EXTENSIONS FOR FINITE CHEVALLEY GROUPS I.

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

1 Introduction

1.1 Let G be a semisimple simply connected algebraic group scheme defined and split over the field 𝔽_{p} with p elements, and k be the algebraic closure of 𝔽_{p}. Moreover, let G(𝔽_{q}) be the finite Chevalley group consisting of 𝔽_{q}-rational points of G where q = p^{r} for a non-negative integer r. Let G_{r} denote the rth Frobenius kernel of G. For any rational G-module V, let V^{(l)} denote the rational G-module obtained by l twists by the Frobenius endomorphism. In 1977, Cline, Parshall, Scott and van der Kallen [CPSK] proved an important stability result which relates the rational cohomology of G with the cohomology of the finite group G(𝔽_{q}). More precisely, they show that for any fixed non-negative integer n, if V is a finite dimensional rational G-module, then H^{n}(G, V^{(l)}) ≅ H^{n}(G(𝔽_{q}), V^{(l)}) ≅ H^{n}(G(𝔽_{q}), V) for sufficiently large r and l. In other words, the restriction map H^{n}(G, V^{(l)}) → H^{n}(G(𝔽_{q}), V^{(l)}) is an isomorphism for large r and l.

The purpose of this paper is to give explicit formulas for the extensions between simple modules of finite Chevalley groups. It is well-known that the simple G-modules are indexed in a natural way by the set of dominant weights X(T)_{+}. For each λ ∈ X(T)_{+}, let L(λ) be the corresponding simple module. Let X_{r}(T) be the set of p^{r}-restricted weights in X(T)_{+}. For λ ∈ X_{r}(T), L(λ) remains simple upon restriction to G(𝔽_{q}) and G_{r}. In fact a complete set of simple modules for G(𝔽_{q}) and G_{r} are obtained in this way.

We will now describe the main results in this paper. In Section 2 we derive the following Ext^{1}-formula. Let p ≥ 3(h − 1) where h is the Coxeter number associated with the root system for G. For λ, μ ∈ X_{r}(T),

Ext_{G(𝔽_{q})}^{1}(L(λ), L(μ)) ≅ ⨁_{ν∈Γ} Ext_{G}^{1}(L(λ) ⊗ L(ν)^{(r)}, L(μ) ⊗ L(ν))  (1.1.1)

where Γ = {ν ∈ X(T)_{+}|⟨ν, α_{0}^{∨}⟩ ≤ h − 1}. The proof relies on previous work by the authors [BNP] that relates extensions between Mod(G(𝔽_{q})), Mod(G), and Mod(G_{r}) via spectral sequences. Here Mod(G(𝔽_{q})) denotes the category of kG(𝔽_{q})-modules, Mod(G) denotes the category of rational G-modules, and Mod(G_{r}) denotes the category of rational G_{r}-modules.

In Section 3, this formula is refined in the case when r ≥ 2. For σ ∈ X_{r}(T), let σ = σ̂ + σ_{r−1}p^{r−1} where σ̂ ∈ X_{r−1}(T). One obtains for p ≥ 3(h − 1) and λ, μ ∈ X_{r}(T):

Ext_{G(𝔽_{q})}^{1}(L(λ), L(μ)) ≅ Ext_{G}^{1}(L(λ), L(μ)) ⊕ R  (1.1.2)

Date: December 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.

where

R = ⨁_{ν∈Γ−{0}} Hom_{G}(L(ν), Ext_{G_{1}}^{1}(L(λ_{r−1}), L(μ_{r−1}))^{(−1)}) ⊗ Hom_{G}(L(λ̂)), L(μ̂) ⊗ L(ν)).

(1.1.3)

Even though this previous result is valid only for r ≥ 2, it is interesting to note that this formula involves the computation of extensions for simple modules over G_{1}. Recall that the rational representations of G_{1} are equivalent to the restricted representations of Lie(G). As a consequence of this result we can show that Ext_{G(𝔽_{q})}^{1}(L(λ), L(μ)) = 0 if Ext_{G}^{1}(L(λ), L(μ)) = 0 with an additional “distance” or “size” condition on the weights λ and μ.

The next section is devoted to looking at the situation when r = 1. In this situation we also can show that the computation of G(𝔽_{p})-extensions of simple modules reduces to the computation of extensions in the categories of G and G_{1}-modules. More precisely, if p ≥ 3(h − 1) and λ, μ ∈ X_{1}(T) then

Ext_{G(𝔽_{p})}^{1}(L(λ), L(μ)) ≅ Ext_{G}^{1}(L(λ), L(μ)) ⊕ R  (1.1.4)

where

R = ⨁_{ν∈Γ−{0}} Hom_{G}(L(ν), Ext_{G_{1}}^{1}(L(λ), L(μ) ⊗ L(ν))^{(−1)}).  (1.1.5)

The formulas given in (1.1.2) and (1.1.4) can be regarded as concrete realizations of the aforementioned stability result given in [CPSK]. These formulas have the advantage that the “remainder” R is explicitly described and indicates how other components (aside from the rational cohomology) contribute to the finite group cohomology.

These formulas are also used to answer long standing questions involving the existence of self-extensions for finite Chevalley groups. In 1985, J.E. Humphreys [Hum] speculated about the existence of self-extensions for G(𝔽_{q}). In particular, he showed that for r = 1 and for the root system of type C_{2}, there is an interesting family of self-extensions (even for large primes). Here we show, as a corollary to the Ext^{1}-formula (1.1.2), that there are no self-extensions for simple G(𝔽_{q})-modules, for r ≥ 2 and p ≥ 3(h − 1). That is, Ext_{G(𝔽_{q})}^{1}(L(λ), L(λ)) = 0 for all λ ∈ X_{r}(T) with r ≥ 2 and p ≥ 3(h − 1).

The determination of self-extensions is much more subtle in the r = 1 case. In Section 4, a careful analysis of line bundle cohomology is used to show that for λ ∈ X_{1}(T), Ext_{G(𝔽_{p})}^{1}(L(λ), L(λ)) = 0 with a few exceptions on λ when the root system is of type A_{1} or C_{n}. Our results provide verification of many of the observations in Humphreys’ paper. In fact, our proofs clearly demonstrate how his family of self-extensions naturally arises. We should remark that H.H. Andersen [And1] earlier proved that self-extensions occur for simple G_{r}-modules if p = 2 and the underlying root system is of type C_{n}. Our methods involve translating the question of self-extensions to computations in the G and G_{1}-categories. Once this takes place, we apply Andersen’s result as well as other computations in his paper to handle several important cases. In the final section of the paper, the self-extensions results are used to obtain criterion for the semisimplicity of finite-dimensional G(𝔽_{q})-modules.

