ON COMPARING THE COHOMOLOGY OF ALGEBRAIC GROUPS, FINITE CHEVALLEY GROUPS AND FROBENIUS KERNELS

CHRISTOPHER P. BENDEL, DANIEL K. NAKANO, AND CORNELIUS PILLEN

1 Introduction

1.1 Let G be a semisimple simply connected algebraic group defined and split over the field 𝔽_{p} with p elements. Let k denote the algebraic closure of 𝔽_{p}. Let F : G → G^{(1)} be the Frobenius map. For a fixed r ≥ 1, let G_{r} be the kernel of the rth iteration of the Frobenius map. Furthermore, let G(𝔽_{q}) where q = p^{r} be the finite Chevalley group consisting of the 𝔽_{q}-rational points of G. For over thirty years, there has been a considerable amount of effort in relating the representation theory of reductive algebraic groups, Frobenius kernels and finite groups of Lie type in the defining characteristic. In this paper we will be primarily interested in the following three questions.

(1.1.1) Can all extensions between G_{r}-modules be found via an extension theory for G?

(1.1.2) Can all extensions between G(𝔽_{q})-modules be found via an extension theory for G?

(1.1.3) Given two rational G-modules M and N, is there a relationship between Ext_{G_{r}}^{•}(M, N) and Ext_{G(𝔽_{q})}^{•}(M, N)?

J.E. Humphreys [H] and H.H. Andersen [And1, p. 388] first posed question (1.1.2) for Ext^{1} between simple G(𝔽_{q})-modules. Andersen was able to answer this question affirmatively for simple G(𝔽_{q})-modules in several generic cases (see [And1, Thm. 3.2]).

The methods employed in this paper will be functorial and different from the ideas used in the past. In order to answer questions (1.1.1) and (1.1.2) we compare the category of G(𝔽_{q})- modules and G_{r}-modules with truncated categories of G-modules which contain enough projective objects. This will be accomplished by constructing two Grothendieck spectral sequences. These truncated categories of G-modules are highest weight categories, and are thus equivalent to module categories for finite-dimensional quasi-hereditary algebras. Our procedure will also demonstrate why it is easier to provide an answer to (1.1.1) as opposed to (1.1.2) for simple modules (see Corollary 6.2). The idea of comparing the cohomology of finite groups with extensions of modules over certain finite-dimensional algebras was also used earlier in work of Doty, Erdmann and the second author [DEN]. In that paper, it was shown that one can construct a spectral sequence involving extensions for the Schur algebra S(n, d) (n ≥ d) to extensions for the symmetric group on d-letters Σ_{d}. Results comparing the cohomology in these two categories involving specific classes of modules are given in [KN]. Applications involving computations of cohomology are provided in [N] [DN]. The question (1.1.3) is the most intractable question of the three. We will show that ideas from [LN] can be used to provide a vanishing criterion of cohomology for finite Chevalley groups in terms of vanishing of Lie algebra cohomology. These results only work in the case when r = 1, but have the advantage that there are no strong restrictions on the size of the prime.

Date: April 2000.

1991 Mathematics Subject Classification. Primary 20C, 20G; Secondary 20J06, 20G10.

Research of the second author was supported in part by NSF grant DMS-9800960.

1.2 Now we will describe the contents of the paper in greater detail. The results in Section 2 and 3 provide qualitative results involving the relationship between the cohomology of G(𝔽_{q}) and G_{r}. In Section 2, we give a criterion for projectivity of modules over G(𝔽_{q}). A criterion for G_{r} was given earlier in [BN, Thm. (4.2.1)]. This criterion involves induced modules, Weyl modules for G and principal series modules for G(𝔽_{q}). In the following section, we investigate sufficient conditions to insure the vanishing of Ext_{G(𝔽_{p})}^{n}(M, N) for M, N ∈ mod(G). This is given in terms of the vanishing of certain weight spaces in the cohomology for U_{1} where U is the unipotent radical of the Borel subgroup B of G.

The remainder of the paper is devoted to providing answers to questions (1.1.1) and (1.1.2). In Section 4, we first define a saturated set of weights π_{s} in X(T)_{+} and let C_{s} be the full subcategory of G-modules whose composition factors all have highest weight in π_{s}. We proceed by constructing two functors G_{s}(−) and H_{s}(−) from the category of G(𝔽_{q})-modules and the category of G_{r}-modules, respectively, into C_{s}. For M ∈ C_{s} and N_{1} ∈ mod(G(𝔽_{q})) and N_{2} ∈ mod(G_{r}) two spectral sequences are obtained:

E_{2}^{i,j} = Ext_{G}^{i}(M, R^{j}G_{s}(N_{1})) ⇒ Ext_{G(𝔽_{q})}^{i+j}(M, N_{1})  (1.2.1)
E_{2}^{i,j} = Ext_{G}^{i}(M, R^{j}H_{s}(N_{2})) ⇒ Ext_{G_{r}}^{i+j}(M, N_{2}).  (1.2.2)

The higher right derived functors R^{j}G_{s} and R^{j}H_{s} are of particular interest since these provide us with vital information about the connection between these categories. The composition factors of these functors when applied to a module can be found by calculating extensions (in G(𝔽_{q}) or G_{r}) between the module and the appropriate projective cover in C_{s}. The constructions in Section 4 allow us to prove stability results in Section 5. In particular Theorem 5.2, Corollary 5.3, and Theorem 5.5 by and large provide an answer to the questions (1.1.1) and (1.1.2) given in Section 1. In Section 6 we look at the image of H_{s} on a simple module and show in Theorem 6.1 that this is a semisimple G-module for large enough primes p. As a byproduct one obtains a formula in Corollary 6.2 which allows one to compute Ext^{1} between two simple modules in mod(G_{r}) by knowing Ext^{1} between simples in mod(G). We demonstrate in Section 7 that the situation for finite Chevalley groups is not as straightforward as in the Frobenius kernel case by looking at G_{s} on simple modules. Our analysis shows that one can recover the isomorphisms given in [And2] (see Corollary 7.1) in a more functorial manner via our approach. Theorem 7.6 generalizes Andersen’s results for the generic case [And2, Theorem 3.2] allowing for a larger class of modules and extensions of higher degree. Furthermore, in Theorem 7.5 we are able to give a complete description of the cohomology of the finite group G(𝔽_{q}) and all its simple modules in terms of cohomology for the algebraic group with simple modules in the truncated category C_{s}, provided the prime is large enough.

1.3 Notation: Unless otherwise stated G will always denote a semisimple simply connected algebraic group defined and split over the finite field 𝔽_{p} with p elements for a prime p. Here k denotes the algebraic closure of 𝔽_{p}. Let Φ be a root system associated to G with respect to a maximal split torus T. Moreover, let Φ^{+} (resp. Φ^{−}) be positive (resp. negative) roots and Δ be a base consisting of simple roots. Let B be a Borel subgroup containing T corresponding to the positive roots and U be the unipotent radical of B.

The Euclidean space associated with Φ will be denoted by 𝔼 and the inner product on 𝔼 will be denoted by ⟨ , ⟩. The Weyl group W is the group generated by reflections associated to the root system Φ and the affine Weyl group W_{p} is the group generated by W and translations by elements in pℤΦ. Let X(T) be the integral weight lattice obtained from Φ. The set X(T) has a partial ordering defined as follows. If λ, μ ∈ X(T) then λ ≥ μ if and only if λ − μ ∈ ∑_{α∈Π} ℕα. If α^{∨} = 2α/⟨α, α⟩ is the coroot corresponding to α ∈ Φ, then the set of dominant integral weights is defined by

X(T)_{+} = {λ ∈ X(T) : 0 ≤ ⟨λ, α^{∨}⟩ for all α ∈ Δ}.

Furthermore, the set of p^{r}-restricted weights is

X_{r}(T) = {λ ∈ X(T) : 0 ≤ ⟨λ, α^{∨}⟩ < p^{r} for all α ∈ Δ}.

The affine Weyl group W_{p} acts on X(T) via the “dot action” given by w · λ = w(λ + ρ) − ρ where w ∈ W (w ∈ W_{p}), λ ∈ X(T), and ρ is the half sum of positive roots.

Let H be an affine algebraic group scheme over k and let H_{r} = ker F^{r}. Here F : H → H^{(1)} is the Frobenius map and F^{r} is the rth iteration of the Frobenius map. We note that there is a categorical equivalence between restricted Lie(H)-modules and H_{1}-modules. For any group scheme H, let mod(H) be the category of finite-dimensional rational H-modules. If r ≥ 1 and q = p^{r}, let H(𝔽_{q}) be the finite group obtained by taking the 𝔽_{q} rational points of H.

For a reductive algebraic group G the simple modules will be denoted by L(λ) and the induced modules by H^{0}(λ) = ind_{B}^{G}λ, where λ ∈ X(T)_{+}. For the infinitesimal group scheme G_{r}, the simple modules will be denoted by L_{r}(λ). If λ ∈ X_{r}(T), then L_{r}(λ) ≅ L(λ). The induced and coinduced modules are given by Z′_{r}(λ) = ind_{B_{r}}^{G_{r}}λ and Z_{r}(λ) = coind_{B_{r}}^{G_{r}}λ for λ ∈ X_{r}(T). If λ ∈ X_{r}(T), the injective hull of L_{r}(λ) in mod(G_{r}) is Q_{r}(λ). This is also the projective cover of L_{r}(λ) as a G_{r}-module [Jan1, II. 11.5(4)]. For the finite group G(𝔽_{q}), the simple modules are L(λ) for λ ∈ X_{r}(T). Moreover, for each λ ∈ X(T (𝔽_{q})), the induced module is given by M_{r}(λ) = ind_{B(𝔽_{q})}^{G(𝔽_{q})}λ.

1.4 Acknowledgments: We would like to acknowledge Jim Humphreys for useful comments and suggestions on an earlier version of this paper. The first author and the third author thank the Department of Mathematics and Statistics at Utah State University for its hospitality during the preparation of this paper. The third author would like to thank the Mathematics Department of the University of Oregon where the author spent his sabbatical while this paper was written.

2 Projectivity results

2.1 It is not known how to construct the simple G_{r}-modules nor how one can compute the dimensions of these modules. So, for practical purposes it does not make sense to test a module for projectivity by showing that extensions between the module and the simple modules are zero. There are other canonical modules like Weyl modules, induced modules and Verma modules whose construction is well-known and whose character can be determined. In [BN, Thm. (4.2.1)], it was shown that one can test projectivity for a module over G_{r} by examining the extensions between the module and a family of canonical modules. The result is presented below.

Theorem . Let G be a reductive algebraic k-group scheme and Λ be a set of representatives for X(T)/p^{r}X(T). Moreover, let M ∈ mod(G). The following statements are equivalent (a) M is projective in mod(G_{r});

(b) Ext_{G_{r}}^{i}(M, ⊕_{λ∈Λ}L_{r}(λ)) = 0 for all i > 0;

(c) Ext_{G_{r}}^{i}(M, ⊕_{λ∈Λ}H^{0}(λ)) = 0 for all i > 0;

(d) (i) Ext_{G_{r}}^{i}(M, ⊕_{λ∈Λ}Z′_{r}(λ)) = 0 for all i > 0 (ii) R^{i} ind_{B}^{G} (Hom_{B_{r}}(M, ⊕_{λ∈Λ}λ)^{(−r)}) = 0 for all i > 0;

(e) Ext_{G_{r}}^{i}(⊕_{λ∈Λ}L_{r}(λ), M) = 0 for all i > 0;

(f) Ext_{G_{r}}^{i}(⊕_{λ∈Λ}V (λ), M) = 0 for all i > 0;

(g) (i) Ext_{G_{r}}^{i}(⊕_{λ∈Λ}Z_{r}(λ), M) = 0 for all i > 0

(ii) R^{i} ind_{B}^{G} (Hom_{B_{r}^{+}}(⊕_{λ∈Λ}λ, M)^{(−r)}) = 0 for all i > 0.

