γ ∈ Δ, \overline{Γ} = Γ\}
Λ_{2} =\{Γ\overline{Γ}|Γ ∈ Δ, \overline{Γ} ≠ Γ\}
Λ =Λ_{1} ∪ Λ_{2}

If Γ ∈ Λ let m_{Γ}(s) be the multiplicity of Γ in the characteristic polynomial of s. So we have the function

Λ → ℕ_{0}
Γ ↦ m_{Γ}(s)

with ∑_{Γ∈Λ} m_{Γ}(s).d_{Γ} = n. Conversely any such function uniquely defines a semisimple conjugacy class in U_{n}(q).

Sp_{2n}(q): A conjugacy class of semisimple elements in Sp_{2n}(q) is also completely defined by their common characteristic polynomial. However, if s ∈ Sp_{2n}(q) and ω is a root of the characteristic polynomial of s then ω^{−1} is also a root occurring with the same multiplicity. With that in mind we define the following involution on Δ:

Γ =(X^{m} + a_{m−1}X^{m−1} + ⋯ + a_{1}X + a_{0}) ↦
Γ̃ =a_{0}^{−1}(a_{0}X^{m} + a_{1}X^{m−1} + ⋯ + a_{m−1}X + 1)

So Γ̃ is the unique monic polynomial with roots the inverses of those of Γ. Next we define the sets:

Φ_{0} ={X − 1, X + 1}
Φ_{1} ={Γ|Γ ∈ Δ\Φ_{0}, Γ̃ = Γ}
Φ_{2} ={ΓΓ̃|Γ ∈ Δ, Γ̃ ≠ Γ}
Φ =Φ_{0} ∪ Φ_{1} ∪ Φ_{2}

If Γ ∈ Φ let m_{Γ}(s) be the multiplicity of Γ in the characteristic polynomial of s. So we have the function

Φ → ℕ_{0}
Γ ↦ m_{Γ}(s)

with m_{X+1}(s) even and ∑_{Γ∈Φ} m_{Γ}(s).d_{Γ} = 2n. Conversely any such function uniquely defines a semisimple conjugacy class in Sp_{2n}(q).

SO_{m}^{(±)}(q): Let SO_{m}^{(±)}(q) be any one of our special orthogonal groups. Once again if s ∈ SO_{m}^{(±)}(q) and ω is a root of the characteristic polynomial of s then ω^{−1} is also a root occurring with the same multiplicity. So again we have the function:

Φ → ℕ_{0}
Γ ↦ m_{Γ}(s)

with ∑_{Γ∈Φ} m_{Γ}(s).d_{Γ} = m and m_{X+1}(s) even. This does not uniquely determine a conjugacy class however. If both m_{X−1}(s) and m_{X+1}(s) are non-zero this function defines 2 conjugacy classes, otherwise we just get 1 conjugacy class. Later on the semisimple elements we consider will all have m_{X+1}(s) = 0.

Cl_{m}^{(±)}(q): [10, Section 2] We have the surjection described in 5.3:

π : Cl_{m}^{(±)}(q) ↠ SO_{m}^{(±)}(q)

An element of Cl_{m}^{(±)}(q) is described as semisimple if its image in SO_{m}^{(±)}(q) is semisimple. Let π(t) = s ∈ SO_{m}^{(±)}(q) be semisimple and C its conjugacy class in SO_{m}^{(±)}(q). If m_{X−1}(s) and m_{X+1}(s) are both non-zero then π^{−1}(C) is the union of \frac{q−1}{2} conjugacy classes in Cl_{m}^{(±)}(q) and t is conjugate to −t. Otherwise π^{−1}(C) is the union of q − 1 conjugacy classes in Cl_{m}^{(±)}(q) and no 2 distinct pre-images of s are conjugate. From now on we will use m_{Γ}(t) to denote m_{Γ}(s).

For any semisimple element s we have a corresponding decomposition of V = 𝔽_{q}^{m}. For each Γ ∈ Φ (or Δ for GL_{n}(q) or Λ for U_{n}(q)) we define V_{Γ} to be the null space of Γ(s) (or Γ(π(s)) in the case of the special Clifford groups).

V = ⊕_{Γ}V_{Γ}

is then an orthogonal decomposition of V.

6.2. Characters of Finite Classical groups. We will describe labels for the ordinary irreducible characters of some of the finite classical groups. We follow P. Fong and B. Srinivasan in[9, Section 1] and[10, Section 2] for the description of the characters. We leave out Sp_{2n}(q) and SO^{±}(q), the appropriate characters for these groups will be described in section 8.4. First of all we will describe a subset of all the ordinary characters called the unipotent characters.

The unipotent characters of GL_{n}(q) and U_{n}(q) are both labeled by partitions of n (for example (n) corresponds to the trivial character of both groups).

The unipotent characters of CSp_{2n}(q) and SO_{2n+1}(q) are both labeled by symbols with rank n and odd defect.

The unipotent characters of CSO_{2n}^{+}(q) are labeled by symbols with rank n and defect ≡ 0 (mod 4) with the added rule that degenerate symbols label 2 characters.

The unipotent characters of CSO_{2n}^{−}(q) are labeled by symbols with rank n and defect ≡ 2 (mod 4).

Consider the following table:

\begin{matrix}G & G^{∗} \\ GL_{n}(q) & GL_{n}(q) \\ U_{n}(q) & U_{n}(q) \\ SO_{2n+1}(q) & Sp_{2n}(q) \\ CSp_{2n}(q) & Cl_{2n+1}(q) \\ CSO_{2n}^{+}(q) & Cl_{2n}^{+}(q) \\ CSO_{2n}^{−}(q) & Cl_{2n}^{−}(q) \end{matrix}

Let s ∈ G^{∗} be semisimple with corresponding decomposition V = ⊕_{Γ}V_{Γ}. We define Ψ_{Γ}(s) as follows: If G^{∗} is a general linear or unitary group then Ψ_{Γ}(s) = {partitions of m_{Γ}(s)}. Now we assume that G^{∗} is not a general linear or unitary group. For Γ ∈ Φ_{0} let Ψ_{Γ}(s) = \{symbols of rank ⌊\frac{m_{Γ}(s)}{2}⌋\} subject to the following conditions:

(1) If the form induced on V_{Γ} is symplectic or orthogonal of odd dimension, then the symbols have odd defect.

(2) If the form induced on V_{Γ} is orthogonal of even dimension and type 1, then the symbols have defect ≡ 0 (mod 4). Moreover, degenerate symbols are counted twice. If λ is such a degenerate symbol then we say λ, λ′ ∈ Ψ_{Γ}(s).

(3) If the form induced on V_{Γ} is orthogonal of even dimension and type -1, then the symbols have ≡ 2 (mod 4).

For Γ ∈ Φ_{1} ∪ Φ_{2} let Ψ_{Γ}(s) = {partitions of m_{Γ}(s)}. Now we set Ψ(s) = ∏_{Γ} Ψ_{Γ}(s).

The characters of G are labeled by a semisimple element s of G^{∗} together with a λ ∈ Ψ(s).

We denote this character χ_{s,λ}. χ_{s,λ} and χ_{t,μ} represent the same character if and only if s is conjugate to t in G^{∗} and λ = μ.

Now if s does not have −1 as an eigenvalue then we can label χ_{s,λ} by a unipotent character of C_{G^{∗}}(s)^{∗}[9, Section 1], [10, Section 4]. We will now describe C_{G^{∗}}(s) for all relevant G to make this labeling clear.

GL_{n}(q): Let G^{∗} = GL_{n}(q) and let s ∈ G^{∗} be semisimple. Then:

C_{G^{∗}}(s) ≅ ∏_{Γ∈Δ} GL_{m_{Γ}(s)}(q^{d_{Γ}})

U_{n}(q): Let G^{∗} = U_{n}(q) and let s ∈ G^{∗} be semisimple. Then:

C_{G^{∗}}(s) ≅ ( ∏_{Γ∈Φ_{1}} U_{m_{Γ}(s)}(q^{d_{Γ}})) × ( ∏_{Γ∈Φ_{2}} GL_{m_{Γ}(s)}(q^{\frac{d_{Γ}}{2}} ))

Sp_{2n}(q), Cl_{2n+1}(q), Cl_{2n}^{±}(q): Let G^{∗} = G_{m}(q) be one of the above groups. Let s ∈ G^{∗} be semisimple.

C_{G^{∗}}(s) ≅ G_{m_{X−1}(s)}(q) × ( ∏_{Γ∈Φ_{1}} U_{m_{Γ}(s)}(q^{d_{Γ}})) × ( ∏_{Γ∈Φ_{2}} GL_{m_{Γ}(s)}(q^{d_{Γ}}))

7. Harish-Chandra Induction

7.1. Harish-Chandra Induction. Let p be an odd prime not dividing q and (K, O, k) a p-modular system as introduced in the introduction 1. Let G = G_{m}(q) be GL_{n}(q), U_{n}(q), Sp_{2n}(q), CSp_{2n}(q), SO_{2n+1}(q), SO_{2n}^{+}(q), SO_{2n}^{−}(q), CSO_{2n}^{+}(q) or CSO_{2n}^{−}(q) and L = L_{m,(n_{1},...,n_{t})}(q) (or L = L_{(n_{1},...,n_{t})}(q) in the case of the general linear group) as described in section 5.4. We describe a functor HCInd_{L}^{G} from OL-mod to OG-mod[8, Example 4.6(iii)].