1.2 Notation: Throughout this paper G is a semisimple simply connected algebraic group defined and split over the finite field 𝔽_{p} with p elements. The field k is the algebraic closure of 𝔽_{p} and G will be considered as an algebraic group scheme over k. 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 ⟨ , ⟩. 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∈Δ} ℕα. Let α^{∨} = 2α/⟨α, α⟩ be the coroot corresponding to α ∈ Φ and α_{0}^{∨} denote the coroot with α_{0} being the highest short root. Moreover, let ρ be the half sum of positive roots and w_{0} denote the long element of the Weyl group. The Coxeter number associated to Φ is h = ⟨ρ, α_{0}^{∨}⟩ + 1. 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 simple modules for G are labelled by the set X(T)_{+} and denoted by L(λ), λ ∈ X(T)_{+}. A complete set of non-isomorphic simple G_{r}-modules and simple G(𝔽_{q})-modules are obtained by taking {L(λ) : λ ∈ X_{r}(T)}.

1.3 Acknowledgments: 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 parts of this paper were written.

2 General Ext^{1}-formula

2.1 Let C be the full subcategory of Mod(G) with objects having composition factors whose high weight lie in π where

π = {λ ∈ X(T)_{+} : ⟨λ + ρ, β^{∨}⟩ < 2p^{r}⟨ρ, β^{∨}⟩ for β short and β ∈ Φ^{+}}.

The category C coincides with the p^{r}-bounded category as defined in [Jan1, p. 360]. Let F_{C} be the truncation functor from Mod(G) to Mod(C) which takes M ∈ Mod(G) to the largest submodule of M in C. Note that this functor takes finite-dimensional modules to finitedimensional modules. Let G = F ∘ ind_{G(𝔽_{q})}^{G} and R^{j}G be the higher right derived functors of G. For M ∈ C and N ∈ Mod(G(𝔽_{q})) there exists a first quadrant spectral sequence [BNP, Thm. 4.4a]

E_{2}^{i,j} = Ext_{G}^{i}(M, R^{j}G(N)) ⇒ Ext_{G(𝔽_{q})}^{i+j}(M, N).  (2.1.1)

By analyzing this spectral sequence the following result [BNP, Cor. 5.3i] was obtained which relates extensions of simple modules for G(𝔽_{q}) with extensions in Mod(G).

(2.1.2) If p ≥ 2(h − 1) and λ, μ ∈ X_{r}(T) then Ext_{G(𝔽_{q})}^{1}(L(λ), L(μ)) ≅ Ext_{G}^{1}(L(λ), G(L(μ))).

Interestingly enough, one can improve this result for all primes by imposing a condition on the high weights of the simple modules [BNP, Thm. 5.5a].

(2.1.3) If λ, μ ∈ X_{r}(T) with μ ≯ λ then Ext_{G(𝔽_{q})}^{1}(L(λ), L(μ)) ≅ Ext_{G}^{1}(L(λ), G(L(μ))).

2.2 The result below is an improved version of statement (2.1.3) because the image G(k) is much easier to describe than G(L(μ)) for μ arbitrary. Indeed, for p ≥ 3(h−1), the module G(k) was seen in [BNP, Thm. 7.4] to be semisimple.

Theorem . Let λ, μ ∈ X_{r}(T). Then Ext_{G(𝔽_{q})}^{1}(L(λ), L(μ)) ≅ Ext_{G}^{1}(L(λ), L(μ) ⊗ G(k)).

Proof. Without loss of generality we may assume that μ ≯ λ because Ext_{G(𝔽_{q})}^{1}(L(λ), L(μ)) ≅ Ext_{G(𝔽_{q})}^{1}(L(μ), L(λ)). According to [P1, Lem. 1.4] we have 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(σ) whose p^{r}-restricted highest weights satisfy σ ≥ λ.

Let M = St_{r} ⊗ L((p^{r} − 1)ρ + w_{0}λ). In particular it follows that if μ ≠ λ (and hence μ ≱ λ), then

Hom_{G}(M ⊗ L(−w_{0}μ), k) ≅ Hom_{G}(M, L(μ))
⊆ Hom_{G(𝔽_{q})}(M, L(μ))
≅ Hom_{G(𝔽_{q})}(M ⊗ L(−w_{0}μ), k)
= 0.

On the other hand,

Hom_{G}(M ⊗ L(−w_{0}λ), k) ≅ Hom_{G}(M, L(λ))
≅ Hom_{G(𝔽_{q})}(M, L(λ))
≅ Hom_{G(𝔽_{q})}(M ⊗ L(−w_{0}λ), k)
≅ k.

Therefore, we have the following short exact sequence of G-modules

0 → R → St_{r} ⊗ L((p^{r} − 1)ρ + w_{0}λ) ⊗ L(−w_{0}μ) → L(λ) ⊗ L(−w_{0}μ) → 0. (2.2.1)

Note that this is a short exact sequence in the category C because μ ≯ λ implies that St_{r} ⊗ L((p^{r} − 1)ρ + w_{0}λ) ⊗ L(−w_{0}μ) is in C. 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(−w_{0}μ), k) → Hom_{G(𝔽_{q})}(St_{r} ⊗ L((p^{r} − 1)ρ + w_{0}λ) ⊗ L(−w_{0}μ), k)
→ Hom_{G(𝔽_{q})}(R, k) → Ext_{G(𝔽_{q})}^{1}(L(λ) ⊗ L(−w_{0}μ), k) → 0.

If μ ≠ λ, we have from above that

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

Consequently, Hom_{G(𝔽_{q})}(L(λ) ⊗ L(−w_{0}μ), k) = 0 and

Hom_{G(𝔽_{q})}(R, k) ≅ Ext_{G(𝔽_{q})}^{1}(L(λ) ⊗ L(−w_{0}μ), k) ≅ Ext_{G(𝔽_{q})}^{1}(L(λ), L(μ)). (2.2.2)

By adjointness we then have

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

Similarly, for μ = λ it follows that

k ≅ Hom_{G(𝔽_{q})}(L(λ) ⊗ L(−w_{0}λ), k) ≅ Hom_{G}(L(λ) ⊗ L(−w_{0}λ), G(k))

and

k ≅ Hom_{G(𝔽_{q})}(M ⊗ L(−w_{0}λ), k) ≅ Hom_{G}(M ⊗ L(−w_{0}λ), G(k))

In either case, from the short exact sequence (2.2.1) one also obtains the following long exact sequence:

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

From the long exact sequence and the preceding isomorphisms, we get an injective map

Hom_{G}(R, G(k)) → Ext_{G}^{1}(L(λ) ⊗ L(−w_{0}μ), G(k)).

But, R ∈ C so by adjointness we have

Hom_{G(𝔽_{q})}(R, k) ≅ Hom_{G}(R, G(k)).

It follows from the isomorphism in (2.2.2) that there exists an injective map

Ext_{G(𝔽_{q})}^{1}(L(λ) ⊗ L(−w_{0}μ), k) → Ext_{G}^{1}(L(λ) ⊗ L(−w_{0}μ), G(k)).