Moreover, statements (a), (b), (c), (e), and (f) are equivalent for M ∈ mod(G_{r}).

2.2 In general the simple G(𝔽_{q})-modules have also not been determined. The following result shows that projectivity over G(𝔽_{q}) can also be checked by using principal series modules, Weyl modules or induced modules.

Theorem . Let Λ be a set of representatives for X(T)/p^{r}X(T), and Γ = X(T (𝔽_{q})). Moreover, let M ∈ mod(G(𝔽_{q})). The following statements are equivalent

(a) M is projective in mod(G(𝔽_{q}));

(b) Ext_{G(𝔽_{q})}^{i}(M, ⊕_{λ∈Λ}L_{r}(λ)) = 0 for all i > 0;

(c) Ext_{G(𝔽_{q})}^{i}(M, ⊕_{λ∈Λ}H^{0}(λ)) = 0 for all i > 0;

(d) Ext_{G(𝔽_{q})}^{i}(⊕_{λ∈Λ}V (λ), M) = 0 for all i > 0;

(e) Ext_{G(𝔽_{q})}^{i}(M, ⊕_{λ∈Γ}M_{r}(λ)) = 0 for all i > 0.

Proof. If M ∈ mod(G(𝔽_{q})) then (a) ⇔ (b). Furthermore, (a) implies (c), (d) and (e). In order to finish the proof we show that (e) ⇒ (a), (c) ⇒ (b) and (d) ⇒ (b).

(e) ⇒ (a): From our assumption and Frobenius reciprocity we have

0 = Ext_{G(𝔽_{q})}^{i}(M, ⊕_{λ∈Γ}M_{r}(λ)) ≅ Ext_{B(𝔽_{q})}^{i}(M, ⊕_{λ∈Γ}λ).

Therefore, M is projective in mod(B(𝔽_{q})), and thus projective in mod(U(𝔽_{q})). Since U(𝔽_{q}) is the p-Sylow subgroup in G(𝔽_{q}), it follows that M is projective in mod(G(𝔽_{q})).

(c) ⇒ (b) and (d) ⇒ (b): Define the following order relation on X(T). Let μ ≤_{ℚ} λ on X(T) if and only if λ − μ is a non-negative rational linear combination of simple roots. Moreover, let

C_{ℤ} = {λ ∈ X(T) : 0 < ⟨λ + ρ, β^{∨}⟩ ≤ p − 1 for all β ∈ Φ^{+}}.

We will prove the statement by using induction on the order relation. Assume first that condition (c) holds. If λ ∈ C_{ℤ} then H^{0}(λ) = L_{r}(λ), so Ext_{G(𝔽_{q})}^{i}(M, L_{r}(λ)) = 0 for all i > 0 for all λ ∈ C_{ℤ}. Now for an arbitrary λ ∈ Λ assume that Ext_{G(𝔽_{q})}^{i}(M, L_{r}(μ)) = 0 for all i > 0 and μ <_{ℚ} λ with μ ∈ Λ. There exists a short exact sequence of G-modules given by

0 → L_{r}(λ) → H^{0}(λ) → N → 0.

All composition factors L(σ) of N satisfy σ < λ. If σ ∈ Λ then by induction Ext_{G(𝔽_{q})}^{i}(M, L_{r}(σ)) = 0 for all i > 0. On the other hand, if σ ∉ Λ then write σ = σ_{0} + p^{r}σ_{1} where σ_{0} ∈ Λ. By using Steinberg’s tensor product theorem, it follows that

Ext_{G(𝔽_{q})}^{i}(M, L_{r}(σ)) = Ext_{G(𝔽_{q})}^{i}(M, L_{r}(σ_{0}) ⊗ L_{r}(σ_{1})).

But σ <_{ℚ} λ so all the composition factors of L_{r}(σ_{0}) ⊗ L_{r}(σ_{1}) have high weights which are less than λ. Consequently, Ext_{G(𝔽_{q})}^{i}(M, L_{r}(σ_{0}) ⊗ L_{r}(σ_{1})) = 0, and Ext_{G(𝔽_{q})}^{i}(M, N) = 0 for all i > 0.

Consider the long exact sequence in cohomology:

0 → Hom_{G(𝔽_{q})}(M, L_{r}(λ)) → Hom_{G(𝔽_{q})}(M, H^{0}(λ)) → Hom_{G(𝔽_{q})}(M, N)
→ Ext_{G(𝔽_{q})}^{1}(M, L_{r}(λ)) → Ext_{G(𝔽_{q})}^{1}(M, H^{0}(λ)) → Ext_{G(𝔽_{q})}^{1}(M, N)
→ Ext_{G(𝔽_{q})}^{2}(M, L_{r}(λ)) → ⋯

We have Ext_{G(𝔽_{q})}^{i}(M, L_{r}(λ)) = 0 for all i ≥ 2. Therefore, by [BN, Prop. 2.4.2(c)] one has Ext_{G(𝔽_{q})}^{i}(M, L_{r}(λ)) = 0 for all i > 0. To show that (d) ⇒ (b), one can use an analogous inductive argument. _{□}

3 Vanishing of cohomology for G(𝔽_{p}) and G_{1}

3.1 Let M be a rational G-module. In [LN] it was shown that if M is projective over G_{1} then M is projective over G(𝔽_{p}). The converse to this statement also holds if one assumes that the composition factors of M have high weights which are not too large (i.e. M is in the p-bounded category). A natural generalization to this question is the following.

(3.1.1) Let M and N be rational G-modules. Under what conditions does Ext_{G_{1}}^{i}(M, N) = 0 imply that Ext_{G(𝔽_{p})}^{i}(M, N) = 0 for all i > 0?

Observe if N = ⊕_{λ∈Λ}L_{1}(λ) or ⊕_{λ∈Λ}H^{0}(λ) then this holds by [LN, Cor. 3.5] [BN, Thm. (4.2.1)] and Theorem 2.2. In order to demonstrate the subtlety of this problem, we present the following example. Let G = SL_{2}(k) with p ≥ 7. There are three blocks for kSL_{2}(𝔽_{p}) and \frac{p+1}{2} blocks for G_{1}. Let L_{1}(λ) be in the principal block for SL_{2}(𝔽_{p}), but not in the principal block for G_{1}. Then for some i > 0, H^{i}(SL_{2}(𝔽_{p}), L_{1}(λ)) ≠ 0, but H^{i}(G_{1}, L_{1}(λ)) = 0 for all i ≥ 0.

3.2 Let M and N be B-modules. According to [LN, Thm. 3.2], there exists a spectral sequence

E_{1}^{i,j} = Ext_{U_{1}}^{i+j}(M, N)_{i} ⇒ Ext_{U(𝔽_{p})}^{i+j}(M, N).  (3.2.1)

The following result demonstrates that the T (𝔽_{p}) invariants on the U_{1}-cohomology are directly related to the calculation of the B(𝔽_{p}) and the G(𝔽_{p}) cohomology.

Theorem . Let n ≥ 0.

(a) If M, N ∈ mod(B) and Ext_{U_{1}}^{n}(M, N)^{T (𝔽_{p})} = 0 then Ext_{B(𝔽_{p})}^{n}(M, N) = 0.

(b) If M, N ∈ mod(G) and Ext_{U_{1}}^{n}(M, N)^{T (𝔽_{p})} = 0 then Ext_{G(𝔽_{p})}^{n}(M, N) = 0.

Proof. (a) First observe that B(𝔽_{p}) = T (𝔽_{p})⋉U(𝔽_{p}) and modules over kT (𝔽_{p}) are semisimple. Therefore, the Lyndon-Hochschild-Serre spectral sequence yields for n ≥ 0:

Ext_{B(𝔽_{p})}^{n}(M, N) ≅ Ext_{U(𝔽_{p})}^{n}(M, N)^{T (𝔽_{p})}.  (3.2.2)

The spectral sequence (3.2.1) for M, N ∈ mod(B) has differentials which are T (𝔽_{p})-homomorphisms. The fixed point functor (−)^{T (𝔽_{p})} is exact. This means that we can obtain another spectral sequence from (3.2.1)

E_{1}^{i,j} = Ext_{U_{1}}^{i+j}(M, N)_{i}^{T (𝔽_{p})} ⇒ Ext_{B(𝔽_{p})}^{i+j}(M, N).  (3.2.3)

Next observe that

Ext_{U_{1}}^{n}(M, N)^{T (𝔽_{p})} = ⨁_{i≤0} ⨁_{j=n−i} Ext_{U_{1}}^{i+j}(M, N)_{i}^{T (𝔽_{p})}.  (3.2.4)

Consequently, if Ext_{U_{1}}^{n}(M, N)^{T (𝔽_{p})} = 0 then Ext_{U_{1}}^{i+j}(M, N)_{i}^{T (𝔽_{p})} = 0 for all i, j such that i + j = n, thus Ext_{B(𝔽_{p})}^{n}(M, N) = 0.

(b) Since U(𝔽_{p}) is the p-Sylow subgroup of G(𝔽_{p}) the restriction map res : Ext_{G(𝔽_{p})}^{n}(M, N) → Ext_{B(𝔽_{p})}^{n}(M, N) is injective for all n ≥ 0. The result now follows from part (a). _{□}

4 Spectral sequences

4.1 Let G be an algebraic k-group scheme and H be a closed subgroup scheme of G. Moreover, let M ∈ mod(G) and N ∈ mod(H). There exists a first quadrant spectral sequence [Jan1, I. 4.5 Prop.]:

E_{2}^{i,j} = Ext_{G}^{i}(M, R^{j}ind_{H}^{G}(N)) ⇒ Ext_{H}^{i+j}(M, N).  (4.1.1)

When G is as defined in 1.3 and H = G(𝔽_{q}) the induction functor ind_{G(𝔽_{q})}^{G}− is exact because H is a finite algebraic k-group. In this case the spectral sequence (4.1.1) collapses and yields the following isomorphism.

(4.1.2) For i ≥ 0, Ext_{G(𝔽_{q})}^{i}(M, N) ≅ Ext_{G}^{i}(M, ind_{G(𝔽_{q})}^{G}N).

Similarly, if G is reductive and H = G_{r} then G/G_{r} is affine. Therefore, the induction functor ind_{G_{r}}^{G}− is exact. The spectral sequence (4.1.1) collapses in this case and yields the following isomorphism.

(4.1.3) For i ≥ 0, Ext_{G_{r}}^{i}(M, N) ≅ Ext_{G}^{i}(M, ind_{G_{r}}^{G}N).

The isomorphisms (4.1.2) and (4.1.3) provide affirmative answers to questions (1.1.1) and (1.1.2). Unfortunately, for practical purposes this answer is not satisfactory because the modules ind_{G(𝔽_{q})}^{G}N and ind_{G_{r}}^{G}N are in general infinite-dimensional and difficult to compute. We will need to modify this construction to provide a better solution to computing cohomology for G(𝔽_{q}) and G_{r} via cohomology for G.

4.2 Let us review some general properties about coalgebras and comodules. The conventions and results will follow those proved in [Don1, Sec. 1]. Let C be any coalgebra, and Mod(C) be the category of right comodules for C. Now suppose that B is a subcoalgebra of C and let Mod_{B}(C) be the full subcategory of Mod(C) whose objects belong to B. If M ∈ Mod(C), then let M_{B} be the unique maximal C-subcomodule of M belonging to B. The functor M ↦ M_{B} is a functor from Mod(C) to Mod_{B}(C). Since the image of the structure map M_{B} → M_{B} ⊗ C is contained in M_{B} ⊗ B, one can regard M_{B} as a B-comodule. The functor F_{B}: M ↦ M_{B}, from Mod(C) to Mod(B), is left exact and takes injectives to injectives.

Let π be a set of simple C-comodules and M ∈ Mod(C). Let O_{π}(M) be the unique maximal C-subcomodule such that all the composition factors of which lie in π. Set C(π) = O_{π}(C). Then C(π) is a subcoalgebra of C. Furthermore, O_{π}(M) is the same as F_{C(π)}(M) if we regard F_{C(π)}(M) in Mod(C) by inflation.