We set U to be the subgroup of G consisting of matrices of the form:

( \begin{matrix}I_{n_{1}} & … & ∗ \\ & ... & ⋮ \\ & & I_{n_{t}} \end{matrix} )

for the general linear group and

( \begin{matrix}I_{n_{1}} & … & ∗ & ∗ & ∗ & … & ∗ \\ & ... & ⋮ & ⋮ & ⋮ & ... & ⋮ \\ & & I_{n_{t}} & ∗ & ∗ & … & ∗ \\ & & & I_{m−2s} & ∗ & … & ∗ \\ & & & & I_{n_{t}} & … & ∗ \\ & & & & & ... & ⋮ \\ & & & & & & I_{n_{1}} \end{matrix} )

otherwise.

Now in all cases U has p′ order. In other words |U| is invertible in O. Let U^{+} be the idempotent \frac{1}{|U|} ∑_{u∈U} u in OG. L normalises U and so commutes with U^{+} and so OGU^{+} is an (OG,OL)-bimodule. Now set HCInd_{L}^{G} to be the functor OGU^{+} ⊗_{OL} −.

If b is a central idempotent of OG and c a central idempotent of OL. We will want the functor OGbU^{+} ⊗_{OLc} − from OLc-mod to OGb-mod. We denote this functor HCInd_{L,c}^{G,b}.

7.2. Characters under Harish-Chandra Induction. Harish-Chandra induction is a functor from OL-mod to OG-mod. However, we want to know what happens at the level of characters viewing Harish-Chandra induction as a functor from KL-mod to KGmod. Let R_{L}^{G} denote Harish-Chandra induction on characters. Again if b is a central idempotent of OG and c a central idempotent of OL then we denote by:

R_{L,c}^{G,b} : Irr(OLc) → Irr(OGb)

the corresponding function on characters.

In this section we only consider L = L_{m,(k)}(q) (or L = L_{(k,n−k)}(q) for the general linear group). The reason for this is that Harish-Chandra induction is transitive. This means that if L has more general linear factors then the functor R_{L}^{G} can be calculated iteratively.

7.2.1. Unipotent Characters. First we will first look at the effect of Harish-Chandra induction on unipotent characters for the groups GL_{n}(q), U_{n}(q), CSp_{2n}(q), SO_{2n+1}(q), CSO_{2n}^{±}(q)[12, Section 5.3].

GL_{n}(q): If G = GL_{n}(q) and L = L_{(k,n−k)} ≅ GL_{k}(q) × GL_{n−k}(q) then the multiplicity of χ_{1,γ} in R_{L}^{G}(χ_{1,α} ⊗ χ_{1,β}) is g_{α,β}^{γ} (see 2.4).

U_{n}(q), CSp_{2n}(q), SO_{2n+1}(q), CSO_{2n}^{−}(q): Let G = G_{m}(q) be one of the above groups and L = L_{m,(k)}(q) ≅ GL_{k}(q) × G_{m−2k}(q). We use the alternative description of partitions and symbols described in section 2.3.

In all four groups the multiplicity of χ_{1,(s,μ,ν)} in R_{L}^{G}(χ_{1,γ}⊗χ_{1,(t,α,β)}) is 0 unless s = t and in this case it is equal to the multiplicity of χ^{(μ,ν)} in Ind_{S_{k}×W_{v−k}}^{W_{v}}(χ^{γ}⊗χ^{(α,β)}) for an appropriate v (see 3).

CSO_{2n}^{+}(q): For CSO_{2n}^{+}(q) we again use the description in 2.3. We then do exactly the same calculation as above unless s = 0. In this case the calculation is then carried out in a Weyl group of type D_{m} instead of B_{m}. In other words the multiplicity of χ_{1,(0,μ,ν)} in R_{L}^{G}(χ_{1,γ} ⊗ χ_{1,(0,α,β)}) is equal to the multiplicity of

χ^{{μ,ν}} in IndW̃_{v} ; S_{k}×W̃_{v−k}(χ^{γ} ⊗ χ^{{α,β}}) for an appropriate v. Note that χ_{1,(0,μ,μ)},

χ′_{1,(0,μ,μ)} correspond to χ^{{μ,μ}}, χ^{′{μ,μ}} respectively.

7.2.2. General Case. Now we drop the assumption that the characters are unipotent. However, we keep the assumption that our semisimple labels don’t have −1 as an eigenvalue. Let s ∈ L^{∗} be semisimple. We can then use 6.2 to calculate R_{L}^{G}(χ_{s,λ}) by passing to Harish-Chandra induction for unipotent characters from C_{L^{∗}}(s)^{∗} to C_{G^{∗}}(s)^{∗}[12, Section 5.4].

GL_{n}(q): Let G = GL_{n}(q) and L = L_{(k,n−k)} ≅ GL_{k}(q)×GL_{n−k}(q). If χ_{s_{1},λ_{1}} ⊗χ_{s_{2},λ_{2}} is a character of L then

C_{L^{∗}}(s_{1} × s_{2}) ≅ ∏_{Γ∈Δ}(GL_{m_{Γ}(s_{1})}(q^{d_{Γ}}) × GL_{m_{Γ}(s_{2})}(q^{d_{Γ}}))

So for each Γ ∈ Δ we do Harish-Chandra induction to get a sum of unipotent character of

GL_{m_{Γ}(s_{1})+m_{Γ}(s_{2})}(q^{d_{Γ}})

We now have a sum of unipotent characters for C_{G^{∗}}(s_{1} × s_{2})^{∗} and hence a character of G. This character is R_{L}^{G}(χ_{s_{1},λ_{1}} ⊗ χ_{s_{2},λ_{2}}).

U_{n}(q): Let G = U_{n}(q) and L = L_{n,(k)} ≅ GL_{k}(q) × U_{n−2k}(q). If χ_{s_{1},λ_{1}} ⊗ χ_{s_{2},λ_{2}} is a character of L then

C_{L^{∗}}(s_{1} × s_{2}) ≅ ∏_{Γ∈Λ_{1}}(GL_{m_{Γ}(s_{1})}(q^{d_{Γ}}) × U_{m_{Γ}(s_{2})}(q^{d_{Γ}}))×
∏_{Γ\overline{Γ}∈Λ_{2}}(GL_{m_{Γ}(s_{1})}(q^{d_{Γ}}) × GL_{m\overline{Γ}(s_{1})}(q^{d_{Γ}}) × GL_{m_{Γ\overline{Γ}}(s_{2})}(q^{d_{Γ}}))

Then for each Γ ∈ Λ_{1} we do Harish-Chandra induction to get a sum of unipotent characters of

U_{2m_{Γ}(s_{1})+m_{Γ}(s_{2})}(q^{d_{Γ}})

and for each Γ ∈ Λ_{2} we do Harish-Chandra induction twice to get a sum of unipotent characters of

GL_{m_{Γ}(s_{1})+m\overline{Γ}(s_{1})+m_{Γ\overline{Γ}}(s_{2})}(q^{d_{Γ}})

We now have a sum of unipotent characters of C_{G^{∗}}(s_{1} × s_{2})^{∗} and hence a character of G. This character is R_{L}^{G}(χ_{s_{1},λ_{1}} ⊗ χ_{s_{2},λ_{2}}).

SO_{2n+1}(q), CSp_{2n}(q), CSO_{2n}^{+}(q), CSO_{2n}^{−}(q): Let G = G_{m}(q) be SO_{2n+1}(q), CSp_{2n}(q), CSO_{2n}^{+}(q) or CSO_{2n}^{−}(q) and L = L_{m,(k)} ≅ GL_{k}(q) × G_{m−2k}(q). If χ_{s_{1},λ_{1}} ⊗ χ_{s_{2},λ_{2}} is a character of L then

C_{L^{∗}}(s_{1} × s_{2}) ≅ (GL_{m_{X−1}(s_{1})}(q) × G_{m_{X−1}(s_{2})}^{∗}(q))×
∏_{Γ∈Ψ_{1}}(GL_{m_{Γ}(s_{1})}(q^{d_{Γ}}) × U_{m_{Γ}(s_{1})}(q^{d_{Γ}}))
∏_{ΓΓ̃∈Ψ_{2}}(GL_{m_{Γ}(s_{1})}(q^{d_{Γ}}) × GL_{mΓ̃(s_{1})}(q^{d_{Γ}}) × GL_{m_{ΓΓ̃}(s_{2})}(q^{d_{Γ}}))

Once again we do Harish-Chandra induction on unipotent characters to obtain a sum of unipotent characters of C_{G^{∗}}(s_{1} × s_{2})^{∗} and hence a character of G. This character is R_{L}^{G}(χ_{s_{1},λ_{1}} ⊗ χ_{s_{2},λ_{2}}).

7.3. Characters with Equal Dimension. We can use Harish-Chandra induction to show that some pairs of characters have the same dimension. This will be useful later when we will be performing some calculations.

Lemma 7.3.1. The following pairs of characters have the same dimensions. (1) χ_{s,λ} and χ_{s^{−q_{0}},λ^{−q_{0}}} of GL_{k}(q) where q = q_{0}^{2} and λ_{Γ}^{−q_{0}} = λ\overline{Γ} (see 6.1).

(2) χ_{s,λ} and χ_{s^{−1},λ^{−1}} of GL_{k}(q) where λ_{Γ}^{−1} = λΓ̃ (see section 6.1).