The five term exact sequence of the spectral sequence in (2.1.1) 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(𝔽_{q})}^{1}(L(λ) ⊗ L(−w_{0}μ), k)
≅ Ext_{G}^{1}(L(λ) ⊗ L(−w_{0}μ), G(k))
≅ Ext_{G}^{1}(L(λ), L(μ) ⊗ G(k)).

□

In Theorem 2.5, we will use the decomposition of G(k) from [BNP] to further identify Ext_{G(𝔽_{q})}^{1}(L(λ), L(μ)). We first present some technical results which will simplify this identification.

2.3 For a rational G-module M, the G-socle of M is contained in the G_{r}-socle of M (with M considered as a G_{r}-module via restriction). We now provide a condition under which the socles are in fact equal.

Proposition . Let p ≥ 2(h − 1) and M be a finite-dimensional rational G-module that is p^{r}-bounded.

(a) If the G-socle of M contains only p^{r}-restricted highest weights, then soc_{G} M = soc_{G_{r}} M.

(b) If λ, μ, ν ∈ X_{r}(T) with ⟨ν, α_{0}^{∨}⟩ < p^{r} then

Hom_{G}(L(λ), L(μ) ⊗ L(ν)) ≅ Hom_{G_{r}}(L(λ), L(μ) ⊗ L(ν)).

Proof. (a) By assumption, soc_{G} M ≅ ⊕_{i}L(σ_{i}) for some σ_{i} which are p^{r}-restricted. The condition on p implies that that the injective hull (and projective cover) Q_{r}(σ_{i}) of each L(σ_{i}) over G_{r} admits the structure of a G-module. Further, each Q_{r}(σ_{i}) is injective in the p^{r}-bounded category (cf. [Jan1, II 11.11]). Hence, there is an embedding of G-modules M ↪ ⊕_{i}Q_{r}(σ_{i}) which is an isomorphism on G-socles. Recall that soc_{G} Q_{r}(σ_{i}) = L(σ_{i}) = soc_{G_{r}}(Q_{r}(σ_{i})) and hence one obtains the assertion via

soc_{G_{r}} M ⊆ ⊕_{i} soc_{G_{r}} Q_{r}(σ_{i}) = ⊕_{i} soc_{G} Q_{r}(σ_{i}) ≅ soc_{G} M ⊆ soc_{G_{r}} M.

(b) It was shown in [BK, Theorem B] that the G-socle of L(μ)⊗L(ν) contains only simple modules with p^{r}-restricted highest weights. The assertion follows immediately from part

(a) because L(μ) ⊗ L(ν) is p^{r}-bounded. _{□}

2.4 The following proposition is a generalization of Lemma 5.1 and Proposition 5.5 in [And1]. It will be used to simplify the formula in Theorem 2.5.

Proposition . Assume p ≥ 3(h − 1), λ, μ ∈ X_{r}(T), and ν_{1}, ν_{2} ∈ X(T)_{+} satisfy ⟨ν_{i}, α_{0}^{∨}⟩ < 2(h − 1), for i = 1, 2. Then the following hold:

(a) Ext_{G}^{1}(L(λ) ⊗ L(ν_{1})^{(r)}, L(μ) ⊗ L(ν_{2})) ≅ Hom_{G/G_{r}}(L(ν_{1})^{(r)}, Ext_{G_{r}}^{1}(L(λ), L(μ) ⊗ L(ν_{2})).

(b) If p^{r}ξ is a weight of Ext_{G_{r}}^{1}(L(λ), L(μ) ⊗ L(ν_{2})) then ⟨ξ, α_{0}^{∨}⟩ ≤ h − 1. In particular, Ext_{G_{r}}^{1}(L(λ), L(μ) ⊗ L(ν_{2})) is semi-simple.

(c) If Ext_{G}^{1}(L(λ) ⊗ L(ν_{1})^{(r)}, L(μ) ⊗ L(ν_{2})) ≠ 0, then ⟨ν_{1}, α_{0}^{∨}⟩ ≤ h − 1.

Proof. (a) The Lyndon-Hochschild-Serre exact sequence for G_{r} ⊴ G yields

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

The first four terms of the five term exact sequence are

0 → Ext_{G/G_{r}}^{1}(L(ν_{1})^{(r)}, Hom_{G_{r}}(L(λ), L(μ) ⊗ L(ν_{2})))
→ Ext_{G}^{1}(L(λ) ⊗ L(ν_{1})^{(r)}, L(μ) ⊗ L(ν_{2}))
→ Hom_{G/G_{r}}(L(ν_{1})^{(r)}, Ext_{G_{r}}^{1}(L(λ), L(μ) ⊗ L(ν_{2})))
→ Ext_{G/G_{r}}^{2}(L(ν_{1})^{(r)}, Hom_{G_{r}}(L(λ), L(μ) ⊗ L(ν_{2})))

From Proposition 2.3b, for i = 1 or 2, we have isomorphisms

Ext_{G/G_{r}}^{i}(L(ν_{1})^{(r)}, Hom_{G_{r}}(L(λ), L(μ) ⊗ L(ν_{2}))) ≅ Ext_{G/G_{r}}^{i}(L(ν_{1})^{(r)}, Hom_{G}(L(λ), L(μ) ⊗ L(ν_{2})))
≅ Ext_{G/G_{r}}^{i}(L(ν_{1})^{(r)}, k) ⊗ Hom_{G}(L(λ), L(μ) ⊗ L(ν_{2}))
≅ Ext_{G}^{i}(L(ν_{1}), k) ⊗ Hom_{G}(L(λ), L(μ) ⊗ L(ν_{2})).

The size of p forces ν_{1} to be in the closure of the bottom alcove, which implies Ext_{G}^{i}(L(ν), k) = 0 and hence from the exact sequence, there is an isomorphism

Ext_{G}^{1}(L(λ) ⊗ L(ν_{1})^{(r)}, L(μ) ⊗ L(ν_{2})) ≅ Hom_{G/G_{r}}(L(ν_{1})^{(r)}, Ext_{G_{r}}^{1}(L(λ), L(μ) ⊗ L(ν_{2}))).

(b) From the proof of Proposition 2.3a, there is an embedding of G-modules L(μ) ⊗ L(ν_{2}) ↪ ⊕_{i}Q_{r}(σ_{i}) for some (finite) set of p^{r}-restricted weights σ_{i}. Therefore, we can define a G-module M via the exact sequence

0 → L(μ) ⊗ L(ν_{2}) → ⊕_{i}Q_{r}(σ_{i}) → M → 0.

Consider the associated long exact sequence

⋯ → Hom_{G_{r}}(L(λ), ⊕_{i}Q_{r}(σ_{i})) → Hom_{G_{r}}(L(λ), M)
→ Ext_{G_{r}}^{1}(L(λ), L(μ) ⊗ L(ν_{2})) → Ext_{G_{r}}^{1}(L(λ), ⊕_{i}Q_{r}(σ_{i})) → ….

