Glauberman’s Correspondence via Canonical Induction Formulae

Robert Boltje Institut für Mathematik Universität Augsburg 86 135 Augsburg email: boltje@math.uni-augsburg.de

Introduction

Let G and S be finite groups of coprime orders and assume that S acts from the left on G. By GS := G ⋊ S we denote the resulting semi-direct product. The action of S on G naturally induces actions of S on the character ring R(G) of G and the set Irr(G) of irreducible characters of G. The set of fixed points of an S-set Ω will be denoted by Ω^{S}, e.g. R(G)^{S}, Irr(G)^{S}, G^{S} = C_{S}(G).

The aim of this article is the construction of a map

gl^{G,S}: R(G)^{S} → R(G^{S})

which is related to the Glauberman correspondence (cf. [G] or [I2, §13]) stating that, if S is solvable, there is a ‘natural’ bijection between Irr(G)^{S} and Irr(G^{S}). The map gl^{G,S} might be a candidate for a more compact definition of this correspondence and also for a unified treatment of Glauberman’s correspondence and Isaacs’ correspondence (cf. [I1]) which is a bijection between the same sets in the case where G is of odd order, thus covering the remaining case. The definition of gl^{G,S} was suggested by Puig and uses the canonical induction formula for G introduced in [Bo] by extending the method which led to definitions of Adams operations (cf. [Bo, Thm. 2.1 (f)]) and Chern classes (cf. [Sy]).

In Sections 1 and 2 we recall the necessary facts about Glauberman’s correspondence and the canonical induction formula. Section 3 is devoted to the definition of the map gl^{G,S} and first properties and examples. Finally, in Section 4 we determine the canonical induction formula for an extraspecial p-group G of exponent p for an odd prime p, which is needed to calculate an example in Section 3.

1. Glauberman’s correspondence

Let G and S be as in the introduction. We first recall the basic facts about Glauberman’s correspondence. If S is a p-group, there is very explicit description of this correspondence by the following lemma.

1.1 Lemma (cf. [I2, §13]) Assume that S is a p-group and let χ ∈ Irr(G)^{S} be an S-invariant irreducible character of G. Then there is a unique constituent λ ∈ Irr(G^{S}) of res_{G^{S}}^{G}(χ) whose multiplicity is not divisible by p. Moreover, this multipicity is congruent to 1 or -1 modulo p.

Using the above Lemma, the Glauberman correspondence can be defined in the following way.

1.2 Theorem (cf. [I2, Thm. 13.1]) If S is solvable, then there is a unique bijection π^{G,S}: Irr(G)^{S} → Irr(G^{S}), called the Glauberman correspondence, satisfying the following conditions

(i) If T ⊴ S, then π^{G,T} (Irr(G)^{S}) = Irr(G^{T})^{S} and π^{G,S} = π^{G^{T} ,S/T} ∘ π^{G,T}.

(ii) If S is a p-group and χ ∈ Irr(G)^{S}, then π^{G,S}(χ) is the unique irreducible constituent λ of res_{G^{S}}^{G}(χ) with p ∤ (λ, res_{G^{S}}^{G}(χ))_{G^{S}} according to Lemma 1.1.

The uniqueness part of the above Theorem is obvious with Lemma 1.1. The map π^{G,S} can be calculated by choosing a sequence of normal subgroups in S such that the quotients of subsequent subgroups are p-groups. In order to prove that the result is independent of the chosen sequence one uses another explicit description of Glauberman’s correspondence in the case where S is a cyclic group (cf. [I2, §13]), but we won’t need this in the sequel.

There is a complement to Glauberman’s correspondence if one drops the solvability hypothesis on S which is due to Isaacs and is proved by totally different methods. Note that if S is not solvable, then G is of odd order by the Feit-Thompson Theorem.

1.3 Theorem (cf. [I1]) If |G| is odd, then there is a ‘natural’ bijection between Irr(G)^{S} and Irr(G^{S}) called Isaacs’ correspondence.

Although Glauberman’s and Isaacs’ correspondences are defined in very different ways, they are related by the following theorem of Wolf.

1.4 Theorem (cf. [Wo]) If S is solvable and |G| is odd, then Glauberman’s correspondence and Isaacs’ correspondence coincide.

1.5 Remark By Wolf’s result we may use a unified notation for both correspondences and write π^{G,S}: Irr(G)^{S} → Irr(G^{S}) for one or both of these correspondences whenever S acts on G with (|S|, |G|) = 1. It is easy to verify (also in the case of Isaacs’ correspondence which we didn’t define) that for a one-dimensional S-invariant character φ of G one has π^{G,S}(φ) = res_{G^{S}}^{G}(φ).

For later use in Section 3 we formulate some consequences of Lemma 1.1 in the next Proposition. For χ ∈ Irr(G) we will denote by Tr_{S}(χ) ∈ R(G)^{S} the S-orbit sum of χ, i.e. if T := Stab_{S}(χ) is the stabilizer of χ, then Tr_{S}(χ) = ∑_{s∈S/T} ^{s}χ. Note that the elements Tr_{S}(χ), where χ runs through a set of representatives of the S-orbits S\Irr(G) of Irr(G), form a ℤ-basis of R(G)^{S}. Moreover, we write R(G)_{<S}^{S} for the ℤ-span of the elements Tr_{S}(χ) ∈ R(G)^{S}, χ ∈ Irr(G) with Stab_{S}(χ) < S. Then we have R(G)^{S} = ℤ·Irr(G)^{S} ⊕R(G)_{<S}^{S}.