Let G be an algebraic k-group scheme and H be a closed subgroup scheme of G. Set C = k[G] and let π be a set of simple C-comodules (or equivalently simple rational Gmodules). Consider the functor T = F_{C(π)} ∘ ind_{H}^{G}. This is a functor from Mod(k[H]) to Mod(C(π)). We can now construct the following spectral sequence.

Theorem . Let G be an algebraic k-group scheme with H a closed subgroup scheme of G. Let M ∈ Mod(C(π)) and N ∈ Mod(H). Then there exists a first quadrant spectral sequence

E_{2}^{i,j} = Ext_{C(π)}^{i}(M, R^{j}T (N)) ⇒ Ext_{H}^{i+j}(M, N).

Proof. The functor S = Hom_{C(π)}(M, −) is left exact. Moreover, the functor T : Mod(H) → Mod(C(π)) takes injectives to injectives because F_{C(π)} and ind_{H}^{G}− have this property. Furthermore, note that since M ∈ Mod(C(π)) we have

S ∘ T (−) = Hom_{C(π)}(M, T (−)) = Hom_{C}(M, ind_{H}^{G}(−)) = Hom_{H}(M, −).

The result now follows by [Jan1, I. Prop. 4.1]. _{□}

4.3 Let G be as defined in 1.3 and let π be a subset of X(T)_{+}. In the context of the previous section, we will abuse notation and also let π denote the set of simple rational G-modules having highest weight in π. Since O_{π} is equivalent to F_{C(π)}, it follows that Mod(C(π)) is equivalent to the full subcategory of G-modules whose composition factors have highest weight in π. Now assume that π is saturated. That is, if λ ∈ π and μ ∈ X(T)_{+} such that μ ≤ λ then μ ∈ π. In this case, Donkin proved the following isomorphism [Don1, (2.1f) Thm.].

(4.3.1) For M, N ∈ Mod(C(π)), Ext_{G}^{n}(M, N) ≅ Ext_{C(π)}^{n}(M, N) for n ≥ 0.

For a fixed r ≥ 1, let q = p^{r} and for all 0 < s ≤ p − 1 let

π_{s} = {λ ∈ X(T)_{+} : ⟨λ + ρ, α_{0}^{∨}⟩ < 2sp^{r}⟨ρ, α_{0}^{∨}⟩},

where α_{0} denotes the highest short root. Observe that if μ ≤ λ then λ − μ = ∑_{α∈Δ} c_{α}α where c_{α} ≥ 0. Then

⟨(λ + ρ) − (μ + ρ), α_{0}^{∨}⟩ = ⟨∑_{α∈Δ} c_{α}α, α_{0}^{∨}⟩ ≥ 0.

Therefore, ⟨λ + ρ, α_{0}^{∨}⟩ ≥ ⟨μ + ρ, α_{0}^{∨}⟩. This computation shows that π_{s} is saturated. Furthermore, let C_{s} be the full subcategory of G-modules all of whose composition factors have highest weights in π_{s}. For s = 1, the category C_{1} coincides with the p^{r}-bounded category as defined in [Jan1, p. 360]. Since the category C_{s} is equivalent to the category Mod(C(π_{s})) (as noted more generally above), the isomorphism in (4.3.1) holds for modules in C_{s}.

4.4 Let G_{s} = F_{C(π_{s})} ∘ ind_{G(𝔽_{q})}^{G} and H_{s} = F_{C(π_{s})} ∘ ind_{G_{r}}^{G}. Theorem 4.2 (with H = G(𝔽_{q}) or H = G_{r} and π = π_{s}) can be combined with the isomorphism in (4.3.1) to construct the following spectral sequences.

Theorem . Let M ∈ C_{s}.

(a) For N ∈ mod(G(𝔽_{q})) there exists a spectral sequence

E_{2}^{i,j} = Ext_{G}^{i}(M, R^{j}G_{s}(N)) ⇒ Ext_{G(𝔽_{q})}^{i+j}(M, N).

(b) For N ∈ mod(G_{r}) there exists a spectral sequence

E_{2}^{i,j} = Ext_{G}^{i}(M, R^{j}H_{s}(N)) ⇒ Ext_{G_{r}}^{i+j}(M, N).

4.5 Given λ ∈ π_{s} with corresponding simple G-module L(λ), let P(λ) denote the projective cover of L(λ) in the category C_{s}. Projective covers exist in the category C_{s} since it is equivalent to the category Mod(C(π_{s})) and C(π_{s}) is a finite-dimensional coalgebra (see the discussion in Section 4.2). The following result describes the composition factors of the higher right derived functors of G_{s} and H_{s}.

Theorem . For j ≥ 0:

(a) If M ∈ mod(G(𝔽_{q})) then

[R^{j}G_{s}(M) : L(λ)] = dim_{k} Ext_{G(𝔽_{q})}^{j}(P(λ), M).

(b) If N ∈ mod(G_{r}) then

[R^{j}H_{s}(N) : L(λ)] = dim_{k} Ext_{G_{r}}^{j}(P(λ), N).

Proof. We will prove part (a). The proof for part (b) is completely analogous. Since R^{j}G_{s}(M) is a module in C_{s}, by definition of P(λ), we have

[R^{j}G_{s}(M) : L(λ)] = dim_{k} Hom_{C_{s}}(P(λ), R^{j}G_{s}(M)).  (4.5.1)

¿From Theorem 4.4 there is a spectral sequence

E_{2}^{i,j} = Ext_{C_{s}}^{i}(P(λ), R^{j}G_{s}(M)) ⇒ Ext_{G(𝔽_{q})}^{i+j}(P(λ), M).

Since P(λ) is projective in C_{s}, the spectral sequence collapses to

E_{2}^{0,j} = Hom_{C_{s}}(P(λ), R^{j}G_{s}(M)) ≅ Ext_{G(𝔽_{q})}^{j}(P(λ), M),

from which the result follows. _{□}

4.6 For a p^{r}-restricted weight λ ∈ X_{r}(T), if p ≥ 2(h − 1), the G_{r} structure on Q_{r}(λ) lifts to a G-structure [Jan2]. Moreover, when considered as a G-module, Q_{r}(λ) is p^{r}-bounded and can be identified as the injective hull (and projective cover) of L(λ) in the p^{r}-bounded category C_{1} [Jan1, p. 360]. In other words, for a restricted weight λ, the module P(λ) may be identified with Q_{r}(λ).

The following proposition, which is an adaptation of Lemma 2.2 of [And2], gives us information about P(λ) for general λ ∈ π_{s}. For convenience, let π_{0} = X_{r}(T). Note that any weight λ ∈ π_{s} may be expressed as λ = λ_{0} + p^{r}λ_{1} with λ_{0} ∈ X_{r}(T), that is with λ_{0} being p^{r}-restricted. Further, for s ≥ 1, λ_{1} must satisfy ⟨λ_{1}, β^{∨}⟩ < 2s(h − 1) (or ⟨λ_{1} + ρ, β^{∨}⟩ < (2s + 1)(h − 1)) for β short and β ∈ Φ ∩ X(T)_{+}. Hence if p ≥ (2s + 1)(h − 1), the weight λ_{1} lies in the bottom alcove \overline{C}_{ℤ}.

Proposition . Assume s ≥ 1, M ∈ C_{s}, and one of the following two conditions holds:

(i) λ = λ_{0} + p^{r}λ_{1} ∈ π_{s} and p ≥ (2s + 1)(h − 1); (ii) λ = λ_{0} + p^{r}λ_{1} ∈ π_{s−1} and p ≥ 2s(h − 1).

Then the following hold:

(a) Ext_{G}^{1}(Q_{r}(λ_{0}) ⊗ L(λ_{1})^{(r)}, M) = 0.

(b) The projective module P(λ) in C_{s} is a quotient of Q_{r}(λ_{0}) ⊗ L(λ_{1})^{(r)}.

Proof. Let S = Q_{r}(λ_{0}) ⊗ L(λ_{1})^{(r)}. For part (a), it suffices to show that Ext_{G}^{1}(S, L(μ)) = 0 for all composition factors L(μ) of M. Since M is in C_{s}, any such μ are in π_{s}, and so may be expressed as μ = μ_{0} +p^{r}μ_{1} with μ_{0} ∈ X_{r}(T). The short exact sequence of group schemes

1 → G_{r} → G → G/G_{r} → 1

gives rise to the Lyndon-Hochschild-Serre spectral sequence

E_{2}^{i,j} = Ext_{G/G_{r}}^{i}(k, Ext_{G_{r}}^{j}(S, L(μ))) ⇒ Ext_{G}^{i+j}(S, L(μ)).

Since Q_{r}(λ_{0}) is projective over G_{r}, S is also. Hence, the spectral sequence collapses and gives an isomorphism

Ext_{G}^{i}(S, L(μ)) ≅ Ext_{G/G_{r}}^{i}(k, Hom_{G}(S, L(μ))

for all i. In particular, we have

Ext_{G}^{1}(S, L(μ)) ≅ Ext_{G/G_{r}}^{1}(k, Hom_{G_{r}}(S, L(μ)))

= Ext_{G/G_{r}}^{1}(k, Hom_{G_{r}}(Q_{r}(λ_{0}) ⊗ L(λ_{1})^{(r)}, L(μ_{0}) ⊗ L(μ_{1})^{(r)}))

= Ext_{G/G_{r}}^{1}(k, Hom_{G_{r}}(Q_{r}(λ_{0}), L(μ_{0})) ⊗ (L(λ_{1})^{(r)})^{∗} ⊗ L(μ_{1})^{(r)})

= \begin{cases}Ext_{G/G_{r}}^{1}(k, k ⊗ (L(λ_{1})^{(r)})^{∗} ⊗ L(μ_{1})^{(r)}) & if μ_{0} = λ_{0} \\ 0 & otherwise \end{cases}

= \begin{cases}Ext_{G/G_{r}}^{1}(L(λ_{1})^{(r)}, L(μ_{1})^{(r)}) & if μ_{0} = λ_{0} \\ 0 & otherwise \end{cases}

= 0

by the Linkage Principle (cf. [Jan1, II. 6.17]). More precisely, under condition (i), as both λ and μ lie in π_{s} and p ≥ (2s + 1)(h − 1), both λ_{1} and μ_{1} lie in the bottom alcove \overline{C}_{ℤ}. And so the claim follows from the Linkage Principle.

On the other hand, under condition (ii), λ lies in π_{s−1} while μ lies in π_{s}. Now, the condition on p implies that ⟨λ_{1}+ρ, β^{∨}⟩ < (2(s−1)+1)(h−1) ≤ p−(h−1) and ⟨μ_{1}+ρ, β^{∨}⟩ < (2s + 1)(h − 1) ≤ p + (h − 1) for short β ∈ Φ ∩ X(T)_{+}. In other words, λ_{1} lies in the bottom alcove and moreover, more than h − 1 beneath the upper wall, while μ either lies in the bottom alcove or no more than h − 1 above the upper wall. Hence, the desired vanishing again follows from the Linkage Principle.

For part (b), by definition of P(λ), there exists a surjection P(λ) → L(λ) of G-modules. On the other hand, since there is a surjection Q_{r}(λ_{0}) → L(λ_{0}), there is also a surjection (of G-modules) S = Q_{r}(λ_{0}) ⊗ L(λ_{1})^{(r)} → L(λ_{0}) ⊗ L(λ_{1})^{(r)} ≅ L(λ). To see that the map S → L(λ) lifts to S → P(λ), consider the short exact sequence

0 → N → P(λ) → L(λ) → 0

and the corresponding long exact sequence of Ext-groups:

0 → Hom_{G}(S, N) → Hom_{G}(S, P(λ)) → Hom_{G}(S, L(λ)) → Ext_{G}^{1}(S, N) → ….