Using the injectivity of Q_{r}(σ_{i}) we obtain a surjection

Hom_{G_{r}}(L(λ), M) → Ext_{G_{r}}^{1}(L(λ), L(μ) ⊗ L(ν_{2})).

Now assume that p^{r}ξ is a weight of the G-module Ext_{G_{r}}^{1}(L(λ), L(μ)⊗L(ν_{2})). Then it is also a weight of the G-module Hom_{G_{r}}(L(λ), M) and λ+p^{r}ξ must be a weight of M. The highest weight of each Q_{r}(σ_{i}) is 2(p^{r} − 1)ρ + ω_{0}σ_{i} while σ_{i} ≤ μ + ω_{0}ν_{2}. The highest weight of M is therefore less than or equal to 2(p^{r} −1)ρ+ω_{0}μ+ν_{2}. We obtain λ+p^{r}ξ ≤ 2(p^{r} −1)ρ+ω_{0}μ+ν_{2} or

p^{r}ξ ≤ 2(p^{r} − 1)ρ − λ + ω_{0}μ + ν_{2}.  (2.4.1)

On the other hand, L(μ) ⊗ L(ν_{2}) has a filtration of simple modules with not necessarily p^{r}-restricted highest weights γ = γ_{0} + p^{r}γ_{1} with γ ≤ μ + ν_{2}. From [And2, Lemma 2.3] we obtain the following: for any γ and any highest weight p^{r}ξ of Ext_{G_{r}}^{1}(L(λ), L(γ_{0})⊗L(γ_{1})^{(r)}) ≅ Ext_{G_{r}}^{1}(L(λ), L(γ_{0})) ⊗ L(γ_{1})^{(r)} there exists a simple root α with p^{r}ξ ≤ −ω_{0}λ + γ_{0} + p^{r}γ_{1} + p^{r−1}α ≤ −ω_{0}λ + μ + ν_{2} + p^{r−1}α. For any weight p^{r}ξ of Ext_{G_{r}}^{1}(L(λ), L(μ) ⊗ L(ν_{2})) we obtain the inequality

p^{r}ξ ≤ −ω_{0}λ + μ + ν_{2} + p^{r−1}α.  (2.4.2)

Adding equations (2.4.1) and (2.4.2) yields

2p^{r}ξ ≤ 2(p^{r} − 1)ρ − λ − ω_{0}λ + μ + ω_{0}μ + 2ν_{2} + p^{r−1}α.

Taking the inner product with α_{0} and dividing both sides by 2 yields p^{r}⟨ξ, α_{0}^{∨}⟩ ≤ p^{r}(h−1)+ (h − 1) + p^{r−1}. Since p ≥ 3(h − 1) we have ⟨ξ, α_{0}^{∨}⟩ ≤ (h − 1), Moreover, Ext_{G_{r}}^{1}(L(λ), L(μ) ⊗ L(ν_{2})) is isomorphic to the rth Frobenius twist of a G-module whose highest weights are all inside the lowest alcove. Hence, it is semi-simple.

(c) This follows immediately from parts (a) and (b). _{□}

2.5 For large primes, we obtain an Ext^{1}-formula between simple G(𝔽_{q})-modules. Set Γ_{s} = {ν ∈ X(T)_{+}|⟨ν, α_{0}^{∨}⟩ < s} and Γ = Γ_{h}.

Theorem . For p ≥ 3(h − 1) and λ, μ ∈ X_{r}(T),

(a) Ext_{G(𝔽_{q})}^{1}(L(λ), L(μ)) ≅ ⨁_{ν∈Γ} Ext_{G}^{1}(L(λ) ⊗ L(ν)^{(r)}, L(μ) ⊗ L(ν)).

(b) In particular for self-extensions, we have

Ext_{G(𝔽_{q})}^{1}(L(λ), L(λ)) ≅ ⨁_{ν∈Γ} Ext_{G}^{1}(L(λ) ⊗ L(ν)^{(r)}, L(λ) ⊗ L(ν)).

Proof. According to [BNP, Thm. 7.4], if p ≥ 3(h − 1), the module G(k) is semisimple and

G(k) = ⨁_{ν∈Γ_{2h−1}} L(ν − ω_{0}p^{r}ν).

Hence, from Theorem 2.2, Ext_{G(𝔽_{q})}^{1}(L(λ), L(μ)) may be identified in terms of a direct sum of G-extensions of the form

Ext_{G}^{1}(L(λ), L(μ) ⊗ L(ν − ω_{0}p^{r}ν)) ≅ Ext_{G}^{1}(L(λ), L(μ) ⊗ L(ν) ⊗ L(−ω_{0}p^{r}ν))
≅ Ext_{G}^{1}(L(λ) ⊗ L(p^{r}ν), L(μ) ⊗ L(ν))
≅ Ext_{G}^{1}(L(λ) ⊗ L(ν)^{(r)}, L(μ) ⊗ L(ν)).

Therefore, by Proposition 2.4c,

Ext_{G(𝔽_{q})}^{1}(L(λ), L(μ)) ≅ ⨁_{ν∈Γ_{2h−1}} Ext_{G}^{1}(L(λ) ⊗ L(ν)^{(r)}, L(μ) ⊗ L(ν))
≅ ⨁_{ν∈Γ_{h}} Ext_{G}^{1}(L(λ) ⊗ L(ν)^{(r)}, L(μ) ⊗ L(ν)).
□

If either λ or μ are (h − 1)-deep inside an alcove, the formula in part (a) of the Theorem immediately implies Theorem 3.2 in [And2] with significantly weaker deepness conditions (see also [BNP, Thm. 7.6]). It explains the observations by Andersen and Ye in [Y] that Andersen’s formula seemed to work for weights much closer to the alcove wall than predicted. It is also interesting to note that our Ext^{1}-formula has similarities to Jantzen’s [Jan2] and Chastkofsky’s [Cha] formula for decomposing projective indecomposable G_{r}-modules as modules over G(𝔽_{q}):

[Q_{r}(λ) : U_{r}(μ)] = ∑_{ν∈Γ}[L(μ) ⊗ L(ν) : L(λ) ⊗ L(ν)^{(r)}]_{G}.

Here U_{r}(μ) denotes the projective cover of L(μ) has a kG(𝔽_{q})-module.

3 Extensions for r ≥ 2

3.1 In this section, we strengthen Theorem 2.5 in the case that r ≥ 2 (see Theorem 3.2) and use this to show the vanishing of self-extensions in Section 3.4. Throughout this section we use the following notation. Any σ ∈ X_{r}(T) may be expressed as σ = σ_{0} + σ_{1}p + ⋯ + σ_{r−1}p^{r−1} where σ_{i} ∈ X_{1}(T) for 0 ≤ i ≤ r −1. For r ≥ 2, set σ̂ = σ_{0} +σ_{1}p+⋯+σ_{r−2}p^{r−2} so that σ = σ̂ + σ_{r−1}p^{r−1}. The following technical result will be used in the proof of Theorem 3.2.