1.6. Proposition Let S be a p-group.

(i) For χ ∈ Irr(G)^{S} with Stab_{S}(χ) < S we have

res_{G^{S}}^{G}(Tr_{S}(χ)) ≡ 0 (mod pR(G^{S})).

(ii) For λ ∈ Irr(G^{S}) we have ind_{G^{S}}^{G}(λ) ∈ R(G)^{S}, and there is a unique χ ∈ Irr(G)^{S} with p ∤ (χ, ind_{G^{S}}^{G}(λ))_{G}; Moreover, (χ, ind_{G^{S}}^{G}(λ))_{G} ≡ ±1 (mod p) for this χ.

(iii) For λ ∈ R(G^{S}) we have

res_{G^{S}}^{G}(ind_{G^{S}}^{G}(λ)) ≡ λ (mod pR(G^{S})).

(iv) For χ ∈ R(G)^{S} we have

ind_{G^{S}}^{G}(res_{G^{S}}^{G}(χ)) ≡ χ (mod pR(G)^{S} + R(G)_{<S}^{S}).

Proof (i) Let T := Stab_{S}(χ), then

res_{G^{S}}^{G}(Tr_{S}(χ)) = ∑_{s∈S/T} res_{G^{S}}^{G}(^{s}χ) = ∑_{s∈S/T} ^{s}(res_{G^{S}}^{G}(χ)) = |S/T | · res_{G^{S}}^{G}(χ).

(ii) Let s ∈ S, then ^{s}(ind_{G^{S}}^{G}(λ)) = ind_{G^{S}}^{G}(^{s}λ) = ind_{G^{S}}^{G}(λ). Furthermore, since π^{G,S}: Irr(G)^{S} → Irr(G^{S}) is surjective, Lemma 1.1 implies that there is some χ ∈ Irr(G)^{S} with p ∤ (λ, res_{G^{S}}^{G}(χ))_{G^{S}} = (ind_{G^{S}}^{G}(λ), χ)_{G}. Moreover, since π^{G,S} is injective, Lemma 1.1 implies that χ is unique.

(iii) It suffices to prove the congruence for λ ∈ Irr(G^{S}). If λ = π^{G,S}(χ) for χ ∈ Irr(G)^{S}, then by part (ii) we have

ind_{G^{S}}^{G}(λ) ≡ ±χ (mod pR(G)^{S} + R(G)_{<S}^{S}).

Hence res_{G^{S}}^{G}(ind_{G^{S}}^{G}(λ)) ≡ λ (mod pR(G^{S})) by part (i).

(iv) It suffices to prove the congruence for Tr_{S}(χ), χ ∈ Irr(G). First we assume that Stab_{S}(χ) < S. Then (i) implies ind_{G^{S}}^{G}(res_{G^{S}}^{G}(Tr_{S}(χ))) ∈ pR(G) ⊆ pR(G)^{S} + R(G)_{<S}^{S}, and we also have Tr_{S}(χ) ∈ pR(G)^{S} + R(G)_{<S}^{S}. If Stab_{S}(χ) = S and λ ∈ Irr(G^{S}) is the Glauberman correspondent of χ, then res_{G^{S}}^{G}(χ) ≡ ±λ (mod pR(G^{S})) and ind_{G^{S}}^{G}(±λ) ≡ χ (mod pR(G)^{S} +R(G)_{<S}^{S}) by part (ii). This yields the result for χ.

2. The canonical induction formula

In this section we recall the definition of the canonical induction formula and some of its basic properties from [Bo].

For a finite group G let Ĝ := Hom(G, ℂ^{×}) ⊆ Irr(G) denote the multiplicative group of its one-dimensional characters and let R^{ab}(G) ⊆ R(G) denote the ℤ-span of Ĝ. Then R^{ab}(G) is naturally isomorphic to R(G^{ab}), where G^{ab} denotes the commutator factor group of G. We form the direct sum ⨁_{H≤G} R^{ab}(H) which has the set M(G) := {(H, φ) | H ≤ G, φ ∈ Ĥ} as an obvious ℤ-basis. The group G acts on ⨁_{H≤G} R^{ab}(H) by conjugation and leaves the basis M(G) invariant inducing the obvious conjugation action on M(G). We set

R_{+}^{ab}(G) := [⨁_{H≤G} R^{ab}(H)]_{G},

where the index G denotes taking G-coinvariants of a G-module, i.e. the largest factor group (modulo a G-submodule) with trivial G-action. In R_{+}^{ab}(G) the basis elements (H, φ) ∈ M(G) of ⨁_{H≤G} R^{ab}(H) become identified according to their distribution into G-orbits. Hence, the set G\M(G) of G-orbits can be considered as a ℤ-basis of R_{+}^{ab}(G). We write [H, φ]_{G} for the G-orbit of an element (H, φ) ∈ M(G) and consider [H, φ]_{G} also as the image of the basis element (H, φ) ∈ ⨁_{H≤G} R^{ab}(H) in R_{+}^{ab}(G). With these identifications we consider R_{+}^{ab}(G) as the free abelian group on the elements [H, φ]_{G} ∈ G\M(G). The induction map ⨁_{H≤G} ind_{H}^{G}: ⨁_{H≤G} R^{ab}(H) → R(G) which is invariant under conjugation induces the map

b_{G}: R_{+}^{ab}(G) → R(G), [H, φ]_{G} ↦ ind_{H}^{G}(φ).