(3) χ_{s,λ} and χ_{s,λ′} of CSO_{2n}^{+}(q) where s ∈ 𝔽_{q} and λ is a degenerate symbol.

Proof. We prove all three results by considering Harish-Chandra induction of a pair of characters from L to G that give the same character and noting that the dimension is always multiplied by [G : LU] (see 7).

(1) We set G = U_{n+2k}(q) and L = GL_{k}(q) × U_{n}(q). We Harish-Chandra induce χ_{s,λ} ⊗ χ and χ_{s^{−q_{0}},λ^{−q_{0}}} ⊗ χ where χ is any unipotent character of U_{(}q).

(2) We set G = CSp_{2(n+k)}(q) and L = GL_{k}(q) × CSp_{2n}(q). We Harish-Chandra induce χ_{s,λ} ⊗ χ and χ_{s^{−1},λ^{−1}} ⊗ χ where χ is any unipotent character of Sp_{2n}(q).

(3) We set G = CSO_{2(n+1)}^{+}(q) and L = GL_{1}(q) × CSO_{2n}^{+}(q). We Harish-Chandra

induce χ_{1,(1)} ⊗ χ_{s,λ} and χ_{1,(1)} ⊗ χ_{s,λ′}.

□

8. Unipotent Blocks of Finite Classical groups

We continue with the assumption that p is an odd prime not dividing q. We will describe a subset of the p-blocks of our finite classical groups called the unipotent blocks. First we need the notion of linear and unitary prime[12, Section 6.1]. Let G = GL_{n}(q), U_{n}(q), Sp_{2n}(q), CSp_{2n}(q), SO_{2n+1}(q), SO_{2n}^{±}(q) or CSO_{2n}^{±}(q).

Let d be the multiplicative order of q mod p and p^{a} the maximum power of p dividing q^{d} − 1. If G = GL_{n}(q) let e = d. If G = U_{n}(q) let e be the multiplicative order of −q_{0} mod p. In this case we have:

• d = e if e is odd

• d = \frac{1}{2}e if e is even

If G is any of the other seven groups define e to be the multiplicative order of q^{2} (mod p). In these cases we have:

• e = d if d is odd

• e = \frac{1}{2}d if d is even When G is not a general linear group we have the notion of p being a linear or unitary

prime. When we work with the unitary group p is unitary if e = d and linear otherwise. For all the other groups p is linear if e = d and unitary otherwise. From now on we will assume p is linear with respect to q.

8.1. Blocks. Let us now restrict our attention to G = GL_{n}(q), U_{n}(q), CSp_{2n}(q), SO_{2n+1}(q) or CSO_{2n}^{±}(q). We follow[12, Section 6.2]. We will describe the unipotent blocks of Sp_{2n}(q) and SO_{2n}^{±}(q) at the end of this section.

First we introduce the notion of a ϕ_{d}-torus where ϕ_{d} is the dth cyclotomic polynomial. A ϕ_{d}-torus of an algebraic group G is an F stable torus T whose polynomial order is (ϕ_{d})^{t} for some t. Now a d-split Levi subgroup K of G is a subgroup of the form C_{G}(T) for such a T. A character of K^{F} is described as d-cuspidal if it does not appear as a summand of a character Harish-Chandra induced up from any K′^{F} where K′ < K is a d-split Levi subgroup of K.

The unipotent blocks of G^{F} are labeled by conjugacy classes (in G) of pairs (K, ψ) where K is a d-split Levi subgroup of G and ψ is a d-cuspidal character of K^{F}.

Now we take G = G_{m}(q) = G^{F} to be one of our six groups and p a linear prime with respect to q where appropriate. A typical unipotent block of G is labeled by (K, ψ) where K^{F} ≅ GL_{1}(q^{d})^{t} × G_{m′}(q) and ψ = (1_{GL_{1}(q^{d})})^{t} ⊗ χ where m′ = m − dt when G is the general linear group and m − 2dt otherwise and χ is a unipotent character of G_{m′}(q) whose label is an e-core.

An alternative way to describe the p-blocks, where p is a linear prime with respect to q, is as follows. This description can be found in more detail in[9, Theorem D] and [10, Theorem 10B, Theorem 11E]. First take a unipotent character of G. Associated to this unipotent character we have either a partition or symbol of which we take the e-core. Our unipotent block is then just labeled by this e-core μ.

We can pair our two descriptions up by letting μ in the second description be the label of χ in the first. We call this block B_{μ}.

Note that for G = CSO_{2n}^{+}(q), χ labeled by a degenerate symbol and t > 0 then μ would be both copies of this degenerate symbol. However, the two corresponding characters of G = CSO_{2(n−dt)}^{+}(q) are conjugate in G and hence both labels label the same block of G.

8.2. Defect Groups and Dual Defect Groups. Consider the block of G labeled by (K, ψ). Any Sylow p-subgroup of C_{G}^{0}([K, K])^{F} is a defect group for the block[12, Section 6.2]. P. Fong and B. Srinivasan describe the concept of dual defect groups. If D ≤ G is a defect group of (K, ψ) then D^{∗} is naturally a subgroup of G^{∗}. For GL_{n}(q) and U_{n}(q) we identify G with G^{∗} and D with D^{∗}. For the other four groups see[10, Section 12, Section 13]. D^{∗} is then described as a dual defect group for (K, ψ).

8.3. Characters in Unipotent Blocks. We now describe the characters in a unipotent block[9, Section 7], [10, Section 12, Section 13]. If Γ ∈ Φ (or Δ for GL_{n}(q) or Λ for U_{n}(q)) then we set e_{Γ} to be the additive of d_{Γ} (mod d). Let G_{m}(q) be one of our groups and let B_{μ} be a unipotent block of G_{m}(q), χ_{t,λ} an irreducible character of G_{m}(q) and fix a dual defect group D^{∗} of B_{μ}. Then χ_{t,λ} lies in B_{μ} if and only if:

(1) t is conjugate to x for some x ∈ D^{∗}

(2) The e-core of λ_{X−1} is a subset of μ

(3) The e_{Γ}-core of λ_{Γ} is empty for all other Γ

8.4. Unipotent Blocks of Sp_{2n}(q) and SO_{2n}^{±}(q). Let G = Sp_{2n}(q) (respectively SO_{2n}^{±}(q)) and Ĝ = CSp_{2n}(q) (respectively CSO_{2n}^{±}(q)). We now deal with the unipotent blocks of OG via the following lemma:

Lemma 8.4.1. The unipotent blocks of OG are in one-to-one correspondence with those of OĜ. Furthermore if i and j are corresponding block idempotents then we have the following isomorphism of O-algebras:

OZ(Ĝ)_{p} ⊗_{O} OGi ≅ OĜj

The isomorphism is given by multiplication by j and the correspondence of characters from OĜj to OZ(Ĝ)_{p} ⊗_{O} OGi is given by restriction. Also if D is a defect group of OGi then Z(Ĝ)_{p} × D is a defect group of OĜj.

Before we prove the above we mention that we will label the block corresponding to i with the same label (symbol) as the block corresponding to j.

Proof. Apply[4, Theorem 12] to G = Sp_{2n}(\overline{𝔽_{q}}) and CSp_{2n}(\overline{𝔽_{q}}) (respectively SO_{2n}^{±}(\overline{𝔽_{q}}) and CSO_{2n}^{±}(\overline{𝔽_{q}})). The lemma then follows from chasing the appropriate character correspondences and then applying[4, Proposition 6]. □

9. Main Theorem

Let G = G_{m}(q) be one of the following groups:

(1) U_{n}(q)

(2) (a) Sp_{2n}(q)

(b) CSp_{2n}(q)

(3) SO_{2n+1}(q)

(4) (a) SO_{2n}^{±}(q)

(b) CSO_{2n}^{±}(q) We continue with our assumption that q is odd in all except case (1) where it can be

even or odd. As in previous sections let p an odd prime not dividing q. We also assume that p is a linear prime with respect to q. d and e will have the meaning given in 8.

For each w (0 ≤ w < p) we wish to find an integer m = m(w) and a unipotent block B_{ρ} of G_{m}(q) such that theorem 1.0.2 holds. We now describe conditions for m and ρ to satisfy in each of the four cases.

1 ρ is an e-core partition with a representation on a 2d-abacus such that the ith runner has at least w−1 fewer beads than the (i+2)th runner for (0 ≤ i ≤ 2d−3). If r is the rank of ρ then m = r + 2dw.

2,3,4 There exists a 2d-linear diagram of ρ such that the ith runner has at least w − 1 fewer beads than the (i + 1)th runner for (0 ≤ i ≤ d − 2) and (d ≤ i ≤ 2d − 2). ρ has non-zero rank r and m = 2(r + dw) + 1 in case 3 and 2(r + dw) in cases 2 and 4.

We now fix m and ρ along with a 2d-abacus representation of ρ such that the above property holds and every position on the first w rows is occupied with a bead.

We need to set up some notation that is required for us to state and prove the main theorem.

