REPRESENTATIONS OF THE DOUBLE BURNSIDE ALGEBRA AND COHOMOLOGY OF THE EXTRASPECIAL p-GROUP

AKIHIKO HIDA AND NOBUAKI YAGITA

Abstract. Let E be the extraspecial p-group of order p^{3} and exponent p where p is an odd prime. We determine the mod p cohomology of summands in the stable splitting of p-completed classifying space BE modulo nilpotence. It is well known that indecomposable summands in the complete stable splitting correspond to simple modules for the mod p double Burnside algebra. We shall use representation theory of the double Burnside algebra and the theory of biset functors.

1. Introduction

Let p be an odd prime and E = p_{+}^{1+2} the extraspecial p-group of order p^{3} and exponent p. The (integral or mod p) cohomology ring of E is widely studied (for example, [9], [10], [11], [12], [17], [18], [20], [21]). In particular, in [21], the second author consider cohomology rings of finite groups with Sylow p-subgroup E, or more generally, cohomology rings of saturated fusion systems on E, and obtained the cohomology of various summands in the stable splitting of BE where BE is a p-completed classifying space of E.

Let P be a finite p-group where p is an arbitrary prime. By the Segal conjecture (Carlsson’s theorem [5]) and the result of Lewis, May and McClure [13], it is well known that, in the stable homotopy category, the endomorphism ring of BE is isomorphic to the completion of the double Burnside ring A_{ℤ}(P, P). Consequently, indecomposable stable summands of BP correspond to primitive idempotents of the double Burnside algebra A_{p}(P, P) over a finite field 𝔽_{p}, or equivalently, simple A_{p}(P, P)-modules (except the one dimensional module with trivial minimal subgroup). Benson and Feshbach [2] and Martino and Priddy [14] classified simple A_{p}(P, P)-modules and studied stable splittings of BP for various p-groups. See Benson’s survey [1] for more details.

In this paper, we shall determine the cohomology of stable summands of BE for the extraspecial p-group E = p_{+}^{1+2} through the action of double Burnside algebra A_{p}(E, E) over 𝔽_{p}. Let

BE = ⋁_{i} ⋁_{1≤j≤m(i)} X_{ij}

be the complete stable splitting of BE such that X_{ij} ∼ X_{mn} if and only if i = m. Let

1 = ∑_{i} ∑_{1≤j≤m(i)} e_{ij}

be a corresponding decomposition of unity into orthogonal primitive idempotents in A_{p}(E, E). Consider the sum X_{i} = ∨_{1≤j≤m(i)}X_{ij} of all equivalent indecomposable summands. Let e_{i} = ∑_{1≤j≤m(i)} e_{ij}. We want to consider the cohomology of X_{i} modulo nilpotence. However, the cohomology of X_{i} does not have ring structure which relates to the structure of the cohomology ring of BE. Hence, in this paper, let

Date: October 2, 2012.

H^{∗}(X_{i}) = H^{∗}(E)e_{i}

where

H^{∗}(E) = (𝔽_{p} ⊗ H^{∗}(BE; ℤ))/\sqrt{0}.

Note that A_{p}(E, E) acts on H^{∗}(E) (see Lemma 3.4 below) and H^{∗}(X_{i}) is actually defined, though it depends on the choice of an idempotent e_{i}. The structure of H^{∗}(E) is known ([10], [18]) and the main purpose of this paper is to determine H^{∗}(X_{i}) for every summand X_{i}, see section 10 below.

For example, let X(𝔽_{p}) be the principal dominant summand of BE, namely, the summand corresponding to the trivial Out(E)-module. In [21], it was shown that H^{∗}(X(𝔽_{p})) is the Dickson subalgebra of H^{∗}(E) for p = 3, 5, 7. These are obtained by the cohomology of the fourth Janko group J_{4} (p = 3)(Green [9]), the Thompson group T h (p = 5) and the exotic saturated fusion systems of Ruiz and Viruel [16] (p = 7). On the other hand, in this paper, we prove this result for every p without use of large sporadic simple groups.

Corollary 10.7. Let X(𝔽_{p}) be the principal dominant summand of BE. Then H^{∗}(X(𝔽_{p})) is isomorphic to the (positive degree part of) Dickson subalgebra 𝔻𝔸 of H^{∗}(E).

Let S be a simple A_{p}(E, E)-module corresponding to X_{ij}. The multiplicity m(i) is equal to the dimension of S over 𝔽_{p}. Note that 𝔽_{p} is a splitting field of A_{p}(E, E). Dietz and Priddy [6] studied the stable splitting of BE, and in particular, determined the multiplicity m(i). Here, we reprove this result using cohomology, see Proposition 10.1.

Note that it may be possible to consider mod p cohomology H^{∗}(BE; 𝔽_{p})/\sqrt{0} instead of H^{∗}(E) = (𝔽_{p} ⊗ H^{∗}(BE; ℤ))/\sqrt{0}. In general, H^{∗}(E) is a subalgebra of H^{∗}(BE; 𝔽_{p})/\sqrt{0} and coincide with H^{∗}(BE; 𝔽_{p})/\sqrt{0} if p > 3. But for p = 3, H^{∗}(E) is strictly smaller than H^{∗}(BE; 𝔽_{p})/\sqrt{0}. Here, we consider only H^{∗}(E) because the structure of H^{∗}(E) is easier than H^{∗}(BE; 𝔽_{p})/\sqrt{0} for p = 3 and we can treat all odd primes p uniformly. Moreover, the A_{p}(E, E)-modules structure of H^{∗}(BE; 𝔽_{p}) can be deduced from that of H^{∗}(E) by [20], [21].

Let P be a finite p-group. Let A_{p}(P, P) be the double Burnside algebra over 𝔽_{p}. Simple A_{p}(P, P)-modules are classified by [2], [14]. In this paper, we shall follow the functorial approach by Webb [19] and Bouc, Stancu and Thévenaz [4]. In section 3, we review some results in [4].

In section 4, we summarize the results on the structure of H^{∗}(E). In section 5, we study the cohomology of a maximal elementary p-subgroup A of E. We compare the GL_{2}(𝔽_{p})-module structures of H^{∗}(A) and H^{∗}(E) using the morphism q^{∗} : H^{∗}(A) ⟶ H^{∗}(E) induced by a surjective morphism q : E ⟶ A. Note that Out(E) ≃ Out(A) ≃ GL_{2}(𝔽_{p}).

In section 6, we calculate the image of the transfer map from the maximal elementary abelian p-subgroup A of E. In section 7, we study some A_{p}(E, E)-submodules of H^{∗}(E). In section 8, we consider simple A_{p}(E, E)-modules with cyclic minimal subgroup. Similarly, in section 9, we consider simple A_{p}(E, E)-modules with minimal subgroup A.

COHOMOLOGY OF EXTRASPECIAL p-GROUP 3

Using these results, in section 10, we determine H^{∗}(X_{i}) = H^{∗}(E)e_{i} for every X_{i}. More precisely, we construct an 𝔽_{p}-subspace W of H^{∗}(E) such that the multiplication by e_{i} induces an isomorphism W ⟶^{∼} W e_{i} = H^{∗}(E)e_{i} = H^{∗}(X_{i}). Note that, in general, neither W nor H^{∗}(X_{i}) is an A_{p}(E, E)-submodule of H^{∗}(E).

2. Finite dimensional algebras and modules

Let Λ be a finite dimensional algebra over a filed k. We denote the Jacobson radical of Λ by rad(Λ), namely, rad(Λ) is the intersection of all maximal ideals of Λ. If e is a primitive idempotent in Λ, then eΛ is a projective indecomposable right Λ-module and eΛ/erad(Λ) is a simple right Λ-module. Let

1 = ∑_{1≤i≤l} ∑_{1≤j≤m(i)} e_{ij}

be a decomposition of unity into primitive orthogonal idempotents in Λ, where

e_{ij}Λ/e_{ij}rad(Λ) ≃ e_{mn}Λ/e_{mn}rad(Λ)

if and only of i = m. Then

S_{i} = e_{i1}Λ/e_{i1}rad(Λ) (1 ≤ i ≤ l)

gives the complete set of representatives of isomorphism classes of simple right Λ-modules. Let e_{i} = ∑_{1≤j≤m(i)} e_{ij} and, in this paper, we call e_{i} an idempotent corresponding to the simple module S_{i}. Multiplication by e_{i} induces the identity map on S_{i} and S_{j}e_{i} = 0 for j ≠ i. On the other hand, we have

S_{i}e_{i1} ≃ Hom_{Λ}(e_{i1}Λ, S_{i}) ≃ End_{Λ}(S_{i})

and the multiplicity m(i) is equal to dim_{End_{Λ}(S_{i})} S_{i}.

Let M be a finite dimensional right Λ-module. Then M has a composition series

0 = M_{0} ⊂ M_{1} ⊂ ⋯ ⊂ M_{n} = M

such that each quotient M_{j}/M_{j−1} is a simple Λ-module. The number of quotients M_{j}/M_{j−1} such that M_{j}/M_{j−1} ≃ S_{i} equals to

dim_{k} Me_{i1}/ dim_{k} End_{Λ}(S_{i}) = dim_{k} Me_{i}/m(i) dim_{k} End_{Λ}(S_{i}) = dim_{k} Me_{i}/ dim_{k} S_{i}.

If every composition factor of M is isomorphic to S_{i}, then M = Me_{i} and the right multiplication by e_{i} induces the identity on M. In general case, to determine Me_{i}, we shall use the following lemma in section 10. Note that H^{∗}(E) is not finite dimensional, but each homogeneous part H^{n}(E) is a finite dimensional A_{p}(E, E)- module and we can apply the lemma.

Lemma 2.1. Let M be a finite dimensional Λ-module and S a simple Λ-module. Let e be an idempotent corresponding to S. Let

0 ⊂ L ⊂ N ⊂ M

be a sequence of Λ-submodules. Suppose that M/N has no composition factor which is isomorphic to S and every simple subqutiont module of N/L is isomorphic to S. If W is a k-subspace of N such that N = L ⊕ W, then the multiplication by e induces an isomorphism

e : W ⊕ Le ≃ W e ⊕ Le = Me.

Proof. Since (M/N)e = 0, we have Me = Ne = W e + Le. Hence the map e : W ⊕ Le ⟶ Me is surjective. On the other hand, since (N/L)e = N/L, this map is injective and Me = W e ⊕ Le. □

3. Biset category and biset functor

In this section, we summarize the theory of biset category and functors in [4]. Here, we concentrate on right free bisets only, though, in [4], Bouc, Stancu and Thévenaz consider the general bisets.

Let G and H be finite groups and let A_{ℤ}(G, H) be the Grothendieck group of the category of the finite right free (G, H)-biset. Let K ≤ G and let φ : K ⟶ H be a group homomorphism. We set

G ×_{(K,φ)} H = G × H/ ∼

where (gk, h) ∼ (g, φ(k)h) for g ∈ G, k ∈ K and h ∈ H. Then every transitive right free (G, H)-set is isomorphic to G ×_{(K,φ)} H for some K and φ : K ⟶ H. Let ζ_{K,φ} be the element of A_{ℤ}(G, H) corresponding to G ×_{(K,φ)} H. Thus A_{ℤ}(G, H) is a free abelian group with one basis element for each conjugacy class of the pair (K, φ) where K ≤ G and φ : K ⟶ H.

There exists a product

A_{ℤ}(G_{2}, G_{3}) × A_{ℤ}(G_{1}, G_{2}) ⟶ A_{ℤ}(G_{1}, G_{3})

induced by the map

(X, Y) ↦ Y ×_{G_{2}} X.

If H_{1} ≤ G_{1}, H_{2} ≤ G_{2}, ϕ_{1} : H_{1} ⟶ G_{2}, ϕ_{2} : H_{2} ⟶ G_{3}, then the product is given by

ζ_{H_{2},ϕ_{2}}ζ_{H_{1},ϕ_{1}} = ∑_{x∈ϕ_{1}(H_{1})\G_{2}/H_{2}} ζ_{ϕ_{1}^{−1}(ϕ_{1}(H_{1})∩^{x}H_{2}),ϕ_{2}c_{x^{−1}} ϕ_{1}}

(see [2, p.160]).

Let k be a field. We set A_{k}(G, H) = k ⊗ A_{ℤ}(G, H). Then A_{k}(G, G) is a finite dimensional k-algebra. We denote by J(G) the k-subspace of A_{k}(G, G) spanned by all ζ_{H,φ} such that φ(H) < G. Then J(G) is an ideal of A_{k}(G, G).

There exists an injective k-algebra homomorphism

ι : kOut(G) ⟶ A_{k}(G, G)

given by φ ↦ ζ_{G,φ} where we denote the automorphism of G which represents φ by the same notation φ. Note that φ_{1}, φ_{2} ∈ Out(G) then ζ_{G,φ_{1}}ζ_{G,φ_{2}} = ζ_{G,φ_{1}φ_{2}} and in fact ι is an algebra homomorphism. On the other hand, there exists a surjective k-algebra homomorphism

π : A_{k}(G, G) ⟶ kOut(G)

such that Ker π = J(G) and πι is the identity map on kOut(G). We view kOut(G)- modules as A_{k}(G, G)-modules via π. On the other hand, if W is an A_{k}(G, G)- module, then we can view W as a kOut(G)-module via ι.

Let C_{k} be the k-linear category whose object are all finite groups and morphisms are given by Hom_{C_{k}}(G, H) = A_{k}(G, H). A contravariant k-linear functor

C_{k} ⟶ k-Mod

is called an inflation functor [19].

Let F be an inflation functor. If we define the right action of A_{k}(H, H) on F (H) by

vϕ = F (ϕ)(v)

for ϕ ∈ A_{k}(H, H) and v ∈ F (H), then F (H) is a right A_{k}(H, H)-module since F is a contravariant functor. If V a k-subspace of F (H), we set

V A_{k}(G, H) = ∑_{ϕ∈A_{k}(G,H)} F (ϕ)(V)

for a finite group G, then V A_{k}(−, H) is a subfunctor of F.

Let W be a simple A_{k}(G, G)-module. Then the inflation functor

L_{G,W} = W ⊗_{A_{k}(G,G)} A_{k}(−, G).

has the unique maximal subfunctor J_{G,W} and

S_{G,W} = L_{G,W} /J_{G,W}

is a simple functor, where

J_{G,W} (K) = ⋂_{ϕ∈A_{k}(G,K)} Ker(L_{G,W} (ϕ) : L_{G,W} (K) ⟶ L_{G,W} (G))

by [4, 2.3. Lemma].

Let S be a simple inflation functor. A minimal group of S is a group H of minimal order such that S(H) ≠ 0. If H is a minimal subgroup of S and V = S(H), then V is a simple A_{k}(H, H)-module by [4, 3.1. Proposition] and S ≃ S_{H,V} by [4, 3.2. Proposition]. Let P be a finite group. If S_{H,V} (P) ≠ 0, then S_{H,V} (P) is a simple A_{k}(P, P)-module by [4, 3.1. Proposition]. In this paper, we write

S(P, H, V) = S_{H,V} (P)

if S_{H,V} (P) ≠ 0 in order to emphasize that this is a nonzero simple A_{k}(P, P)- module. Conversely, let W be a simple A_{k}(P, P)-module. The minimal group of W is a group H of minimal order such that

W A_{k}(H, P)A_{k}(P, H) ≠ 0.

If H is a minimal subgroup of W then there exists a simple kOut(H)-module V such that S_{P,W} ≃ S_{H,V} and

W ≃ S_{V,H}(P) = S(P, H, V)

as A_{k}(P, P)-modules by [4, 5.1. Proposition]. The following lemmas are almost same as [4, 3.5. Proposition]. We include proofs for completeness.

Lemma 3.1. Let F be an inflation functor. Let H be a subgroup of P and V a simple A_{k}(H, H)-module. Suppose that W = S_{H,V} (P) ≠ 0. Suppose that V_{2} ⊂ V_{1} are A_{k}(H, H)-submodules of F (H) and V_{1}/V_{2} ≃ V as A_{k}(H, H)-modules. Let F_{i} = V_{i}A_{k}(−, H) be the subfunctor of F generated by V_{i}. Then M = F_{1}/F_{2} has the unique maximal subfunctor M′ such that M/M′ ≃ S_{H,V} where

M′(X) = ⋂_{ϕ∈A_{k}(H,X)} Ker M(ϕ).

In particular, M(P)/M′(P) ≃ W.

Proof. Since we have an isomorphism,

Hom(L_{H,V} , F_{1}/F_{2}) ≃ Hom_{A_{k}(H,H)}(V, F_{1}(H)/F_{2}(H))

by [4, 2.2. Lemma] or section 2 in [3], there exists a nonzero morphism

θ : L_{H,V} ⟶ M.