Proposition . Let p ≥ 3(h − 1), λ, μ ∈ X_{r}(T), and ν_{1}, ν_{2} ∈ X(T)_{+}. If r ≥ 2, ν_{1} ≠ 0, and ν_{2} ∈ Γ, then

Hom_{G/G_{r−1}}(L(λ_{r−1})^{(r−1)} ⊗ L(ν_{1})^{(r)}, Ext_{G_{r−1}}^{1}(L(λ̂), L(μ̂) ⊗ L(ν_{2})) ⊗ L(μ_{r−1})^{(r−1)}) = 0.

Proof. Since Hom_{G/G_{r−1}}(L(λ_{r−1})^{(r−1)}⊗L(ν_{1})^{(r)}, Ext_{G_{r−1}}^{1}(L(λ̂), L(μ̂)⊗L(ν_{2}))⊗L(μ_{r−1})^{(r−1)}) ≅ Hom_{G/G_{r−1}}(L(−ω_{0}μ_{r−1})^{(r−1)} ⊗ L(ν_{1})^{(r)}, Ext_{G_{r−1}}^{1}(L(λ̂), L(μ̂) ⊗ L(ν_{2})) ⊗ L(−ω_{0}λ_{r−1})^{(r−1)}), we may assume that ⟨μ_{r−1}, α_{0}^{∨}⟩ ≤ ⟨λ_{r−1}, α_{0}^{∨}⟩.

By Proposition 2.4b, any highest weight ξ of Ext_{G_{r−1}}^{1}(L(λ̂), L(μ̂)⊗L(ν_{2})) satisfies ⟨ξ, α_{0}^{∨}⟩ ≤ p^{r−1}(h − 1). Any highest weight η of Ext_{G_{r−1}}^{1}(L(λ̂), L(μ̂) ⊗ L(ν_{2})) ⊗ L(μ_{r−1})^{(r−1)} therefore satisfies ⟨η, α_{0}^{∨}⟩ ≤ p^{r−1}(⟨μ_{r−1}, α_{0}^{∨}⟩+h−1). By Steinberg’s Tensor Product Theorem, the Gmodule L(λ_{r−1})^{(r−1)}⊗L(ν_{1})^{(r)} may be identified with the simple module L(p^{r−1}λ_{r−1}+p^{r}ν_{1}). Hence, the desired Hom-group is clearly zero unless

⟨p^{r−1}λ_{r−1} + p^{r}ν_{1}, α_{0}^{∨}⟩ ≤ p^{r−1}(⟨μ_{r−1}, α_{0}^{∨}⟩ + h − 1).

Using the assumptions ⟨μ_{r−1}, α_{0}^{∨}⟩ ≤ ⟨λ_{r−1}, α_{0}^{∨}⟩ and ν_{1} ≠ 0, one obtains p^{r} ≤ p^{r}⟨ν_{1}, α_{0}^{∨}⟩ ≤ p^{r−1}(h − 1). Hence, with p ≥ 3(h − 1), one obtains a contradiction. _{□}

3.2 The theorem below shows that Theorem 2.5 can be significantly strengthened for r ≥ 2. In particular, extensions of simple G(𝔽_{q}) modules are determined by G-extensions of the same simple modules, the socle of extensions of simple G_{1}-modules, and decomposition of tensor products of simple G-modules.

Theorem . Let G be a semisimple simply connected algebraic group with p ≥ 3(h − 1). If r ≥ 2 and λ, μ ∈ X_{r}(T) then

Ext_{G(𝔽_{q})}^{1}(L(λ), L(μ)) ≅ Ext_{G}^{1}(L(λ), L(μ)) ⊕ R  (3.2.1)

where

R = ⨁_{ν∈Γ−{0}} Hom_{G}(L(ν), Ext_{G_{1}}^{1}(L(λ_{r−1}), L(μ_{r−1}))^{(−1)}) ⊗ Hom_{G}(L(λ̂), L(μ̂) ⊗ L(ν)).

(3.2.2)

Proof. Without loss of generality we may assume that ⟨μ, α_{0}^{∨}⟩ ≤ ⟨λ, α_{0}^{∨}⟩ because

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

From Theorem 2.5, the remainder term R may be identified as

R = ⨁_{ν∈Γ−{0}} Ext_{G}^{1}(L(λ) ⊗ L(ν)^{(r)}, L(μ) ⊗ L(ν)).

We proceed to identify Ext_{G}^{1}(L(λ) ⊗ L(ν)^{(r)}, L(μ) ⊗ L(ν)) for ν in Γ − {0}. We begin by applying the Lyndon-Hochschild-Serre spectral sequence for G_{r−1} ⊴ G:

E_{2}^{i,j} = Ext_{G/G_{r−1}}^{i}(L(λ_{r−1})^{(r−1)} ⊗ L(ν)^{(r)}, Ext_{G_{r−1}}^{j}(L(λ̂), L(μ̂) ⊗ L(ν)) ⊗ L(μ_{r−1})^{(r−1)})
⇒ Ext_{G}^{i+j}(L(λ) ⊗ L(ν)^{(r)}, L(μ) ⊗ L(ν)).

We have E^{1} = Ext_{G}^{1}(L(λ) ⊗ L(ν)^{(r)}, L(μ) ⊗ L(ν)). By Proposition 3.1, it follows that E_{2}^{0,1} = 0 and hence E^{1} ≅ E_{2}^{1,0} (from the beginning of the corresponding five term exact sequence 0 → E_{2}^{1,0} → E^{1} → E_{2}^{0,1} → E_{2}^{2,0} → E^{2}). Therefore, with Proposition 2.3b, we have the following isomorphisms:

E^{1} ≅ Ext_{G/G_{r−1}}^{1}(L(λ_{r−1})^{(r−1)} ⊗ L(ν)^{(r)}, Hom_{G_{r−1}}(L(λ̂), L(μ̂) ⊗ L(ν)) ⊗ L(μ_{r−1})^{(r−1)})
≅ Ext_{G/G_{r−1}}^{1}(L(λ_{r−1})^{(r−1)} ⊗ L(ν)^{(r)}, Hom_{G}(L(λ̂), L(μ̂) ⊗ L(ν)) ⊗ L(μ_{r−1})^{(r−1)})
≅ Ext_{G/G_{r−1}}^{1}(L(λ_{r−1})^{(r−1)} ⊗ L(ν)^{(r)}, L(μ_{r−1})^{(r−1)}) ⊗ Hom_{G}(L(λ̂), L(μ̂) ⊗ L(ν))
≅ Ext_{G}^{1}(L(λ_{r−1}) ⊗ L(ν)^{(1)}, L(μ_{r−1})) ⊗ Hom_{G}(L(λ̂), L(μ̂) ⊗ L(ν)).