The groups R_{+}^{ab}(H), H ≤ G, form a Mackey functor on G, a notion we don’t need in the sequel. We only need one particular feature of this structure, namely, for K ≤ H ≤ G, the existence of restriction maps

res_{+}_{K}^{H}: R_{+}^{ab}(H) → R_{+}^{ab}(K), [U, μ]_{H} ↦ ∑_{h∈K\H/U}[K ∩ ^{h}U, res_{K∩^{h}U}^{^{h}U}(^{h}μ)]_{K},

where h runs through a set of representatives in H of the double cosets K\H/U. The maps b_{H}: R_{+}^{ab}(H) → R(H), H ≤ G, form a morphism of Mackey functors; in particular, they commute with restriction maps.

If G′ denotes the commutator subgroup of G, the G′-fixed points of a G-module form naturally a G^{ab}-module, and this construction defines a map

p_{G}: R(G) → R^{ab}(G), Irr(G) ∋ χ ↦ \begin{cases}χ, & if χ ∈ Ĝ, \\ 0, & if χ ∉ Ĝ. \end{cases}

Finally, let π_{G}: R_{+}^{ab}(G) → R^{ab}(G) be the map induced by the projection ⨁_{H≤G} R^{ab}(H) → R^{ab}(G).

The next Theorem states that the family of maps b_{G}: R_{+}^{ab}(G) → R(G), where G runs through all finite groups G, which by the way are surjective according to Brauers induction theorem (cf. [Br]), has a unique functorial section a_{G}: R(G) → R_{+}^{ab}(G), called the canonical induction formula.

2.1 Theorem (cf. [Bo, Thm. 2.1]) There is a unique family of maps a_{G}: R(G) → R_{+}^{ab}(G), G running over all finite groups, such that

(i) res_{+}_{H}^{G} ∘ a_{G} = a_{H} ∘ res_{H}^{G}, whenever H ≤ G, and

(ii) π_{G} ∘ a_{G} = p_{G}.

Moreover, if for (H, φ) ∈ M(G) and χ ∈ R(G) the coefficient of a_{G}(χ) at the basis element [H, φ]_{G} of R_{+}^{ab}(G) is denoted by α_{(H,φ)}(χ) ∈ ℤ, i.e.

a_{G}(χ) = ∑_{[H,φ]_{G}∈G\M(G)} α_{(H,φ)}(χ)[H, φ]_{G} ∈ R_{+}^{ab}(G),

then:

(a) b_{G} ∘ a_{G} = id_{R(G)}, i.e.

χ = ∑_{[H,φ]_{G}∈G\M(G)} α_{(H,φ)}(χ)ind_{H}^{G}(φ)

for all χ ∈ R(G).

(b) For all φ ∈ Ĝ one has a_{G}(φ) = [G, φ]_{G}.

(c) For a genuine character χ and (H, φ) ∈ M(G) one has α_{(H,φ)}(χ) = 0 unless (res_{H}^{G}(χ), φ) ≠ 0.

(d) For χ ∈ R(G) and (H, φ) ∈ M(G) one has α_{(H,φ)}(χ) = 0 unless Z(G) ≤ H, where Z(G) denotes the centre of G.

(e) For any automorphism σ: G → G and any χ ∈ R(G) one has a_{G}(^{σ}χ) = ^{σ}(a_{G}(χ)), where ^{σ}χ := χ ∘ σ^{−1} and ^{σ}[H, φ]_{G} := [σ(H), φ ∘ σ^{−1}]_{G}.

(f) For any χ ∈ R(G) one has

∑_{[H,φ]_{G}∈G\M(G)} α_{(H,φ)}(χ) = χ(1).

3. The map gl^{G,S}

We keep the notation of the introduction. In particular, G and S are finite groups of coprime orders and S acts on G. In this Section we will define the map gl^{G,S}: R(G)^{S} → R(G^{S}). First we recall a preliminary lemma on coprime action (cf. [I2, 13.8 and 13.9]).

3.1 Lemma If Ω is a finite left GS-set such that G acts transitively on Ω, then

(i) there exists an S-stable element in Ω, i.e. Ω^{S} ≠ ∅, and

(ii) the set Ω^{S} is a transitive G^{S}-set.