We define two subfunctors F′_{1} and F′_{2} of F by

F′_{1}/F_{2} = θ(L_{H,V}), F′_{2}/F_{2} = θ(J_{H,V}).

Then F′_{2}/F_{2} is a unique maximal subfunctor of F′_{1}/F_{2} and

F′_{1}/F′_{2} ≃ L_{H,V} /J_{H,V} ≃ S_{H,V} .

Since (F_{1}/F′_{1})(H) = 0 and F_{1}(H) = V_{1}, we have F′_{1}(H) = V_{1}. Hence we have that F_{1} = F′_{1} since F_{1} is generated by V_{1}. Hence M′ = F′_{2}/F_{2} is a unique maximal subfunctor of M. The description of M′ follows from the following commutative diagram:

L_{H,V} (X) ⟶^{θ(X)} M(X)
L_{H,V} (ϕ)↓  ↓^{M(ϕ)}
L_{H,V} (H) ⟶^{≃} M(H).
□

Lemma 3.2. Let F be an inflation functor. Let W be a simple A_{k}(P, P)-module with minimal subgroup H. Let V be a simple kOut(H)-module such that W = S_{H,V} (P). Suppose that W_{2} ⊂ W_{1} are A_{k}(P, P)-submodules of F (P) such that W_{1}/W_{2} ≃ W. Let V_{1} = W_{1}A_{k}(H, P). Then there exists an A_{k}(H, H)-submodules V_{2} of V_{1} such that V_{1}/V_{2} ≃ V as A_{k}(H, H)-modules and

V_{1}A_{k}(P, H)/(V_{1}A_{k}(P, H) ∩ W_{2}) ≃ W

as A_{k}(P, P)-modules.

Proof. By Lemma 3.1, W_{1}A_{k}(−, P) has a maximal subfunctor E′ containing the subfunctor W_{2}A_{k}(−, P) such that

W_{1}A_{k}(−, P)/E′ ≃ S_{P,W} ≃ S_{H,V} .

Let V_{2} = E′(H). Then V_{1}/V_{2} ≃ S_{H,V} (H) ≃ V. Since W ≃ W_{1}/W_{2} and W A_{k}(H, P)A_{k}(P, H) = W, it follows that

V_{1}A_{k}(P, H) + W_{2} = W_{1}A_{k}(H, P)A_{k}(P, H) + W_{2} = W_{1}.

Hence we have

V_{1}A_{k}(P, H)/(V_{1}A_{k}(P, H) ∩ W_{2}) ≃ W_{1}/W_{2} ≃ W.

This completes the proof. □

Corollary 3.3. Let F be an inflation functor. Let H be a subgroup of P and V a simple kOut(H)-module. Suppose that W = S_{H,V} (P) ≠ 0. Let N be a submodule of F (H). If F (H)/N has no subquotient module isomorphic to V, then F (P)/NA_{k}(P, H) has no subquotient module isomorphic to W.

Proof. Let F̄ = F/NA_{k}(−, H). If F (P)/NA_{k}(P, H) = F̄(P) has subquotionet module which is isomorphic to W, then F̄(H) = F (H)/N has a subquotient which is isomorphic to V by Lemma 3.2. This contradicts the assumption. □

Now assume that p is a prime number and k = 𝔽_{p}. We set

A_{p}(P, Q) = A_{𝔽_{p}}(P, Q) = 𝔽_{p} ⊗ A_{ℤ}(P, Q)

for finite group P, Q. Let H^{∗}(P) = (𝔽_{p} ⊗ H^{∗}(BP; ℤ))/\sqrt{0}. Let H ≤ P and let φ : H ⟶ Q be a group morphism. The action of ζ_{H,φ} ∈ A_{p}(P, Q) on cohomology is given by

Tr_{H}^{P}φ^{∗} : H^{∗}(BQ; 𝔽_{p}) ⟶ H^{∗}(BP; 𝔽_{p})

and H(B(−); 𝔽_{p}) is an inflation functor. Similarly, 𝔽_{p} ⊗ H(B(−); ℤ) is an inflation functor. Moreover H^{∗}(−) is an inflation functor by the following lemma.

Lemma 3.4. Let Q be a subgroup of P. If x ∈ H^{∗}(BQ; 𝔽_{p}) is nilpotent, then Tr_{Q}^{P}(x) is nilpotent.

Proof. We will prove y = Tr_{Q}^{P}(x) is nilpotent. By Quillen’s theorem [15], we only need to prove that res_{A}^{P}(y) is nilpotent for each elementary abelian p-subgroup A. By the double coset formula, res_{A}^{P}(y) is a sum of

(i) elements in the image of transfer map from the proper subgroup of A. (ii) elements of the form res_{A}^{P}(x^{t}) where x^{t} ∈ H^{∗}(BQ^{t}, 𝔽_{p}) is the t-conjugate

of y for some t ∈ P such that A ⊂ Q^{t}. Since A is elementary abelian, elements in (i) are zero. Since conjugation and restriction maps are of course algebra morphisms, elements in (ii) are also nilpotent.

□

4. Cohomology of E = p^{1+2}

Let p be an odd prime. The extraspecial p-group E = p_{+}^{1+2} has a presentation as

E = ⟨a, b, c | [a, b] = c, a^{p} = b^{p} = c^{p} = [a, c] = [b, c] = 1⟩.

The cohomology of E is well known. In particular ([10], [18], [21]), H^{∗}(E) is generated by

y_{1}, y_{2}, C, v

subject to the following relation:

y_{1}^{p}y_{2} − y_{1}y_{2}^{p} = 0, Cy_{i} = y_{i}^{p}, C^{2} = y_{1}^{2p−2} + y_{2}^{2p−2} − y_{1}^{p−1}y_{2}^{p−1}

where |y_{i}| = 2, |C| = 2p − 2 and |v| = 2p. We write v^{p−1} by V.

Let R be a subalgebra of H^{∗}(E) and x_{1}, …, x_{r} elements of H^{∗}(E). We set

R{x_{1}, …, x_{r}} = ∑_{i=1}^{r} Rx_{i}

if x_{1}, …, x_{r} are linear independent over R. Moreover, if W = ∑_{i=1}^{r} 𝔽_{p}x_{i} is a 𝔽_{p}-vector space spanned by x_{1}, …, x_{r}, then we set

R{W } = R{x_{1}, …, x_{r}}.

Lemma 4.1 ([21, section 3]). We have the following expression,

H^{∗}(E) = 𝔽_{p}[C, v]{y_{1}^{i}y_{2}^{j}|0 ≤ i, j ≤ p − 1, (i, j) ≠ (p − 1, p − 1)}.

The maximal elementary abelian p-subgroups of E are

A_{i} = ⟨c, ab^{i}⟩ for 0 ≤ i ≤ p − 1, A_{∞} = ⟨c, b⟩.

Let A(E) be the set of all elementary abelian p-subgroup of E, thus,

A(E) = {A_{0}, …, A_{p−1}, A_{∞}}.

Letting H^{∗}(A) ≅ 𝔽_{p}[y, u] and writing i_{A}^{∗}(x) = x|A for the inclusion i_{A}: A ⊂ E, the images of restriction maps are given by

y_{1}|A_{i} = y for i ∈ 𝔽_{p}, y_{1}|A_{∞} = 0, y_{2}|A_{i} = iy for i ∈ 𝔽_{p}, y_{2}|A_{∞} = y,
C|A_{i} = y^{p−1}, v|A_{i} = u^{p} − y^{p−1}u for all i.

The action of g = ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) ∈ GL_{2}(𝔽_{p}) = Out(E) is given by

g(a) = a^{α}b^{γ}, g(b) = a^{β}b^{δ}, g(c) = c^{det(g)}.

and the action of g on the cohomology is given by ([10], [18, p.491])

g^{∗}C = C, g^{∗}y_{1} = αy_{1} + βy_{2}, g^{∗}y_{2} = γy_{1} + δy_{2}, g^{∗}v = (det(g))v.

We consider the Out(E)-module decomposition of H^{∗}(E). Let S^{i} be the homogeneous part of degree 2i in 𝔽_{p}[y_{1}, y_{2}]. Thus, for 0 ≤ i ≤ p − 1,

S^{i} = 𝔽_{p}{y_{1}^{i}, y_{1}^{i−1}y_{2}, …, y_{2}^{i}}.

Recall that 𝔽_{p}{v} ≃ det as Out(E)-modules where det is the one dimensional determinant representation. Then p(p − 1) simple 𝔽_{p}Out(E)-modules

S^{i}v^{q} ≃ S^{i} ⊗ (det)^{q} (0 ≤ i ≤ p − 1, 0 ≤ q ≤ p − 2)

give the complete set of representatives of nonisomorphic simple 𝔽_{p}Out(E)-modules. Let us write

ℂ𝔸 = 𝔽_{p}[C, V]

and

𝔻𝔸 = 𝔽_{p}[D_{1}, D_{2}]

where D_{1} = C^{p} + V, D_{2} = CV. Then

ℂ𝔸 = H^{∗}(E)^{Out(E)}

the Out(E)-invariants, and

𝔻𝔸|A = H^{∗}(A)^{Out(A)}

for all A ∈ A(E). Moreover, we have

ℂ𝔸 = 𝔻𝔸{1, C, …, C^{p}}.

Next, we study the Out(E)-module structure of S^{j} for j = (p − 1) + i with 1 ≤ i ≤ p − 2. Let

T^{i} = 𝔽_{p}{y_{1}^{p−1}y_{2}^{i}, y_{1}^{p−2}y_{2}^{i+1}, ..., y_{1}^{i}y_{2}^{p−1}}

for 0 ≤ i ≤ p − 2. In particular, T^{0} = S^{p−1}. Then we have

S^{j} = 𝔽_{p}{y_{1}^{j}, y_{1}^{j−1}y_{2}, ..., y_{1}^{p}y_{2}^{i−1}, y_{2}^{j}, T^{i}}

since, in H^{∗}(E), we have the relation y_{1}^{p}y_{2} − y_{1}y_{2}^{p} = 0, and hence (for i ≥ 2)

y_{1}^{j−1}y_{2} = y_{1}^{i−1}y_{2}^{p}, y_{1}^{j−2}y_{2}^{2} = y_{1}^{i−2}y_{2}^{p+1}, ..., y_{1}^{p}y_{2}^{i−1} = y_{1}y_{2}^{j−1}.

Since y_{1}^{j} = Cy_{1}^{i}, ..., y_{1}^{p}y_{2}^{i−1} = Cy_{1}y_{2}^{i−1}, y_{2}^{j} = Cy_{2}^{i}, we can write S^{j} = CS^{i} + T^{i}. In general, we have the following.

Proposition 4.2. Let 0 ≤ i ≤ p − 2 and m ≥ 1. Then

S^{m(p−1)+i} = C^{m−1}(CS^{i} + T^{i})

where S^{0} = 𝔽_{p} and T^{0} = S^{p−1}. In particular, the subalgebra of H^{∗}(E) generated by y_{1} and y_{2} is written as

∑_{i=0}^{p−1} S^{i} + 𝔽_{p}[C]C(𝔽_{p}C + S^{p−1}) + ∑_{i=1}^{p−2} 𝔽_{p}[C](CS^{i} + T^{i}).

The 𝔽_{p}-subspace CS^{i} is a GL_{2}(𝔽_{p})-submodule of C^{p−1+i} = CS^{i} + T^{i}. Let T̄^{i} = S^{p−1+i}/(CS^{i}) = (CS^{i} + T^{i})/CS^{i} for 1 ≤ i ≤ p − 2.

Lemma 4.3 (Glover [8, (5.7)]). As GL_{2}(𝔽_{p})-modules,

T̄^{i} ≅ (S^{p−1−i} ⊗ det^{i}).

Proof. Let g = (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}), g′ = (\begin{matrix}1 & 0 \\ 1 & 1 \end{matrix}) ∈ GL_{2}(𝔽_{p}). We set \overline{y_{1}^{i}y_{2}^{p−1}} = y_{1}^{i}y_{2}^{p−1} +

CS^{i} ∈ T̄^{i}. Then 𝔽_{p}\{\overline{y_{1}^{i}y_{2}^{p−1}}\} is the unique nonzero minimal 𝔽_{p}⟨g⟩-submodule of T̄^{i}. On the other hand, \overline{y_{1}^{i}y_{2}^{p−1}} generates T̄^{i} as an 𝔽_{p}⟨g′⟩-module, hence it follows that T̄^{i} is a simple 𝔽_{p}GL_{2}(𝔽_{p})-module. Since dim T̄^{i} = p − i, it is isomorphic to S^{p−i−1} ⊗det^{q} for some 0 ≤ q ≤ p−2. Since 𝔽_{p}{y_{2}^{p−i−1} ⊗det^{q}} is the unique nonzero minimal 𝔽_{p}⟨g⟩-submodule of S^{p−i−1}⊗det^{q}, there exists an 𝔽_{p}GL_{2}(𝔽_{p})-isomorphism

T̄^{i} ⟶^{∼} S^{p−i−1} ⊗ det^{q} sending \overline{y_{1}^{i}y_{2}^{p−1}} to y_{2}^{p−i−1} ⊗ det^{q}.

Now consider the action of t = diag(λ_{1}, λ_{2}) ∈ GL_{2}(𝔽_{p}). Since

t^{∗}(\overline{y_{1}^{i}y_{2}^{p−1}}) = λ_{1}^{i}(\overline{y_{1}^{i}y_{2}^{p−1}})

and

t^{∗}(y_{2}^{p−i−1} ⊗ det^{q}) = λ_{2}^{p−i−1}(λ_{1}λ_{2})^{q}(y_{2}^{p−i−1} ⊗ det^{q}),

we have i = q. This completes the proof of the lemma. □

From Lemma 4.1, we have the following.

Theorem 4.4.

H^{∗}(E) = 𝔽_{p}[C, v]{(⨁_{i=0}^{p−1} S^{i}) ⊕ (⨁_{i=1}^{p−2} T^{i})} = ℂ𝔸{⨁_{i=0}^{p−2} ⨁_{q=0}^{p−2}(S^{i}v^{q} ⊕ T^{i}v^{q}))}.

Let e_{i,q} be an idempotent in 𝔽_{p}GL_{2}(𝔽_{p}) corresponding to the simple module S^{i} ⊗ det^{q}, namely, (S^{i} ⊗ det^{q})e_{i,q} = S^{i} ⊗ det^{q} and Se_{i,q} = 0 for a simple module which is not isomorphic to S^{i} ⊗ det^{q}. Let H_{i,q} = H^{∗}(E)e_{i,q}.

Corollary 4.5. We have the following isomorphisms as 𝔽_{p}-vector spaces.

(1) For 0 ≤ q ≤ p − 2,

H_{0,q} = ℂ𝔸{v^{q}}, H_{p−1,q} = ℂ𝔸{S^{p−1}v^{q}}.

(2) Let 1 ≤ i ≤ p − 2 and 0 ≤ q ≤ p − 2. Assume i + q ≡ m mod p − 1 where 0 ≤ m ≤ p − 2, then

H_{i,q} ≃ ℂ𝔸{S^{i}v^{q} ⊕ T^{p−1−i}v^{m}}.

We shall need the following lemma in the proof of Proposition 9.2.

Lemma 4.6. Let r = n(p − 1) − 2 for 0 ≤ n ≤ p − 1. If 0 ≤ m ≤ p − 2, then vS^{m+1}H^{2r}(E) has no composition factor isomorphic to S^{m} as a GL_{2}(𝔽_{p})-module.

Proof. Let R be the subalgebra of H^{∗}(E) generated by y_{1}, y_{2} and C. Then H^{∗}(E) ≃ R ⊗ 𝔽_{p}[v]. Set

K(m) = ⨁_{i≡m mod (p−1)}(R{v, …, v^{p−2}} ∩ H^{2i}(E)).

First, we claim that vS^{m+1}H^{2r}(E) ⊂ K(m). Since r − (p − 2)p < 0, we have

H^{2r}(E) ⊂ ⨁_{0≤i≤p−3}(R ∩ H^{2(r−ip)})v^{i} ⊂ ⨁_{0≤i≤p−3} Rv^{i}.

Hence vS^{m+1}H^{2r}(E) ⊂ R{v, …, v^{p−2}}. Then we have vS^{m+1}H^{2r}(E) ⊂ K(m) since

deg vS^{m+1}H^{2r}(E) = 2(p + m + 1 + r) ≡ m mod 2(p − 1).

Now, we show that K(m) has no composition factor isomorphic to S^{m}. If m = 0, then the result holds since H_{0,0} = ℂ𝔸 by Corollary 4.5 and ℂ𝔸∩R{v, …, v^{p−2}} = 0. Next assume that 1 ≤ m ≤ p − 2. Then H_{m,0} = ℂ𝔸{S^{m} ⊕ T^{p−1−m}v^{m}} and the degree of each nonzero homogeneous part of ℂ𝔸{T^{p−1−m}v^{m}} is equivalent to 0 modulo 2(p − 1). On the other hand, since ℂ𝔸{S^{m}} ⊂ R[V], we have ℂ𝔸{S^{m}} ∩ K(m) = 0. Hence K(m) has no simple subquoitent submodule isomorphic to S^{m}. This completes the proof. □

We consider the image of the inflation map from a cyclic quotient.