We set G̃ = G̃_{m}(q) to be O_{2n+1}(q) (respectively O_{2n}^{±}(q), CO_{2n}^{±}(q)) when G = SO_{2n+1}(q) (respectively SO_{2n}^{±}(q), CSO_{2n}^{±}(q)) and G otherwise. Let T_{i} be the subgroup of G̃ generated by

( \begin{matrix}I_{d(i−1)} \\ & & & I_{d} \\ & & I_{m−2di} \\ & −I_{d} \\ & & & & I_{d(i−1)} \end{matrix} )

in case 2 and by

( \begin{matrix}I_{d(i−1)} \\ & & & I_{d} \\ & & I_{m−2di} \\ & I_{d} \\ & & & & I_{d(i−1)} \end{matrix} )

in cases 1,3 and 4.

Let G = G_{0} > G_{1} > ⋯ > G_{w} = L be a sequence of Levi subgroups of G where G_{i} = L_{m,(d^{i})}(q) ≅ GL_{d}(q)_{1} × ⋯ × GL_{d}(q)_{i} × G_{m−2di}(q) as described in 5.4.

We denote by H the subgroup (GL_{d}(q)_{1}.T_{1})×⋯×(GL_{d}(q)_{w}.T_{w})×G̃_{m−2dw}(q) of G̃_{m}(q). Note that if we adopt our notation from the introduction we have H is naturally ismorphic to (GL_{d}(q).2)^{w} × G̃_{m−2dw}(q).

Now consider the Levi subgroup L_{m,(dw)}(q) ≅ GL_{dw}(q) × G_{m−2dw}(q) ≥ L of G. Let S be the subgroup of permutation matrices of GL_{dw}(q) ≤ L_{m,(dw)}(q) whose conjugation action permutes the GL_{d}(q)_{i}s. Clearly S normalises H and intersects it trivially so H.S ≅ ((GL_{d}(q).2) ≀ S_{w}) × G̃_{m−2dw}(q). We set N = H.S ∩ G and have in all cases that |N| = 2^{w}w!|L|.

For (0 < i ≤ w) let a_{i} be the principal block idempotent of OGL_{d}(q)_{i} and for (0 ≤ i ≤ w) let f_{w−i} be the unipotent block idempotent of OG_{m−2di}(q) associated with the partition ρ. We set b_{i} to be the block idempotent a_{1} ⊗ ⋯ ⊗ a_{i} ⊗ f_{w−i} of OG_{i}. We set b = b_{0} and f = b_{w}. When ρ is degenerate in case 4 we denote by f_{0} and f′_{0} the 2 blocks of G_{m−2dw}(q) labeled by ρ and by f = b_{w} and f′ = b′_{w} the block idempotents a_{1} ⊗ ⋯ ⊗ a_{w} ⊗ f_{0} and a_{1} ⊗ ⋯ ⊗ a_{w} ⊗ f′_{0} of OL respectively.

Fix a Sylow p-subgroup R of GL_{d}(q) (note that |R| = p^{a} see 8 for the meaning of a) and let P_{1}×⋯×P_{w} be w copies of R, one in each GL_{d}(q)_{i} of GL_{d}(q)_{1}×⋯×GL_{d}(q)_{w}. We also set Z to be the subgroup of G consisting of scalar matrices and P = Z_{p}.(P_{1} × ⋯ × P_{w}). P is a defect group for OGb (see 8.2) and N_{G}(P) ≤ N.

Additionally for (1 ≤ i ≤ w) let U_{i} be the subgroup of matrices of G_{i−1} of the form:

( \begin{matrix}I_{d(i−1)} \\ & I_{d} & ∗ & ∗ \\ & & I_{m−di} & ∗ \\ & & & I_{d} \\ & & & & I_{d(i−1)} \end{matrix} )

And set U_{i}^{+} = \frac{1}{|U_{i}|} ∑_{u∈U_{i}} u.

9.1. Non-Degenerate Case. In this subsection we assume that ρ is non-degenerate. We restrict our attention to cases 1, 2(b), 3 and 4(b). We will later prove the corresponding theorem for cases 2(a) and 4(a) using 8.4.1.

Theorem 9.1.1. ONf is a block of ONf and is Morita equivalent to OGb.

For the the proof of this theorem, which will fill this section, we follow W. Turner[19, Section 2]. We will need a number of lemmas first.

Lemma 9.1.2.

(1) P is defect group for OG_{i}b_{i} for (0 ≤ i ≤ w).

(2) Br_{P}^{G}(b_{i}) = a′_{1} ⊗ ⋯ ⊗ a′_{w} ⊗ f_{0} where a′_{i} is the principal block idempotent of C_{GL_{d}(q)_{i}}(P_{i}) (0 ≤ i ≤ w). Also Br_{P}^{G}(U_{i}^{+}) = 1.

(3) N stabilizes f and as an O(N × L)-module, ONf is indecomposable with vertex Δ(P). In particular, ONf is a block of N.

(4) OGb and ONf both have defect group P and are Brauer correspondents.

Proof.

(1) P_{i} is defect group for OGL_{d}(q)_{i}a_{i} and (P_{i+1} × ⋯ × P_{w}).Z_{p} is a defect group for OG_{m−2di}(q)f_{w−i} (see 8.2).

(2) C_{G}(P) < G_{i} so Br_{P}^{G}(b_{i}) = Br_{P}^{G_{i}}(b_{i})

Br_{P}^{G_{i}}(b_{i}) = Br_{P_{1}}^{GL_{d}(q)_{1}}(a_{1})⊗⋯⊗Br_{P^{i}}^{GL_{d}(q)_{i}}(a_{i})⊗Br_{(P_{i+1}×···×P_{w}).Z_{p}}^{GL_{d}(q)_{i+1}×···×GL_{d}(q)_{w}×G_{m−2dw}(q)}(f_{w−i}) and from[3, Theorem 3.2] we see that we must get a block of the form: a′_{1} ⊗ ⋯ ⊗ a′_{w} ⊗ ε where ε is a sum of unipotent block idempotents of G_{m−2dw}(q) of defect zero. Secondly[5, Lemma 4.5] tells us that a′_{1} ⊗ ⋯ ⊗ a′_{w} ⊗ f_{0} must appear as a constituent. Finally we see that no other block idempotent of C_{G}(P) can appear as otherwise we would have two distinct block idempotents α and β of G_{i} with defect group P and Br_{P}^{G_{i}}(α)Br_{P}^{G_{i}}(β) ≠ 0. This is of course a contradiction as Br_{P}^{G_{i}} is an algebra homomorphism. The second part is clear.

(3) N clearly stabilizes f in all cases except case 4(b). The only thing to check in this case is that CO_{m−2dw}^{±}(q) stabilizes f_{0}. This is clear however, by looking at part (2) and noting that conjugation by N_{G}(P) commutes with Br_{P}^{G}. By part(1), OLf has vertex Δ(P). Since C_{G}(P) ≤ L, the conjugate of Δ(P) by an element of N × L outside L × L is never conjugate to Δ(P) in L × L. Consequently, the stabilizer of OLf in N × L is exactly L × L. So if ONf = Ind_{L×L}^{N×L}(OLf) were decomposable then Ind_{L×L}^{N×L}(OLf) would have an indecomposable summand, as a O(N × L)-module, whose restriction to L × L has every summand without a vertex contained in Δ(P). Thus this indecomposable O(N × L)-module does not have a vertex contained in Δ(P). This is of course a contradiction and so ONf is indecomposable as a O(N × L)-module. Its vertex is clearly contained in Δ(P) and its restriction to L×L has a summand with vertex Δ(P). So ONf has vertex Δ(P) as a O(N × L)-module.

(4) P is a defect group for OGb by part (1). Secondly we note that any p-subgroup of N is contained in L (L ◁ N and p ∤ [N : L]). This tells us that OLf has the same defect group as ONf. So we have that ONf has defect group P. Finally we have that Br_{P}^{N}(f) = Br_{P}^{G}(b) by (2) and so ONf is the Brauer correspondent of OGb in N.

□

By Alperin’s description of the Brauer correspondence[1, Chapter 14, Theorem 2] the O(G×G)-module OGb and the O(N ×N)-module ONf both have vertex Δ(P) and are Green correspondents. Let X be the Green correspondent of OGb in G × N. Then X is the unique indecomposable summand of Res_{G×N}^{G×G}(OGb) with vertex Δ(P) and ONf is the unique indecomposable summand of Res_{N×N}^{G×N}(X) with vertex Δ(P). It is then clear that bX = X and that Xf ≠ 0, and so Xf = X and X is an (OGb,ONf)-bimodule.

Let Y be _{G}Y_{L} = OGb_{0}U_{1}^{+}b_{1} … U_{w}^{+}b_{w}, an (OGb,OLf)-bimodule. So the functor Y ⊗_{OL}− from OL-mod to OG-mod is HCInd_{G_{1}b_{1}}^{G_{0}b_{0}} … HCInd_{G_{1}b_{w}}^{G_{0}b_{w−1}}.

Proposition 9.1.3. There is a sequence of O-split monomorphisms of algebras

ONf ↪ End_{OG}(X) ↪ End_{OG}(Y)

Also the left OGb-module X is a progenerator for OGb.

Proof. _{G}OGb_{G} is isomorphic to a direct summand of Ind_{G×N}^{G×G}(_{G}X_{N}) as they are Green correspondents. Thus _{G}OGb is a direct summand of [G : N] copies of _{G}X and _{G}X is a progenerator for OGb.

Now there is an O-split homomorphism of algebras ONf → End_{OG}(X) given by multiplying on the right of X. Since _{N}ONf_{N} is a direct summand of Res_{N×N}^{G×N}(_{G}X_{N}) this homomorphism is an O-split monomorphism.