By part (a), Ext_{G}^{1}(S, N) = 0 and so the map Hom_{G}(S, P(λ)) → Hom_{G}(S, L(λ)) is a surjection, giving the desired lifting. _{□}

4.7 The preceding proposition allows us to obtain some information about the higher right derived functors of G_{s} and H_{s}.

Corollary . Let s ≥ 1 and p ≥ 2s(h − 1) with M ∈ mod(G(𝔽_{q})) and N ∈ mod(G_{r}). If j > 0 then

(a) R^{j}G_{s}(M) contains no composition factors with high weight in π_{s−1};

(b) R^{j}H_{s}(N) contains no composition factors with high weight in π_{s−1}.

Proof. Let λ = λ_{0} + p^{r}λ_{1} be a weight in π_{s−1}. By Proposition 4.6(b), the projective cover P(λ) in C_{s} is a quotient of Q_{r}(λ_{0}) ⊗ L(λ_{1})^{(r)}. But, the highest weight of Q_{r}(λ_{0}) ⊗ L(λ_{1})^{(r)} is 2(p^{r} − 1)ρ + w_{0}λ_{0} + p^{r}λ_{1}. This weight is in π_{s} and so Q_{r}(λ_{0}) ⊗ L(λ_{1})^{(r)} lies in C_{s}. By the projectivity of P(λ) in C_{s}, P(λ) must be a G-summand of Q_{r}(λ_{0}) ⊗ L(λ_{1})^{(r)}. The latter module is projective as a G_{r}-module and G(𝔽_{q})-module. The assertion now follows by Theorem 4.5. _{□}

For s ≥ 1 and λ ∈ π_{s−1}, the proof in fact shows that the module P(λ) in C_{s} may be identified with Q_{r}(λ_{0}) ⊗ L(λ_{1})^{(r)} since the G-head of each module is L(λ).

5 Stability of Extensions

5.1 The first result of this section is a vanishing result involving the extensions of certain G-modules with the higher right derived functors of G_{s} and H_{s}.

Proposition . Let s ≥ 1, M ∈ mod(G(𝔽_{q})) and N ∈ mod(G_{r}) and let L ∈ C_{s} be such that L has only p^{r}-restricted composition factors in its head. If p ≥ 2s(h − 1) then

(a) Ext_{G}^{i}(L, R^{j}G_{s}(M)) = 0;

(b) Ext_{G}^{i}(L, R^{j}H_{s}(M)) = 0 for 0 ≤ i ≤ s − 1 and j > 0.

Proof. (a) Let P_{0} be the projective cover of L in C_{s} and R_{1} be the kernel of the map P_{0} → L. Inductively, set P_{i} to be the projective cover of R_{i} and R_{i+1} to be the kernel of P_{i} → R_{i}. With this procedure we have constructed a projective resolution of L in C_{s}:

⋯ → P_{2} → P_{1} → P_{0} → L → 0.

Since the head of L contains only p^{r}-restricted composition factors, it follows by Proposition 4.6(b) that the composition factors of P_{0} (and R_{1}) are in π_{1}. An inductive argument shows that the composition factors of P_{i} are in π_{i+1} and the composition factors in the head of P_{i} are in π_{i} for 0 ≤ i ≤ s − 1.

Therefore, by Corollary 4.7, Hom_{G}(P_{i}, R^{j}G_{s}(M)) = 0 for j > 0 and 0 ≤ i ≤ s − 1. Consequently, Ext_{G}^{i}(L, R^{j}G_{s}(M)) = 0 for j > 0 and 0 ≤ i ≤ s − 1. Part (b) follows by a similar argument. _{□}

5.2 With the preceding results, one can provide positive answers to questions (1.1.1) and (1.1.2) by identifying certain extensions over G(𝔽_{q}) or G_{r} with extensions between finite-dimensional G-modules.

Theorem . Let s ≥ 1, M ∈ mod(G(𝔽_{q})) and N ∈ mod(G_{r}) and let L ∈ C_{s} be such that L has only p^{r}-restricted composition factors in its head (as G-module). If p ≥ 2s(h − 1) then there are the following isomorphisms and embeddings:

(a) Ext_{G(𝔽_{q})}^{i}(L, M) ≅ Ext_{G}^{i}(L, G_{s}(M)) for all 0 ≤ i ≤ s;

(b) Ext_{G}^{s+1}(L, G_{s}(M)) ↪ Ext_{G(𝔽_{q})}^{s+1}(L, M);

(c) Ext_{G_{r}}^{i}(L, N) ≅ Ext_{G}^{i}(L, H_{s}(N)) for all 0 ≤ i ≤ s;

(d) Ext_{G}^{s+1}(L, H_{s}(N)) ↪ Ext_{G_{r}}^{s+1}(L, N).

Proof. The proofs of the two cases are analogous and we prove parts (a) and (b). From Theorem 4.4 there is a spectral sequence

E_{2}^{i,j} = Ext_{G}^{i}(L, R^{j}G_{s}(M)) ⇒ Ext_{G(𝔽_{q})}^{i+j}(L, M).

By Propostion 5.1, we have Ext_{G}^{i}(L, R^{j}G_{s}(M)) = 0 for j > 0 and 0 ≤ i ≤ s − 1. The spectral sequence has differential with bidegree (r, 1 − r). Consequently, Ext_{G(𝔽_{q})}^{i}(L, M) ≅

Ext_{G}^{i}(L, G_{s}(M)) for 0 ≤ i ≤ s. Furthermore, since E_{2}^{i,j} = 0 for j > 0 and 0 ≤ i ≤ s − 1, the image of the differentials are zero in E_{2}^{s+1,0}, thus E_{2}^{s+1,0} ↪ Ext_{G(𝔽_{q})}^{s+1}(L, M). _{□}

5.3 The succeeding corollaries are immediate applications of the theorem to determining extensions between simple modules and Weyl modules.

Corollary . Let p ≥ 2(h − 1) and λ, μ ∈ X_{r}(T). Then there exists the following isomorphisms and embeddings:

(i) Ext_{G(𝔽_{q})}^{1}(L(λ), L(μ)) ≅ Ext_{G}^{1}(L(λ), G_{1}(L(μ))); (ii) Ext_{G}^{2}(L(λ), G_{1}(L(μ))) ↪ Ext_{G(𝔽_{q})}^{2}(L(λ), L(μ)); (iii) Ext_{G(𝔽_{q})}^{1}(V (λ), L(μ)) ≅ Ext_{G}^{1}(V (λ), G_{1}(L(μ))); (iv) Ext_{G}^{2}(V (λ), G_{1}(L(μ))) ↪ Ext_{G(𝔽_{q})}^{2}(V (λ), L(μ));

(v) Ext_{G_{r}}^{1}(L(λ), L(μ)) ≅ Ext_{G}^{1}(L(λ), H_{1}(L(μ))); (vi) Ext_{G}^{2}(L(λ), H_{1}(L(μ))) ↪ Ext_{G_{r}}^{2}(L(λ), L(μ)); (vii) Ext_{G_{r}}^{1}(V (λ), L(μ)) ≅ Ext_{G}^{1}(V (λ), H_{1}(L(μ))); (viii) Ext_{G}^{2}(V (λ), H_{1}(L(μ))) ↪ Ext_{G_{r}}^{2}(V (λ), L(μ)).

5.4 The following examples demonstrate that in order to study higher extensions between simple modules it is not sufficient to truncate the induction functor from the finite group (or the first Frobenius kernel) to the algebraic group at the level of twice the Steinberg weight. The methods for computing the functors G_{1}(−) and H_{1}(−) will be described in Chapters 6 and 7.

Example. Let G = SL_{2}(k) and G(𝔽_{q}) = SL_{2}(𝔽_{p}) with p ≥ 7. By a direct computation we have

G_{1}(L(p − 2)) ≅ L(p − 2) ⊕ L(2p − 3).

The simple module L(p − 4) is not in the same G-block as L(p − 2) or L(2p − 3), thus Ext_{G}^{2}(L(p − 4), G_{1}(L(p − 2))) = 0. On the other hand, the structure of the projective indecomposable modules are known for G(𝔽_{q}) = SL_{2}(𝔽_{p}). One can construct the minimal projective resolution of the simple module L(p−4) to show that Ext_{G(𝔽_{p})}^{2}(L(p−4), L(p−2)) ≠ 0. Consequently, Ext_{G(𝔽_{p})}^{2}(L(p − 4), L(p − 2)) ≠ Ext_{G}^{2}(L(p − 4), G_{1}(L(p − 2))).

Example. Let G = SL_{2}(k) and G_{1} = (SL_{2})_{1} with p ≥ 7. We have

H_{1}(L(p − 2)) ≅ L(p − 2) ⊕ L(2p − 2) ⊕ L(2p − 2).

We have Ext_{G_{1}}^{j}(L(p − 2), L(p − 2)) = 0 for j odd and Ext_{G_{1}}^{2}(L(p − 2), L(p − 2)) ≅ L(2)^{(1)} as a G-module. There exists a spectral sequence

E_{2}^{i,j} = Ext_{G/G_{1}}^{i}(k, Ext_{G_{1}}^{j}(L(p − 2), L(2p − 2)) ⇒ Ext_{G}^{i+j}(L(p − 2), L(2p − 2)).

Observe by the Steinberg tensor product theorem that

Ext_{G_{1}}^{j}(L(p − 2), L(2p − 2)) ≅ Ext_{G_{1}}^{j}(L(p − 2), L(p − 2)) ⊗ L(1)^{(1)}.

Therefore, E_{2}^{1,1} = 0, E_{2}^{2,0} = Ext_{G/G_{1}}^{2}(k, L(1)^{(1)}) = 0 and E_{2}^{0,2} = Hom_{G/G_{1}}(k, L(2)^{(1)} ⊗ L(1)^{(1)}) = 0. This shows that Ext_{G}^{2}(L(p − 2), L(2p − 2)) = 0. A similar spectral sequence argument can be used to show that Ext_{G}^{2}(L(p−2), L(p−2)) = 0. It follows that Ext_{G}^{2}(L(p− 2), H_{1}(L(p − 2))) = 0, and Ext_{G_{1}}^{2}(L(p − 2), L(p − 2)) ≠ Ext_{G}^{2}(L(p − 2), H_{1}(L(p − 2))).

5.5 Small primes: The results in Section 5.3 rely on the fact that the prime p is not too small. In this section we obtain a slightly weaker version of Corollary 5.3 for arbitrary primes.

Theorem . Let λ, μ ∈ X_{r}(T) with μ ≯ λ. Then

(a) Ext_{G(𝔽_{q})}^{1}(L(λ), L(μ)) ≅ Ext_{G}^{1}(L(λ), G_{1}(L(μ)));

(b) Ext_{G_{r}}^{1}(L(λ), L(μ)) ≅ Ext_{G}^{1}(L(λ), H_{1}(L(μ))).

Proof. (a) From [P1, Lem. 1.4] one obtains the following statement. Let λ ∈ X_{r}(T) and let M be a G(𝔽_{q})-module that contains only simple composition factors whose p^{r}-restricted highest weights γ satisfy γ ≤ (p^{r} −1)ρ+w_{0}λ where w_{0} is the long element in the Weyl group. Then the G(𝔽_{q})-head of St_{r} ⊗ M contains only simple modules L(μ) whose p^{r}-restricted highest weights satisfy μ ≥ λ. In particular it follows that for μ ≱ λ

Hom_{G}(St_{r} ⊗ L((p^{r} − 1)ρ + w_{0}λ), L(μ)) ⊆ Hom_{G(𝔽_{q})}(St_{r} ⊗ L((p^{r} − 1)ρ + w_{0}λ), L(μ)) = 0.

On the other hand,

Hom_{G}(St_{r} ⊗ L((p^{r} − 1)ρ + w_{0}λ), L(λ)) ≅ Hom_{G(𝔽_{q})}(St_{r} ⊗ L((p^{r} − 1)ρ + w_{0}λ), L(λ)) ≅ k.