Lemma 4.7. Let 0 ≤ i ≤ p − 1. Then the elements

y_{1}^{i}, (y_{1} + y_{2})^{i}, …, (y_{1} + iy_{2})^{i}

span S^{i}.

Proof. If 0 ≤ k ≤ i, then

(y_{1} + ky_{2})^{i} = ∑_{j=0}^{i} (\begin{matrix}i \\ j \end{matrix}) y_{1}^{i−j}k^{j}y_{2}^{j} = ∑_{j=0}^{i} k^{j}((\begin{matrix}i \\ j \end{matrix}) y_{1}^{i−j}y_{2}^{j}).

Since

(\begin{matrix}i \\ j \end{matrix}) y_{1}^{i−j}y_{2}^{j}, 0 ≤ j ≤ i

is a basis of S^{i} and the (i + 1) × (i + 1)-matrix (k^{j})_{k,j} is invertible, we have that

(y_{1} + ky_{2})^{i}, 0 ≤ k ≤ i

is a basis of S^{i}. □

Lemma 4.8. Let p ≤ n and λ_{1}, λ_{2} ∈ 𝔽_{p}. Then

(λ_{1}y_{1} + λ_{2}y_{2})^{n} = C(λ_{1}y_{1} + λ_{2}y_{2})^{n−p+1}.

Proof. Since

(λ_{1}y_{1} + λ_{2}y_{2})^{p} = λ_{1}y_{1}^{p} + λ_{2}y_{2}^{p} = C(λ_{1}y_{1} + λ_{2}y_{2})

we have

(λ_{1}y_{1} + λ_{2}y_{2})^{n} = (λ_{1}y_{1} + λ_{2}y_{2})^{p}(λ_{1}y_{1} + λ_{2}y_{2})^{n−p} = C(λ_{1}y_{1} + λ_{2}y_{2})^{n−p+1}.
□

Corollary 4.9. Let Q ≤ E be a subgroup of order p. Then

∑_{φ:E↠Q} φ^{∗}H^{+}(Q) = 𝔽_{p}[C]( ⨁_{1≤i≤p−1} S^{i}).

Proof. Since

∑_{φ:E↠Q} φ^{∗}(H^{2n}(Q)) = ∑_{λ_{1},λ_{2}∈𝔽_{p}}(λ_{1}y_{1} + λ_{2}y_{2})^{n}

for n > 0, the results follows from Lemma 4.7 and Lemma 4.8. □

5. Cohomology of A

In this section, we study the structure of H^{∗}(A), where A is a rank 2 elementary abelian p-subgroup of E so that H^{∗}(A) = 𝔽_{p}[y, u]. We fix a quotient map

q : E ⟶ E/⟨c⟩ ⟶^{∼} A

and consider the induced map q^{∗} : H^{∗}(A) ⟶ H^{∗}(E). We may assume that q^{∗}(y) = y_{1} and q^{∗}(u) = y_{2}. This map is a GL_{2}(𝔽_{p})-module morphism. Here, we identify Out(E), Out(A) and GL_{2}(𝔽_{p}). The kernel of q^{∗} is given by

Ker q^{∗} = H^{∗}(A)d_{2} = 𝔽_{p}[y, u]d_{2}

where d_{2} = yu^{p} − y^{p}u. The image of q^{∗} is a subalgebra of H^{∗}(E) generated by y_{1} and y_{2}.

Let S(A)^{i} = H^{2i}(A). Then q^{∗} induces a GL_{2}(𝔽_{p})-isomorphism S(A)^{i} ≃ S^{i} for 0 ≤ i ≤ p−1. Moreover S(A)^{i} ⊗det^{q}, 0 ≤ i ≤ p−2, 0 ≤ q ≤ p−2 give the complete set of representatives of isomorphism classes of simple GL_{2}(𝔽_{p})-modules.

Next we described simple A_{p}(A, A)-modules. Let Q ≤ A be a subgroup of order p. Then Out(Q) is a cyclic group of order p − 1. Let U_{i} = 𝔽_{p}u_{i} (0 ≤ i ≤ p − 2) be the simple right Out(Q)-module defined by u_{i}σ = λ^{i}u_{i} where Out(Q) = ⟨σ⟩ and 𝔽_{p}^{×} = ⟨λ⟩. Note that if n ≡ i (mod p − 1) where 0 ≤ i ≤ p − 1, then

H^{2n}(Q) ≃ U_{i}

as right kQ-modules. Since Tr_{Q}^{A}(H^{∗}(Q)) = 0, we have

H^{∗}(Q)A_{p}(A, Q) = ∑_{φ:A↠Q} φ^{∗}(H^{∗}(Q)) = ∑_{n≥0} ∑_{λ_{1},λ_{2}∈𝔽_{p}} 𝔽_{p}(λ_{1}y + λ_{2}u)^{n}.

In particular, if 1 ≤ n ≤ p − 1, we have H^{2n}(Q)A_{p}(A, Q) = H^{2n}(A). Since

⋂_{R<A} Ker(res_{R}^{A} : H^{∗}(A) ⟶ H^{∗}(R)) = d_{2}H^{∗}(A),

it follows that

⋂_{ϕ∈A(Q,A)} Ker(ϕ : H^{n}(A) ⟶ H^{n}(Q)) = 0

for 1 ≤ n ≤ p − 1. Hence H^{2n}(A) ≃ S(A, Q, U_{n}) for 1 ≤ n ≤ p − 2 and H^{2(p−1)}(A) ≃ S(A, Q, U_{0}) by Proposition 3.1, and these modules give all simple A_{p}(A, A)-module with minimal subgroup Q by Proposition 3.2. Hence we have the following classification of simple A_{p}(A, A)-modules.

Proposition 5.1 (Harris and Kuhn [7, Example 8.1.]). Up to isomorphisms, the simple A_{p}(A, A)-modules are given as follows.

(1) S(A, A, S(A)^{i} ⊗ det^{q}) (0 ≤ i ≤ p − 1, 0 ≤ q ≤ p − 2).

(2) S(A, Q, U_{i}) (0 ≤ i ≤ p − 2),

dim S(A, Q, U_{i}) = \{ \begin{matrix}p & (i = 0) \\ i + 1 & (1 ≤ 0 ≤ p − 2). \end{matrix}

(3) One dimensional module with trivial minimal subgroup.

Now we consider the structure of H^{∗}(A) as an A_{p}(A, A)-module. First, we see GL_{2}(𝔽_{p})-module structure more closely and specify all simple submodules of H^{∗}(A) isomorphic to S^{p−1} ⊗ det^{i} for 0 ≤ i ≤ p − 2.

Since C^{j}S^{p−1} is contained in Im q^{∗} for any j ≥ 0, we have the following short exact sequence of GL_{2}(𝔽_{p})-modules,

0 ⟶ d_{2}H^{∗}(A) ⟶ (q^{∗})^{−1}(C^{j}S^{p−1}) ⟶ C^{j}S^{p−1} ⟶ 0.

Since C^{j}S^{p−1} is projective as an 𝔽_{p}GL_{2}(𝔽_{p})-module, this exact sequence splits. Hence, in particular, for 0 ≤ j ≤ p − 1, there exist GL_{2}(𝔽_{p})-submodules W_{j} such that

(q^{∗})^{−1}(C^{j}S^{p−1}) = W_{j} ⊕ d_{2}H^{∗}(A).

Note that since

y_{1}^{(j+1)(p−1)}, y_{1}^{(j+1)(p−1)−1}y_{2}, …, y_{1}^{(j+1)(p−1)−(p−2)}y_{2}^{p−2}, y_{2}^{(j+1)(p−1)}

is a basis of C^{j}S^{p−1}, it follows that

q^{∗}(y^{(j+1)(p−1)}), q^{∗}(y^{(j+1)(p−1)−1}u), …, q^{∗}(y^{(j+1)(p−1)−(p−2)}u^{p−2}), q^{∗}(u^{(j+1)(p−1)})

is a basis of C^{j}S^{p−1}.

Next, we consider Dickson subalgebra of H^{∗}(A). Let D̃_{1} = res_{A}^{E}(D_{1}) = y^{p(p−1)} + res_{A}^{E}(V) and D̃_{2} = res_{A}^{E}(D_{2}) = d_{2}^{p−1}. Let 𝔻̃𝔸 = 𝔽_{p}[D̃_{1}, D̃_{2}]. Then 𝔻̃𝔸 = H^{∗}(A)^{Out(A)} and the restriction map induces an isomorphism

res_{A}^{E} : 𝔻𝔸 ⟶^{∼} 𝔻̃𝔸.

Lemma 5.2. We have

D̃_{1}y^{(j+1)(p−1)−l}u^{l} ≡ y^{(p+j+1)(p−1)−l}u^{l} (mod d_{2}H^{∗}(A))

for any 0 ≤ l ≤ p − 2 and

D̃_{1}u^{(j+1)(p−1)} ≡ u^{(p+j+1)(p−1)} (mod d_{2}H^{∗}(A)).

In particular,

(q^{∗})^{−1}(C^{j+mp}S^{p−1}) = D̃_{1}^{m}W_{j} ⊕ d_{2}H^{∗}(A)

for any m ≥ 0 and 0 ≤ j ≤ p − 1.

Proof. First, we have

D̃_{1}y^{(j+1)(p−1)−l}u^{l} = (y^{p(p−1)} + res_{A}^{E}(v)^{p−1})y^{(j+1)(p−1)−l}u^{l}
≡ y^{(p+j+1)(p−1)−l}u^{l}

for 0 ≤ l ≤ p − 2 since res_{A}^{E}(v)y = d_{2}. On the other hand, modulo d_{2}H^{∗}(A),

y^{p(p−1)}u^{(j+1)(p−1)} ≡ (y^{p}u)^{p−1}u^{j(p−1)} ≡ (yu^{p})^{p−1}u^{j(p−1)} = y^{p−1}u^{(p+j)(p−1)}
(u^{p} − y^{p−1}u)^{p−1} ≡ (u^{p} − y^{p−1}u)u^{p(p−2)}

COHOMOLOGY OF EXTRASPECIAL p-GROUP 13

and

(u^{p} − y^{p−1}u)^{p−1}u^{(j+1)(p−1)} ≡ (u^{p} − y^{p−1}u)u^{(p+j)(p−1)−1}
= u^{(p+j+1)(p−1)} − y^{p−1}u^{(p+j)(p−1)}.

Hence we have

D̃_{1}u^{(j+1)(p−1)} = (y^{p(p−1)} + (u^{p} − y^{p−1}u)^{p−1})u^{(j+1)(p−1)} ≡ u^{(p+j+1)(p−1)}.

□

There exists a sequence of GL_{2}(𝔽_{p})-submodules,

H^{∗}(A) ⊃ d_{2}H^{∗}(A) ⊃ d_{2}^{2}H^{∗}(A) ⊃ · · · .

We shall consider each factor module d_{2}^{m}H^{∗}(A)/d_{2}^{m+1}H^{∗}(A). Note that, since d_{2}g = g^{∗}(d_{2}) = (det g)d_{2} for any g ∈ GL_{2}(𝔽_{p}) and 𝔻̃𝔸 = H^{∗}(A)^{Out(A)}, it follows that

D̃_{1}^{i}D̃_{2}^{j}d_{2}^{m}W_{n} ≃ S(A)^{p−1} ⊗ det^{m}

for any m. First, we consider the factor module H^{∗}(A)/d_{2}H^{∗}(A). Lemma 5.3. There exists a sequence of GL_{2}(𝔽_{p})-submodules,

H^{∗}(A) ⊃ (𝔽_{p}[D̃_{1}]{⨁_{n=0}^{p−1} W_{n}} ⊕ d_{2}H^{∗}(A)) ⊃ d_{2}H^{∗}(A).

Moreover,

H^{∗}(A)/(𝔽_{p}[D̃_{1}]{⨁_{n=0}^{p−1} W_{n}} ⊕ d_{2}H^{∗}(A))

has no simple subquotient module which is isomorphic to S(A)^{p−1} ⊗ det^{i} for any i. Proof. By Lemma 5.2, q^{∗} induces isomorphisms,

H^{∗}(A)/(𝔽_{p}[D̃_{1}]{⨁_{n=0}^{p−1} W_{n}} ⊕ d_{2}H^{∗}(A)) ≃ 𝔽_{p}[y_{1}, y_{2}]/𝔽_{p}[C]S^{p−1}

and

𝔽_{p}[D̃_{1}]{⨁_{n=0}^{p−1} W_{n}} ≃ (𝔽_{p}[D̃_{1}]{⨁_{n=0}^{p−1} W_{n}} ⊕ d_{2}H^{∗}(A))/d_{2}H^{∗}(A) ≃ 𝔽_{p}[C]S^{p−1}.

Hence the result follows from Proposition 4.2. □

Next we consider the factor module d_{2}^{m}H^{∗}(A)/d_{2}^{m+1}H^{∗}(A) for general m ≥ 0. Since d_{2}^{m} induces an isomorphism,

(H^{∗}(A)/d_{2}H^{∗}(A)) ⊗ det^{m} ≃ d_{2}^{m}H^{∗}(A)/d_{2}^{m+1}H^{∗}(A),

we have the following. Lemma 5.4. There exists a sequence of GL_{2}(𝔽_{p})-modules,

d_{2}^{m}H^{∗}(A) ⊃ (𝔽_{p}[D̃_{1}]{⨁_{n=0}^{p−1} d_{2}^{m}W_{n}} ⊕ d_{2}^{m+1}H^{∗}(A)) ⊃ d_{2}^{m+1}H^{∗}(A).

Moreover,

d_{2}^{m}H^{∗}(A)/(𝔽_{p}[D̃_{1}]{⨁_{n=0}^{p−1} d_{2}^{m}W_{n}} ⊕ d_{2}^{m+1}H^{∗}(A))

has no simple subquotient module which is isomorphic to S(A)^{p−1} ⊗ det^{i} for any i.

Since d_{2}^{p−1} = D̃_{2} and 𝔻̃𝔸 = 𝔽_{p}[D̃_{1}, D̃_{2}], we have the following by Lemma 5.3 and 5.4.

Proposition 5.5. For any 0 ≤ m ≤ p − 2, the submodule

𝔻̃𝔸{⨁_{n=0}^{p−1} d_{2}^{m}W_{n}}

is a sum of all simple submodules of H^{∗}(A) isomorphic to S(A)^{p−1} ⊗ det^{m}. In particular,

H^{∗}(A)/𝔻̃𝔸{⨁_{n=0}^{p−1} d_{2}^{m}W_{n}}

has no simple subquotient module isomorphic to S(A)^{p−1} ⊗ det^{m}.

Now, we consider A_{p}(A, A)-module structure of H^{∗}(A). Let J(A) be the ideal of A_{p}(A, A) generated by bisets which factor through a proper subgroup of A. If L^{∗} = ⊕_{n≥0}L^{n} is a graded vector subspace of H^{∗}(E), we set L^{+} = ⊕_{n>0}L^{n}, the positive degree part of L^{∗}.

Proposition 5.6. Let Q ≤ A be a subgroup of order p. Let

L^{n}(Q) = H^{n}(Q)A_{p}(A, Q) + (d_{2}H^{∗}(A) ∩ H^{n}(A))

and

L^{∗}(Q) = ⨁_{n≥0} L^{n}(Q) = H^{∗}(Q)A_{p}(A, Q) + d_{2}H^{∗}(A).

Then we have the following.

(1) We have a sequence of A_{p}(A, A)-submodules,

H^{∗}(A) ⊃ L^{∗}(Q) ⊃ d_{2}H^{∗}(A).

(2) We have H^{∗}(A)J(A) ⊂ H^{∗}(Q)A_{p}(A, Q). In particular, every simple subquotient module of H^{∗}(A)/L^{∗}(Q) has minimal subgroup A. Moreover H^{∗}(A)/L^{∗}(Q) has no simple subquotient module isomorphic to S(A, A, S(A)^{p−1} ⊗ det^{i}) for any i.

(3) The factor module L^{+}(Q)/d_{2}H^{∗}(A) is a direct sum of A_{p}(A, A)-modules with minimal subgroup Q. More precisely, if n > 0 and n ≡ i (mod p − 1) where 0 ≤ i ≤ p − 2, then

L^{2n}(Q)/(d_{2}H^{∗}(A) ∩ H^{2n}(A)) ≃ S(A, Q, U_{i}).

(4) We have d_{2}H^{∗}(A)J(A) = 0. In particular, every simple subquotinet module of d_{2}H^{∗}(A) has minimal subgroup A.