Next Res_{G×L}^{G×N}(_{G}X_{N}) is indecomposable with vertex Δ(P). First, G × L contains Δ(P) so _{G}X_{N} is a direct summand of Ind_{G×L}^{G×N}(Res_{G×L}^{G×N}(_{G}X_{N})). Since (G×L)⊴(G×N) there exists an indecomposable summand M of Res_{G×L}^{G×N}(_{G}X_{N}) such that Res_{G×L}^{G×N}(_{G}X_{N}) is a direct sum of conjugates of M in G × N. It is possible to pick a set of coset representatives of (G × L) in (G × N) that all normalise Δ(P). So Res_{G×L}^{G×N}(_{G}X_{N}) is the sum of indecomposable modules all with vertex Δ(P). Secondly, Res_{G×L}^{G×N}(_{G}X_{N}) =_{G} X_{N} ⊗_{ON} ONf_{L} is a direct summand of Ind_{N×L}^{G×L}(ONf) which by Green correspondence has exactly one summand with vertex Δ(P). So _{G}X_{L} is indecomposable with vertex Δ(P).

Now we claim that _{G}X_{L} is the only summand of Res_{G×L}^{G×G}(_{G}OGb_{G}) with vertex containing Δ(P). In a direct decomposition of Res_{G×N}^{G×G}(_{G}OGb_{G}) every summand is either GX_{N}, has a vertex strictly smaller than Δ(P) or has a vertex that is conjugate to Δ(P) in (G × G) but not in (G × N). So when we restrict down to (G × L) every summand is either _{G}X_{L}, has a vertex strictly smaller than Δ(P) in size or has a vertex that is conjugate to Δ(P) in (G × G) but not in (G × N) so certainly not in (G × L). So _{G}X_{L} is the only summand of Res_{G×L}^{G×G}(_{G}OGb_{G}) with vertex containing Δ(P).

Let _{G}Y_{L} = OGb_{0}U_{1}^{+}b_{1} … U_{w}^{+}b_{w}. Each b_{i} and each U_{i}^{+} is an idempotent in (OG)^{L}. These idempotents commute with each other and so their product is an idempotent contained in (OG)^{L}. Hence, _{G}Y_{L} is also a direct summand of _{G}OGb_{L}. We show that it has as direct summands all summands of _{G}OGb_{L} with vertex containing Δ(P), using the Brauer homomorphism. This tells us _{G}Y_{L} =_{G} X_{L} ⊕ ∗ and consequently we have an O-split monomorphism End_{OG}(X) ↪ End_{OG}(Y). The calculation goes:

Y (Δ(P)) = OGb_{0}U_{1}^{+}b_{1} … U_{w}^{+}b_{w}(Δ(P))
= kC_{G}((P))Br_{P}^{G}(b_{0}U_{1}^{+}b_{1} … U_{w}^{+}b_{w})
= kC_{G}((P))Br_{P}^{G}(b_{0})Br_{P}^{G}(U_{1}^{+})Br_{P}^{G}(b_{1}) … Br_{P}^{G}(U_{w}^{+})Br_{P}^{G}(b_{w})
= kC_{G}((P))Br_{P}^{G}(b_{0})Br_{P}^{G}(b_{1}) … Br_{P}^{G}(b_{w})
= kC_{G}((P))Br_{P}^{G}(b)
= OGb(Δ(P))

So Y has indeed as summands all summands of _{G}OGb_{L} with vertex containing Δ(P).

□

Let φ be the character of KLf as a representation of L.

Proposition 9.1.4. The O-rank of ONf and End_{OG}(Y) are both equal to 2^{w}w! dim_{K}(KLf).

Before we prove the above proposition we state the following lemma of Chuang and Kessar[6, Lemma 4.2] with out proof.

Lemma 9.1.5. Let d be a positive integer and σ a partition equal to its own d-core with a d-abacus representation such that the ith runner has at least w − 1 fewer beads than the (i + 1)th runner for 0 ≤ i ≤ d − 2 and fix this abacus representation of this core. Let λ be a partition with d-core σ and weight v ≤ w. Let μ be a partition such that μ_{i} ≤ λ_{i} for all i with d-core σ and weight v − 1. Then μ is obtained by removing a d-hook from λ. If this removal occurs on the αth runner then the complement of the Young diagram of μ in that of λ is the Young diagram of the hook partition (α + 1, 1^{(d−α−1)}).

Proof. (of 9.1.4) Since O is a principal ideal domain, proving the proposition is equivalent to showing that K ⊗_{O} ONf and K ⊗_{O} End_{OG}(Y) have the same dimension over K. Now K ⊗_{O} ONf ≅ KNf ≅ Ind_{L}^{N}(KLf). So dim_{K}(K ⊗_{O} ONf) = 2^{w}w! dim_{K}(KLf). Also K ⊗_{O} End_{OG}(Y) ≅ End_{KG}(K ⊗_{O} Y). Additionally we have K ⊗_{O} Y ≅ (K ⊗_{O} Y) ⊗_{KLf} KLf. Let φ be the character of KLf as a character of L. We will calculate R_{G_{1},b_{1}}^{G_{0},b_{0}} … R_{G_{w},b_{w}}^{G_{w−1},b_{w−1}}(φ).

Let ρ^{0} and ρ^{1} be the two partitions associated to ρ as in 2.3. Our condition 9 on ρ demands that ρ^{0} and ρ^{1} both satisfy the conditions of σ in 9.1.5 for our d and w. We will now explain what this means in terms of Harish-Chandra induction 7:

Let 0 ≤ i ≤ w − 1 and χ_{1,τ} be a unipotent character of G_{m−2d(i+1)}(q) such that τ has e-core ρ. Then:

R_{GL_{d}(q)_{i}×G_{m−2d(i+1)}(q),a_{i+1}⊗f_{w−i−1}}^{G_{m−2di}(q),f_{w−i}}(χ_{1,(α+1,1^{(d−α−1)})} ⊗ χ_{1,τ})

is equal to the sum of the χ_{1,λ}s where λ is a partition obtained from τ by sliding a bead 1 place down the 2αth or (2α + 1)th runner in case 1 and the αth or (α + d)th runner in cases 2,3 and 4.

Let us count the number of ways of sliding single beads down the lth runner of a core j times, so that on the resulting runner the bottom bead has been moved down σ_{1}^{l} times, the second bottom bead has been moved down σ_{2}^{l} times, etc, so that σ_{1}^{l} ≥ σ_{2}^{l} ≥ … and ∑_{i} σ_{i}^{l} = j. It is equal to the number of ways of writing the numbers 1, …, j in the Young diagram of σ_{1}^{l}, σ_{2}^{l}, … so that numbers increase across rows and down columns, that is, the degree of the character ζ^{σ^{l}} of the symmetric group S_{j}.

The irresducible characters in the block KLf are of the form χ_{s_{1},λ_{1}} ⊗ ⋯ ⊗ χ_{s_{w},λ_{w}} ⊗ χ_{s,ρ} where either s_{i} is 1 and λ_{i} is an d-hook partition or s_{i} is a non-trivial p-element of GL_{d}(q) and λ_{i} is the partition (1) 8.3. Note that s is always 1 or something whose image under π is 1 in cases 2(b) and 4(b). In other words, in the latter case, s is just a p-element of the underlying field in the special Clifford group.

We can now describe R_{G_{1},b_{1}}^{G_{0},b_{0}} … R_{G_{w},b_{w}}^{G_{w−1},b_{w−1}}(χ_{s_{1},λ_{1}} ⊗ ⋯ ⊗ χ_{s_{w},λ_{w}} ⊗ χ_{s,ρ}). Suppose that the s_{i}s are grouped together with like elements such that s_{1},…,s_{r_{0}} are all equal to 1 and the remaining elements are grouped together into conjugacy classes as follows:

s_{r_{0}+1} ∼ ⋯ ∼ s_{r_{0}+α_{1}} ∼ t_{1},
s_{r_{0}+α_{1}+1} ∼ ⋯ ∼ s_{r_{0}+α_{1}+β_{1}} ∼ \overline{t_{1}},
s_{r_{0}+α_{1}+β_{1}+1} ∼ ⋯ ∼ s_{r_{0}+α_{1}+β_{1}+α_{2}} ∼ t_{2},
s_{r_{0}+α_{1}+β_{1}+α_{2}+1} ∼ ⋯ ∼ s_{r_{0}+α_{1}+β_{1}+α_{2}+β_{2}} ∼ \overline{t_{2}},
…

where \overline{t} means t^{−1} (or t^{−q_{0}} in case 1). Additionally set r_{i} = α_{i} + β_{i} giving ∑_{i} r_{i} = w and:

λ_{1} = ⋯ = λ_{l_{0}} = (1^{d}),
λ_{l_{0}+1} = ⋯ = λ_{l_{0}+l_{1}} = (2, 1^{d−2}),
…,
λ_{l_{0}+···+l_{d−2}+1} = ⋯ = λ_{l_{0}+···+l_{d−1}} = (d)

where ∑_{i} l_{i} = r_{0}.