Therefore, we have the following short exact sequence

0 → R → St_{r} ⊗ L((p^{r} − 1)ρ + w_{0}λ) → L(λ) → 0.  (5.5.1)

Note that this is a short exact sequence in the category C_{1} because St_{r}⊗L((p^{r}−1)ρ+w_{0}λ) ∈ C_{1}. By using the fact that the Steinberg module is projective over G(𝔽_{q}) we obtain the following exact sequence:

0 → Hom_{G(𝔽_{q})}(L(λ), L(μ)) → Hom_{G(𝔽_{q})}(St_{r} ⊗ L((p^{r} − 1)ρ + w_{0}λ), L(μ))
→ Hom_{G(𝔽_{q})}(R, L(μ)) → Ext_{G(𝔽_{q})}^{1}(L(λ), L(μ)) → 0.

If μ ≯ λ then by our previous observations we have

Hom_{G(𝔽_{q})}(L(λ), L(μ)) ≅ Hom_{G(𝔽_{q})}(St_{r} ⊗ L((p^{r} − 1)ρ + w_{0}λ), L(μ)).

Consequently,

Hom_{G(𝔽_{q})}(R, L(μ)) ≅ Ext_{G(𝔽_{q})}^{1}(L(λ), L(μ)).  (5.5.2)

From the short exact sequence (5.5.1) one obtains the following long exact sequence:

0 → Hom_{G}(L(λ), G_{1}(L(μ))) → Hom_{G}(St_{r} ⊗ L((p^{r} − 1)ρ + w_{0}λ), G_{1}(L(μ)))
→ Hom_{G}(R, G_{1}(L(μ))) → Ext_{G}^{1}(L(λ), G_{1}(L(μ)))
→ Ext_{G}^{1}(St_{r} ⊗ L((p^{r} − 1)ρ + w_{0}λ), G_{1}(L(μ))) → · · · .

Observe that by adjointness we have for μ ≱ λ

0 = Hom_{G(𝔽_{q})}(L(λ), L(μ)) ≅ Hom_{G}(L(λ), G_{1}(L(μ)));

0 = Hom_{G(𝔽_{q})}(St_{r} ⊗ L((p^{r} − 1)ρ + w_{0}λ), L(μ)) ≅ Hom_{G}(St_{r} ⊗ L((p^{r} − 1)ρ + w_{0}λ), G_{1}(L(μ))).

Similarly, for μ = λ it follows that

Hom_{G(𝔽_{q})}(L(λ), L(λ)) ≅ Hom_{G}(L(λ), G_{1}(L(λ))) ≅ k;

Hom_{G(𝔽_{q})}(St_{r} ⊗ L((p^{r} − 1)ρ + w_{0}λ), L(λ)) ≅ Hom_{G}(St_{r} ⊗ L((p^{r} − 1)ρ + w_{0}λ), G_{1}(L(λ))) ≅ k.

The long exact sequence and these isomorphisms now define an injective map

Hom_{G}(R, G_{1}(L(μ))) → Ext_{G}^{1}(L(λ), G_{1}(L(μ))).

But, R ∈ C_{1} so by adjointness we have

Hom_{G(𝔽_{q})}(R, L(μ)) ≅ Hom_{G}(R, G_{1}(L(μ))).

It follows by (5.5.2) that there exists an injective map

Ext_{G(𝔽_{q})}^{1}(L(λ), L(μ)) → Ext_{G}^{1}(L(λ), G_{1}(L(μ))).

The five term exact sequence of the spectral sequence in 4.4 makes the base map E^{1,0} → E^{1} an injective map in the other direction. Therefore, one obtains the desired isomorphism

Ext_{G(𝔽_{q})}^{1}(L(λ), L(μ)) ≅ Ext_{G}^{1}(L(λ), G_{1}(L(μ))).

A similar argument can be used to prove part (b). _{□}

6 Cohomology for Frobenius kernels

6.1 In the previous section, extensions of simple modules over G(𝔽_{q}) and G_{r} were identified with certain extensions over G. These G extensions involve the “induction” functors G_{s} and H_{s}. In this and the following section, we study G_{s}(L(μ)) and H_{s}(L(μ)) for μ ∈ X_{r}(T), in order to improve this identification to involve only simple modules. We begin by showing that the module H_{s}(L(μ)) is semisimple for μ ∈ X_{r}(T).

Theorem . Let p ≥ (2s + 1)(h − 1) and μ ∈ X_{r}(T). Then H_{s}(L(μ)) is semisimple. Moreover,

H_{s}(L(μ)) ≅ ⨁_{ν∈π_{s}} Hom_{G}(L(ν), H_{s}(L(μ))) ⊗ L(ν) ≅ ⨁_{ν∈π_{s}} L(ν)^{dim_{k} Hom_{Gr} (L(ν),L(μ))}.

Proof. First, consider the socle of H_{s}(L(μ)) over G (or equivalently in the category C_{s}):

soc_{G} H_{s}(L(μ)) = ⨁_{ν∈π_{s}} Hom_{G}(L(ν), H_{s}(L(μ))) ⊗ L(ν)

= ⨁_{ν∈π_{s}} L(ν)^{dim_{k} Hom_{Gr} (L(ν),L(μ))}

by adjointness. So, if H_{s}(L(μ)) is semisimple, then it has the claimed form.

Theorem 4.5 can be used to obtain information about the composition factors of H_{s}(L(μ)). Specifically we have

[H_{s}(L(μ)) : L(ν)] = dim_{k} Hom_{G_{r}}(P(ν), L(μ))
≤ dim_{k} Hom_{G_{r}}(Q_{r}(ν_{0}) ⊗ L(ν_{1})^{(r)}, L(μ)) (by Proposition 4.6(b)).

As a G_{r}-module, L(ν_{1})^{(r)} is trivial and so Q_{r}(ν_{0}) ⊗ L(ν_{1})^{(r)} = ⊕Q_{r}(ν_{0}). Further, since ν_{0} and μ are both p^{r}-restricted weights, Hom_{G_{r}}(Q_{r}(ν_{0}), L(μ)) = Hom_{G_{r}}(L(ν_{0}), L(μ)), and this will be non-zero if and only if ν_{0} = μ. Hence, the only possible composition factors of H_{s}(L(μ)) are those L(ν) with ν = μ + p^{r}σ. As noted in Section 4.6, since ν is in π_{s} and μ is p^{r}-restricted, σ must lie in the bottom alcove.

To show that H_{s}(L(μ)) is semisimple, it now suffices to show that Ext_{G}^{1}(L(ν_{1}), L(ν_{2})) = 0 for any such weights ν_{1} = μ + p^{r}σ_{1} and ν_{2} = μ + p^{r}σ_{2}. The argument is similar to that of Proposition 4.6. The short exact sequence of group schemes

1 → G_{r} → G → G/G_{r} → 1

gives rise to the Lyndon-Hochschild-Serre spectral sequence

E_{2}^{i,j} = Ext_{G/G_{r}}^{i}(k, Ext_{G_{r}}^{j}(L(ν_{1}), L(ν_{2}))) ⇒ Ext_{G}^{i+j}(L(ν_{1}), L(ν_{2})).

The beginning of the 5-term exact sequence is

0 → E^{1,0} → E^{1} → E^{0,1} = Hom_{G/G_{r}}(k, Ext_{G_{r}}^{1}(L(ν_{1}), L(ν_{2}))) → ….

However,

Ext_{G_{r}}^{1}(L(ν_{1}), L(ν_{2})) = Ext_{G_{r}}^{1}(L(μ)⊗L(σ_{1})^{(r)}, L(μ)⊗L(σ_{2})^{(r)}) = Ext_{G_{r}}^{1}(⊕L(μ), ⊕L(μ)) = 0

since there are no self-extensions over G_{r} (cf. [Jan1, II. 12.9]). Hence there is an isomorphism

Ext_{G}^{1}(L(ν_{1}), L(ν_{2})) = E^{1} ≅ E^{1,0} = Ext_{G/G_{r}}^{1}(k, Hom_{G_{r}}(L(ν_{1}), L(ν_{2}))).

Finally, we have

Ext_{G/G_{r}}^{1}(k, Hom_{G_{r}}(L(ν_{1}), L(ν_{2}))) = Ext_{G/G_{r}}^{1}(k, Hom_{G_{r}}(L(μ) ⊗ L(σ_{1})^{(r)}, L(μ) ⊗ L(σ_{2})^{(r)}))
= Ext_{G/G_{r}}^{1}(k, Hom_{G_{r}}(L(μ), L(μ)) ⊗ (L(σ_{1})^{(r)})^{∗} ⊗ L(σ_{2})^{(r)})
= Ext_{G/G_{r}}^{1}(k, k ⊗ (L(σ_{1})^{(r)})^{∗} ⊗ L(σ_{2})^{(r)})
= Ext_{G/G_{r}}^{1}(L(σ_{1})^{(r)}, L(σ_{2})^{(r)}) = 0

by the Linkage Principle since σ_{1} and σ_{2} lie in the same alcove. _{□}

6.2 The following result shows that one can completely determine the extensions of simple modules in mod(G_{r}) by knowing the extension theory of simple modules in mod(G).

Corollary . Let p ≥ (2s + 1)(h − 1) and λ, μ ∈ X_{r}(T). Moreover, let L ∈ C_{s} be such that L has only p^{r}-restricted composition factors in its head. Then

(a) For 0 ≤ i ≤ s there exists the following isomorphism of k-vector spaces:

Ext_{G_{r}}^{i}(L, L(μ)) ≅ ⨁_{ν∈π_{s}} Hom_{G_{r}}(L(ν), L(μ)) ⊗ Ext_{G}^{i}(L, L(ν)).

(b) In particular for 0 ≤ i ≤ s, we have

Ext_{G_{r}}^{i}(L(λ), L(μ)) ≅ ⨁_{ν∈π_{s}} Hom_{G_{r}}(L(ν), L(μ)) ⊗ Ext_{G}^{i}(L(λ), L(ν)).

(c) For p ≥ 3(h − 1), we have

Ext_{G_{r}}^{1}(L(λ), L(μ)) ≅ ⨁_{ν∈π_{1}} Hom_{G_{r}}(L(ν), L(μ)) ⊗ Ext_{G}^{1}(L(λ), L(ν))

where π_{1} is the set of all p^{r}-bounded weights.

Proof. The result follows by combining the isomorphism

Ext_{G_{r}}^{i}(L, L(μ)) ≅ Ext_{G}^{i}(L, H_{s}(L(μ)))

for 0 ≤ i ≤ s of Theorem 5.2 with the preceding identification of H_{s}(L(μ)) given in Theorem 6.1. _{□}

Observe that the restriction on the prime indicates that the direct sums appearing here may be taken over those ν ∈ π_{s} with ν = μ + p^{r}σ and σ ∈ \overline{C}_{ℤ}.

7 Cohomology for finite Chevalley groups

7.1 In this section, we use the module G_{s}(L(μ)) to recover and extend the results of [And2] on extensions of G(𝔽_{q})-modules. The hypotheses of these results depend on the type of G. This is because when G is not of type A_{1}, ⟨α, α_{0}^{∨}⟩ ≤ 1 for all simple roots α, while ⟨α, α_{0}^{∨}⟩ = 2 for the simple root α = α_{0} in type A_{1}. First, consider the embedding L(λ) ↪ G_{s}(L(λ)) of G-modules, which corresponds to the identity map under the isomorphism Hom_{G}(L(λ), G_{s}(L(λ)) ≅ Hom_{G(𝔽_{q})}(L(λ), L(λ)).

Proposition . Let s ≥ 1 and p ≥ (2s+1)(h−1). Suppose λ, μ ∈ X_{r}(T) satisfy ⟨λ+μ, α_{0}^{∨}⟩ < p^{r} − ϵp^{r−1} − 1 where ϵ = 2 if G is of type A_{1} and ϵ = 1 in all other cases. Then

Ext_{G}^{1}(L(λ), G_{s}(L(μ))/L(μ)) = 0.