Proof. We have d_{2}H^{∗}(A)J(A) = 0 since the restriction of d_{2} to any proper subgroup is zero. Hence d_{2}H^{∗}(A) is an A_{p}(A, A)-submodule and the claims in (1) and (4) are proved.

Since A is an elementary abelian p-group, any transfer from proper subgroup to A is zero. Hence it follows that

H^{∗}(A)J(A) ⊂ H^{∗}(Q)A_{p}(A, Q) = ∑_{φ:A↠Q} φ^{∗}H^{∗}(Q) = ∑_{n≥0} ∑_{λ_{1},λ_{2}∈𝔽_{p}}(λ_{1}y + λ_{2}u)^{n}.

Then q^{∗} induces an isomorphism of GL_{2}(𝔽_{p})-modules,

H^{∗}(A)/L^{∗}(Q) ≃ 𝔽_{p}[y_{1}, y_{2}]/𝔽_{p}[C]{∑_{i=1}^{p−1} S^{i}}

and

L^{∗}(Q)/d_{2}H^{∗}(A) ≃ 𝔽_{p}[C]{∑_{i=1}^{p−1} S^{i}}

by Corollary 4.9, where 𝔽_{p}[y_{1}, y_{2}] is a subalgebra of H^{∗}(E) generated by y_{1} and y_{2}. It follows that no simple subquotient module of H^{∗}(A)/L^{∗}(Q) is isomorphic to S(A, A, S(A)^{p−1} ⊗ det^{i}) by Proposition 4.2.

On the other hand, since ⋂_{R<A} Ker res_{R}^{A} = d_{2}H^{∗}(A), it follows that

⋂_{ϕ∈A(Q,A)} Ker(ϕ : H^{∗}(Q)A_{p}(A, Q) ⟶ H^{∗}(Q)) = H^{∗}(Q)A_{p}(A, Q) ∩ d_{2}H^{∗}(A).

Since H^{+}(Q) is a direct sum of simple A(Q, Q)-modules with minimal subgroup Q, it follows that

L^{+}(Q)/d_{2}H^{∗}(A) ≃ H^{+}(Q)A_{p}(A, Q)/(H^{+}(Q)A_{p}(A, Q) ∩ d_{2}H^{∗}(A))

is a direct sum of simple A_{p}(A, A)-modules with minimal subgroup Q by Lemma 3.2. This completes the proof of (2) and (3). □

Finally, we shall specify simple A_{p}(A, A)-submodules of H^{∗}(A) isomorphic to S(A, A, S(A)^{p−1} ⊗ det^{i}).

Proposition 5.7. Let i, j ≥ 0, 0 ≤ m ≤ p − 2 and 0 ≤ n ≤ p − 1. If j + m > 0, then D̃_{1}^{i}D̃_{2}^{j}d_{2}^{m}W_{n} is an A_{p}(A, A)-submodule with minimal subgroup A, namely,

D̃_{1}^{i}D̃_{2}^{j}d_{2}^{m}W_{n} ≃ S(A, A, S(A)^{p−1} ⊗ det^{m})

as A_{p}(A, A)-modules. The quotient module

H^{∗}(A)/(𝔻̃𝔸{⨁_{n=0}^{p−1} D̃_{2}W_{n}})

has no simple subquotient module which is isomorphic to S(A, A, S(A)^{p−1}). On the other hand, if m > 0, then the quotient module

H^{∗}(A)/(𝔻̃𝔸{⨁_{n=0}^{p−1} d_{2}^{m}W_{n}})

has no simple subquotient module which is isomorphic to S(A, A, S(A)^{p−1} ⊗ det^{m}).

Proof. First, note that D̃_{1}^{i}D̃_{2}^{j}d_{2}^{m}W_{n} ≃ S(A)^{p−1} ⊗ det^{m} as GL_{2}(𝔽_{p})-modules. By Lemma 5.6(4), D̃_{1}^{i}D̃_{2}^{j}d_{2}^{m}W_{n} is an A_{p}(A, A)-submodule and

D̃_{1}^{i}D̃_{2}^{j}d_{2}^{m}W_{n} ≃ S(A, A, S(A)^{p−1} ⊗ det^{m}).

If 1 ≤ m ≤ p − 2, then, as a GL_{2}(𝔽_{p})-module,

H^{∗}(A)/(𝔻̃𝔸{⨁_{n=0}^{p−1} d_{2}^{m}W_{n}})

has no simple subquotient module which is isomorphic to S(A)^{p−1}⊗det^{m} by Proposition 5.5. Hence, as an A_{p}(A, A)-module,

H^{∗}(A)/(𝔻̃𝔸{⨁_{n=0}^{p−1} d_{2}^{m}W_{n}})

has no simple subquotient module which is isomorphic to S(A, A, S(A)^{p−1} ⊗det^{m}).

Next suppose m = 0. By Lemma 5.6 (2)(3), H^{∗}(A)/d_{2}H^{∗}(A) has no subquotient module which isomorphic to S(A, A, S(A)^{p−1}). Hence it follows that 𝔻̃𝔸{⊕_{n=0}^{p−1}W_{n}}/𝔻̃𝔸{⊕_{n=0}^{p−1}D̃_{2}W_{n}} has no subquotient module which is isomorphic to S(A, A, S(A)^{p−1}) since

𝔻̃𝔸{⨁_{n=0}^{p−1} W_{n}} ∩ d_{2}H^{∗}(A) = 𝔻̃𝔸{⨁_{n=0}^{p−1} D̃_{2}W_{n}}.

On the other hand, H^{∗}(A)/𝔻̃𝔸{⊕_{n=0}^{p−1}W_{n}} has no subquotient module which is isomorphic to S(A, A, S(A)^{p−1}) by Proposition 5.5. This completes the proof of the proposition. □

6. Images of transfer maps

We study the image of transfer from maximal elementary abelian p-subgroups. Let M_{i} = CS^{i} + T^{i} for 0 ≤ i ≤ p − 2 where S^{0} = 𝔽_{p}, T^{0} = S^{p−1}. Moreover, let ŷ_{A} = y_{1} if A = A_{i}, i ∈ 𝔽_{p} and ŷ_{A} = y_{2} if A = A_{∞}. Note that res_{A}^{E}(ŷ_{A}) = y for any A ∈ A(E).

Lemma 6.1. Let A ∈ A(E).

(1) We have

Tr_{A}^{E}(u^{p−1}) = \{ \begin{matrix}(iy_{1} − y_{2})^{p−1} − C & (A = A_{i}, 0 ≤ i ≤ p − 1) \\ y_{1}^{p−1} − C & (A = A_{∞}). \end{matrix}

(2) Suppose that 0 ≤ j ≤ p − 2 and 0 ≤ m ≤ p, then

Tr_{A}^{E}(u^{m(p−1)+j}) = \{ \begin{matrix}0 & (0 ≤ m ≤ j ≤ p − 2) \\ (\begin{matrix}m − 1 \\ j \end{matrix}) v^{j}C^{m−j−1}Tr_{A}^{E}(u^{p−1}) & (0 ≤ j < m, j ≤ p − 2). \end{matrix}

In particular,

Tr_{A}^{E}(u^{m(p−1)+j}) ∈ 𝔽_{p}[C]{v^{j}M_{0}}.

Moreover, for any l ≥ 1,

C^{l−1}Tr_{A}^{E}(u^{p−1}) = (−1)^{l−1}(Tr(u^{p−1}))^{l}.

(3) If 0 ≤ m ≤ p − 2 then ŷ_{A}^{m}Tr_{A}^{E}(u^{p−1}) (A ∈ A(E)) is a basis of M_{m}.

(4) For 1 ≤ n ≤ p,

Tr_{A}^{E}((y^{p−1} − u^{p−1})^{n}) = −(C + Tr(u^{p−1}))^{n} + C^{n}
= \{ \begin{matrix}−(iy_{1} − y_{2})^{n(p−1)} + C^{n} & (A = A_{i}, 0 ≤ i ≤ p − 1) \\ −y_{1}^{n(p−1)} + C^{n} & (A = A_{∞}). \end{matrix}

In particular, Tr_{A}^{E}((y^{p−1} − u^{p−1})^{n}) ∈ 𝔽_{p}[C]{M_{0}}.

Proof. Since res_{A}^{E}(v) = u^{p} − y^{p−1}u and res_{A}^{E}(C) = y^{p−1}, we have u^{p} = res_{A}^{E}(v) + (res_{A}^{E}(C))u.

(1) First, note that by the double coset formula,

(6.1.1)  res_{A_{i}}^{E}Tr_{A_{j}}^{E}(u^{p−1}) = \{ \begin{matrix}−y^{p−1} & (i = j) \\ 0 & (i ≠ j) \end{matrix}

for i, j = 0, 1, …, p − 1, ∞. Let 0 ≤ i, j ≤ p − 1. Then

res_{A_{i}}^{E}((jy_{1} − y_{2})^{p−1} − C) = (jy − iy)^{p−1} − y^{p−1} = ((j − i)^{p−1} − 1)y^{p−1}
= \{ \begin{matrix}−y^{p−1} & (i = j) \\ 0 & (i ≠ j) \end{matrix}

and

res_{A_{∞}}^{E}((jy_{1} − y_{2})^{p−1} − C) = (−y)^{p−1} − y^{p−1} = 0.

On the other hand,

res_{A_{i}}^{E}(y_{1}^{p−1} − C) = \{ \begin{matrix}y^{p−1} − y^{p−1} = 0 & (0 ≤ i ≤ p − 1) \\ −y^{p−1} & (i = ∞). \end{matrix}

By Quillen’s theorem [15], we get the equation.

(2) First assume that 0 ≤ m ≤ j ≤ p − 2. We proceed by induction on m. The case m = 0 is trivial. Let 0 < m ≤ j ≤ p − 2. Then

u^{m(p−1)+j} = u^{p}u^{(m−1)(p−1)+(j−1)}
= (res_{A}^{E}(v) + res_{A}^{E}(C)u)u^{(m−1)(p−1)+(j−1)}
= res_{A}^{E}(v)u^{(m−1)(p−1)+(j−1)} + res_{A}^{E}(C)u^{(m−1)(p−1)+j}.

Hence

Tr_{A}^{E}(u^{m(p−1)+j}) = vTr_{A}^{E}(u^{(m−1)(p−1)+(j−1)}) + CTr_{A}^{E}(u^{(m−1)(p−1)+j}) = 0

by induction. Next assume 0 ≤ j < m ≤ p, j ≤ p − 2. If m = 1 then j = 0 and the result is trivial. Assume 1 < m ≤ p. If j = 0 then,

u^{m(p−1)} = u^{p}u^{(m−2)(p−1)+(p−2)}
= (res_{A}^{E}(v) + res_{A}^{E}(C)u)u^{(m−2)(p−1)+(p−2)}
= res_{A}^{E}(v)u^{(m−2)(p−1)+(p−2)} + res_{A}^{E}(C)u^{(m−1)(p−1)}.

Hence

Tr_{A}^{E}(u^{m(p−1)}) = CTr_{A}^{E}(u^{(m−1)(p−1)}) = C^{m−1}Tr_{A}^{E}(u^{p−1}).

Note that m − 2 ≤ p − 2 since m ≤ p. On the other hand, if 0 < j < m ≤ p, j ≤ p − 2, then

u^{m(p−1)+j} = u^{p}u^{(m−1)(p−1)+(j−1)}
= (res_{A}^{E}(v) + res_{A}^{E}(C)u)u^{(m−1)(p−1)+(j−1)}
= res_{A}^{E}(v)u^{(m−1)(p−1)+(j−1)} + res_{A}^{E}(C)u^{(m−1)(p−1)+j}.

Hence, if j < m − 1, then

Tr_{A}^{E}(u^{m(p−1)+j})
= v (\begin{matrix}m − 2 \\ j − 1 \end{matrix}) v^{j−1}C^{m−j−1}Tr_{A}^{E}(u^{p−1}) + C (\begin{matrix}m − 2 \\ j \end{matrix}) v^{j}C^{m−j−2}Tr_{A}^{E}(u^{p−1})
= (\begin{matrix}m − 1 \\ j \end{matrix}) v^{j}C^{m−j−1}Tr_{A}^{E}(u^{p−1}).

If j = m − 1, then

Tr_{A}^{E}(u^{m(p−1)+j}) = v (\begin{matrix}m − 2 \\ j − 1 \end{matrix}) v^{j−1}C^{m−j−1}Tr_{A}^{E}(u^{p−1}) = v^{j}C^{m−j−1}Tr_{A}^{E}(u^{p−1}).

On the other hand, since

res_{A′}^{E}(Tr_{A}^{E}(u^{p−1})) = \{ \begin{matrix}−y^{p−1} & (A′ = A) \\ 0 & (A′ ≠ A), \end{matrix}

we have

res_{A′}^{E}(Tr_{A}^{E}(u^{p−1}))^{l} = res_{A′}^{E}((−1)^{l−1}C^{l−1}Tr_{A}^{E}(u^{p−1}))

for any A′.

(3) By (1), ŷ_{A}^{m}Tr_{A}^{E}(u^{p−1}) ∈ M_{m}. On the other hand, ŷ_{A}^{m}Tr_{A}^{E}(u^{p−1}), A ∈ A(E) are linearly independent by (6.1.1). Hence the results follows since dim M_{m} = p + 1.

(4) By (2), we have

Tr_{A}^{E}((y^{p−1} − u^{p−1})^{n}) = Tr_{A}^{E}(∑_{j=0}^{n} (\begin{matrix}n \\ j \end{matrix}) (−1)^{j}y^{(p−1)(n−j)}u^{(p−1)j})

= ∑_{j=1}^{n} (\begin{matrix}n \\ j \end{matrix}) (−1)^{j}C^{n−j}(−1)^{j−1}(Tr_{A}^{E}(u^{p−1}))^{j}

= − ∑_{j=0}^{n} (\begin{matrix}n \\ j \end{matrix}) C^{n−j}(Tr_{A}^{E}(u^{p−1}))^{j} + C^{n}
= −(C + Tr_{A}^{E}(u^{p−1}))^{n} + C^{n}.
□

Lemma 6.2. (1) If m ≥ 0 and 1 ≤ i ≤ p − 2, then

Tr_{A}^{E}(u^{m(p−1)+i}) ∈ ℂ𝔸{v^{i}M_{0}}.

(2) If m ≥ 1, then

Tr_{A}^{E}(u^{m(p−1)}) ≡ C^{m−1}Tr_{A}^{E}(u^{p−1}) mod ℂ𝔸{V M_{0}}.

In particular,

Tr_{A}^{E}(u^{m(p−1)}) ∈ ℂ𝔸{M_{0}}.

(3) For any n ≥ 0, we have

Tr_{A}^{E}((u^{p−1} − y^{p−1})^{n}) ∈ ℂ𝔸{M_{0}}.

Proof. Since res_{A}^{E}(v) = u^{p} − y^{p−1}u, we have

Tr_{A}^{E}(u^{k}) = CTr_{A}^{E}(u^{k−(p−1)}) + vTr_{A}^{E}(u^{k−p})

for k ≥ p.

We prove (1) and (2) by induction on m. If m = 0 then Tr_{A}^{E}(u^{i}) = 0. If m = 1 then the results follows by Lemma 6.1. Assume that m > 1. If 1 ≤ i ≤ p − 2, then

Tr_{A}^{E}(u^{m(p−1)+i}) = CTr_{A}^{E}(u^{(m−1)(p−1)+i}) + vTr(u^{(m−1)(p−1)+i−1})

and the right hand side is contained in ℂ𝔸{v^{i}M_{0}} by induction.

On the other hand, modulo ℂ𝔸{V M_{0}},

Tr_{A}^{E}(u^{m(p−1)}) = CTr_{A}^{E}(u^{(m−1)(p−1)}) + vTr(u^{(m−2)(p−1)+(p−2)})
≡ CTr_{A}^{E}(u^{(m−1)(p−1)})
≡ CC^{m−2}Tr_{A}^{E}(u^{p−1})
= C^{m−1}Tr_{A}^{E}(u^{p−1})

by induction. Since Tr_{A}^{E}(u^{m(p−1)}y^{k(p−1)}) = C^{k}Tr_{A}^{E}(u^{m(p−1)}), (3) follows from (2). □

Lemma 6.3. Suppose 1 ≤ q ≤ p − 2. Let φ : A ⟶ A′ ≤ E be an isomorphism. Then

Tr_{A}^{E}φ^{∗}(res_{A′}^{E}(v)^{q}) ∈ C^{q−1}M_{q}.