Now we have an expression for R_{G_{1},b_{1}}^{G_{0},b_{0}} … R_{G_{w},b_{w}}^{G_{w−1},b_{w−1}}(χ_{s_{1},λ_{1}} ⊗ ⋯ ⊗ χ_{s_{w},λ_{w}} ⊗ χ_{s,ρ}):

∑_{\begin{matrix}l_{0}+···+l_{d}=r_{0} \\ |σ^{i}|+|τ^{i}|=l_{i},|ν^{i}|=r_{i} \end{matrix}} \binom{l_{0}}{|σ^{0}|} dim ζ^{σ^{0}} dim ζ^{τ^{0}} … \binom{l_{d−1}}{|σ^{d−1}|} dim ζ^{σ^{d−1}} dim ζ^{τ^{d−1}}

dim ζ^{ν^{1}} dim ζ^{ν^{2}} …
χ(s × s_{1} × \overline{s_{1}} × ⋯ × s_{w} × \overline{s_{w}}, μ)

Where μ_{X−1} is the partition/symbol whose e-core is ρ and whose e-quotients are [σ^{0}, …, σ^{d−1}] and [τ^{0}, …, τ^{d−1}] (or just [σ^{0}, τ^{0}, …, σ^{d−1}, τ^{d−1}] in case 1) with respect to the 2d-abacus representation of ρ already fixed 9 and μ_{ψ_{i}} = ν^{i} where ψ_{i} is the minimal polynomial of t_{i} × \overline{t_{i}}.

Now we can permute the λ_{i}s and still get the same character of G when we do Harish-Chandra induction. There are exactly w!/(l_{0}!l_{1}! … l_{d−1}!α_{1}!β_{1}!α_{2}!β_{2}! …) such permutations. Also each irreducible character appears in φ with multiplicity equal to the dimension of said character. So we have the following expression for R_{G_{1},b_{1}}^{G_{0},b_{0}} … R_{G_{w},b_{w}}^{G_{w−1},b_{w−1}}(φ):

∑_{\begin{matrix}l_{0}+···+l_{d−1}+r_{1}+r_{2}+···=w \\ α_{i}+β_{i}=r_{i},|σ^{i}|+|τ^{i}|=l_{i},κ \end{matrix}} \frac{w!}{l_{0}!l_{1}! … l_{d−1}!α_{1}!β_{1}!α_{2}!β_{2}! …}

dim(χ_{s_{1},λ_{1}} ⊗ ⋯ ⊗ χ_{s_{w},λ_{w}} ⊗ χ_{s,ρ})

\binom{l_{0}}{|σ^{0}|} dim ζ^{σ^{0}} dim ζ^{τ^{0}} … \binom{l_{d−1}}{|σ^{d−1}|} dim ζ^{σ^{d−1}} dim ζ^{τ^{d−1}}

dim ζ^{ν^{1}} dim ζ^{ν^{2}} … χ(s × s_{1} × \overline{s_{1}} × ⋯ × s_{w} × \overline{s_{w}}, μ)

Where s is the scalar matrix κI. So κ runs over (𝔽_{q}^{×})_{p} in cases 2(b) and 4(b) and just 1 otherwise. So we get:

∑_{\begin{matrix}l_{0}+···+l_{d−1}+r_{1}+r_{2}+···=w \\ |σ^{i}|+|τ^{i}|=l_{i},κ \end{matrix}} \frac{w!}{l_{0}!l_{1}! … l_{d−1}!r_{1}!r_{2}! …}

\binom{l_{0}}{|σ^{0}|} dim ζ^{σ^{0}} dim ζ^{τ^{0}} … \binom{l_{d−1}}{|σ^{d−1}|} dim ζ^{σ^{d−1}} dim ζ^{τ^{d−1}}

dim ζ^{ν^{1}} dim ζ^{ν^{2}} … χ(s × s_{1} × \overline{s_{1}} × ⋯ × s_{w} × \overline{s_{w}}, μ)

[ ∑_{α_{i}+β_{i}=r_{i}} \binom{r_{1}}{α_{1}}\binom{r_{2}}{α_{2}} … dim(χ_{s_{1},λ_{1}} ⊗ ⋯ ⊗ χ_{s_{w},λ_{w}} ⊗ χ_{s,ρ})]

So the dimension of End_{KG}((K ⊗_{O} Y) ⊗_{K} KLf) over K is:

∑_{\begin{matrix}l_{0}+···+r_{1}+···=w \\ |σ^{i}|+|τ^{i}|=l_{i},κ \end{matrix}} (\frac{w!}{l_{0}!l_{1}! … l_{d−1}!r_{1}!r_{2}! …})^{2}

\binom{l_{0}}{|σ^{0}|}^{2}(dim ζ^{σ^{0}})^{2}(dim ζ^{τ^{0}})^{2} … \binom{l_{d−1}}{|σ^{d−1}|}^{2}(dim ζ^{σ^{d−1}})^{2}(dim ζ^{τ^{d−1}})^{2}

(dim ζ^{ν^{1}})^{2}(dim ζ^{ν^{2}})^{2} …

[ ∑_{α_{i}+β_{i}=r_{i}} \binom{r_{1}}{α_{1}}\binom{r_{2}}{α_{2}} … dim(χ_{s_{1},λ_{1}} ⊗ ⋯ ⊗ χ_{s_{w},λ_{w}} ⊗ χ_{s,ρ})]^{2}

= ∑_{\begin{matrix}l_{0}+···+r_{1}+···=w \\ σ_{i}+τ_{i}=l_{i},κ \end{matrix}} (\frac{w!}{l_{0}!l_{1}! … l_{d−1}!r_{1}!r_{2}! …})^{2}\binom{l_{0}}{σ_{0}}^{2} … \binom{l_{d−1}}{σ_{d−1}}^{2}

[ ∑_{σ^{i}⊢σ_{i},τ^{i}⊢τ_{i}}(dim ζ^{σ^{0}})^{2}(dim ζ^{τ^{0}})^{2} … (dim ζ^{σ^{d−1}})^{2}(dim ζ^{τ^{d−1}})^{2}]

(dim ζ^{ν^{1}})^{2}(dim ζ^{ν^{2}})^{2} …

[ ∑_{α_{i}+β_{i}=r_{i}} \binom{r_{1}}{α_{1}}\binom{r_{2}}{α_{2}} … dim(χ_{s_{1},λ_{1}}) … dim(χ_{s_{w},λ_{w}}) dim(χ_{s,ρ})]^{2}

Now using the fact that ∑_{σ⊢h} dim(ζ^{σ})^{2} = h! we get:

∑_{\begin{matrix}l_{0}+···+r_{1}+···=w \\ σ_{i}+τ_{i}=l_{i},κ \end{matrix}} (\frac{w!}{l_{0}!l_{1}! … l_{d−1}!r_{1}!r_{2}! …})^{2}\binom{l_{0}}{σ_{0}}^{2} … \binom{l_{d−1}}{σ_{d−1}}^{2}

σ_{0}!τ_{0}! … σ_{d−1}!τ_{d−1}!r_{1}!r_{2}! …

[ ∑_{α_{i}+β_{i}=r_{i}} \binom{r_{1}}{α_{1}}\binom{r_{2}}{α_{2}} … dim(χ_{s_{1},λ_{1}}) … dim(χ_{s_{w},λ_{w}}) dim(χ_{s,ρ})]^{2}

Now dim(χ_{t,(1)}) = dim(χ_{\overline{t},(1)}) 7.3. This means that for fixed (r_{1}, r_{2}, …) the choice of the α_{i}s does not affect dim(χ_{s_{1},λ_{1}}) … dim(χ_{s_{w},λ_{w}}) dim(χ_{s,ρ}). Using this and the fact that ∑_{i=0}^{r} \binom{r}{i} = 2^{r} we get:

w! ∑_{\begin{matrix}l_{0}+···+r_{1}+···=w \\ σ_{i}+τ_{i}=l_{i},κ \end{matrix}} \frac{w!}{l_{0}!l_{1}! … l_{d−1}!r_{1}!r_{2}! …}\binom{l_{0}}{σ_{0}} … \binom{l_{d−1}}{σ_{d−1}}

2^{∑ r_{i}}[ ∑_{α_{i}+β_{i}=r_{i}} \binom{r_{1}}{α_{1}}\binom{r_{2}}{α_{2}} … dim(χ_{s_{1},λ_{1}} ⊗ ⋯ ⊗ χ_{s_{w},λ_{w}} ⊗ χ_{s,ρ})^{2}]

= w! ∑_{\begin{matrix}l_{0}+···+r_{1}+···=w \\ κ \end{matrix}} \frac{w!}{l_{0}!l_{1}! … l_{d−1}!r_{1}!r_{2}! …}

2^{∑ r_{i}}2^{∑ l_{i}}( ∑_{α_{i}+β_{i}=r_{i}} \binom{r_{1}}{α_{1}} … dim(χ_{s_{1},λ_{1}} ⊗ ⋯ ⊗ χ_{s_{w},λ_{w}} ⊗ χ_{s,ρ})^{2})

= 2^{w}w! ∑_{\begin{matrix}l_{0}+···+l_{d−1}+α_{1}+β_{1}+α_{2}+β_{2}···=w \\ κ \end{matrix}} \frac{w!}{l_{0}!l_{1}! … l_{d−1}!α_{1}!β_{1}!α_{2}!β_{2}! …}

dim(χ_{s_{1},λ_{1}} ⊗ ⋯ ⊗ χ_{s_{w},λ_{w}} ⊗ χ_{s,ρ})^{2}

= 2^{w}w! dim_{K}(KLf)
□

Proof. (of 9.1.1) By 9.1.4 ONf and End_{OG}(Y) have the same O-rank. As a consequence we have that all the monomorphisms in 9.1.3 become isomorphisms. Therefore, since X is a progenerator as a left OGb-module, _{OGb}X_{ONf} induces a Morita equivalence between OGb and ONf. □

9.2. Degenerate Case. We now assume that ρ is degenerate and restrict out attention to case 4(b). Again we will later prove the corresponding theorem for case 4(a) using 8.4.1.