Proof. From Theorem 4.5, L(μ) is the only restricted composition factor of G_{s}(L(μ)) and moreover appears only once. Hence, the composition factors of G_{s}(L(μ))/L(μ) are not restricted. Let L(ν) be a non-restricted composition factor of G_{s}(L(μ)) and write ν = ν_{0} + p^{r}ν_{1} as usual. Since ν_{0} is restricted and ν is not, ν_{1} is necessarily non-zero. According to Theorem 4.5 and Proposition 4.6(b), we have

[G_{s}(L(μ)) : L(ν)] = dim_{k} Hom_{G(𝔽_{q})}(P(ν), L(μ))
≤ dim_{k} Hom_{G(𝔽_{q})}(Q_{r}(ν_{0}) ⊗ L(ν_{1}), L(μ))
= dim_{k} Hom_{G(𝔽_{q})}(Q_{r}(ν_{0}), L(μ) ⊗ L(−w_{0}ν_{1})).

So for L(ν) to be a composition factor, we must have Hom_{G(𝔽_{q})}(Q_{r}(ν_{0}), L(ω)) ≠ 0 for some composition factor L(ω) of L(μ) ⊗ L(−w_{0}ν_{1}). By [Jan3] or [Cha] (see also [And2, 1(2)]), Q_{r}(ν_{0}) decomposes as a direct sum U_{r}(ν_{0}) ≅ ⨁_{σ} U_{r}(σ) (of projective indecomposables for G(𝔽_{q})) where each σ satisfies ⟨σ, α_{0}^{∨}⟩ ≥ p^{r} − 1 + ⟨ν_{0}, α_{0}^{∨}⟩. Hence, there is at least one ω with ω = ν_{0} or ⟨ω, α_{0}^{∨}⟩ ≥ p^{r} − 1 + ⟨ν_{0}, α_{0}^{∨}⟩.

On the other hand, by Lemma 2.5 of [And2], if Ext_{G}^{1}(L(λ), L(ν)) ≠ 0, then ⟨λ+ν_{0}, α_{0}^{∨}⟩ ≥ p^{r}⟨ν_{1}, α_{0}^{∨}⟩ − ϵp^{r−1}. Hence, for such ν and any weight ω of L(μ) ⊗ L(−w_{0}ν_{1}), we have

⟨ω, α_{0}^{∨}⟩ ≤ ⟨μ + ν_{1}, α_{0}^{∨}⟩

< p^{r} − ϵp^{r−1} − 1 − ⟨λ + μ, α_{0}^{∨}⟩ + ⟨μ + ν_{1}, α_{0}^{∨}⟩ ( by original assumption )

≤ (p^{r} − 1)⟨ν_{1}, α_{0}^{∨}⟩ − ϵp^{r−1} − ⟨λ + μ, α_{0}^{∨}⟩ + ⟨μ + ν_{1}, α_{0}^{∨}⟩
= p^{r}⟨ν_{1}, α_{0}^{∨}⟩ − ϵp^{r−1} − ⟨λ, α_{0}^{∨}⟩
≤ ⟨ν_{0}, α_{0}^{∨}⟩ ( by Lemma 2.5 of [And2] ).

Hence, Ext_{G}^{1}(L(λ), L(ν)) = 0 for all composition factors L(ν) of G_{s}(L(μ))/L(μ). _{□}

Corollary . Assume p ≥ 3(h − 1). Let λ, μ ∈ X_{r}(T) satisfy ⟨λ + μ, α_{0}^{∨}⟩ < p^{r} − ϵp^{r−1} − 1 where ϵ = 2 if G is of type A_{1} and ϵ = 1 in all other cases. Then

(a) (Theorem 2.8 of [And2]) Ext_{G(𝔽_{q})}^{1}(L(λ), L(μ)) ≅ Ext_{G}^{1}(L(λ), L(μ)).

(b) Moreover, there is an embedding Ext_{G}^{2}(L(λ), L(μ)) ↪ Ext_{G(𝔽_{q})}^{2}(L(λ), L(μ)).

Proof. For (a), consider the the short exact sequence

0 → L(μ) → G_{1}(L(μ)) → G_{1}(L(μ))/L(μ) → 0.

This short exact sequence induces a long exact sequence

⋯ → Hom_{G}(L(λ), G_{1}(L(μ))/L(μ)) → Ext_{G}^{1}(L(λ), L(μ))
→ Ext_{G}^{1}(L(λ), G_{1}(L(μ))) → Ext_{G}^{1}(L(λ), G_{1}(L(μ))/L(μ)) → …

The first term is zero since G_{1}(L(μ))/L(μ) has no restricted composition factors, and the last term is zero by Proposition 7.1. Hence there is an isomorphism

Ext_{G}^{1}(L(λ), L(μ)) ≅ Ext_{G}^{1}(L(λ), G_{1}(L(μ))),

and the claimed isomorphism follows from Corollary 5.3.

For (b), continuing the above long exact sequence, we have

⋯ → Ext_{G}^{1}(L(λ), G_{1}(L(μ))/L(μ)) → Ext_{G}^{2}(L(λ), L(μ)) → Ext_{G}^{2}(L(λ), G_{1}(L(μ)) → ….

Again, since the first term is zero by the Proposition, we have an embedding

Ext_{G}^{2}(L(λ), L(μ)) ↪ Ext_{G}^{2}(L(λ), G_{1}(L(μ))).

The result follows by combining this with the embedding

Ext_{G}^{2}(L(λ), G_{1}(L(μ))) ↪ Ext_{G(𝔽_{q})}^{2}(L(λ), L(μ))

of Corollary 5.3. _{□}

7.2 More generally, semisimplicity of G_{s}(L(μ)) can be used to identify G(𝔽_{q})-extensions with G-extensions.

Theorem . Let s ≥ 1 and μ ∈ X_{r}(T). Then G_{s}(L(μ)) is semisimple if and only if

G_{s}(L(μ)) ≅ ⨁_{ν∈π_{s}} Hom_{G(𝔽_{q})}(L(ν), L(μ)) ⊗ L(ν).

Proof. Consider the socle of G_{s}(L(μ)) over G (or equivalently in the category C_{s}):

soc_{G} G_{s}(L(μ)) = ⨁_{ν∈π_{s}} Hom_{G}(L(ν), G_{s}(L(μ))) ⊗ L(ν)

= ⨁_{ν∈π_{s}} Hom_{G(𝔽_{q})}(L(ν), L(μ)) ⊗ L(ν).

So G_{s}(L(μ)) is semisimple if and only if it has the claimed form. _{□}

The previous result together with Theorem 5.2 yields the following corollary.

Corollary . Let s ≥ 1, p ≥ 2s(h − 1), μ ∈ X_{r}(T), and let L ∈ C_{s} be such that L has only p^{r}-restricted composition factors in its head (as G-module). Assume that G_{s}(L(μ)) is semisimple. Then for 0 ≤ i ≤ s there exists the following isomorphism of vector spaces:

Ext_{G(𝔽_{q})}^{i}(L, L(μ)) ≅ ⨁_{ν∈π_{s}} Hom_{G(𝔽_{q})}(L(ν), L(μ)) ⊗ Ext_{G}^{i}(L, L(ν)).

In particular, for p ≥ 2(h − 1), we have

Ext_{G(𝔽_{q})}^{1}(L, L(μ)) ≅ ⨁_{ν∈π_{1}} Hom_{G(𝔽_{q})}(L(ν), L(μ)) ⊗ Ext_{G}^{1}(L, L(ν)),

where π_{1} is the set of all p^{r}-bounded weights.

7.3 We will now refine the result given in Corollary 7.2 by identifying homomorphisms over G(𝔽_{q}) with homomorphisms over G.

Lemma . Let p ≥ 2(h − 1), and λ, μ, ν ∈ X_{r}(T) with ⟨ν, α_{0}^{∨}⟩ < p^{r} − 1. Then

Hom_{G}(L(λ), L(μ) ⊗ L(ν)) ≅ Hom_{G(𝔽_{q})}(L(λ), L(μ) ⊗ L(ν)).

Proof. First it is shown that the G-socle of L(μ) ⊗ L(ν) contains only simple modules with p^{r}-restricted highest weights. Assume that

0 ≠ Hom_{G}(L(γ_{0}) ⊗ L(γ_{1})^{(r)}, L(μ) ⊗ L(ν)),

where γ_{0} ∈ X_{r}(T). Without loss of generality we may assume that ⟨γ_{0}, α_{0}^{∨}⟩ ≥ ⟨μ, α_{0}^{∨}⟩ otherwise replace γ_{0} by −w_{0}μ and μ by −w_{0}γ_{0}. For the simple module L(γ_{0} + p^{r}γ_{1}) to appear as a composition factor in L(μ) ⊗ L(ν) it is necessary that

⟨γ_{0} + p^{r}γ_{1}, α_{0}^{∨}⟩ ≤ ⟨μ + ν, α_{0}^{∨}⟩ ≤ ⟨γ_{0} + ν, α_{0}^{∨}⟩,

which implies that p^{r}⟨γ_{1}, α_{0}^{∨}⟩ ≤ ⟨ν, α_{0}^{∨}⟩ < p^{r} − 1 and forces γ_{1} = 0.

The above argument shows that soc_{G} L(μ)⊗L(ν) ≅ ⊕_{i}L(σ_{i}) where all σ_{i} are p^{r}-restricted and satisfy ⟨μ − ν, α_{0}^{∨}⟩ ≤ ⟨σ_{i}, α_{0}^{∨}⟩ ≤ ⟨μ + ν, α_{0}^{∨}⟩. Moreover, the G-socle of L(μ) ⊗ L(ν) is contained in the G(𝔽_{q})-socle. Using the injectivity of Q_{r}(γ) in the p^{r}-bounded category one can embed L(μ) ⊗ L(ν) in ⊕_{i}Q_{r}(σ_{i}). The G(𝔽_{q})-socle of L(μ) ⊗ L(ν) is therefore contained in the G(𝔽_{q})-socle of ⊕_{i}Q_{r}(σ_{i}).

Now assume that dim_{k} Hom_{G}(L(λ), L(μ) ⊗ L(ν)) < dim_{k} Hom_{G(𝔽_{q})}(L(λ), L(μ) ⊗ L(ν)). Without loss of generality we may assume this time that ⟨μ, α_{0}^{∨}⟩ ≥ ⟨λ, α_{0}^{∨}⟩. It follows that dim_{k} Hom_{G}(L(λ), ⊕_{i}Q_{r}(σ_{i})) < dim_{k} Hom_{G(𝔽_{q})}(L(λ), ⊕_{i}Q_{r}(σ_{i})). That implies that, for at least one σ_{i}, dim_{k} Hom_{G}(L(λ), Q_{r}(σ_{i})) < dim_{k} Hom_{G(𝔽_{q})}(L(λ), Q_{r}(σ_{i})).

The module Q_{r}(σ_{i}) are also injective as G(𝔽_{q})-modules. A formula by Jantzen [Jan3] amd Chastkofsky [Cha] tells us how the Q_{r}(σ_{i}) split into indecomposable injective G(𝔽_{q})- modules, denoted here by U_{r}(ξ). The module Q_{r}(σ_{i}) contains U_{r}(σ_{i}) exactly once, while any U_{r}(ξ) with ξ ≠ σ_{i} that appears as a summand forces ⟨σ_{i}, α_{0}^{∨}⟩ + p^{r} − 1 ≤ ⟨ξ, α_{0}^{∨}⟩ (see also [And2]). Clearly λ ≠ σ_{i}. Therefore we obtain the following sequence of inequalities:

⟨μ, α_{0}^{∨}⟩ − ⟨ν, α_{0}^{∨}⟩ + p^{r} − 1 = ⟨μ − ν, α_{0}^{∨}⟩ + p^{r} − 1 ≤ ⟨σ_{i}, α_{0}^{∨}⟩ + p^{r} − 1 ≤ ⟨λ, α_{0}^{∨}⟩ ≤ ⟨μ, α_{0}^{∨}⟩.