Proof. For any φ, φ^{∗}res_{A′}^{E}(v) = λμ where λ ∈ 𝔽_{p}^{×} and μ equals to res_{A}^{E}(v) or some E-conjugate of y(u^{p−1} − y^{p−1}). If φ^{∗}res_{A′}^{E}(v) = λres_{A}^{E}(v) then Tr_{A}^{E}φ^{∗}res_{A′}^{E}(v) = 0. On the other hand, if φ^{∗}res_{A′}^{E}(v) = λμ and μ is E-conjugate of y(u^{p−1} −y^{p−1}), then Tr_{A}^{E}φ^{∗}(res_{A′}^{E}(v)^{q}) ∈ 𝔽_{p}Tr_{A}^{E}(y^{q}(u^{p−1} − y^{p−1})^{q}) ⊂ ŷ_{A}^{q}C^{q−1}M_{0} ⊂ C^{q−1}M_{q} by Lemma 6.1(4). □

Lemma 6.4. Let φ : A ⟶ A′ ≤ E be an isomorphism. Then

Tr_{A}^{E}φ^{∗}(res_{A′}^{E}(v^{j}S^{j})) = 0

for any j ≥ 0.

Proof. Since φ^{∗}(res_{A′}^{E}(v^{j}S^{j})) ⊂ 𝔽_{p}d_{2}^{j}, the image of transfer map is zero. □

For A′ ∈ A(E), let y′ ∈ H^{2}(A′) be the element corresponding to y ∈ H^{2}(A), that is, y′ = res_{A′}^{E}(ŷ_{A′}). Note that res_{A′}^{E}(S^{i}) = 𝔽_{p}(y′)^{i} and res_{A′}^{E}(C) = (y′)^{p−1}.

Lemma 6.5. Suppose that 0 ≤ m ≤ p and 0 ≤ i, q ≤ p − 2. Let φ : A ⟶ A′ ≤ E be an isomorphism. Then the element

Tr_{A}^{E}φ^{∗}(res_{A′}^{E}(C^{m}v^{q})(y′)^{i})

is contained in the following 𝔽_{p}-subspace W:

condition  W
q ≤ i, m + q < i + 1  0
q ≤ i, m + q ≥ i + 1  C^{m+q−i−1}v^{i}M_{q}
i < q, m > 0, m + q < p + i + 1  0
i < q, m > 0, m + q ≥ p + i + 1 C^{m+q−p−i−1}V v^{i}M_{q}
i < q, m = 0  C^{q−i−1}v^{i}M_{q}

Proof. Assume q ≤ i. Then

Tr_{A}^{E}φ^{∗}(res_{A′}^{E}(C^{m}v^{q})(y′)^{i}) = Tr_{A}^{E}φ^{∗}((res_{A′}^{E}(v)y′)^{q}res_{A′}^{E}(C^{m})(y′)^{i−q})
⊂ 𝔽_{p}(vŷ_{A})^{q}Tr_{A}^{E}φ^{∗}((y′)^{m(p−1)+(i−q)})
⊂ 𝔽_{p}(vŷ_{A})^{q}Tr_{A}^{E}(u^{m(p−1)+(i−q)})

and the result follows from Lemma 6.1. Next, assume i < q and m > 0. Then

Tr_{A}^{E}φ^{∗}(res_{A′}^{E}(C^{m}v^{q})(y′)^{i}) = Tr_{A}^{E}φ^{∗}((res_{A′}^{E}(v)y′)^{q}res_{A′}^{E}(C^{m−1})(y′)^{p−1−(q−i)})
⊂ 𝔽_{p}(vŷ_{A})^{q}Tr_{A}^{E}φ^{∗}((y′)^{(m−1)(p−1)+(p−1−(q−i))})
⊂ 𝔽_{p}(vŷ_{A})^{q}Tr_{A}^{E}(u^{(m−1)(p−1)+(p−1−(q−i))})

and the result follows from Lemma 6.1. Finally, assume i < q and m = 0. Then

Tr_{A}^{E}φ^{∗}(res_{A′}^{E}(v^{q})(y′)^{i}) = Tr_{A}^{E}φ^{∗}((res_{A′}^{E}(v)y′)^{i}res_{A′}^{E}(v)^{q−i})
⊂ 𝔽_{p}(vŷ_{A})^{i}Tr_{A}^{E}φ^{∗}(res_{A′}^{E}(v)^{q−i})
⊂ (vŷ_{A})^{i}C^{q−i−1}M_{q−i}
⊂ C^{q−i−1}v^{i}M_{q}

by Lemma 6.3. □

Lemma 6.6. Suppose that m ≥ 0 and 0 ≤ i, q ≤ p − 2. Let φ : A ⟶ A′ ≤ E be an isomorphism. Then the element

Tr_{A}^{E}φ^{∗}(res_{A′}^{E}(C^{m}v^{q})(y′)^{i})

is contained in the following 𝔽_{p}-subspace W:

condition  W
q ≤ i  ℂ𝔸{v^{i}M_{q}}
i < q, m > 0 ℂ𝔸{V v^{i}M_{q}}

Proof. The proof of this lemma is similar to the proof of the previous lemma using Lemma 6.2 instead of Lemma 6.1. Assume q ≤ i. Then

Tr_{A}^{E}φ^{∗}(res_{A′}^{E}(C^{m}v^{q})(y′)^{i}) ⊂ 𝔽_{p}(vŷ_{A})^{q}Tr_{A}^{E}(u^{m(p−1)+(i−q)})

and the result follows from Lemma 6.2. Next, assume i < q and m > 0. Then

Tr_{A}^{E}φ^{∗}(res_{A′}^{E}(C^{m}v^{q})(y′)^{i}) ⊂ 𝔽_{p}(vŷ_{A})^{q}Tr_{A}^{E}(u^{(m−1)(p−1)+(p−1−(q−i))})

and the result follows from Lemma 6.2. □

7. A_{p}(E, E)-submodules of H^{∗}(E)

Let A_{p}(E, E) = 𝔽_{p} ⊗ A_{ℤ}(E, E) be the double Burnside algebra of E over 𝔽_{p}. In this section, we consider some A_{p}(E, E)-submodules of H^{∗}(E). Note that A_{p}(E, E) = 𝔽_{p}Out(E) ⊕ J(E). First we shall show that the ideal J(E) is generated three kinds of bisets.

Definition 7.1. (1) Let I_{0} be the 𝔽_{p}-subspace of A_{p}(E, E) spanned by all bisets corresponding to E ≥ A ⟶^{φ} E, A ∈ A(E) and φ is an injective morphism.

(2) Let I_{1} be the 𝔽_{p}-subspace of A_{p}(E, E) spanned by all bisets corresponding to ϕ : E ⟶ E and ϕ(E) ∈ A(E).

Lemma 7.2. The ideal J(E) is generated by I_{0}, I_{1} and the elements in A_{p}(E, E) corresponding to the bisets which factor through the trivial subgroup.

Proof. Let A ∈ A(E), ψ : A ⟶ E and ψ′ : E ⟶ E and assume that |ψ(A)| = |ψ′(E)| = p. It suffices to show that ζ_{A,ψ} and ζ_{E,ψ′} are contained in the ideal generated by I_{0} and I_{1}.

First consider ζ_{A,ψ}. There exist an automorphism φ_{1} : A ⟶^{∼} A and a surjective morphism ψ_{1} : A ⟶ ψ(A) such that ψ = ψ_{1}φ_{1} and Ker ψ_{1} = ⟨c⟩. Then ψ_{1} extends to ϕ_{1} : E ⟶ E such that |ϕ_{1}(E)| = p^{2} and we have ψ = ϕ_{1}φ_{1}. Hence

ζ_{A,ψ} = ζ_{E,ϕ_{1}}ζ_{A,φ_{1}} ∈ I_{1}I_{0}.

Next we consider ζ_{E,ψ′}. There exist morphism ϕ_{2}, ϕ_{3} : E ⟶ E such that |ϕ_{2}(E)| = |ϕ_{3}(E)| = p^{2} and ϕ_{3}ϕ_{2} = ψ′. Then it follows that

ζ_{E,ψ′} = ζ_{E,ϕ_{3}}ζ_{E,ϕ_{2}} ∈ I_{1}^{2}

and this completes the proof. □

Next, we consider the effect of I_{0} on some subspaces of H^{∗}(E). Recall that M_{i} = CS^{i} + T^{i} for 0 ≤ i ≤ p − 2 where S^{0} = 𝔽_{p} and T^{0} = S^{p−1}.

Lemma 7.3. (1) If 0 ≤ j ≤ p − 1 and 0 ≤ i ≤ p − 2, then

(C^{j}v^{i}M_{i})I_{0} ⊂ C^{j}v^{i}M_{i}

and

v^{i}I_{0} ⊂ C^{i−1}M_{i}.

(2) If j ≥ 0 and 0 ≤ i, q ≤ p − 2, then

(C^{j}v^{q}M_{i})I_{0} ⊂ ℂ𝔸{v^{i}M_{q}}.

(3) If j ≥ 0 and 1 ≤ q ≤ p − 2, then

(C^{j}v^{q}M_{0})I_{0} ⊂ ℂ𝔸{V M_{q}}.

(4) If j ≥ 0 and 1 ≤ i ≤ p − 2, then

(C^{j}V M_{i})I_{0} ⊂ ℂ𝔸{v^{i}M_{0}}.

Proof. (1) follows from Lemma 6.5. (2) and (3) follow from Lemma 6.6. Let us prove (4). Since

res_{A}^{E}(C^{j}V M_{i}) ⊂ 𝔽_{p}res_{A}^{E}(C^{j}V)y^{p−1+i}
= 𝔽_{p}res_{A}^{E}(C^{j+1}V)y^{i}
= 𝔽_{p}D̃_{2}res_{A}^{E}(C^{j}y^{i})

for every A ∈ A(E), it follows that (C^{j}V M_{i})I_{0} ⊂ ℂ𝔸{v^{i}M_{0}} by Lemma 6.6. □

Now, we consider certain A_{p}(E, E)-submodules of H^{∗}(E).

Theorem 7.4. H^{∗}(E) is a direct sum of the following A_{p}(E, E)-submodules.

(7.4.1)  ℂ𝔸{𝔽_{p} + S^{p−1}}
(7.4.2)  ℂ𝔸{S^{i} + T^{i} + 𝔽_{p}v^{i} + v^{i}S^{p−1}} (1 ≤ i ≤ p − 2)
(7.4.3)  ℂ𝔸{v^{i}(S^{i} ⊕ T^{i})} (1 ≤ i ≤ p − 2)
(7.4.4)  ∑_{1≤i≠q≤p−2} ℂ𝔸{v^{q}(S^{i} ⊕ T^{i})}.

Proof. By Theorem 4.4, as a GL_{2}(𝔽_{p})-module, H^{∗}(E) is a direct sum of these submodules. We shall prove that these are in fact A_{p}(E, E)-submodules in Proposition 7.6, 7.9, 7.11 and 7.13 below. □

Let us show that these modules are A_{p}(E, E)-submodules and analyze their detailed structure. We shall prove Corollary 7.7, 7.10, 7.12 and 7.14 using Proposition 8.2, 8.4 and Corollary 9.3. Note that above four corollaries are not used in section 8 and section 9.

Let W be an 𝔽_{p}Out(E)-submodule of H^{∗}(E). In order to prove that W is an A_{p}(E, E)-submodule, it suffices to show that W I_{0} and W I_{1} are contained in W by Lemma 7.2. Note that (𝔻𝔸x)I_{0} ⊂ 𝔻𝔸(xI_{0}) for any x ∈ H^{∗}(E). Moreover (vS^{i})I_{1} = (vT^{i})I_{1} = 0 for i > 0 and D_{2}I_{1} = 0 since φ^{∗}(d_{2}) = 0 for any surjective morphism φ : E ⟶ A.

Definition 7.5. We set

N_{i} = ∑_{j=0}^{p−1} C^{j}M_{i}

for 0 ≤ i ≤ p − 2.

First, we consider the module in (7.4.1).

Proposition 7.6. The following is a series of A_{p}(E, E)-submodules of H^{∗}(E),

ℂ𝔸{𝔽_{p} + S^{p−1}} ⊃ ℂ𝔸{M_{0}} ⊃ ℂ𝔸{V M_{0}}.

Moreover we have a decomposition into A_{p}(E, E)-submodules,

ℂ𝔸{V M_{0}} = 𝔻𝔸{D_{2}N_{0}} ⊕ 𝔻𝔸{D_{2}} ⊕ 𝔻𝔸{V S^{p−1}}.

Proof. First we show that 𝔻𝔸{D_{2}N_{0}} is an A_{p}(E, E)-submodule of H^{∗}(E). We have 𝔻𝔸{D_{2}N_{0}}I_{1} = 0. On the other hand, since C^{j}M_{0}I_{0} ⊂ C^{j}M_{0} for 0 ≤ j ≤ p − 1 by Lemma 7.3(1), 𝔻𝔸{D_{2}N_{0}}I_{0} ⊂ 𝔻𝔸{D_{2}N_{0}}.

Next, since 𝔻𝔸{D_{2}}I_{0} = 𝔻𝔸{D_{2}}I_{1} = 0, 𝔻𝔸{D_{2}} is an A_{p}(E, E)-submodule. Similarly, 𝔻𝔸{V S^{p−1}}I_{1} = 0. Moreover 𝔻𝔸{V S^{p−1}}I_{0} = 0 by Lemma 6.4 and 𝔻𝔸{V S^{p−1}} is an A_{p}(E, E)-submodule. We have a direct sum decomposition as an 𝔽_{p}-vector space,

ℂ𝔸{V M_{0}} = 𝔻𝔸{∑_{j=0}^{p}(C^{j}V M_{0})}

= 𝔻𝔸{C(∑_{j=0}^{p−1}(C^{j}V M_{0})} ⊕ 𝔻𝔸{V M_{0}}
= 𝔻𝔸{D_{2}N_{0}} ⊕ 𝔻𝔸{D_{2}} ⊕ 𝔻𝔸{V S^{p−1}}

and this is a decomposition as an A_{p}(E, E)-module.

Next we consider ℂ𝔸{M_{0}}. By Proposition 4.2, ℂ𝔸{M_{0}}I_{1} ⊂ ℂ𝔸{M_{0}}. By Lemma 7.3(2),

ℂ𝔸{M_{0}}I_{0} = 𝔻𝔸{∑_{j=0}^{p} C^{j}M_{0}}I_{0} ⊂ ℂ𝔸{M_{0}}.

Hence ℂ𝔸{M_{0}} is an A_{p}(E, E)-submodule. Finally we consider ℂ𝔸{𝔽_{p}+S^{p−1}}. By Proposition 4.2, (ℂ𝔸{𝔽_{p} + S^{p−1}}^{+})I_{1} ⊂ ℂ𝔸{M_{0}}. Note that, as 𝔽_{p}-vector spaces, ℂ𝔸{𝔽_{p} + S^{p−1}} = 𝔽_{p}[D_{1}] ⊕ ℂ𝔸{M_{0}}. Since 𝔽_{p}[D_{1}]I_{0} = 0, we have ℂ𝔸{𝔽_{p} + S^{p−1}}I_{0} ⊂ ℂ𝔸{M_{0}}. □

Corollary 7.7. (1) ℂ𝔸{𝔽_{p} + S^{p−1}} = 𝔽_{p}[D_{1}] ⊕ ℂ𝔸{M_{0}} as GL_{2}(𝔽_{p})-modules and

ℂ𝔸{𝔽_{p} + S^{p−1}}^{+}/ℂ𝔸{M_{0}} ≃ ⨁ S(E, E, 𝔽_{p})

as A_{p}(E, E)-modules.

(2) ℂ𝔸{M_{0}} = 𝔽_{p}[C]{M_{0}} ⊕ ℂ𝔸{V M_{0}} as GL_{2}(𝔽_{p})-modules and

ℂ𝔸{M_{0}}/ℂ𝔸{V M_{0}} ≃ ⨁ S(E, Q, U_{0})

as A_{p}(E, E)-modules, where Q ≤ P, |Q| = p and U_{0} is the trivial 𝔽_{p}Out(Q)-module.

(3) As A_{p}(E, E)-modules,

𝔻𝔸{D_{2}N_{0}} ≃ ⨁ S(E, A, S(A)^{p−1})
𝔻𝔸{D_{2}} ≃ ⨁ S(E, E, 𝔽_{p})

and

𝔻𝔸{V S^{p−1}} ≃ ⨁ S(E, E, S^{p−1}).

Proof. (1) Since ℂ𝔸{𝔽_{p} + S^{p−1}}I_{0} and (ℂ𝔸{𝔽_{p} + S^{p−1}}^{+})I_{1} are contained in ℂ𝔸{M_{0}} as in the proof of Proposition 7.6, it follows that

(ℂ𝔸{𝔽_{p} + S^{p−1}}^{+}/ℂ𝔸{M_{0}})J(E) = 0.

Since GL_{2}(𝔽_{p}) acts on D_{1} trivially, we have

ℂ𝔸{𝔽_{p} + S^{p−1}}^{+}/ℂ𝔸{M_{0}} ≃ ⨁ S(E, E, 𝔽_{p})

(2) This follows from Proposition 8.2.