Theorem 9.2.1. ON(f + f′) is a block of N and is Morita equivalent to OGb.

The proof of this theorem will closely resemble that of 8.4.1. We will need all the corresponding lemmas first.

Lemma 9.2.2.

(1) P is defect group for OG_{i}b_{i} for (0 ≤ i ≤ w − 1) and also for OLf and OLf′.

(2) Br_{P}^{G}(b_{i}) = Br_{P}^{G}(f) = Br_{P}^{G}(f′) = a′_{1} ⊗⋯⊗a′_{w} ⊗(f_{0} +f′_{0}) where a′_{i} is the principal block of C_{GL_{d}(q)_{i}}(P_{i}) for (0 ≤ i ≤ w − 1). In addition we have Br_{P}^{G}(U_{i}^{+}) = 1.

(3) N stabilizes (f + f′), ONf and ONf′ are both indecomposable with vertex Δ(P) and ON(f + f′) is a block of N.

(4) OGb and ON(f + f′) both have defect group P and are Brauer correspondents.

Proof.

(1) P_{i} is defect group for OGL_{d}(q)_{i}a_{i} and (P_{i+1} × ⋯ × P_{w}).Z_{p} is a defect group for OG_{m−di}(q)f_{w−i}. Similarly for OLf and OLf′.

(2) Identical to 9.1.2 part (2).

(3) As with the non-degenerate case the only thing to check for the first part is that CO_{m−2dw}^{+}(q) stabilizes (f +f′). This is again clear as in the non-degenerate case. Also as in the non-degenerate case we have both ONf and ONf′ are indecomposable as O(N × L)-modules with vertex Δ(P). Note that ONf and ONf′ lie in different blocks of O(N × L). This implies that if ON(f + f′) were decomposable as a O(N × N)-module we would have to get the same decomposition ON(f + f′) ≅ ONf ⊕ ONf′. However N_{G}(P) ≤ N is transitive on the blocks of C_{G}(P) appearing in the image of b under Br_{P}^{G}. This means f_{0} is conjugate to f′_{0} and hence f to f′. Thus ON(f + f′) is one block of N.

(4) P is a defect group for OGb by part (1). As in the non-degenerate case every p-subgroup of N lies in L so ON(f +f′) has defect group P. Also Br_{P}^{N}(f +f′) = Br_{P}^{G}(b) by (2) and so ON(f + f′) is the Brauer correspondent of OGb in N.

□

We again let X be the Green correspondent of OGb in G × N. This time we set Y =_{G} Y_{L} = OGb_{0}U_{1}^{+}b_{1} … U_{w}^{+}(b_{w} + b′_{w}). Recall that f = b_{w} and f′ = b′_{w}.

Proposition 9.2.3. There is a sequence of O-split monomorphisms of algebras

ON(f + f′) ↪ End_{OG}(X) ↪ End_{OG}(Y)

Also the left OGb-module X is a progenerator for OGb.

During the proof we will refer back to the corresponding proposition for the nondegenerate case 9.1.3.

Proof. As in the non-degenerate case X is clearly a progenerator for OGb. It is also clear that we have the O-split monomorphism ON(f + f′) ↪ End_{OG}(X).

This time _{G}X_{L} is not indecomposable. In instead we have that it is the direct sum of 2 indecomposable modules Xf and Xf′ each with vertex Δ(P). Next, as with the non-degenerate case, we have that _{G}X_{L} is the direct sum of indecomposable modules all with vertex Δ(P). Finally we see that _{G}X_{L}f =_{G} X_{N} ⊗_{ON} ONf_{L} is a direct summand of Ind_{N×L}^{G×L}(ONf) which by Green correspondence has exactly one summand with vertex Δ(P). So _{G}X_{L}f is indecomposable with vertex Δ(P). Similarly for _{G}X_{L}f′.

The proof that we have an O-split monomorphism End_{OG}(X) ↪ End_{OG}(Y) is again as in the non-degenerate case with b_{w} replaced by b_{w} + b′_{w}. □

Let φ be the character of KL(f + f′) as a representation of L.

Proposition 9.2.4. The O-rank of ON(f + f′) and End_{OG}(Y) are both equal to 2^{w}w! dim_{K}(KLf).

Proof. We obtain a combinatorial rule for Harish-Chandra induction in this case by comparing with the non-degenerate case where everything is calculated in the Weyl group of type B. We first want to calculate

R_{G_{w−v+1},b_{w−v+1}}^{G_{w−v},b_{w−v}} … R_{G_{w},(b_{w}+b′_{w})}^{G_{w−1},b_{w−1}}(χ_{1,λ_{1}} ⊗ ⋯ ⊗ χ_{1,λ_{v}} ⊗ (χ_{s,ρ} + χ′_{s,ρ}))

where v ≤ w and s is a p-element of the underlying field 𝔽_{q}.

Let τ = {X, Y } be a non-degenerate symbol of weight w − v. Then the multiplicity of χ_{s,τ} is just the number of ways of obtaining τ from ρ by sliding beads down the appropriate runners. (Appropriate means the runners determined by the λ_{i}s. Also obtaining τ from ρ means obtaining τ so that X is on the left or the right.)

Now let τ = {X, X} be a degenerate symbol of weight w − v. Then again the multiplicity of χ_{s,τ} is just the number of ways of obtaining τ from ρ by sliding beads down the appropriate runners. The same is true of χ′_{s,τ}.

Adopting the same notation as in the non-degenerate case we let φ be the character of KL(f + f′). We have the following expression for R_{G_{1},b_{1}}^{G_{0},b_{0}} … R_{G_{w},(b_{w}+b′_{w})}^{G_{w−1},b_{w−1}}(φ):

∑_{\begin{matrix}l_{0}+···+l_{d−1}+r_{1}+r_{2}+···=w \\ α_{i}+β_{i}=r_{i},|σ^{i}|+|τ^{i}|=l_{i},κ \end{matrix}} \frac{w!}{l_{0}!l_{1}! … l_{d−1}!α_{1}!β_{1}!α_{2}!β_{2}! …}

dim(χ_{s_{1},λ_{1}} ⊗ ⋯ ⊗ χ_{s_{w},λ_{w}} ⊗ χ_{s,ρ})

\binom{l_{0}}{|σ^{0}|} dim ζ^{σ^{0}} dim ζ^{τ^{0}} … \binom{l_{d−1}}{|σ^{d−1}|} dim ζ^{σ^{d−1}} dim ζ^{τ^{d−1}}

dim ζ^{ν^{1}} dim ζ^{ν^{2}} …

χ(s × s_{1} × \overline{s_{1}} × ⋯ × s_{w} × \overline{s_{w}}, μ)

Compare with the non-degenerate case. χ(s×s_{1} ×\overline{s_{1}} ×⋯×s_{w} ×\overline{s_{w}}, μ) means the sum of both characters when μ_{X−1} is degenerate. Note that dim(χ_{s,ρ}) = dim(χ′_{s,ρ}) (see 7.3).

= ∑_{\begin{matrix}l_{0}+···+l_{d−1}+r_{1}+r_{2}+···=w \\ |σ^{i}|+|τ^{i}|=l_{i},κ \end{matrix}} \frac{w!}{l_{0}!l_{1}! … l_{d−1}!r_{1}!r_{2}! …}

\binom{l_{0}}{|σ^{0}|} dim ζ^{σ^{0}} dim ζ^{τ^{0}} … \binom{l_{d−1}}{|σ^{d−1}|} dim ζ^{σ^{d−1}} dim ζ^{τ^{d−1}}

dim ζ^{ν^{1}} dim ζ^{ν^{2}} … χ(s × s_{1} × \overline{s_{1}} × ⋯ × s_{w} × \overline{s_{w}}, μ)

[ ∑_{α_{i}+β_{i}=r_{i}} \binom{r_{1}}{α_{1}}\binom{r_{2}}{α_{2}} … dim(χ_{s_{1},λ_{1}} ⊗ ⋯ ⊗ χ_{s_{w},λ_{w}} ⊗ χ_{s,ρ})]

Note that if we swap all the σ^{i}s and τ^{i}s over then we get the same character. Recalling also that χ(s × s_{1} × \overline{s_{1}} × ⋯ × s_{w} × \overline{s_{w}}, μ) means the sum of both characters when μ_{X−1} is degenerate the dimension of End_{KG}((K ⊗_{O} Y) ⊗_{K} KLf) over K is:

2 ∑_{\begin{matrix}l_{0}+···+r_{1}+···=w \\ |σ^{i}|+|τ^{i}|=l_{i},κ \end{matrix}} (\frac{w!}{l_{0}!l_{1}! … l_{d−1}!r_{1}!r_{2}! …})^{2}

\binom{l_{0}}{|σ^{0}|}^{2}(dim ζ^{σ^{0}})^{2}(dim ζ^{τ^{0}})^{2} … \binom{l_{d−1}}{|σ^{d−1}|}^{2}(dim ζ^{σ^{d−1}})^{2}(dim ζ^{τ^{d−1}})^{2}