Applying Proposition 2.4a to Ext_{G}^{1}(L(λ_{r−1}) ⊗ L(ν)^{(1)}, L(μ_{r−1})), for any ν ∈ Γ − {0}, we then have

E^{1} ≅ Ext_{G}^{1}(L(λ_{r−1}) ⊗ L(ν)^{(1)}, L(μ_{r−1})) ⊗ Hom_{G}(L(λ̂), L(μ̂) ⊗ L(ν))
≅ Hom_{G/G_{1}}(L(ν)^{(1)}, Ext_{G_{1}}^{1}(L(λ_{r−1}), L(μ_{r−1}))) ⊗ Hom_{G}(L(λ̂), L(μ̂) ⊗ L(ν))
≅ Hom_{G}(L(ν), Ext_{G_{1}}^{1}(L(λ_{r−1}), L(μ_{r−1}))^{(−1)}) ⊗ Hom_{G}(L(λ̂), L(μ̂) ⊗ L(ν)).

The theorem now follows from Theorem 2.5. _{□}

3.3 It was shown in [CPSK, Thm. 7.4] and in [And2, Prop 2.7] that the restriction map in cohomology res : Ext_{G}^{1}(L(λ), L(μ)) → Ext_{G(𝔽_{q})}^{1}(L(λ), L(μ)) is an injective map . In [And2, Rem. 2.10(iii)] Andersen observed that for fixed λ and μ and for sufficiently large r this map is an isomorphism. Our formula in Theorem 3.2 captures this fact. For r large, and λ, μ fixed, λ_{r−1} = μ_{r−1} = 0, thus R = 0. Indeed, from Theorem 3.2, we deduce the following sufficient conditions for extensions between simple G(𝔽_{q})-modules to vanish.

Corollary . Let G be a semisimple simply connected algebraic group and p ≥ 3(h − 1). Moreover, let λ, μ ∈ X_{r}(T) with r ≥ 2. If

(a) λ_{r−1} = μ_{r−1} or

(b) there exists a root α such that |⟨λ̂ − μ̂, α^{∨}⟩| > h − 1, then Ext_{G(𝔽_{q})}^{1}(L(λ), L(μ)) ≅ Ext_{G}^{1}(L(λ), L(μ)). Hence, if in addition Ext_{G}^{1}(L(λ), L(μ)) = 0, then Ext_{G(𝔽_{q})}^{1}(L(λ), L(μ)) = 0.

Proof. If λ_{r−1} = μ_{r−1}, then Ext_{G_{1}}^{1}(L(λ_{r−1}), L(μ_{r−1})) = 0 by [And1, Thm. 4.5]. Hence the remainder term R in Theorem 3.2 vanishes giving the claimed isormorhism. In the second case, for any ν ∈ Γ−{0}, the assumption on λ̂ and ν̂ implies that Hom_{G}(L(λ̂), L(μ̂)⊗L(ν)) = 0. Hence, R = 0 and the result follows. _{□}

3.4 Self-extensions A particular consequence of Corollary 3.3 is that self-extensions do not exist for simple G(𝔽_{q})-modules when r ≥ 2 and p ≥ 3(h − 1).

Corollary . Let G be a semisimple simply connected algebraic group with p ≥ 3(h − 1) and r ≥ 2. For all λ ∈ X_{r}(T), Ext_{G(𝔽_{q})}^{1}(L(λ), L(λ)) = 0.

Proof. Since it is well known that Ext_{G}^{1}(L(λ), L(λ)) = 0 (cf. [Jan1, II 2.12]), this follows immediately from Corollary 3.3a. _{□} 4 Extensions for r = 1

4.1 Throughout this section, assume that G is a simple algebraic group defined and split over 𝔽_{p}. We begin by proving a formula for the extensions of simple modules over G(𝔽_{p}). As in the case when r ≥ 2 (Theorem 3.2), this formula reduces the computation of these extension groups to computing extensions for the algebraic group G and for the first Frobenius kernel G_{1}.

Theorem . Let p ≥ 3(h − 1) and λ, μ ∈ X_{1}(T). Then

Ext_{G(𝔽_{p})}^{1}(L(λ), L(μ)) ≅ Ext_{G}^{1}(L(λ), L(μ)) ⊕ R  (4.1.1)

where

R = ⨁_{ν∈Γ−{0}} Hom_{G}(L(ν), Ext_{G_{1}}^{1}(L(λ), L(μ) ⊗ L(ν))^{(−1)}).  (4.1.2)

Proof. By Theorem 2.5, the remainder term R may be identified as

R = ⨁_{ν∈Γ−{0}} Ext_{G}^{1}(L(λ) ⊗ L(ν)^{(1)}, L(μ) ⊗ L(ν)).

By Proposition 2.4a, we have

Ext_{G}^{1}(L(λ) ⊗ L(ν)^{(1)}, L(λ) ⊗ L(ν)) ≅ Hom_{G}(L(ν), Ext_{G_{1}}^{1}(L(λ), L(μ) ⊗ L(ν))^{(−1)}).
□

4.2 For the remainder of this section, we will analyze self-extensions in the case when r = 1, with the goal of determining when Ext_{G(𝔽_{q})}^{1}(L(λ), L(λ)) = 0. The culminating statement is Theorem 4.10. As above, according to Theorem 2.5b, for p ≥ 3(h − 1) and λ ∈ X_{1}(T),

Ext_{G(𝔽_{p})}^{1}(L(λ), L(λ)) ≅ ⨁_{ν∈Γ} Ext_{G}^{1}(L(λ) ⊗ L(ν)^{(1)}, L(λ) ⊗ L(ν)).

Since Ext_{G}^{1}(L(λ), L(λ)) = 0 (cf. [Jan1, II 2.12]), to show that Ext_{G(𝔽_{q})}^{1}(L(λ), L(λ)) = 0, it suffices to show that Ext_{G}^{1}(L(λ) ⊗ L(ν)^{(1)}, L(λ) ⊗ L(ν)) = 0 for all ν ∈ Γ − {0}. As such, these latter Ext-groups are the focus of our study.

4.3 We begin by making an estimate on Ext_{G}^{1}(L(λ) ⊗ L(ν)^{(1)}, L(λ) ⊗ L(ν)) with an Extgroup involving induced modules H^{0}(λ) = ind_{B}^{G} λ and Weyl modules V (λ) ≅ H^{0}(−ω_{0}λ)^{∗}.

Proposition . Let λ ∈ X_{1}(T), 0 ≠ ν_{1} ∈ X(T)_{+}, ν_{2} ∈ Γ, and M ∈ {L(λ), V (λ)}. There exist the following embeddings

(a) Ext_{G}^{1}(M ⊗ L(ν_{1})^{(1)}, L(λ) ⊗ L(ν_{2})) ↪ Ext_{G}^{1}(M ⊗ L(ν_{1})^{(1)}, H^{0}(λ) ⊗ L(ν_{2}));

(b) Ext_{G}^{1}(L(λ) ⊗ L(ν_{1})^{(1)}, L(λ) ⊗ L(ν_{2})) ↪ Ext_{G}^{1}(V (λ) ⊗ L(ν_{1})^{(1)}, H^{0}(λ) ⊗ L(ν_{2})).