(3) By Corollary 9.3,

𝔻𝔸{D_{2}N_{0}} ≃ ⨁ S(E, A, S(A)^{p−1}).

Since 𝔻𝔸{D_{2}}I_{0} = 𝔻𝔸{D_{2}}I_{1} = 0 and GL_{2}(𝔽_{p}) acts on D_{2} trivially, it follows that

𝔻𝔸{D_{2}} ≃ ⨁ S(E, E, 𝔽_{p}).

Similarly, since

𝔻𝔸{V S^{p−1}}I_{0} = 𝔻𝔸{V S^{p−1}}I_{1} = 0,

it follows that

𝔻𝔸{V S^{p−1}} ≃ ⨁ S(E, E, S^{p−1}).
□

Next, we consider the module in (7.4.2).

Lemma 7.8. Let 1 ≤ i ≤ p − 2.

(1) ℂ𝔸{S^{i} + T^{i}}I_{0} ⊂ ℂ𝔸{v^{i}M_{0}}.

(2) ℂ𝔸{S^{i} + T^{i}}I_{1} ⊂ 𝔽_{p}[C]{S^{i}}.

Proof. (1) First, note that

ℂ𝔸{S^{i} + T^{i}} ⊂ 𝔻𝔸{S^{i}} + 𝔻𝔸[C]{M_{i}}.

Since 𝔻𝔸{S^{i}}I_{0} = 0 and 𝔻𝔸[C]{M_{i}}I_{0} ⊂ ℂ𝔸{v^{i}M_{0}} by Lemma 7.3(2), the proof is completed.

(2) We have that ℂ𝔸{V (S^{i} + T^{i})}I_{1} = 0. Hence it follows that

ℂ𝔸{S^{i} + T^{i}}I_{1} ⊂ 𝔽_{p}[C]{S^{i} + T^{i}}I_{1} ⊂ 𝔽_{p}[C]{S^{i}}

by Corollary 4.9 and the proof is completed. □

Proposition 7.9. Let 1 ≤ i ≤ p − 2. Let Z_{i} = 𝔽_{p}[C]{S^{i}} + ℂ𝔸{v^{i}M_{0} + V M_{i}}. Then we have the following A_{p}(E, E)-submodules:

ℂ𝔸{S^{i} + T^{i} + 𝔽_{p}v^{i} + v^{i}S^{p−1}}
|
ℂ𝔸{S^{i} + T^{i} + v^{i}M_{0}}
⧸ ⧹
𝔽_{p}[V]{V S^{i}} + Z_{i} 𝔽_{p}[C]{T^{i}} + Z_{i}
⧹ ⧸
Z_{i}
|
ℂ𝔸{v^{i}M_{0} + V M_{i}}

Proof. First, we consider ℂ𝔸{v^{i}M_{0} + V M_{i}}. We have ℂ𝔸{v^{i}M_{0} + V M_{i}}I_{1} = 0. Since ℂ𝔸 = 𝔻𝔸{1, C, …, C^{p}}, in order to show that ℂ𝔸{v^{i}M_{0} + V M_{i}}I_{0} ⊂ ℂ𝔸{v^{i}M_{0} +V M_{i}}, it suffices to show that C^{j}{v^{i}M_{0} +V M_{i}}I_{0} ⊂ ℂ𝔸{v^{i}M_{0} +V M_{i}} for j ≥ 0. By Lemma 7.3(3), we have (C^{j}v^{i}M_{0})I_{0} ⊂ ℂ𝔸{V M_{i}}. On the other hand, (C^{j}V M_{i})I_{0} ⊂ ℂ𝔸{v^{i}M_{0}} by Lemma 7.3(4). This shows that ℂ𝔸{v^{i}M_{0} + V M_{i}} is an A_{p}(E, E)-submodule.

Next, consider Z_{i} = 𝔽_{p}[C]{S^{i}} + ℂ𝔸{v^{i}M_{0} + V M_{i}}. Since res_{A}^{E}(C^{j}S^{i}) = 𝔽_{p}y^{j(p−1)+i}, it follows that (𝔽_{p}[C]{S^{i}})I_{1} ⊂ 𝔽_{p}[C]{S^{i}} by Corollary 4.9. On the other hand, we have that 𝔽_{p}[C]{S^{i}}I_{0} ⊂ ℂ𝔸{v^{i}M_{0}} by Lemma 7.3(2). Hence Z_{i} is an A_{p}(E, E)-submodule.

Next we consider ℂ𝔸{S^{i}+T^{i}+v^{i}M_{0}}, 𝔽_{p}[V]{V S^{i}}+Z_{i} and 𝔽_{p}[C]{T^{i}}+Z_{i}. Since these are GL_{2}(𝔽_{p})-submodules, it suffices to show that ℂ𝔸{S^{i} +T^{i} +v^{i}M_{0}}J(E) ⊂ Z_{i}. But this follows from Lemma 7.8 since ℂ𝔸{v^{i}M_{0}} ⊂ Z_{i}.

Finally, we consider ℂ𝔸{S^{i} + T^{i} + 𝔽_{p}v^{i} + v^{i}S^{p−1}}. Note that

ℂ𝔸{S^{i} + T^{i} + 𝔽_{p}v^{i} + v^{i}S^{p−1}} = ℂ𝔸{v^{i}} + ℂ𝔸{S^{i} + T^{i} + v^{i}M_{0}}.

We claim that ℂ𝔸{v^{i}}I_{0} ⊂ ℂ𝔸{S^{i} + T^{i}}. It suffices to show that (C^{j}v^{i})I_{0} ⊂ ℂ𝔸{S^{i} + T^{i}} for j ≥ 0. If j = 0, then v^{i}I_{0} ⊂ C^{i−1}M_{i} by Lemma 7.3(1). If j > 0, then

(C^{j}v^{i})I_{0} ⊂ (C^{j−1}v^{i}M_{0})I_{0} ⊂ ℂ𝔸{V M_{i}} ⊂ ℂ𝔸{S^{i} + T^{i}}

by Lemma 7.3(3). On the other hand, ℂ𝔸{v^{i}}I_{1} ⊂ ℂ𝔸{S^{i}, T^{i}} by Proposition 4.2. Hence

ℂ𝔸{S^{i} + T^{i} + v^{i} + 𝔽_{p}v^{i} + S^{p−1}}J(E) ⊂ ℂ𝔸{S^{i} + T^{i} + v^{i}M_{0}}

and this completes the proof. □

Corollary 7.10. Let 1 ≤ i ≤ p − 2 and set Z_{i} = 𝔽_{p}[C]{S^{i}} + ℂ𝔸{v^{i}M_{0} + V M_{i}} as in Proposition 7.9. The we have the following.

(1) ℂ𝔸{S^{i} + T^{i} + 𝔽_{p}v^{i} + v^{i}S^{p−1}} = 𝔽_{p}[V]{v^{i}} ⊕ ℂ𝔸{S^{i} + T^{i} + v^{i}M_{0}} as GL_{2}(𝔽_{p})- modules and

ℂ𝔸{S^{i} + T^{i} + 𝔽_{p}v^{i} + v^{i}S^{p−1}}/ℂ𝔸{S^{i} + T^{i} + v^{i}M_{0}} ≃ ⨁ S(E, E, det^{i})

as A_{p}(E, E)-modules.

(2) ℂ𝔸{S^{i} + T^{i} + v^{i}M_{0}} = 𝔽_{p}[C]{T^{i}} ⊕ 𝔽_{p}[V]{V S^{i}} ⊕ Z_{i} as 𝔽_{p}-spaces and we have ℂ𝔸{S^{i} + T^{i} + v^{i}M_{0}}J(E) ⊂ Z_{i}. In particular,

ℂ𝔸{S^{i} + T^{i} + v^{i}M_{0}}/(𝔽_{p}[C]{T^{i}} + Z_{i}) ≃ (𝔽_{p}[V]{V S^{i}} + Z_{i})/Z_{i} ≃ ⨁ S(E, E, S^{i}) and

ℂ𝔸{S^{i} + T^{i} + v^{i}M_{0}}/(𝔽_{p}[V]{V S^{i}} + Z_{i}) ≃ (𝔽_{p}[C]{T^{i}} + Z_{i})/Z_{i} ≃ ⨁ S(E, E, T̄^{i})

as A_{p}(E, E)-modules, where T̄^{i} = (CS^{i} + T^{i})/CS^{i}.

(3) Z_{i}/ℂ𝔸{v^{i}M_{0} +V M_{i}} ≃ ⊕S(E, Q, U_{i}) where Q ≤ E, |Q| = p and U_{i} is a simple 𝔽_{p}Out(Q)-module. (See the first part of section 5.)

(4) Every composition factor of ℂ𝔸{v^{i}M_{0} + V M_{i}} as an A_{p}(E, E)-module has minimal subgroup E.

Proof. (1) Since there is a decomposition into 𝔽_{p}-vector subspaces

ℂ𝔸{𝔽_{p} + S^{p−1}} = 𝔽_{p}[V] ⊕ ℂ𝔸{C} ⊕ ℂ𝔸{S^{p−1}},

we have

ℂ𝔸{S^{i} + T^{i} + 𝔽_{p}v^{i} + v^{i}S^{p−1}} = 𝔽_{p}[V]{v^{i}} ⊕ ℂ𝔸{S^{i} + T^{i} + v^{i}M_{0}}.

Moreover

(ℂ𝔸{S^{i} + T^{i} + 𝔽_{p}v^{i} + v^{i}S^{p−1}}/ℂ𝔸{S^{i} + T^{i} + v^{i}M_{0}})J(E) = 0

by the proof of Proposition 7.9. Hence the result follows since GL_{2}(𝔽_{p}) acts on v^{i} as det^{i}.

(2) Since there exist decompositions into 𝔽_{p}-subspaces,

ℂ𝔸{S^{i}} = 𝔽_{p}[C]{S^{i}} ⊕ 𝔽_{p}[V]{V S^{i}} ⊕ ℂ𝔸{V CS^{i}}
ℂ𝔸{T^{i}} = 𝔽_{p}[C]{T^{i}} ⊕ ℂ𝔸{V T^{i}},

we have

ℂ𝔸{S^{i} + T^{i}} = 𝔽_{p}[C]{T^{i}} ⊕ 𝔽_{p}[V]{V S^{i}} ⊕ 𝔽_{p}[C]{S^{i}} ⊕ ℂ𝔸{V M_{i}}

and

ℂ𝔸{S^{i} + T^{i} + v^{i}M_{0}} = 𝔽_{p}[C]{T^{i}} ⊕ 𝔽_{p}[V]{V S^{i}} ⊕ Z_{i}.

Hence the result follows since

ℂ𝔸{S^{i} + T^{i} + v^{i}M_{0}}J(E) ⊂ Z_{i}

by the proof of Proposition 7.9.

(3) This follows from Proposition 8.4.

(4) This follows from Proposition 8.2, 8.4 and Corollary 9.3. □

Next, we consider the submodule in (7.4.3).

Proposition 7.11. Let 1 ≤ i ≤ p − 2. The 𝔽_{p}-subspace ℂ𝔸{v^{i}S^{i} + v^{i}T^{i}} is an A_{p}(E, E)-submodule of H^{∗}(E) and we have the following decomposition into A_{p}(E, E)-submodules,

ℂ𝔸{v^{i}S^{i} + v^{i}T^{i}} = 𝔻𝔸{v^{i}N_{i}} ⊕ 𝔻𝔸{v^{i}(S^{i} + V T^{i})}.

Proof. Since

ℂ𝔸{v^{i}S^{i} + v^{i}T^{i}} = 𝔻𝔸{v^{i}N_{i}} ⊕ 𝔻𝔸{v^{i}(S^{i} + V T^{i})}

as 𝔽_{p}-vector spaces, it suffices to show that 𝔻𝔸{v^{i}N_{i}} and 𝔻𝔸{v^{i}(S^{i} + V T^{i})} are A_{p}(E, E)-submodules. First, these are GL_{2}(𝔽_{p})-submodules since V (CS^{i} + T^{i}) = D_{2}S^{i} + V T^{i}. We have ℂ𝔸{v^{i}S^{i} + v^{i}T^{i}}I_{1} = 0. Since C^{j}v^{i}M_{i}I_{0} ⊂ C^{j}v^{i}M_{i} for 0 ≤ j ≤ p − 1 by Lemma 7.3(1), we have 𝔻𝔸{v^{i}N_{i}}I_{0} ⊂ 𝔻𝔸{v^{i}N_{i}}. On the other hand, 𝔻𝔸{v^{i}(S^{i} + V T^{i})}I_{0} = 0 by Lemma 6.4 and this completes the proof. □

Corollary 7.12. (1) As A(E, E)-modules,

𝔻𝔸{v^{i}N_{i}} ≃ ⨁ S(E, A, S^{p−1}(A) ⊗ det^{i}).

(2) As an A(E, E)-module, every composition factor of 𝔻𝔸{v^{i}(S^{i} + V T^{i})} has minimal subgroup E.

Proof. (1) This follows from Corollary 9.3.

(2) Since 𝔻𝔸{v^{i}(S^{i} + V T^{i})}J(E) = 0 as in the proof of Proposition 7.11, every composition factor of 𝔻𝔸{v^{i}(S^{i} + V T^{i})} has minimal subgroup E. □

Finally, let us consider the submodule in (7.4.4).

Proposition 7.13. The 𝔽_{p}-subspace

∑_{1≤i≠q≤p−2} ℂ𝔸{v^{q}S^{i} + v^{q}T^{i}}

is an A_{p}(E, E)-submodule of H^{∗}(E).

Proof. Let W = ∑_{1≤i≠q≤p−2} ℂ𝔸{v^{q}S^{i} + v^{q}T^{i}}. We have W I_{1} = 0. Since ℂ𝔸 = ∑_{j=0}^{p} 𝔻𝔸{C^{j}}, it suffices to show that C^{j}(v^{q}S^{i} + v^{q}T^{i})I_{0} ⊂ W for 0 ≤ j ≤ p. By Lemma 6.5, C^{j}(v^{q}S^{i} + v^{q}T^{i})I_{0} ⊂ ℂ𝔸{v^{i}M_{q}} ⊂ W, and this completes the proof. □

Corollary 7.14. As an A(E, E)-module, every composition factor of

∑_{1≤i≠q≤p−2} ℂ𝔸{v^{q}S^{i} + v^{q}T^{i}}

has minimal subgroup E.

Proof. This follows from Proposition 8.2, 8.4 and Corollary 9.3. □

8. Simple modules with cyclic minimal subgroup

In this section, we consider simple A_{p}(E, E)-modules induced from cyclic subgroup of order p. Recall that M_{i} = CS^{i} + T^{i} for 0 ≤ i ≤ p − 2 where S^{0} = 𝔽_{p} and T^{0} = S^{p−1}.

Let Q ≤ E be a subgroup of order p. Let U_{i} = 𝔽_{p}u_{i} (0 ≤ i ≤ p−2) be the simple right 𝔽_{p}Out(Q)-module defined by u_{i}σ = λ^{i}u_{i} where Out(Q) = ⟨σ⟩ and 𝔽_{p}^{×} = ⟨λ⟩. Let η be a generator of H^{2}(Q). If n = m(p − 1) + i where m ≥ 0 and 0 ≤ i ≤ p − 2, then H^{2n}(Q) ≃ U_{i} as 𝔽_{p}Out(Q)-modules.

First, we consider the simple A_{p}(E, E)-module corresponding to U_{0}, namely, S(E, Q, U_{0}).

Lemma 8.1. Let n = m(p − 1) > 0. Then we have

H^{2n}(Q)A(E, Q) + ℂ𝔸{V M_{0}} = C^{m−1}M_{0} + ℂ𝔸{V M_{0}}.

Proof. H^{2n}(Q)A(E, Q) is generated by ∑_{ϕ:E↠Q} ϕ^{∗}H^{2n}(Q) and

∑_{A∈A(E)} ∑_{φ:A↠Q} Tr_{A}^{E}φ^{∗}(H^{2n}(Q)).

The former 𝔽_{p}-subspace is equal to C^{m−1}S^{p−1} by Corollary 4.9. On the other hand, if φ : A ↠ Q then φ^{∗}(η) is λy or E-conjugate of λu for some λ ∈ 𝔽_{p}. Hence the later 𝔽_{p}-subspace is equal to ∑_{A∈A(E)} 𝔽_{p}Tr_{A}^{E}(u^{n}). By Lemma 6.2,

Tr_{A}^{E}(u^{n}) ≡ C^{m−1}Tr_{A}^{E}(u^{p−1}) (mod ℂ𝔸{V M_{0}}).