(dim ζ^{ν^{1}})^{2}(dim ζ^{ν^{2}})^{2} …

[ ∑_{α_{i}+β_{i}=r_{i}} \binom{r_{1}}{α_{1}}\binom{r_{2}}{α_{2}} … dim(χ_{s_{1},λ_{1}} ⊗ ⋯ ⊗ χ_{s_{w},λ_{w}} ⊗ χ_{s,ρ})]^{2}

Following all the step through as before we get:

2^{w+1}w! ∑_{\begin{matrix}l_{0}+···+α_{1}+β_{1}+α_{2}+β_{2}···=w \\ κ \end{matrix}} \frac{w!}{l_{0}!l_{1}! … l_{d−1}!α_{1}!β_{1}!α_{2}!β_{2}! …}

dim(χ_{s_{1},λ_{1}} ⊗ ⋯ ⊗ χ_{s_{w},λ_{w}} ⊗ χ_{s,ρ})^{2}

Recall that dim(χ_{s_{1},λ_{1}} ⊗ ⋯ ⊗ χ_{s_{w},λ_{w}} ⊗ χ_{s,ρ}) = dim(χ_{s_{1},λ_{1}} ⊗ ⋯ ⊗ χ_{s_{w},λ_{w}} ⊗ χ′_{s,ρ}) to obtain:

= 2^{w}w! dim_{K}(KLf)
□

Proof. (of 9.2.1) Again 9.2.4 and 9.2.3 give us the relevant information to obtain that OGbX_{ON(f+f′)} induces a Morita equivalence between OGb and ONf. □

9.3. Sp_{2n}(q) and SO_{2n}^{±}(q). We are now is a position to prove 9.1.1 and 9.2.1 our main theorem for cases 2(a) and 4(a).

Corollary 9.3.1.

(1) If ρ is non-degenerate then ONf is a block of N and is Morita equivalent to OGb.

(2) If ρ is degenerate then ON(f + f′) is a block of N and is Morita equivalent to OGb.

Of course part (2) only happens in case 4(a). We will use the proof of the corresponding theorems for cases 2(b) and 4(b) (see 9.1.1 and 9.2.1) as well as 8.4.1.

Proof. Adopting the notation of 8.4.1 we have the correspondence between characters of OĜj and OGi given by restriction but with each character of OGi appearing as a restriction of (q − 1)_{p} distinct characters of OĜj. Given that this correspondence commutes with Harish-Chandra induction we have that the proofs of 9.1.1 and 9.2.1 run through for G. Note that the character sums have to be divided by (q − 1)_{p}. □

All that remains to prove theorem 1.0.2 is the following lemma. We continue with the notation from this section as well as that from 1.0.2. We will consider all 4 cases.

Lemma 9.3.2.

(1) If ρ is non-degenerate then ONf is Morita equivalent to the principal block of O((GL_{d}(q).2 ≀ S_{w}) × Z_{p}).

(2) If ρ is degenerate then ON(f + f′) is Morita equivalent to the principal block of O(M × Z_{p}).

Proof. We prove for cases 1,2(a),3 and 4(a). The statements for cases 2(b) and 4(b) will then follow from the corresponding statements for cases 2(a) and 4(a) and 8.4.1.

(1) This is clear for cases 1 and 2(a). For cases 3 and 4(a) we take the definition of G̃ from the beginning of the section. First consider the unique character of OG_{m−2dw}(q)f_{0}. Since ONf is a block this character is invariant under conjugation by G̃_{m−2dw}(q) as OG_{m−2dw}(q)f_{0} is. Therefore this character induces to 2 different characters of G̃_{m−2dw}(q) and hence OG̃_{m−2dw}(q)f_{0} is not a block and is in fact the direct sum of 2 blocks, OG̃_{m−2dw}(q)f′_{0} and OG̃_{m−2dw}(q)f′′_{0} both Morita equivalent to OG_{m−2dw}(q)f_{0} due to[4, Proposition 6].

ONf is Morita equivalent to the block of O((GL_{d}(q).2 ≀ S_{w}) × G̃_{m−2dw}(q)) with block idempotent a_{1} ⊗ ⋯ ⊗ a_{w} ⊗ f′_{0} by[4, Proposition 6] which is in turn clearly Morita equivalent to the principal block of O(GL_{d}(q).2 ≀ S_{w}).

(2) Let a′ be the principal block idempotent of OM. Then OMa′ is Morita equivalent

to OMa′ ⊗_{O} OG_{m−2dw}f_{0} which is in turn Morita equivalent to ON(f + f′).

□

10. Application to Broué’s Conjecture

We are now in a position to show that OGb is derived equivalent to its Brauer correspondent in N_{G}(P). We will use h to denote the corresponding block idempotent of ON_{G}(P).

Corollary 10.0.3. OGb is derived equivalent to its Brauer correspondent in N_{G}(P).

We will need a couple of lemmas to prove the above corollary but first we define M′ analogously to how we defined M in the introduction.

We have a natural homomorphism N_{GL_{d}(q).2}(R) → {±1} with kernel N_{GL_{d}(q)}(R). This extends to a map N_{GL_{d}(q).2}(R) ≀ S_{w} → {±1} and we use M′ to denote the kernel of this map.

Lemma 10.0.4.

(1) If ρ is non-degenerate then ON_{G}(P)h is Morita equivalent to the principal block of O((N_{GL_{d}(q).2}(R) ≀ S_{w}) × Z_{p}).

(2) If ρ is degenerate then ON_{G}(P)h is Morita equivalent to the principal block of O(M′ × Z_{p}).

Proof. See proof of 9.3.2. □

We will need the following theorem due to A. Marcus[17, Theorem 3.2(b)].

Theorem 10.0.5. Let X, X̃, Y and Ỹ be finite groups with X ◁ X̃, Y ◁ Ỹ, X ≤ Y and X̃ ≤ Ỹ. We also require that X̃ ∩ Y = X and that the natural homomorphism X̃/X → Ỹ/Y is in fact an isomorphism of p′-groups. Now let i (respectively j) be a block idempotent of OX (respectively OY) which is fixed by conjugation by X̃ (respectively Ỹ) and (C, C^{∗}) a pair of complexes giving a derived equivalence between OXi and OY j. Now consider A the subalgebra of OX̃i ⊗_{O} (OỸj)^{op} generated by OXi ⊗_{O} (OY j)^{op} and xi⊗x^{−1}j for all x ∈ X̃. Suppose that C extends to complex of A-modules, then (OX̃i⊗_{OXi} C, OỸj ⊗_{OY i} C^{∗}) is a pair of complexes giving a derived equivalence between OX̃i and OỸj.

If C extends to complex of A-modules we will say that C has a consistent diagonal action of X̃.

We are now in a position to prove corollary 10.0.3.

Proof. 10.0.3 We already have that OGb is Morita equvalent to ONf 9.1.1 (or ON(f + f′) in the degenerate case 9.2.1) which is in turn Morita equivalent to the principal block of O((GL_{d}(q).2≀S_{w})×Z_{p}) (respectively O(M ×Z_{p})) 9.3.2. On the other hand ON_{G}(P)h is Morita equivalent to the principal block of O((N_{GL_{d}(q).2}(R) ≀ S_{w}) × Z_{p}) (respectively O(M′ × Z_{p})) 10.0.4. Therefore all that remains to prove 10.0.3 is that the principal blocks of O(GL_{d}(q).2 ≀ S_{w}) and O(N_{GL_{d}(q).2}(R) ≀ S_{w}) (respectively OM and OM′) are derived equvalent.

R is cyclic so the principal block of OGL_{d}(q).2 is derived equivalent to the principal block of ON_{GL_{d}(q).2}(R). Then by [17, Theorem 4.3(b)] the principal block of O(GL_{d}(q).2 ≀ S_{w}) is derived equivalent to the principal block of O(N_{GL_{d}(q).2}(R) ≀ S_{w}).

By[17, Example 5.5] there exists a pair of complexes (C, C^{∗}) that induce a derived equivalence between the principal blocks of ON_{GL_{d}(q)}(R) and OGL_{d}(q) such that C has a consistent diagonal action of N_{GL_{d}(q).2}(R).

By[17, Theorem 4.3(b)] we can construct a derived equivalence between the principal blocks of O(N_{GL_{d}(q)}(R) ≀ S_{w}) and O(GL_{d}(q) ≀ S_{w}). Let (D, D^{∗}) be the pair of complexes giving this equivalence. Note that from the construction of (D, D^{∗}) and the previous paragraph that D has a consistent diagonal action of (N_{GL_{d}(q).2}(R)) ≀ S_{w} so certainly of M′. Therefore by 10.0.5 we can construct a derived equivalence between the principal blocks of OM and OM′. □

Note that lemma 10.0.4 still holds without our condition 9. Therefore we have the following corollary for all four cases.

Corollary 10.0.6. Let B be a unipotent block of OG_{m′}(q) of weight w with abelian defect group P′. If B′ is the Brauer correspondent of B in N_{G_{m′}(q)}(P′) then the Morita equivalence class of B′ depends only on w and whether B is degenerate or not.

The above corollary is an analogy of part (3) of the proof of Broué’s conjecture for the symmetric group given in the introduction.

Remark 10.0.7. We note that a corresponding theorem to 1.0.2 for the case of q being even has yet to be proven for all but the unitary group.

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