Now we can define gl^{G,S} as the following composition

gl^{G,S}: R(G)^{S}⟶^{a_{G}}R_{+}^{ab}(G)^{S}⟶^{gl_{+}^{G,S}}R_{+}^{ab}(G^{S})⟶^{b_{G^{S}}}R(G^{S}),

where we still have to define the map gl_{+}^{G,S}. First note that by Theorem 2.1 (e), each S-fixed point of R(G) is mapped under a_{G} to an S-fixed point of R_{+}^{ab}(G), where S acts in the obvious way on R_{+}^{ab}(G). In order to define gl_{+}^{G,S} we note that the basis elements [H, φ]_{G} of R_{+}^{ab}(G), which correspond to G\M(G), are permuted by the S-action. Therefore, the S-orbit sums

∑_{s∈S/Stab_{S}([H,φ]_{G})}[^{s}H, ^{s}φ]_{G}, (H, φ) ∈ GS\M(G),

form a ℤ-basis of R_{+}^{ab}(G)^{S}. Let (H, φ) ∈ M(G) and let T := Stab_{S}([H, φ]_{G}) be the stabilizer of [H, φ]_{G} in S. Then GT acts G-transitively on the G-orbit of (H, φ). Hence, by Lemma 3.1 (i), there exists a T -fixed point (H′, φ′) in the G-orbit of (H, φ), i.e. T = Stab_{S}([H′, φ′]_{G}) = Stab_{S}((H′, φ′)). With this notation we now define

gl_{+}^{G,S}( ∑_{s∈S/T}[^{s}H, ^{s}φ]_{G}) := \begin{cases}[G^{S} ∩ H′, res_{G^{S}∩H′}^{H′}(φ′)]_{G^{S}}, & if T = S, \\ 0, & if T < S. \end{cases}

If T = S, we can write H′^{S} instead of G^{S} ∩ H′. By Lemma 3.1 (ii), the definition of gl_{+}^{G,S}, does not depend on the choice of the S-stable representative (H′, φ′) within the G-orbit [H, φ]_{G}.

3.2 Proposition For φ ∈ Ĝ we have

gl^{G,S}(Tr_{S}(φ)) = \begin{cases}res_{G^{S}}^{G}(φ) = π^{G,S}(φ), & if Stab_{S}(φ) = S, \\ 0, & if Stab_{S}(φ) < S. \end{cases}

Proof This is immediate from Theorem 2.1 (b).

3.3 Proposition Assume that S is a p-group.

(i) For χ ∈ R(G)^{S} we have

gl^{G,S}(χ) ≡ res_{G^{S}}^{G}(χ) (mod pR(G^{S})).

(ii) For χ ∈ R(G)_{<S}^{S} we have

gl^{G,S}(χ) ≡ 0 (mod pR(G^{S})).

Proof (i) Since gl^{G,S} = b_{G^{S}} ∘ gl_{+}^{G,S} ∘ a_{G} and res_{G^{S}}^{G} = b_{G^{S}} ∘ res_{+}_{G^{S}}^{G} ∘ a_{G}, it suffices to show that

b_{G^{S}} ∘ gl_{+}^{G,S} ≡ b_{G^{S}} ∘ res_{+}_{G^{S}}^{G} (mod pR(G^{S})).

Let (H, φ) ∈ M(G) with T := Stab_{S}([H, φ]_{G}) = Stab_{S}(H, φ). If T < S, then gl_{+}^{G,S}(∑_{s∈S/T} [^{s}H, ^{s}φ]_{G}) = 0 and

res_{+}_{G^{S}}^{G}( ∑_{s∈S/T}[^{s}H, ^{s}φ]_{G}) = ∑_{s∈S/T} ^{s}(res_{+}_{G^{S}}^{G}([H, φ]_{G})) = |S/T | · res_{+}_{G^{S}}^{G}([H, φ]_{G})

so that the congruence holds in this case. From now on we assume that (H, φ) is S-stable. Then we have by Proposition 1.6 (iii):

(b_{G^{S}} ∘ gl_{+}^{G,S})([H, φ]_{G}) ≡ (res_{G^{S}}^{G} ∘ ind_{G^{S}}^{G} ∘ b_{G^{S}} ∘ gl_{+}^{G,S})([H, φ]_{G}) (mod pR(G^{S}))

= (res_{G^{S}}^{G} ∘ ind_{G^{S}}^{G} ∘ b_{G^{S}})([H^{S}, res_{H^{S}}^{H}(φ)]_{G^{S}})
= (res_{G^{S}}^{G} ∘ ind_{G^{S}}^{G} ∘ ind_{H^{S}}^{G^{S}})(res_{H^{S}}^{H}(φ))
= (res_{G^{S}}^{G} ∘ ind_{H}^{G})(ind_{H^{S}}^{H}(res_{H^{S}}^{H}(φ))).

By Proposition 1.6 (iv) we have

ind_{H^{S}}^{H}(res_{H^{S}}^{H}(φ)) ≡ φ (mod pR(H) + R(H)_{<S}^{S}).

Since ind_{H}^{G}(R(H)_{<S}^{S}) ⊆ pR(G)^{S} +R(G)_{<S}^{S}, we obtain from Proposition 1.6 (i) that

(b_{G^{S}} ∘ gl_{+}^{G,S})([H, φ]_{G}) ≡ res_{G^{S}}^{G}(ind_{H}^{G}(φ)) (mod pR(G^{S})).