Proof. (a) Consider the short exact sequence

0 → L(λ) ⊗ L(ν_{2}) → H^{0}(λ) ⊗ L(ν_{2}) → Q → 0

and the corresponding long exact sequence

⋯ → Hom_{G}(M ⊗ L(ν_{1})^{(1)}, Q) → Ext_{G}^{1}(M ⊗ L(ν_{1})^{(1)}, L(λ) ⊗ L(ν_{2}))
→ Ext_{G}^{1}(M ⊗ L(ν_{1})^{(1)}, H^{0}(λ) ⊗ L(ν_{2})) → …

It suffices to show that Hom_{G}(M ⊗ L(ν_{1})^{(1)}, Q) = 0. Since λ lies in X_{1}(T), by Steinberg’s tensor product theorem, L(λ) ⊗ L(ν_{1})^{(1)} ≅ L(λ + pν_{1}). If M = L(λ) and Hom_{G}(L(λ) ⊗ L(ν_{1})^{(1)}, Q) ≠ 0 then L(λ+pν_{1}) is a composition factor of Q. However, for any composition factor L(θ) of Q, θ < λ+ν_{2}. Since ν_{1} ≠ 0, it follows that L(λ+pν_{1}) cannot be a composition factor of Q and hence there are no such homomorphisms. On the other hand, if M = V (λ), then Hom_{G}(V (λ) ⊗ L(ν_{1})^{(1)}, Q) ≅ Hom_{B}(λ ⊗ L(ν_{1})^{(1)}, Q). If Hom_{B}(λ ⊗ L(ν_{1})^{(1)}, Q) ≠ 0 then Q has λ + pν_{1} as a weight because the head of L(ν_{1})^{(1)} as a B-module is pν_{1}. But, this cannot happen since all weights of Q are less than λ+ν_{2}. (b) This part follows by using the duality H^{0}(λ)^{∗} ≅ V (−w_{0}λ) and applying part (a) twice; first with M = L(λ) and second with M = V (−w_{0}λ). _{□}

4.4 The following proposition will allow us to make further reductions in analyzing the existence of self-extensions.

Proposition . Let p ≥ 3(h − 1), λ ∈ X_{1}(T), ν ∈ Γ, and γ a weight of L(ν). Then the following hold:

(a) Hom_{B_{1}}(V (λ + γ), λ) = 0 unless γ = 0.

(b) As a B-module, Hom_{B_{1}}(V (λ), λ ⊗ L(ν)) is a direct sum of trivial modules.

(c) Ext_{G_{1}}^{1}(V (λ), H^{0}(λ) ⊗ L(ν))^{(−1)} ≅ ind_{B}^{G}(Ext_{B_{1}}^{1}(V (λ), λ ⊗ L(ν))^{(−1)}).

Proof. (a) Suppose that Hom_{B_{1}}(V (λ + γ), λ) ≠ 0. Then there exists σ ∈ X(T) such that Hom_{B/B_{1}}(pσ, Hom_{B_{1}}(V (λ + γ), λ)) ≠ 0. But,

Hom_{B/B_{1}}(pσ, Hom_{B_{1}}(V (λ + γ), λ)) ≅ Hom_{B/B_{1}}(k, Hom_{B_{1}}(V (λ + γ), λ − pσ))
≅ Hom_{B}(V (λ + γ), λ − pσ)
≅ Hom_{G}(V (λ + γ), H^{0}(λ − pσ)).

Hence, by [Jan1, II Prop. 4.13], λ + γ = λ − pσ, so γ = −pσ. Since γ is a weight of L(ν) and ν ∈ Γ, this forces σ = 0 and γ = 0.

(b) Let Z_{1}(λ) = coind_{B^{+}T}^{G_{1}B^{+}}λ as defined in [Jan1, II 9.1]. First observe that

Hom_{G_{1}B^{+}}(Z_{1}(λ), V (λ)) ≅ Hom_{B^{+}}(λ, V (λ)) ≠ 0.

Therefore, there exists a non-zero G_{1}B^{+}-homomorphism ϕ : Z_{1}(λ) → V (λ). Since λ ∈ X_{1}(T), V (λ) has simple head L(λ) as G_{1}-module. The head of Z_{1}(λ) is also L(λ) and all composition factors in V (λ) have weights less than λ, so ϕ must be surjective. Consequently, there exists an embedding

Hom_{B_{1}}(V (λ), λ ⊗ L(ν)) ↪ Hom_{B_{1}}(Z_{1}(λ), λ ⊗ L(ν)).

Since Z_{1}(λ) is projective as a B_{1}-module, the dimension of Hom_{B_{1}}(Z_{1}(λ), λ ⊗ L(ν)) equals the sum of the dimensions of the weight spaces of the form λ + pη in λ ⊗ L(ν). But the size of ν forces any contributing weight η to be zero. Thus Hom_{B_{1}}(Z_{1}(λ), λ ⊗ L(ν)) equals Hom_{B_{1}T} (Z_{1}(λ), λ ⊗ L(ν)) and, as a T -module, it is isomorphic to dim L(ν)_{0} copies of the trivial module. Now Hom_{B_{1}}(V (λ), λ ⊗ L(ν)) is a T -submodule of Hom_{B_{1}}(Z_{1}(λ), λ ⊗ L(ν)), and thus also a direct sum of trivial modules. Since p ≥ 3(h − 1), there are no non-trivial B-extensions of the trivial module by itself, and Hom_{B_{1}}(V (λ), λ⊗L(ν)) is also a direct sum of trivial modules when regarded as a B-module.

(c) This part is a generalization of [Jan1, II 12.8]. We apply the spectral sequence in [Jan1, II 12.7]:

R^{i}ind_{B}^{G}(Ext_{B_{1}}^{j}(V (λ), λ ⊗ L(ν))^{(−1)}) ⇒ Ext_{G_{1}}^{i+j}(V (λ), H^{0}(λ) ⊗ L(ν))^{(−1)}.

It is enough to show that all E_{2}^{i,0}-terms vanish for i > 0. Part (b) shows that E_{2}^{i,0} = R^{i}ind_{B}^{G}(Hom_{B_{1}}(V (λ), λ ⊗ L(ν))^{(−1)}) ≅ R^{i}ind_{B}^{G}(⊕k) = 0. _{□}

4.5 We now indicate how the non-vanishing of the remainder term R in Theorem 2.5 implies a non-vanishing statement in terms of B-cohomology groups.

Lemma . Let p ≥ 3(h − 1), λ ∈ X_{1}(T), and ν_{1}, ν_{2} ∈ Γ with ν_{1} ≠ 0. Suppose that Ext_{G}^{1}(L(λ) ⊗ L(ν_{1})^{(1)}, L(λ) ⊗ L(ν_{2})) ≠ 0. Then there exists a weight σ of L(ν_{1}) such that

Hom_{B/B_{1}}(pσ, Ext_{B_{1}}^{1}(V (λ), λ ⊗ L(ν_{2}))) ≠ 0.