Now p^{r} − 1 ≤ ⟨ν, α_{0}^{∨}⟩ < p^{r} − 1 gives us the desired contradiction. _{□}

For large enough primes one obtains the following result:

Theorem . Let s ≥ 1, p^{r} > 2s(h − 1), and μ ∈ X_{r}(T) and assume that G_{s}(L(μ)) is semisimple. For any weight ν ∈ π_{s} we set ν = ν_{0} + p^{r}ν_{1} with ν_{0} ∈ X_{r}(T). Then

G_{s}(L(μ)) ≅ ⨁_{ν∈π_{s}} Hom_{G}(L(ν_{0}) ⊗ L(ν_{1}), L(μ)) ⊗ L(ν).

Proof. According to Theorem 7.2 we have

G_{s}(L(μ)) ≅ ⨁_{ν∈π_{s}} Hom_{G(𝔽_{q})}(L(ν), L(μ)) ⊗ L(ν).

For any weight ν ∈ π_{s} we set ν = ν_{0} + p^{r}ν_{1} with ν_{0} ∈ X_{r}(T). Then L(ν) ≅ L(ν_{0}) ⊗ L(ν_{1}) as G(𝔽_{q})-modules and so

G_{s}(L(μ)) ≅ ⨁_{ν∈π_{s}} Hom_{G(𝔽_{q})}(L(ν_{0}) ⊗ L(ν_{1}), L(μ)) ⊗ L(ν).

Moreover, p^{r}⟨ν_{1}, α_{0}^{∨}⟩ ≤ ⟨ν, α_{0}^{∨}⟩ < 2sp^{r}(h − 1) implies that ⟨ν_{1}, α_{0}^{∨}⟩ < 2s(h − 1) ≤ p^{r} − 1. Therefore, the theorem now follows by applying Lemma 7.3 to Hom_{G(𝔽_{q})}(L(ν_{0}), L(μ) ⊗ L(−w_{0}ν_{1})). _{□}

Corollary . Let s ≥ 1, p ≥ 2s(h − 1), μ ∈ X_{r}(T), and let L ∈ C_{s} be such that L has only p^{r}-restricted composition factors in its head (as G-module). We assume that G_{s}(L(μ)) is semisimple. For any weight ν ∈ π_{s} we set ν = ν_{0} + p^{r}ν_{1} with ν_{0} ∈ X_{r}(T). Then for 0 ≤ i ≤ s,

Ext_{G(𝔽_{q})}^{i}(L, L(μ)) ≅ ⨁_{ν∈π_{s}} Hom_{G}(L(ν_{0}) ⊗ L(ν_{1}), L(μ)) ⊗ Ext_{G}^{i}(L, L(ν)).

In particular, for p ≥ 2(h − 1), we have

Ext_{G(𝔽_{q})}^{1}(L, L(μ)) ≅ ⨁_{ν∈π_{1}} Hom_{G}(L(ν_{0}) ⊗ L(ν_{1}), L(μ)) ⊗ Ext_{G}^{1}(L, L(ν)),

where π_{1} is the set of all p^{r}-bounded weights.

7.4 We now identify some μ for which G_{s}(L(μ)) is indeed semisimple. Let X[s] = {ν ∈ X(T)_{+}|⟨ν, α_{0}^{∨}⟩ < 2s(h − 1)}.

Theorem . Let s ≥ 1, p ≥ (2s + 1)(h − 1). Then G_{s}(k) is semisimple. Moreover,

G_{s}(k) ≅ ⨁_{ν∈X[s]} L((p^{r} − w_{0})ν).

Proof. Theorem 4.5 can be used to obtain information about the composition factors of G_{s}(k). Specifically we have

[G_{s}(k) : L(ν)] = dim_{k} Hom_{G(𝔽_{q})}(P(ν), k)
≤ dim_{k} Hom_{G(𝔽_{q})}(Q_{r}(ν_{0}) ⊗ L(ν_{1})^{(r)}, k) (by Proposition 4.6(b)).

As a G(𝔽_{q})-module, L(ν_{1})^{(r)} is isomorphic to L(ν_{1}) and so

[G_{s}(k) : L(ν)] ≤ dim_{k} Hom_{G(𝔽_{q})}(Q_{r}(ν_{0}), L(−w_{0}ν_{1})).

Let us assume that dim_{k} Hom_{G(𝔽_{q})}(Q_{r}(ν_{0}), L(−w_{0}ν_{1})) = 0 unless ν_{0} = −w_{0}ν_{1}. Then, for any ν ∈ π_{s}, we have

[G_{s}(k) : L(ν)] ≤ dim_{k} Hom_{G(𝔽_{q})}(Q_{r}(ν_{0}), L(−w_{0}ν_{1}))
= dim_{k} Hom_{G(𝔽_{q})}(L(ν_{0}), L(−w_{0}ν_{1}))
= dim_{k} Hom_{G(𝔽_{q})}(L(ν), k)
= dim_{k} Hom_{G}(L(ν), G_{s}(k)).

Under our assumption it then follows from Theorem 7.2 that G_{s}(k) is semisimple. Moreover, all composition factors have highest weight of the form ν = ν_{0} + p^{r}ν_{1} = −w_{0}ν_{1} + p^{r}ν_{1} and appear with multiplicity one. An easy weight computation shows that ν = −w_{0}ν_{1}+p^{r}ν_{1} ∈ π_{s} if and only if ⟨ν_{1}, α_{0}^{∨}⟩ < 2s(h − 1).

The module Q_{r}(ν_{0}) is projective over G(𝔽_{q}) and dim_{k} Hom_{G(𝔽_{q})}(Q_{r}(ν_{0}), L(−w_{0}ν_{1})) equals the multiplicity of the indecomposable projective G(𝔽_{q})-module U_{r}(−w_{0}ν_{1}) as a summand of Q_{r}(ν_{0}). It remains to be shown that this multiplicity [Q_{r}(ν_{0}), U_{r}(−w_{0}ν_{1})] = 0 unless ν_{0} = −w_{0}ν_{1}. As pointed out in the proof of Lemma 7.3, [Q_{r}(ν_{0}), U_{r}(−w_{0}ν_{1})] ≠ 0 and ν_{0} ≠ −w_{0}ν_{1} imply that p^{r} − 1 ≤ ⟨ν_{0}, α_{0}^{∨}⟩ + p^{r} − 1 ≤ ⟨−w_{0}ν_{1}, α_{0}^{∨}⟩ ≤ ⟨ν_{1}, α_{0}^{∨}⟩. But this is impossible because ⟨ν_{1}, α_{0}^{∨}⟩ < 2s(h − 1). _{□}

We denote by T_{γ}^{δ} the translation functor as defined in [Jan1, II. 7].

Corollary . Let s ≥ 1, p ≥ (2s + 1)(h − 1), and let λ ∈ X_{r}(T) with 0 ≤ i ≤ s.

(a) If λ is p-regular then

H^{i}(G(𝔽_{q}), L(λ)) ≅ ⨁_{ν∈X[s]} H^{i}(G, T_{ν}^{0}L(λ + p^{r}ν)).

(b) If λ is p-singular then

H^{i}(G(𝔽_{q}), L(λ)) ≅ 0.

Proof. From the preceding theorem one can see that all the composition factors of G_{s}(k) have p-regular highest weight. Therefore there can be no nonzero cohomology for simple modules with p-singular highest weight. We may assume that λ is p-regular.

H^{i}(G(𝔽_{q}), L(λ)) ≅ Ext_{G(𝔽_{q})}^{i}(k, L(λ))
≅ Ext_{G(𝔽_{q})}^{i}(L(−w_{0}λ), k)
≅ Ext_{G}^{i}(L(−w_{0}λ), G_{s}(k))  (by Theorem 5.2 )
≅ ⨁_{ν∈X[s]} Ext_{G}^{i}(L(−w_{0}λ), L((p^{r} − w_{0})ν))

(by Theorem 7.4)

≅ ⨁_{ν∈X[s]} Ext_{G}^{i}(L(ν), L(λ + p^{r}ν)) ([Jan1, I. 4.4])

≅ ⨁_{ν∈X[s]} Ext_{G}^{i}(T_{0}^{ν}(k), L(λ + p^{r}ν))

≅ ⨁_{ν∈X[s]} Ext_{G}^{i}(k, T_{ν}^{0}L(λ + p^{r}ν)) ([Jan1, II. 7.6])

≅ ⨁_{ν∈X[s]} H^{i}(G, T_{ν}^{0}L(λ + p^{r}ν)).
□

For i = 1 we can do slightly better. It follows from [And2, Lemma 2.5] that 0 ≠ H^{1}(G, T_{ν}^{0}L(λ + p^{r}ν)) forces ⟨ν, α_{0}^{∨}⟩ ≤ h − 1. Therefore for p ≥ 3(h − 1) one obtains:

H^{1}(G(𝔽_{q}), L(λ)) ≅ ⨁_{{ν∈X(T)_{+}|⟨ν,α_{0}^{∨}⟩≤h−1}} H^{1}(G, T_{ν}^{0}L(λ + p^{r}ν)).

7.5 For large primes we can now prove the existence of a one-to-one correspondence between the ith cohomology for certain simple G-modules and the ith cohomology for the simple G(𝔽_{q})-modules. Following [Jan4] we define W̃_{p} to be the affine group generated by the ordinary Weyl group and the translations by elements of pX(T). For large p any p-regular weight has trivial stabilizer in W̃_{p} [Jan4, Lemma 5].

Theorem . Let s ≥ 1, p ≥ (4s+1)(h−1), ν ∈ X_{r}(T) with ⟨ν, α_{0}^{∨}⟩ < 2s(h−1), and u ∈ W̃_{p} such that u · ν is p^{r}-restricted. Then for 0 ≤ i ≤ s,

H^{i}(G(𝔽_{q}), L(u · ν)) ≅ H^{i}(G, L(u · 0 + p^{r}ν)).

Moreover, all the nonzero ith cohomology for simple modules of the finite group can be obtained in this fashion.

Proof. Assume that 0 ≠ H^{i}(G(𝔽_{q}), L(λ)). By the previous corollary there exists a ν ∈ X(T)_{+} with ⟨ν, α_{0}^{∨}⟩ < 2s(h − 1) and 0 ≠ H^{i}(G, T_{ν}^{0}L(λ + p^{r}ν)). Let γ be the highest weight of the simple module T_{ν}^{0}L(λ + p^{r}ν). By the Linkage Principle there exists a unique u ∈ W̃_{p} such that γ = u · 0 + p^{r}ν. 0 and ν are inside the same alcove. Therefore T_{0}^{ν}T_{ν}^{0}L(λ + p^{r}ν) = L(λ + p^{r}ν). Now clearly λ = u · ν. It remains to be shown that there exists no other weight ν̃ ∈ X(T)_{+} with ⟨ν̃, α_{0}^{∨}⟩ < 2s(h − 1) and 0 ≠ H^{i}(G, T 0 ; ν̃L(λ + p^{r}ν̃)) such that λ = v · ν̃ for some v ∈ W̃_{p}.

Suppose that u · ν = λ = v · ν̃. Then ν̃ is contained in the W̃_{p}-orbit of ν. Both ν and ν̃ are p-regular and inside the lowest alcove. By [Jan4, Lemma 1] there exists a unique element w of the Weyl group and a unique weight ρ_{w} that is the sum of some distinct fundamental weights such that ν̃ = w · ν + pρ_{w}. If ρ_{w} ≠ 0, then there exists a simple root β such that ⟨ν̃, β^{∨}⟩ − ⟨w(ν + ρ), β^{∨}⟩ + 1 = p. But the left hand side is less than or equal to 2s(h − 1) − 1 + (2s + 1)(h − 1) − 1 + 1 = (4s + 1)(h − 1) − 1 < p. Therefore, ρ_{w} = 0. Both ν and ν̃ are p-regular dominant weights. Thus, w is the identity. Hence, u = v and ν = ν̃. _{□}

For groups whose root lattice is identical to the weight lattice the condition on p can be weakened to p ≥ (2s+1)(h−1). The following example illustrates how one can use Theorem 7.5 to compute the G(𝔽_{q})-cohomology via the G-cohomology.