On the other hand

(b_{G^{S}} ∘ res_{+}_{G^{S}}^{G})([H, φ]_{G}) = (res_{G^{S}}^{G} ∘ b_{G})([H, φ]_{G}) = res_{G^{S}}^{G}(ind_{H}^{G}(φ)).

(ii) This follows from part (i) and Proposition 1.6 (i).

3.4 Remark Proposition 3.3 (i) together with Theorem 1.2 shows that the map gl^{G,S} can be used as a definition for the Glauberman correspondence in the case, where S is a p-group. But note, that gl^{G,S} is defined for arbitrary S and G of relatively prime orders. It might be that the Glauberman correspondence can be defined directly from gl^{G,S} without going step by step through a composition series of S. But we have no good evidence for this at the moment. Interesting examples tend to be too big even for computers. And without interesting examples we don’t even know what we would like to prove about gl^{G,S} in the general case. The few examples we know, still allow that gl^{G,S}(χ) is a multiple of a single λ ∈ Irr(G^{S}) for χ ∈ Irr(G)^{S}, but we don’t believe in this property in general. Moreover, it would be very nice if the map gl^{G,S} allowed a unified construction of these two corrspondences. But again we don’t have any serious results relating Isaacs’ correspondence to the map gl^{G,S}.

3.5 Example (i) Let G =< a, b | a^{4} = b^{4} = 1, a^{2} = b^{2}, aba^{−1} = b^{−1} > be the quaternion group of order 8 and let S =< s > be the cyclic group of order 3 acting on G by ^{s}a = b, ^{s}b = ab. Let H_{1} :=< a >, H_{2} :=< b >, H_{3} :=< ab > denote the subgroups of order 4 and Z :=< a^{2} > the centre of G. Then G^{S} = Z and Irr(G)^{S} = {χ} for the irreducible character χ of degree 2. If φ_{i} ∈ Ĥ_{i} and τ ∈ Ẑ denote faithful one-dimensional characters, then the canonical induction formula of G on χ ∈ Irr(G)^{S} is given by

a_{G}(χ) = ∑_{i=1}^{3}[H_{i}, φ_{i}]_{G} − [Z, τ]_{G}.

Applying the definition of gl^{G,S} to χ we obtain

gl^{G,S}(χ) = −τ,

which is exactly the Glauberman correspondent endowed with the right sign. Note that Glauberman’s correspondence not only gives a bijection of irreducible characters but also associates to each χ ∈ Irr(G)^{S} a sign (see [I2] for more about the definition of this sign).

(ii) Let G be an extraspecial group of order p^{2n+1} and exponent p for some odd prime p. Assume furthermore that S is a finite group acting on G with (|S|, |G|) = 1 such that G^{S} = Z, where Z denotes the centre of G. Then all p − 1 non-linear irreducible characters of G are fixed by S. Let χ ∈ Irr(G) be one of those and let res_{Z}^{G}(χ) = p^{n}λ for some λ ∈ Ẑ. Then π^{G,S}(χ) = λ.

We will compare this with gl^{G,S}(χ). In Proposition 4.2 we will show that

a_{G}(χ) = ∑_{d=0}^{n} ∑_{{Z≤H abelian \atop |H/Z|=p^{n−d}}}(−1)^{d}p^{d(d−1)}[H, φ]_{G},

where φ ∈ Ĥ is an arbitrary extension of λ ∈ Irr(Z). Let H be an abelian subgroup of G with Z ≤ H and let φ be an extension of λ. Then [H, φ]_{G} is S-stable if and only if H is S-stable, since ^{s}φ is again an extension of λ ∈ Irr(Z), and therefore G-conjugate to φ, cf. the proof of Proposition 4.2. Since gl_{+}^{G,S} is trivial on S-orbit sums of [H, φ]_{G} with Stab_{S}([H, φ]_{G}) < S, we obtain

gl^{G,S}(χ) = (∑_{d=0}^{n} ∑_{Z≤H abelian H S-invariant |H/Z|=p^{n−d}} (−1)^{d}p^{d(d−1)}) · λ.

If we specialize further, and assume that Z is the only S-stable abelian subgroup containing Z, then we obtain

gl^{G,S}(χ) = (−1)^{n}p^{n(n−1)}λ.

At the other extreme, if all abelian subgroups of G containing Z are Sinvariant, we obtain

gl^{G,S}(χ) = p^{n}λ.

In fact,

∑_{d=0}^{n}(−1)^{d}p^{d(d−1)}#{Z ≤ H abelian | |H/Z| = p^{n−d}} =

= ∑_{[H,φ]_{G}∈G\M(G)} α_{(H,φ)}(χ) = χ(1) = p^{n}

by Theorem 2.1 (f), where α_{(H,φ)}(χ) ∈ ℤ.

4. The canonical induction formula for extraspecial pgroups of exponent p.

In this section we will compute the map a_{G}: R(G) → R_{+}^{ab}(G) for the extraspecial p-group G of exponent p and order p^{2n+1}, where n ≥ 1 and p is an odd prime. For details about extraspecial p-groups see for example [H, III.13] and [Su, IV.4, Exerc. 4], for symplectic forms see [H, II.9], and for the representation theory of G see [H, V.16.14].