Since C^{m−1}Tr_{A}^{E}(u^{p−1}), A ∈ A(E) is a basis of C^{m−1}M_{0}, it follows that

∑_{A∈A(E)} 𝔽_{p}Tr_{A}^{E}(u^{n}) + ℂ𝔸{V M_{0}} = C^{m−1}M_{0} + ℂ𝔸{V M_{0}}

and this completes the proof. □

Proposition 8.2. As A_{p}(E, E)-modules, we have

ℂ𝔸{M_{0}}/ℂ𝔸{V M_{0}} ≃ ⨁ S(E, Q, U_{0}).

Moreover, H^{∗}(E)/ℂ𝔸{M_{0}} and ℂ𝔸{V M_{0}} have no simple subquotient module which is isomorphic to S(E, Q, U_{0}).

Proof. First, note that ℂ𝔸{M_{0}} and ℂ𝔸{V M_{0}} are A_{p}(E, E)-submodules by Proposition 7.6. Let F_{m} = H^{2m(p−1)}(Q)A(−, Q) for m ≥ 1. By Lemma 8.1,

F_{m}(E) + ℂ𝔸{V M_{0}} = C^{m−1}M_{0} + ℂ𝔸{V M_{0}}.

In particular,

( ⋂_{R≤E,|R|=p} Ker res_{R}^{E}) ∩ F_{m}(E) = ℂ𝔸{V M_{0}} ∩ F_{m}(E).

Hence F_{m}(E)/(F_{m}(E) ∩ ℂ𝔸{V M_{0}}) ≃ S(E, Q, U_{0}) by Lemma 3.1, and we have

ℂ𝔸{M_{0}}/ℂ𝔸{V M_{0}} ≃ ( ∑_{m≥0} F_{m}(E) + ℂ𝔸{V M_{0}})/ℂ𝔸{V M_{0}}
≃ ⨁ F_{m}(E)/(F_{m}(E) ∩ ℂ𝔸{V M_{0}})
≃ ⨁ S(E, Q, U_{0}).

Let F = ⊕_{m≥1}F_{m}. Since H^{∗}(Q)/F (Q) has no simple subquotient module which is isomorphic to S(Q, Q, U_{0}), it follows that H^{∗}(E)/F (E) has no simple subquotient module which is isomorphic to S(E, Q, U_{0}) by Corollary 3.3. Since F (E) ⊂ ℂ𝔸{M_{0}}, it follows that H^{∗}(E)/ℂ𝔸{M_{0}} has no simple subquotient module which is isomorphic to S(E, Q, U_{0}). Moreover, since the restriction of ℂ𝔸{V M_{0}} to any subgroup of order p is zero, we have (ℂ𝔸{V M_{0}})A_{p}(Q, E) = 0. Hence ℂ𝔸{V M_{0}} has no simple subquotient module which is isomorphic to S(E, Q, U_{0}) by Lemma 3.2. □

Next, we consider the simple A_{p}(E, E)-module corresponding to U_{i} for 1 ≤ i ≤ p − 2, namely, S(E, Q, U_{i}).

Lemma 8.3. Let 1 ≤ i ≤ p − 2 and n = m(p − 1) + i. Then we have

H^{2n}(Q)A(E, Q) + ℂ𝔸{v^{i}M_{0} + V M_{i}} = C^{m}S^{i} + ℂ𝔸{v^{i}M_{0} + V M_{i}}.

Proof. By Corollary 4.9, H^{2n}(Q)A(E, Q) is spanned by C^{m}S^{i} and

∑_{A∈A(E)} ∑_{φ:A↠Q} Tr_{A}^{E}φ^{∗}(H^{2n}(Q)).

Since φ^{∗}(η) is λy or E-conjugate of λu for some λ ∈ 𝔽_{p}, it follows that

H^{2n}(Q)A(E, Q) + ℂ𝔸{v^{i}M_{0} + V M_{i}} = C^{m}S^{i} + ℂ𝔸{v^{i}M_{0} + V M_{i}}

by Lemma 6.2 and this completes the proof. □

Proposition 8.4. Let 1 ≤ i ≤ p − 2. As A_{p}(E, E)-modules, we have

(𝔽_{p}[C]{S^{i}} + ℂ𝔸{v^{i}M_{0} + V M_{i}})/ℂ𝔸{v^{i}M_{0} + V M_{i}} ≃ ⨁ S(E, Q, U_{i}).

Moreover, H^{∗}(E)/(𝔽_{p}[C]{S^{i}} + ℂ𝔸{v^{i}M_{0} + V M_{i}}) and ℂ𝔸{v^{i}M_{0} + V M_{i}} have no simple subquotient module which is isomorphic to S(E, Q, U_{i}).

Proof. We set Z_{i} = 𝔽_{p}[C]{S^{i}} + ℂ𝔸{v^{i}M_{0} + V M_{i}} as in Proposition 7.9. Then Z_{i} and ℂ𝔸{v^{i}M_{0} + V M_{i}} are A_{p}(E, E)-submodules by Lemma 7.9. Let

F_{m,i} = H^{2m(p−1)+i}(Q)A(−, Q)

for m ≥ 0. By Lemma 8.3,

F_{m,i}(E) + ℂ𝔸{v^{i}M_{0} + V M_{i}} = C^{m}S^{i} + ℂ𝔸{v^{i}M_{0} + V M_{i}}.

In particular,

( ⋂_{R≤E,|R|=p} Ker res_{R}^{E}) ∩ F_{m,i}(E) = ℂ𝔸{v^{i}M_{0} + V M_{i}} ∩ F_{m,i}(E).

Hence F_{m,i}(E)/(F_{m,i}(E) ∩ ℂ𝔸{v^{i}M_{0} + V M_{i}}) ≃ S(E, Q, U_{i}) by Lemma 3.1, and we have

Z_{i}/ℂ𝔸{v^{i}M_{0} + V M_{i}} ≃ ( ∑_{m≥0} F_{m,i}(E) + ℂ𝔸{v^{i}M_{0} + V M_{i}})/ℂ𝔸{v^{i}M_{0} + V M_{i}}
≃ ⨁_{m≥0} F_{m,i}(E)/(F_{m,i}(E) ∩ ℂ𝔸{v^{i}M_{0} + V M_{i}})
≃ ⨁ S(E, Q, U_{i}).

Let F = ⊕_{m≥0}F_{m,i}. Since H^{∗}(Q)/F (Q) has no simple subquotient module which is isomorphic to S(Q, Q, U_{i}), it follows that H^{∗}(E)/F (E) has no simple subquotient module which is isomorphic to S(E, Q, U_{i}) by Corollary 3.3. Since F (E) ⊂ Z_{i}, it follows that H^{∗}(E)/Z_{i} has no simple subquotient module which is isomorphic to S(E, Q, U_{i}). Moreover, since the restriction of ℂ𝔸{v^{i}M_{0}+V M_{i}} to any subgroup of order p is zero, we have (ℂ𝔸{v^{i}M_{0} +V M_{i}})A_{p}(Q, E) = 0. Hence ℂ𝔸{v^{i}M_{0} +V M_{i}} has no simple subquotient module which is isomorphic to S(E, Q, U_{i}) by Lemma 3.2.

□

9. Simple modules with minimal subgroup A

In this section, we consider A_{p}(E, E)-modules induced from A_{p}(A, A)-modules where A is a maximal elementary abelian p-subgroup of E.

Lemma 9.1. Let i, j ≥ 0, 0 ≤ m ≤ p − 2 and 0 ≤ n ≤ p − 1. If j + m > 0, then D̃_{1}^{i}D̃_{2}^{j}d_{2}^{m}W_{n}A_{p}(E, A) is a simple A_{p}(E, E)-submodule of H^{∗}(E) and isomorphic to S(E, A, S(A)^{p−1} ⊗ det^{m}).

Proof. Let F = (D̃_{1}^{i}D̃_{2}^{j}d_{2}^{m}W_{n})A_{p}(−, A). Note that

D̃_{1}^{i}D̃_{2}^{j}d_{2}^{m}W_{n} ≃ S(A, A, S(A)^{p−1} ⊗ det^{m})

by Proposition 5.7. Since F (E) ≠ 0, it suffices to show that

⋂_{ϕ∈A_{p}(A,E)} Ker F (ϕ) = 0

by Lemma 3.1. If α ∈ ∩_{ϕ∈A_{p}(A,E)} Ker F (ϕ) then res_{A′}^{E}(α) = 0 for every maximal elementary abelian subgroup A′ of E. Then α = 0 by Quillen’s theorem [15]. □

Proposition 9.2. Let i, j ≥ 0, 0 ≤ m ≤ p − 2 and 0 ≤ n ≤ p − 1. If j + m > 0, then

(D̃_{1}^{i}D̃_{2}^{j}d_{2}^{m}W_{n})A_{p}(E, A) = D_{1}^{i}D_{2}^{j}v^{m}C^{n}M_{m}

where M_{m} = CS^{m} + T^{m}.

Proof. First, we claim that D_{1}^{i}D_{2}^{j}v^{m}C^{n}M_{m} is an A_{p}(E, E)-submodule of H^{∗}(E). We have (D_{1}^{i}D_{2}^{j}v^{m}C^{n}M_{m})I_{1} = 0. Since v^{m}C^{n}M_{m}I_{0} ⊂ v^{m}C^{n}M_{m} by Lemma 7.3(1), D_{1}^{i}D_{2}^{j}v^{m}C^{n}M_{m} is an A_{p}(E, E)-submodule.

By Lemma 6.1(3), {D_{1}^{i}D_{2}^{j}v^{m}ŷ_{A′}^{m}C^{n}Tr(u^{p−1})}_{A′∈A(E)} is a basis of D_{1}^{i}D_{2}^{j}v^{m}C^{n}M_{m}. Since C^{n}Tr_{A′}^{E}(u^{p−1}) = Tr_{A′}^{E}(u^{(n+1)(p−1)}) by Lemma 6.1(2), we have

D_{1}^{i}D_{2}^{j}v^{m}ŷ_{A′}^{m}C^{n}Tr(u^{p−1}) = Tr_{A′}^{E}(D̃_{1}^{i}D̃_{2}^{j}d_{2}^{m}u^{(n+1)(p−1)})

and it follows that

D_{1}^{i}D_{2}^{j}v^{m}C^{n}M_{m} ⊂ (D̃_{1}^{i}D̃_{2}^{j}d_{2}^{m}W_{n})A_{p}(E, A) + (D̃_{1}^{i}D̃_{2}^{j}d_{2}^{m+1}H^{∗}(A))A_{p}(E, A).

Since

(D̃_{1}^{i}D̃_{2}^{j}d_{2}^{m+1}H^{∗}(A))A_{p}(E, A) ⊂ ∑_{A′∈A(E)} ∑_{φ:A′⟶^{∼}A} Tr_{A′}^{E}φ^{∗}(D̃_{1}^{i}D̃_{2}^{j}d_{2}^{m+1}H^{∗}(A))

⊂ D_{1}^{i}D_{2}^{j}v^{m+1}S^{m+1}H^{∗}(E),

we have

D_{1}^{i}D_{2}^{j}v^{m}C^{n}M_{n} ∩ (D̃_{1}^{i}D̃_{2}^{j}d_{2}^{m+1}H^{∗}(A))A_{p}(E, A) = 0.

Then it follows that

(D̃_{1}^{i}D̃_{2}^{j}d_{2}^{m}W_{n})A_{p}(E, A) + (D̃_{1}^{i}D̃_{2}^{j}d_{2}^{m+1}H^{∗}(A))A_{p}(E, A)
= D_{1}^{i}D_{2}^{j}v^{m}C^{n}M_{m} + (D̃_{1}^{i}D̃_{2}^{j}d_{2}^{m+1}H^{∗}(A))A_{p}(E, A).

and

D_{1}^{i}D_{2}^{j}v^{m}C^{n}M_{n} ≃ (D̃_{1}^{i}D̃_{2}^{j}d_{2}^{m}W_{n})A_{p}(E, A)

since (D̃_{1}^{i}D̃_{2}^{j}d_{2}^{m}W_{n})A_{p}(E, A) is a simple A_{p}(E, E)-module by Lemma 9.1.

Next, we prove that (D̃_{1}^{i}D̃_{2}^{j}d_{2}^{m+1}H^{∗}(A))A_{p}(E, A) does not have a submodule which is isomorphic to D_{1}^{i}D_{2}^{j}v^{m}C^{n}M_{m}. In particular this implies that

(D̃_{1}^{i}D̃_{2}^{j}d_{2}^{m}W_{n})A_{p}(E, A) = D_{1}^{i}D_{2}^{j}v^{m}C^{n}M_{m}.

Assume that there exists an A_{p}(E, E)-submodule of (D̃_{1}^{i}D̃_{2}^{j}d_{2}^{m+1}H^{∗}(A))A_{p}(E, A) isomorphic to D_{1}^{i}D_{2}^{j}v^{m}C^{n}M_{m}. Then some submodules of D_{1}^{i}D_{2}^{j}v^{m+1}S^{m+1}H^{∗}(E) is isomorphic to D_{1}^{i}D_{2}^{j}v^{m}C^{n}M_{m}. Then, as a GL_{2}(𝔽_{p})-module, vS^{m+1}H^{2r}(E) has a submodule which is isomorphic to S^{m} where

r = \frac{1}{2}(deg C^{n}M_{m} − deg vS^{m+1})
= n(p − 1) + (p + m − 1) − (p + m + 1) = n(p − 1) − 2.

But this contradicts to Lemma 4.6 and so the proof is completed. □

Corollary 9.3. (1) We have

𝔻𝔸{D_{2}N_{0}} ≃ ⨁ S(E, A, S(A)^{p−1})

and the quotient module

H^{∗}(E)/𝔻𝔸{D_{2}N_{0}}

has no simple subquotient module which is isomorphic to S(E, A, S(A)^{p−1}).

(2) Assume that 1 ≤ m ≤ p − 2. Then

𝔻𝔸{v^{m}N_{m}} ≃ ⨁ S(E, A, S(A)^{p−1} ⊗ det^{m})

and the quotient module

H^{∗}(E)/𝔻𝔸{v^{m}N_{m}}

has no simple subquotient module which is isomorphic to S(E, A, S(A)^{p−1} ⊗ det^{m}).

Proof. (1) By Lemma 9.1 and Proposition 9.2, 𝔻𝔸{D_{2}N_{0}} is isomorphic to a direct sum of S(E, A, S(A)^{p−1}). Since H^{∗}(A)/𝔻̃𝔸{⊕_{n=0}^{p−1}D̃_{2}W_{n}} has no simple subquotient which is isomorphic to S(A, A, S(A)^{p−1}) by Proposition 5.7, it follows that H^{∗}(E)/𝔻𝔸{D_{2}N_{0}} has no simple subquotient module which is isomorphic to S(E, A, S(A)^{p−1}) by Corollary 3.3. This completes the proof of (1) since

𝔻̃𝔸{⨁_{n=0}^{p−1} D̃_{2}W_{n}}A_{p}(E, A) = 𝔻𝔸{D_{2}N_{0}}

by Proposition 9.2. The proof of (2) is similar to the proof of (1). □

Remark 9.4. We consider d_{2}^{q}S(A)^{i} for 0 ≤ i ≤ p − 2 and q > 0. This is a simple A_{p}(A, A)-submodule of H^{∗}(A) with minimal subgroup A. Let F = d_{2}^{q}S(A)^{i}A(−, A). Since

Tr_{A}^{E}(d_{2}^{q}S(A)^{i}) = 0,

we have that F (E) = 0 and in particular, S_{A,S(A)^{i}⊗det^{q}}(E) = 0. Hence there exists no simple A_{p}(E, E)-module corresponding to S(A)^{i} ⊗ det^{q} for 0 ≤ i, q ≤ p − 2.

10. Cohomology of stable summands

By results in section 8 and section 9, we have the following classification of simple A_{p}(E, E)-modules.

Proposition 10.1 ([6]). Up to isomorphisms, the simple A_{p}(E, E)-modules are given as follows.

(1) S(E, Q, U_{i}) (0 ≤ i ≤ p − 2),

dim S(E, Q, U_{i}) = \{ \begin{matrix}p + 1 & (i = 0) \\ i + 1 & (1 ≤ i ≤ p − 2). \end{matrix}

(2) S(E, A, S(A)^{p−1} ⊗ det^{q}) (0 ≤ q ≤ p − 2),

dim S(E, A, S(A)^{p−1} ⊗ det^{q}) = p + 1.

(3) Simple 𝔽_{p}Out(E)-modules, namely,

S(E, E, S^{i} ⊗ det^{i}) (0 ≤ i ≤ p − 1, 0 ≤ q ≤ p − 2).

(4) The one dimensional module with trivial minimal subgroup.

Recall that M_{i} = CS^{i} + T^{i} and N_{i} = ⊕_{0≤j≤p−1}C^{j}M_{i} for 0 ≤ i ≤ p − 2, where S^{0} = 𝔽_{p} and T^{0} = S^{p−1}.