Example . Let G be of type G_{2} and q = p > 13. The simple roots are denoted by α_{1} and α_{2}, where α_{1} is short. Let ω_{1} and ω_{2} denote the corresponding fundamental weights. Write s_{β} for the reflection belonging to a positive root β. A pair (r, s) will denote the weight rω_{1} + sω_{2}.

It follows from the remark after Corollary 7.4 that H^{1}(G, L(u·0+pν)) ≠ 0 is only possible for ⟨ν, α_{0}^{∨}⟩ ≤ h − 1. This forces all weights u · 0 + pν that are of interest to be p-bounded. From [LY, Table 1] we obtain that H^{1}(G, L(u · 0 + pν)) = 0 for all p-bounded dominant weights u · 0 + pν, unless u · 0 + pν is one of the following list:

{(p − 5, 0), (3, p − 2), (p + 4, p − 4), (3, 2p − 2), (2p − 2, 1)}.

For these weights dim_{k} H^{1}(G, L(u · 0 + pν)) = 1.

The first two weights in the list are p-restricted and will appear in the corresponding list of G(𝔽_{p}) unchanged. The first non-restricted weight is

(p + 4, p − 4) = p(1, 0) + (4, p − 4) = p(1, 0) + (s_{α_{1}+α_{2}} · (0, 0) + p(0, 1)).

Here u ∈ W̃_{p} consists of the reflection s_{α_{1}+α_{2}} followed by the translation by p(0, 1) and ν = (1, 0). Under Theorem 7.5 the weight corresponding to (p+4, p−4) = u·(0, 0)+p(1, 0) is

u · (1, 0) = s_{α_{1}+α_{2}} · (1, 0) + p(0, 1) = (6, p − 5).

Similarly one obtains from

(3, 2p − 2) = p(0, 1) + ((s_{α_{2}} · (0, 0) + p(0, 1)),

the corresponding weight

(6, p − 3) = s_{α_{2}} · (0, 1) + p(0, 1),

and from

(2p − 2, 1) = p(1, 0) + (s_{α_{1}} · (0, 0) + p(1, 0)),

the corresponding weight

(p − 3, 2) = s_{α_{1}} · (1, 0) + p(1, 0).

We conclude that H^{1}(G(𝔽_{p}), L(λ)) = 0 for all p-restricted weights λ, unless λ is one of the following list:

{(p − 5, 0), (3, p − 2), (6, p − 5), (6, p − 3), (p − 3, 2)}.

For these weights dim_{k} H^{1}(G(𝔽_{p}), L(λ)) = 1.

For a complete list of Ext^{1} between simple modules for type G_{2} and q = p ≥ 13 see [Y].

7.6 The following theorem will give a sufficient condition for the semisimplicity of G_{s}(L(μ)). As a consequence we are able to prove a stronger version of [And2, Theorem 3.2]. Following Andersen, we call a weight, μ, m-deep if it has distance at least m from the nearest alcove wall (i.e., if p divides ⟨μ + ρ, α_{0}^{∨}⟩ + c for some c ∈ ℤ, then |c| ≥ m).

Theorem . Let s ≥ 1, p ≥ (2s + 1)(h − 1), and let μ ∈ X_{r}(T) be 2s(h − 1)-deep. Then G_{s}(L(μ)) is semisimple. Moreover, for any L ∈ C_{s} that has only p^{r}-restricted composition factors in its head (as G-module), one obtains for 0 ≤ i ≤ s the following isomorphism of vector spaces:

Ext_{G(𝔽_{q})}^{i}(L, L(μ)) ≅ ⨁_{ν∈π_{s}} Hom_{G}(L(ν_{0}) ⊗ L(ν_{1}), L(μ)) ⊗ Ext_{G}^{i}(L, L(ν)).

Proof. We will show that G_{s}(L(μ)) is semisimple. Arguing as in the proof of Theorem 7.4 we have for any weight ν = ν_{0} + p^{r}ν_{1} ∈ π_{s}:

[G_{s}(L(μ)) : L(ν)] ≤ dim_{k} Hom_{G(𝔽_{q})}(Q_{r}(ν_{0}), L(μ) ⊗ L(−w_{0}ν_{1})).

ν ∈ π_{s} and μ has distance at least 2s(h − 1) from the closest alcove wall. Hence, the tensor product L(μ) ⊗ L(−w_{0}ν_{1}) is semisimple (both as a G-module and a G(𝔽_{q})-module) and all summands have p^{r}-restricted highest weight. Let [Q_{r}(ν_{0}) : U_{r}(γ)] denote the multiplicity of the projective indecomposable G(𝔽_{q})-module U_{r}(γ) as a summand of Q_{r}(ν_{0}). Then

[G_{s}(L(μ)) : L(ν)] ≤ ∑_{γ∈X(T)_{+}}[Q_{r}(ν_{0}) : U_{r}(γ)] · dim_{k} Hom_{G}(L(γ), L(μ) ⊗ L(−w_{0}ν_{1})).

On the other hand the multiplicity of L(ν) in the G-socle of G_{s}(L(μ)) is given by

dim_{k} Hom_{G(𝔽_{q})}(L(ν_{0}), L(μ) ⊗ L(−w_{0}ν_{1})) = dim_{k} Hom_{G}(L(ν_{0}), L(μ) ⊗ L(−w_{0}ν_{1})).

Therefore it is sufficient to prove the following:

[Q_{r}(ν_{0}) : U_{r}(γ)] · dim_{k} Hom_{G}(L(γ), L(μ) ⊗ L(−w_{0}ν_{1})) is zero unless γ = ν_{0}.

Let us assume that there exists γ ≠ ν_{0} with

[Q_{r}(ν_{0}) : U_{r}(γ)] · dim_{k} Hom_{G}(L(γ), L(μ) ⊗ L(−w_{0}ν_{1})) ≠ 0.

By [Jan3] or [Cha] there exists a dominant weight σ with ⟨σ, α_{0}^{∨}⟩ ≤ h − 1 such that [L(γ) ⊗ L(σ) : L(ν_{0} + p^{r}σ)]_{G} ≠ 0. One obtains immediately:

⟨γ + ρ, α_{0}^{∨}⟩ ≥ ⟨γ + σ, α_{0}^{∨}⟩ ≥ p^{r}⟨σ, α_{0}^{∨}⟩.  (7.6.1)

At the same time

[L(γ) ⊗ L(σ) : L(ν_{0} + p^{r}σ)]_{G} = dim_{k} Hom_{G}(L(γ) ⊗ L(σ), Q_{r}(ν_{0}) ⊗ L(σ)^{(r)})
= dim_{k} Hom_{G}(L(ν_{0} + p^{r}(−w_{0}σ)), Q_{r}(ν_{0}) ⊗ L(−w_{0}σ)).

Here we were using the fact that Q_{r}(ν_{0}) ⊗ L(σ)^{(r)}) is the injective hull of L(ν_{0} + p^{r}σ) in the category C_{s} (see the remark after Corollary 4.7). Should ν_{0} have distance greater than ⟨σ, α_{0}^{∨}⟩ from the nearest alcove wall then an easy dimension argument shows that the G-socle of Q_{r}(ν_{0}) ⊗ L(−w_{0}σ) contains only p^{r}-restricted weights (see [P3, Lemma 5.1(b)]). This is impossible. Thus, the weights ν_{0} and ν have distance at most ⟨σ, α_{0}^{∨}⟩ from the closest alcove wall. Our assumption implies that L(ν) is a composition factor of G_{s}(L(μ)). Its linkage class has to be represented in the socle of G_{s}(L(μ)). All weights in a linkage class have the same distance from the closest wall. Therefore, there exists a weight κ = κ_{0} + p^{r}κ_{1} in π_{s} with dim_{k} Hom_{G}(L(κ_{0}), L(μ) ⊗ L(−w_{0}κ_{1})) ≠ 0. κ_{0} has distance at most ⟨σ, α_{0}^{∨}⟩ from the nearest alcove wall while μ has distance at least 2s(h − 1) from the closest wall. Therefore,

⟨κ_{1}, α_{0}^{∨}⟩ ≥ 2s(h − 1) − ⟨σ, α_{0}^{∨}⟩.  (7.6.2)

Of course the fact κ ∈ π_{s} gives the upper bound ⟨κ_{1}, α_{0}^{∨}⟩ < 2s(h − 1). The distance of μ from the alcove walls forces the weights μ, γ, and κ_{0} to be p-regular and lie inside the same alcove. Therefore, there exists an integer b with

(b − 1)p^{r} < ⟨ξ + ρ, α_{0}^{∨}⟩ < bp^{r} for all ξ ∈ {μ, γ, κ_{0}}.

Now (7.6.1) implies that ⟨σ, α_{0}^{∨}⟩ < b. Together with (7.6.2) one obtains that ⟨κ_{1}, α_{0}^{∨}⟩ > 2s(h − 1) − b, or equivalently ⟨κ_{1}, α_{0}^{∨}⟩ ≥ 2s(h − 1) − b + 1.

But now

p^{r}(b − 1) + p^{r}(2s(h − 1) − b + 1) < ⟨κ_{0} + ρ + p^{r}κ_{1}, α_{0}^{∨}⟩ = ⟨κ + ρ, α_{0}^{∨}⟩ < 2sp^{r}(h − 1)

gives the desired contradiction. _{□}

7.7 By the tensor identity we have ind_{G(𝔽_{q})}^{G}L(μ) ≅ L(μ) ⊗ ind_{G(𝔽_{q})}^{G}k. The truncation of L(μ) ⊗ G_{s}(k) in C_{s} is therefore contained in G_{s}(L(μ)). For G_{s}(L(μ)) to be semisimple it is therefore necessary that the truncation of L(μ)⊗G_{s}(k) be semisimple. For p ≥ (2s+1)(h−1), it follows from Theorem 7.4 that

G_{s}(k) ≅ ⨁_{ν∈X[s]} L((p^{r} − w_{0})ν).

The truncation of L(μ) ⊗ G_{s}(k) will therefore contain the tensor products of the form L(μ) ⊗ L(−w_{0}ν) where μ + p^{r}ν ∈ π_{s}. Define b such that (b − 1)p^{r} < ⟨μ + ρ, α_{0}^{∨}⟩ < bp^{r}. It becomes necessary for μ to have distance at least 2s(h − 1) − b from any upper alcove wall. Otherwise, the tensor product would result in a translation across an upper wall and destroy the semisimplicity of G_{s}(L(μ)).

One can actually show that if G_{s}(L(μ)) satisfies the conditions of the previous theorem then G_{s}(L(μ)) is a submodule of the semisimple module L(μ) ⊗ G_{s}(k).

Example . Let G = SL(2, k), G(𝔽_{q}) = SL(2, q) and p ≥ 3. Then

(a) G_{1}(k) ≅ k ⊕ L(q + 1).

(b) If μ < q − 2, then G_{1}(L(μ)) ≅ L(μ) ⊗ G_{1}(k) ≅ L(μ) ⊕ (L(μ) ⊗ L(q + 1)).

(c) G_{1}(L(q − 2)) ≅ L(q − 2) ⊕ L(2q − 3).

(d) G_{1}(St_{r}) ≅ St_{r} ⊕ H^{0}(2q − 2). Notice that for SL(2, k) the module G_{1}(L(μ)) is semisimple if and only if μ is p-regular.

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Department of Mathematics and Statistics and Computer Science, University of Wisconsin, Stout, Menomonie, WI 54751, USA

E-mail address: bendelc@UWSTOUT.EDU

Department of Mathematics and Statistics, Utah State University, Logan, UT 84322-3900, USA

E-mail address: nakano@sunfs.math.usu.edu

Department of Mathematics and Statistics, University of South Alabama, Mobile, AL 36688, USA

E-mail address: pillen@mathstat.usouthal.edu