4.1 The centre Z of G has order p and G/Z is a symplectic 𝔽_{p}-vector space of dimension 2n, where the symplectic form is given by the commutator, if one identifies Z with 𝔽_{p} once and for all. Note that Z equals the Frattini subgroup of G and also the commutator subgroup of G. A subgroup H ≥ Z of G is abelian if and only if H/Z is a totally isotropic subspace of G/Z. The automorphism group Aut(G) has the normal subgroup Inn(G) of inner automorphisms which consists precisely of those automorphisms which act trivially on Z and G/Z. Each automorphism of G induces an automorphism of Z. This yields a surjective group homomorphism β: Aut(G) → Aut(Z) ≅ 𝔽_{p}^{×}, and we obtain a series

1 < Inn(G) < ker(β) < Aut(G)

with Aut(G)/ ker(β) ≅ 𝔽_{p}^{×} and ker(β)/Inn(G) ≅ Sp_{2n}(𝔽_{p}), the group of symplectic 2n by 2n matrices over 𝔽_{p}, since each element of ker(β) respects the symplectic structure of G/Z, and since each element in Sp_{2n}(𝔽_{p}) can be realized as an automorphism of G. Therefore we will denote ker(β) also by S̃p_{2n} ≤ Aut(G). Note that, by a theorem of Witt, Sp_{2n}(𝔽_{p}) acts transitively on the set of totally isotropic subspaces of G/Z of fixed dimension. Hence, S̃p_{2n} acts transitively on the set of abelian subgroups H ≥ Z of G of fixed order. The maximal abelian subgroups of G are all conjugate under S̃p_{2n}, since they all contain Z. Moreover, a maximal abelian subgroup of G has order p^{n+1}.

The set Irr(G) consists of p^{2n} one-dimensional characters and p − 1 characters of degree p^{n}. Let λ be one of the p−1 non-trivial irreducible characters of Z, let λ̃ be an extension of λ to a maximal abelian subgroup H ≤ G, then ind_{H}^{G}(λ̃) is irreducible and depends only on λ and not on H and λ̃. Since ind_{H}^{G}(λ̃) is irreducible, the stabilizer of λ̃ in G is H, and therefore, λ̃ has exactly p^{n} G-conjugates which have to be precisely the p^{n} different extensions of λ to H, since λ is stable under G-conjugation.

4.2 Proposition Let χ ∈ Irr(G) be of degree p^{n} and let λ ∈ Irr(Z) with res_{Z}^{G}(χ) = p^{n} · λ. Then

a_{G}(χ) = ∑_{d=0}^{n}(−1)^{d}p^{d(d−1)} ∑_{{Z ≤ H abelian \atop |H/Z|=p^{n−d}}}[H, φ]_{G},

where φ ∈ Ĥ is an arbitrary extension of λ to H.

Proof We write

a_{G}(χ) = ∑_{[H,φ]_{G}∈G\M(G)} α_{(H,φ)}(χ)[H, φ]_{G},

with α_{(H,φ)}(χ) ∈ ℤ as in Theorem 2.1. By Theorem 2.1 (c) and (d) we obtain α_{(H,φ)}(χ) = 0 unless Z ≤ H and λ = res_{Z}^{H}(φ). But in this case H is abelian. In fact, if the commutator subgroup H′ of H were non-trivial, then H′ ≤ G′ = Z, hence H′ = Z, and res_{H′}^{H}(φ) = 1, which is a contradiction to res_{Z}^{H}(φ) = λ. For a given abelian subgroup H with Z ≤ H, the extensions φ of λ form a full G-orbit, since they do in the case of a maximal abelian subgroup of G. Therefore, we obtain

a_{G}(χ) = ∑_{Z ≤ H abelian} α_{H}[H, φ]_{G},

where α_{H} ∈ ℤ and φ is an extension of λ to H. Let X denote the poset of abelian subgroups of G containing Z. For each d ∈ {0, …, n}, the subgroup S̃p_{2n} ∈ Aut(G) (which fixes λ and hence χ) acts transitively on the set of elements H ∈ X with |H/Z| = p^{n−d}. Hence, by Theorem 2.1 (e) we obtain α_{H} = α_{H′} for H, H′ ∈ X with |H| = |H′|. Therefore, we can write

a_{G}(χ) = ∑_{d=0}^{n} α_{d} ∑_{{H∈X \atop |H/Z|=p^{n−d}}} [H, φ]_{G},  (1)

where α_{d} ∈ ℤ and φ is an extension of λ to H. Now it suffices to show that

α_{d} = (−1)^{d}p^{d(d−1)}

for d ∈ {0, …, n}. Let H_{0} ∈ X be a maximal abelian subgroup of G, and for d ∈ {0, …, n} let H_{d} ∈ X be a subgroup of H_{0} of index p^{d}. We fix e ∈ {0, …, n}. Then, after applying π_{H_{e}} ∘ res_{+}_{H_{e}}^{G} to both sides of Equation (1), Theorem 2.1 (i) and (ii) implies