Let S be a simple A_{p}(E, E)-module and let e be an idempotent in A_{p}(E, E) such that Se = S and S′e = 0 for any simple A_{p}(E, E)-module S′ which is not isomorphic to S. We shall obtain an 𝔽_{p}-subspace W of H^{∗}(E) such that the multiplication by e induces isomorphism

W ⟶^{∼} W e = H^{∗}(E)e.

In general, W is not necessarily an A_{p}(E, E)-submodule. But for simple modules with minimal subgroup A, we see that W is an A_{p}(E, E)-submodule and the equality W = W e holds.

By Proposition 8.2 and 8.4, we have the following.

Theorem 10.2. Let Q ≤ E, |Q| = p and U_{i} (0 ≤ i ≤ p − 2) be simple kQ-modules. Let e be an idempotent corresponding to the simple A_{p}(E, E)-module S(E, Q, U_{i}). Then

W ⟶^{∼} W e = H^{∗}(E)e

where

W = \{ \begin{matrix}𝔽_{p}[C]\{M_{0}\} = 𝔽_{p}[C]\{𝔽_{p}C + S^{p−1}\} & (i = 0) \\ 𝔽_{p}[C]\{S^{i}\} & (1 ≤ i ≤ p − 2). \end{matrix}

Proof. Since we have the following decompositions into 𝔽_{p}-subspaces,

ℂ𝔸{M_{0}} = 𝔽_{p}[C]{M_{0}} ⊕ ℂ𝔸{V M_{0}}

and

Z_{i} = 𝔽_{p}[C]{S^{i}} ⊕ ℂ𝔸{v^{i}M_{0} + V M_{i}}

the result follows from Proposition 8.2 and 8.4. □

Similarly, by Corollary 9.3, we have the following.

Theorem 10.3. Let e be an idempotent in A_{p}(E, E) corresponding to the simple A_{p}(E, E)- module S(E, A, S^{p−1} ⊗ det^{q}). Then

H^{∗}(E)e = \{ \begin{matrix}𝔻𝔸\{D_{2}N_{0}\} & (q = 0) \\ 𝔻𝔸\{v^{q}N_{q}\} & (1 ≤ q ≤ p − 2). \end{matrix}

Next, we consider simple A_{p}(E, E)-modules which corresponds to simple 𝔽_{p}Out(E)- modules. First, we consider simple modules det^{q} and S^{p−1} ⊗ det^{q}.

Theorem 10.4. (1) Let e be an idempotent corresponding to the trivial A_{p}(E, E)- module. Then

H^{∗}(E)e = 𝔻𝔸^{+}e ≃ 𝔻𝔸^{+}.

(2) Let 1 ≤ q ≤ p − 2. Then there exists an idempotent e corresponding to the simple A_{p}(E, E)-module S(E, E, det^{q}) such that

H^{∗}(E)e = ℂ𝔸{v^{q}}e ≃ ℂ𝔸{v^{q}}.

(3) Let e be an idempotent corresponding to the simple A_{p}(E, E)-module S(E, E, S^{p−1}). Then

H^{∗}(E)e = 𝔻𝔸{V S^{p−1}}e ≃ 𝔻𝔸{V S^{p−1}}.

(4) Let 1 ≤ q ≤ p − 2. Then there exists an idempotent e corresponding to the simple A_{p}(E, E)-module S(E, E, S^{p−1} ⊗ det^{q}) such that

H^{∗}(E)e = ℂ𝔸{v^{q}S^{p−1}}e ≃ ℂ𝔸{v^{q}S^{p−1}}.

Proof. (1) Let e be an idempotent corresponding to the trivial A_{p}(E, E)-module. Since direct summands in (7.4.2), (7.4.3) and (7.4.4) have no simple subquotient module which is isomorphic to S(E, E, 𝔽_{p}) by Corollary 7.10, 7.12 and 7.14, it follows that H^{∗}(E)e = ℂ𝔸{𝔽_{p} +S^{p−1}}e. By Corollary 7.7 and Lemma 2.1, we have

H^{∗}(E)e = 𝔽_{p}[D_{1}]^{+}e ⊕ 𝔻𝔸{D_{2}}e ≃ 𝔽_{p}[D_{1}]^{+} ⊕ 𝔻𝔸{D_{2}} = 𝔻𝔸^{+}.

(2) If S is a simple A_{p}(E, E)-module such that S has a composition factor isomorphic to det^{q} as a GL_{2}(𝔽_{p})-module, then S ≃ S(E, E, det^{q}). Hence if e is an idempotent in 𝔽_{p}GL_{2}(𝔽_{p}) corresponding to the simple 𝔽_{p}GL_{2}(𝔽_{p})-module det^{q}, then e is an idempotent corresponding to the simple A_{p}(E, E)-module S(E, E, det^{q}). It follows that

H^{∗}(E)e = H_{0,q} = ℂ𝔸{v^{q}}

by Corollary 4.5.

(3) Let e be an idempotent corresponding to the simple A_{p}(E, E)-module S(E, E, S^{p−1}). Then, by Corollary 7.7, 7.10, 7.12, 7.14 and Lemma 2.1, we have

H^{∗}(E)e = 𝔻𝔸{V S^{p−1}}e = 𝔻𝔸{V S^{p−1}}.

(4) As in the proof of (2), if e is an idempotent in 𝔽_{p}GL_{2}(𝔽_{p}) corresponding to S^{p−1} ⊗ det^{q}, then e is an idempotent corresponding to S(E, E, S^{p−1} ⊗ det^{q}). Hence it follows that

H^{∗}(E)e = H_{p−1,q} = ℂ𝔸{v^{q}S^{p−1}}.

by Corollary 4.5. □

Let us consider remaining simple 𝔽_{p}Out(E)-modules, that is, S^{i} ⊗ det^{q} for 1 ≤ i ≤ p − 2, 0 ≤ q ≤ p − 2.

Theorem 10.5. Let 1 ≤ i ≤ p − 2 and 0 ≤ q ≤ p − 2. Let

S = S^{i}v^{q}, T = T^{p−i−1}v^{s}

where s ≡ i + q (mod p − 1), 0 ≤ s ≤ p − 2. Then there exists an idempotent e in A_{p}(E, E) which corresponds to S(E, E, S^{i} ⊗ det^{q}) such that H^{∗}(E)e = W e ≃ W for the following 𝔽_{p}-subspace W:

\begin{matrix}ℂ𝔸\{V S\} & ⊕ & 𝔻𝔸\{V T \} & (q ≡ 2i ≡ 0) \\ ℂ𝔸\{V S\} & ⊕ & ℂ𝔸\{T \} & (q ≡ 0, 2i ≢ 0) \\ 𝔻𝔸\{S\} & ⊕ & 𝔻𝔸\{V T \} & (i = q, 3i ≡ 0) \\ 𝔻𝔸\{S\} & ⊕ & ℂ𝔸\{T \} & (i = q, 3i ≢ 0) \\ ℂ𝔸\{S\} & ⊕ & 𝔻𝔸\{V T \} & (q ≠ 0, i ≠ q, q + 2i ≡ 0) \\ ℂ𝔸\{S\} & ⊕ & ℂ𝔸\{T \} & (q ≠ 0, i ≠ q, q + 2i ≢ 0) \end{matrix}

where ≡ means equivalent modulo p − 1.

Proof. Let f be an idempotent in 𝔽_{p}GL_{2}(𝔽_{p}) which corresponds to the simple 𝔽_{p}GL_{2}(𝔽_{p})-module S^{i} ⊗ det^{q}. Since

S(E, E, S^{i} ⊗ det^{q}) ≃ S^{i} ⊗ det^{q}

as 𝔽_{p}GL_{2}(𝔽_{p})-modules, there exists an idempotent e in A_{p}(E, E) which corresponds to the simple A_{p}(E, E)-module S(E, E, S^{i} ⊗ det^{q}) such that fe = ef = e. Note that S(E, E, L)(f − e) = 0 for any simple 𝔽_{p}GL_{2}(𝔽_{p})-module L. Since

H^{∗}(E)f = ℂ𝔸{S}f ⊕ ℂ𝔸{T }f ≃ ℂ𝔸{S} ⊕ ℂ𝔸{T }

by Corollary 4.5, it follows that

H^{∗}(E)e = ℂ𝔸{S}e ⊕ ℂ𝔸{T }e.

First, we consider ℂ𝔸{S}e. Assume q = 0. Then

ℂ𝔸{S} ⊂ 𝔽_{p}[V]{V S} + Z_{i}

where Z_{i} = 𝔽_{p}[C]{S} + ℂ𝔸{v^{i}M_{0} + V M_{i}} as in Proposition 7.9. By Corollary 7.10 and Lemma 2.1, we have

(𝔽_{p}[V]{V S} + Z_{i})e = 𝔽_{p}[V]{V S}e + ℂ𝔸{v^{i}M_{0} + V M_{i}}e
≃ 𝔽_{p}[V]{V S} + ℂ𝔸{v^{i}M_{0} + V M_{i}}e.

Since every composition factor of ℂ𝔸{v^{i}M_{0} + V M_{i}} has minimal subgroup E by Corollary 7.10, it follows that

ℂ𝔸{v^{i}M_{0} + V M_{i}}e = ℂ𝔸{v^{i}M_{0} + V M_{i}}f
= ℂ𝔸{CV S}f = ℂ𝔸{CV S}.

Hence, if q = 0, then

ℂ𝔸{S}e ⊂ 𝔽_{p}[V]{V S}e ⊕ ℂ𝔸{CV S}
= ℂ𝔸{V S}e ≃ ℂ𝔸{V S}

and

ℂ𝔸{S}e = ℂ𝔸{V S}e ≃ ℂ𝔸{V S}.

Assume i = q. Then

ℂ𝔸{S} = 𝔻𝔸{∑_{j=0}^{p−1} C^{j+1}S} ⊕ 𝔻𝔸{S}

and we have

𝔻𝔸{∑_{j=0}^{p−1} C^{j+1}S} ⊂ 𝔻𝔸{v^{i}N_{i}}

𝔻𝔸{S} ⊂ 𝔻𝔸{v^{i}(S^{i} + V T^{i})}.

Since 𝔻𝔸{v^{i}N_{i}}e = 𝔻𝔸{v^{i}(S^{i} +V T^{i})}(f −e) = 0 by Corollary 7.12 and ℂ𝔸{S}f = ℂ𝔸{S}, we have

ℂ𝔸{S}e = 𝔻𝔸{S}e = 𝔻𝔸{S}.

Assume q ≠ 0 and i ≠ q. Then ℂ𝔸{S}(f − e) = 0 by Corollary 7.14. Hence we have

ℂ𝔸{S}e = ℂ𝔸{S}f = ℂ𝔸{S}.

Next, we consider ℂ𝔸{T }e. Assume q + 2i ≡ 0. Then

ℂ𝔸{T } = 𝔻𝔸{∑_{j=0}^{p−1} C^{j}T } ⊕ 𝔻𝔸{V T }

and we have

𝔻𝔸{∑_{j=0}^{p−1} C^{j}T } ⊂ 𝔻𝔸{v^{p−1−i}N_{p−1−i}}

𝔻𝔸{V T } ⊂ 𝔻𝔸{v^{p−1−i}(S^{p−1−i} + V T^{p−1−i})}

since s = p − 1 − i. Since

𝔻𝔸{v^{p−1−i}N_{p−1−i}}e = 𝔻𝔸{v^{p−1−i}(S^{p−1−i} + V T^{p−1−i})}(f − e) = 0

by Corollary 7.12 and ℂ𝔸{T }f ≃ ℂ𝔸{T }, we have

ℂ𝔸{T }e = 𝔻𝔸{V T }e ≃ 𝔻𝔸{V T }.

Assume q + 2i ≢ 0 and i + q ≡ 0. Then by Corollary 7.10 and Lemma 2.1,

(𝔽_{p}[C]{T } + Z_{p−1−i})e = 𝔽_{p}[C]{T }e ⊕ ℂ𝔸{v^{p−1−i}M_{0} + V M_{p−1−i}}e
= 𝔽_{p}[C]{T }e ⊕ ℂ𝔸{V T }e
≃ 𝔽_{p}[C]{T } ⊕ ℂ𝔸{V T }.

Hence we have

ℂ𝔸{T }e ≃ ℂ𝔸{T }.

Finally, assume q + 2i ≢ 0 and i + q ≢ 0. Then

ℂ𝔸{T } ⊂ ∑_{1≤j≠r≤p−2} ℂ𝔸{v^{r}(S^{j} + T^{j})}

since p − 1 − i ≠ s. Hence ℂ𝔸{T }(f − e) = 0 by Corollary 7.14 and in particular, we have

ℂ𝔸{T }e = ℂ𝔸{T }f ≃ ℂ𝔸{T }.

This completes the proof. □

By these results, we can describe the cohomology of stable summands of BE.

Definition 10.6. Let X_{M} be an indecomposable stable summand of BE corresponding to a simple right A_{p}(E, E)-module M. Let X(M) be the sum of indecomposable summands equivalent to X_{M} in the complete stable splitting of BE. Let e be an idempotent in A_{p}(E, E) corresponding to M, namely, M = Me and M′e = 0 for a simple A_{p}(E, E)-module which is not isomorphic to M. Let

H^{∗}(X(M)) = H^{∗}(E)e.

Let X_{i,q} = X_{S(E,E,S^{i}⊗det^{q})} and X(i, q) = X(S(E, E, S^{i} ⊗det^{q})) for 0 ≤ i ≤ p−1 and 0 ≤ q ≤ p − 2. Note that X(i, q) ∼ (i + 1)X_{i,q}. Moreover we write

X(E, A, q) = X(S(E, A, S^{p−1} ⊗ det^{q}))

and

X(E, Q, q) = X(S(E, Q, U_{q})).

Corollary 10.7. We have the following isomorphisms:

H^{∗}(X(0, 0)) ≃ 𝔻𝔸^{+}
H^{∗}(X(0, q)) ≃ ℂ𝔸{v^{q}}
H^{∗}(X(p − 1, 0)) ≃ 𝔻𝔸{V S^{p−1}}
H^{∗}(X(p − 1, q)) ≃ ℂ𝔸{v^{q}S^{p−1}} (1 ≤ q ≤ p − 2).

Proof. This follows from Theorem 10.4. □

Corollary 10.8. Let 1 ≤ i ≤ p − 2 and 0 ≤ q ≤ p − 2. Let

S = S^{i}v^{q}, T = T^{p−i−1}v^{s}

where s ≡ i + q (mod p − 1), 0 ≤ s ≤ p − 2. Then H^{∗}(X(i, q)) is isomorphic to the following subspace:

\begin{matrix}ℂ𝔸\{V S\} & ⊕ & 𝔻𝔸\{V T \} & (q ≡ 2i ≡ 0) \\ ℂ𝔸\{V S\} & ⊕ & ℂ𝔸\{T \} & (q ≡ 0, 2i ≢ 0) \\ 𝔻𝔸\{S\} & ⊕ & 𝔻𝔸\{V T \} & (i = q, 3i ≡ 0) \\ 𝔻𝔸\{S\} & ⊕ & ℂ𝔸\{T \} & (i = q, 3i ≢ 0) \\ ℂ𝔸\{S\} & ⊕ & 𝔻𝔸\{V T \} & (q ≠ 0, i ≠ q, q + 2i ≡ 0) \\ ℂ𝔸\{S\} & ⊕ & ℂ𝔸\{T \} & (q ≠ 0, i ≠ q, q + 2i ≢ 0) \end{matrix}

where ≡ means equivalent modulo p − 1.

Proof. This follows from Theorem 10.5. □

Corollary 10.9. We have the following isomorphisms:

H^{∗}(X(E, A, q)) ≃ \{ \begin{matrix}𝔻𝔸\{D_{2}(⨁_{j=0}^{p−1} C^{j}(𝔽_{p}C + S^{p−1}))\} & (q = 0) \\ 𝔻𝔸\{v^{q}(⨁_{j=0}^{p−1} C^{j}(CS^{q} + T^{q}))\} & (1 ≤ q ≤ p − 2) \end{matrix}

H^{∗}(X(E, Q, i)) ≃ \{ \begin{matrix}𝔽_{p}[C]\{𝔽_{p}C + S^{p−1}\} & (i = 0) \\ 𝔽_{p}[C]\{S^{i}\} & (1 ≤ i ≤ p − 2). \end{matrix}

Proof. This follows from Theorem 10.2 and 10.3. □

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Akhiko Hida, Faculty of Education, Saitama University, Shimo-okubo 255, Sakura-ku, Saitama-city, Saitama, Japan

E-mail address: ahida@mail.saitama-u.ac.jp

Nobuaki Yagita, Department of Mathematics, Faculty of Education, Ibaraki University, Mito, Ibaraki, Japan

E-mail address: yagita@mx.ibaraki.ac.jp