p_{H_{e}}(res_{H_{e}}^{G}(χ)) = ∑_{d=0}^{n} α_{d} ∑_{{H∈X \atop |H/Z|=p^{n−d}}} π_{H_{e}}(res_{+}_{H_{e}}^{G}([H, φ]_{G}))  (2)

in R^{ab}(H_{e}). Now we fix an extension λ̃_{e} ∈ Irr(H_{e}) of λ and compare the multiplicities of λ̃_{e} in both sides of the above equation. Since res_{H_{0}}^{G}(χ) is the sum of all different extensions of λ to H_{0}, we have res_{H_{e}}^{G}(χ) = p^{e} ∑ λ̃, where the sum runs over all extensions λ̃ of λ to H_{e}. Hence, the multiplicity of λ̃_{e} in the left hand side of the above equation equals p^{e}. Recall that

res_{+}_{H_{e}}^{G}([H, φ]_{G}) = ∑_{g∈H_{e}\G/H}[H_{e} ∩ ^{g}H, res_{H_{e}∩^{g}H}^{^{g}H}(^{g}φ)]_{H_{e}},

and that the multiplicity of λ̃_{e} in (π_{H_{e}} ∘ res_{+}_{H_{e}}^{G})([H, φ]_{G}) is the coefficient of [H_{e}, λ̃_{e}]_{H_{e}} in res_{+}_{H_{e}}^{G}([H, φ]_{G}). If H_{e} ≰ H, this coefficient is zero. If H_{e} ≤ H, then each of the |H/Z| different G-conjugates (extensions of λ) of φ occur (G : H)/(H : Z) times in the above sum. Among these |H/Z| different Gconjugates of φ, precisely (H : H_{e}) lie above λ̃_{e}. Thus, the multiplicity of λ̃_{e} in (π_{H_{e}} ∘ res_{+}_{H_{e}}^{G})([H, φ]_{G}) equals

\frac{|H|}{|H_{e}|} · \frac{|G|}{|H|} · \frac{|Z|}{|H|} = p^{e+d},

if |H/Z| = p^{n−d}. Hence by comparing the multiplicities of λ̃_{e} in both sides of Equation (2) we obtain (after dividing by p^{e})

1 = ∑_{d=0}^{e} α_{d} · p^{d} · #{H ∈ X | H_{e} ≤ H, |H/Z| = p^{n−d}}

for every e ∈ {0, …, n}.

Since these equations determine α_{d} inductively, it suffices to prove that for given U ∈ X we have

1 = ∑_{U≤H∈X}(−1)^{d(H)}p^{d(H)^{2}},

where d(H) is defined by |H/Z| = p^{n−d(H)}. Now, Möbius inversion in the poset X (cf. [R, Cor. 1 to Prop. 2]) shows that the above equations for U ∈ X are equivalent to the equations

(−1)^{d(U)}p^{d(U)^{2}} = ∑_{U≤H∈X} μ_{X} (U, H),

for U ∈ X. Next we fix U ∈ X. Note that each H ∈ X with U ≤ H is contained in the centralizer C_{G}(U), and C_{G}(U)/Z = (U/Z)^{⊥} with respect to the symplectic form. The original symplectic form on G/Z induces a non-degenerate symplectic form on (C_{G}(U)/Z)/(U/Z) ≅ C_{G}(U)/U, and the poset {H ∈ X | U ≤ H} is isomorphic to the poset of totally isotropic subspaces in (C_{G}(U)/Z)/(U/Z). Now the above equation follows from the following Lemma, noting that μ_{X} (U, H) is the number of chains σ connecting U and H counted with multiplicity (−1)^{|σ|}, and that the above sum equals −μ_{X ∪^{.}ˆ1}(U, ˆ1), if we add an element ˆ1 to X which is bigger than every element in X.

4.3 Lemma Let V be a non-degenerate symplectic 𝔽_{p}-vector space of dimension 2d, and let X_{V} denote the poset of all totally isotropic subspaces of V together with a greatest element ˆ1. Then we have

μ_{X_{V}} (0, ˆ1) = (−1)^{d+1}p^{d^{2}}.

Proof We prove the above equation by induction on d. The cases d = 0 and d = 1 can be verified easily. Let d ≥ 2 and consider a decomposition

V = W ⊥ H

of V into a non-degenerate symplectic subspace W ≠ 0 of dimension 2d − 2 and a hyperbolic plane H. We consider the two maps

f: X_{W} → X_{V}^{op}, U ↦ U,  ˆ1 ↦ ˆ1,
g: X_{V}^{op} → X_{W} , U ↦ U ∩ W, ˆ1 ↦ ˆ1,

where X_{V}^{op} denotes the opposite poset of X_{V}, i.e. the elements are the same and the relations are reversed. Obviously f and g are order reversing, and they form a Galois connection, i.e. f(g(v)) ≥ v and g(f(w)) ≥ w for v ∈ X_{V}^{op} and w ∈ X_{V}. A fundamental property of Galois connections (cf. [Wa, Thm. 4.1] for example) is the fact that, for each v ∈ X_{V}^{op} and w ∈ X_{W}, we have

∑_{t∈f^{−1}({v})} μ_{X_{W}} (w, t) = ∑_{s∈g^{−1}({w})} μ_{X_{V}^{op}}(v, s).

We apply this to v = ˆ1 ∈ X_{V}^{op} and w = 0 ∈ X_{W}, and obtain

μ_{X_{W}} (0, ˆ1) = ∑_{{U∈X_{V} \\{ˆ1\} \atop U∩W =0}} μ_{X_{V}} (U, ˆ1).

For arbitrary U, U′ ∈ X_{V} \{ˆ1}, the relation U ≤ U′ implies U′ ≤ U^{⊥}. The factor space U^{⊥}/U is again a non-degenerate symplectic 𝔽_{p}-vector space of dimension 2(d−dim U), and we have, for given U ∈ X_{V} \{ˆ1}, an isomorphism of posets

{U′ ∈ X_{V} | U ≤ U′} → X_{U^{⊥}/U}, U′ ↦ U′/U, ˆ1 ↦ ˆ1.

If U ∩ W = 0, we have dim U ≤ 2, and the last equation can be written as

a_{d−1} = n_{0}a_{d} + n_{1}a_{d−1} + n_{2}a_{d−2},  (3)

where a_{i} := μ_{X_{V′}}(0, ˆ1) for a non-degenerate symplectic 𝔽_{p}-vector space V′ of dimension 2i, i ∈ ℕ, and where

n_{i} := |{U ∈ X_{V} \ {ˆ1} | dim U = i, U ∩ W = 0}|

for i ∈ {0, 1, 2}. Obviously, n_{0} = 1, and

n_{1} = \frac{p^{2d} − p^{2d−2}}{p − 1} = p^{2d−1} + p^{2d−2},

since we may generate U by a vector in V \ W and have p − 1 choices. The computation of n_{2} is more difficult. First we note that there are

\frac{(p^{2d−1} − 1)(p^{2d−2} − 1)}{(p^{2} − 1)(p − 1)}

elements U ∈ X_{V} \ {ˆ1} with dim U = 2. In fact, there are (p^{2d} − 1)(p^{2d−1} − p) ordered pairs (v, v′) of linearly independent vectors v, v′ ∈ V with v′ ⊥ v. On the other hand each such U has (p^{2} − 1)(p^{2} − p) different ordered bases.

Among these U ∈ X_{V} \ {ˆ1} with dim U = 2 there are by the same argument

\frac{(p^{2(d−1)} − 1)(p^{2(d−1)−2} − 1)}{(p^{2} − 1)(p − 1)}

elements with dim(U ∩ W) = 2, and there are

\frac{(p^{2d−2} − 1)(p^{2d−2} − p^{2d−4})}{(p − 1)^{2}}

elements with dim(U ∩ W) = 1, since there are (p^{2d−2} − 1)(p^{2d−1} − p^{2d−3}) pairs (w, v) of linearly independent vectors w ∈ W and v ∈ W^{⊥} \ (W ∩ W^{⊥}), and each such U has exactly (p−1)(p^{2} −p) ordered bases (w, v) with w ∈ W.

Hence, we have

n_{2} = \frac{(p^{2d} − 1)(p^{2d−2} − 1) − (p^{2d−2} − 1)(p^{2d−4} − 1)}{(p^{2} − 1)(p − 1)} −

− \frac{(p + 1)(p^{2d−2} − 1)(p^{2d−2} − p^{2d−4})}{(p^{2} − 1)(p − 1)}

= \frac{p^{2d−2} − 1}{(p^{2} − 1)(p − 1)}[p^{2d} − 1 − p^{2d−4} + 1 − p^{2d−1} + p^{2d−3} − p^{2d−2} + p^{2d−4}]
= \frac{p^{2d−2} − 1}{(p^{2} − 1)(p − 1)}[p^{2d} − p^{2d−2} − p^{2d−1} + p^{2d−3}]
= \frac{p^{2d−2} − 1}{(p^{2} − 1)(p − 1)}(p^{2} − 1)(p^{2d−2} − p^{2d−3})
= p^{2d−3}(p^{2d−2} − 1).

Using the induction hypothesis for a_{d−1} and a_{d−2}, we obtain from Equation (3)

a_{d} = (1 − n_{1})a_{d−1} − n_{2}a_{d−2}
= (1 − (p + 1)p^{2d−2})(−1)^{d}p^{(d−1)^{2}} − p^{2d−3}(p^{2d−2} − 1)(−1)^{d−1}p^{(d−2)^{2}}
= (−1)^{d+1}(−p^{d^{2}−2d+1} + p^{d2} + p^{d^{2}−1} − p^{4d−5+(d−2)^{2}} + p^{2d−3+(d−2)^{2}})
= (−1)^{d+1}(−p^{d^{2}−2d+1} + p^{d^{2}} + p^{d^{2}−1} − p^{d^{2}−1} + p^{d^{2}−2d+1})
= (−1)^{d+1}p^{d^{2}},

which completes the proof of the lemma.

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