Proof. By Proposition 4.3b, it follows that

Ext_{G}^{1}(V (λ) ⊗ L(ν_{1})^{(1)}, H^{0}(λ) ⊗ L(ν_{2})) ≠ 0.

Consider the Lyndon-Hochschild-Serre spectral sequence for G_{1} ⊴ G:

E_{2}^{i,j} = Ext_{G/G_{1}}^{i}(L(ν_{1})^{(1)}, Ext_{G_{1}}^{j}(V (λ), H^{0}(λ)⊗L(ν_{2}))) ⇒ Ext_{G}^{i+j}(V (λ)⊗L(ν_{1})^{(1)}, H^{0}(λ)⊗L(ν_{2}))

and the beginning of the five-term exact sequence

0 → Ext_{G/G_{1}}^{1}(L(ν_{1})^{(1)}, Hom_{G_{1}}(V (λ), H^{0}(λ) ⊗ L(ν_{2})))
→ Ext_{G}^{1}(V (λ) ⊗ L(ν_{1})^{(1)}, H^{0}(λ) ⊗ L(ν_{2}))
→ Hom_{G/G_{1}}(L(ν_{1})^{(1)}, Ext_{G_{1}}^{1}(V (λ), H^{0}(λ) ⊗ L(ν_{2}))) → ….

For λ ∈ X_{1}(T), H^{0}(λ) has simple G_{1}-socle L(λ). Let Z′_{1}(λ) = ind_{B^{+}}^{G_{1}B^{+}}λ. A similar argument to the one in Proposition 4.4b shows that there exists an injective G_{1}B^{+}- homomorphism ψ : H^{0}(λ) ↪ Z′_{1}(λ). This induces the following homomorphisms of B/B_{1}- modules:

Hom_{G_{1}}(V (λ), H^{0}(λ) ⊗ L(ν_{2})) ↪ Hom_{G_{1}}(V (λ), Z′_{1}(λ) ⊗ L(ν_{2})) ≅ Hom_{B_{1}}(V (λ), λ ⊗ L(ν_{2})).

Now it follows from Proposition 4.4b that, as a B-module, Hom_{G_{1}}(V (λ), H^{0}(λ) ⊗ L(ν_{2})) is a direct sum of trivial modules. Hence as a G/G_{1}-module, Hom_{G_{1}}(V (λ), H^{0}(λ) ⊗ L(ν_{2})) is a direct sum of trivial modules. Since ν_{1} is in the lowest alcove, we have Ext_{G}^{i}(L(ν_{1}), k) = 0 for i ≥ 1 and may conclude that E_{2}^{i,0} = 0 for i ≥ 1. Consequently,

0 ≠ E^{1} ≅ E_{2}^{0,1} = Hom_{G/G_{1}}(L(ν_{1})^{(1)}, Ext_{G_{1}}^{1}(V (λ), H^{0}(λ) ⊗ L(ν_{2})))
≅ Hom_{G/G_{1}}(L(ν_{1})^{(1)}, ind_{B}^{G}(Ext_{B_{1}}^{1}(V (λ), λ ⊗ L(ν_{2}))))
≅ Hom_{B/B_{1}}(L(ν_{1})^{(1)}, Ext_{B_{1}}^{1}(V (λ), λ ⊗ L(ν_{2}))).

Therefore, there exists a weight σ of L(ν_{1}) such that Hom_{B/B_{1}}(pσ, Ext_{B_{1}}^{1}(V (λ), λ⊗L(ν_{2}))) ≠ 0. _{□}

The problem of showing the vanishing of self-extensions for r = 1 can now be reduced to the following statement.

Corollary . Let p ≥ 3(h−1), λ ∈ X_{1}(T) and suppose that Ext_{G(𝔽_{p})}^{1}(L(λ), L(λ)) ≠ 0. Then there exists a weight ν ∈ Γ and a weight σ of L(ν) such that

Hom_{B/B_{1}}(pσ, Ext_{B_{1}}^{1}(V (λ), λ ⊗ L(ν))) ≠ 0.

Proof. As remarked in Section 4.2, if Ext_{G(𝔽_{p})}^{1}(L(λ), L(λ)) ≠ 0, then by Theorem 2.5 there is some ν ∈ Γ such that

Ext_{G}^{1}(L(λ) ⊗ L(ν)^{(1)}, L(λ) ⊗ L(ν)) ≠ 0.

By [And1, Theorem 4.5] we have ν ≠ 0. The assertion follows from the previous lemma. _{□}

4.6 To make use of the preceding reduction criterion, we begin by showing that if Ext_{B}^{1}(V (λ+ γ), λ − pσ) ≠ 0 then there are important conditions imposed on λ, γ and σ.

Theorem . Let p ≥ 3(h − 1), λ ∈ X_{1}(T), ν ∈ Γ, and γ a weight of L(ν). If there exists a weight σ with Ext_{B}^{1}(V (λ + γ), λ − pσ) ≠ 0, then one of the following holds:

(a) λ ↑ λ + γ, and σ = p^{m}α, where α denotes a simple root and m a non-negative integer,

(b) the underlying root system of G is of type A_{1} or C_{n}, and (i) σ = \frac{α_{n}}{2} , where α_{n} is the unique long (last) simple root, (ii) ⟨λ, α_{n}^{∨}⟩ = \frac{p−2−c}{2}, where c is odd and 0 < |c| ≤ h − 1, (iii) γ = \frac{c}{2}α_{n}.

Proof. It follows from [Jan1, II 4.13] that λ − pσ ∉ X(T)_{+}. Using the spectral sequence [And1, 1.1]

E_{2}^{i,j} = Ext_{G}^{i}(V (λ + γ), R^{j}ind_{B}^{G}(λ − pσ)) ⇒ Ext_{B}^{i+j}(V (λ + γ), λ − pσ),

we obtain the isomorphism

Ext_{B}^{1}(V (λ + γ), λ − pσ) ≅ Hom_{G}(V (λ + γ), H^{1}(λ − pσ)).

Now we apply [Jan1, II 5.15] and obtain two possibilities.

Case 1: There exists a dominant weight η and a simple root α such that λ − pσ = s_{α} · η and ⟨η, α^{∨}⟩ = ap^{n} − 1 where 0 < a < p. Moreover, H^{1}(λ − pσ) ≅ H^{0}(η).

Since Hom_{G}(V (λ + γ), H^{0}(η)) ≠ 0 it follows that η = λ + γ. Thus, the above equation becomes λ − pσ = s_{α} · (λ + γ). Multiplying both sides by s_{α} yields s_{α} · λ − ps_{α}(σ) = λ + γ. Solving for γ yields

γ = −ps_{α}(σ) − ⟨λ + ρ, α^{∨}⟩α.  (4.6.1)

We take the inner product of equation (4.6.1) with α^{∨} and obtain

⟨γ, α^{∨}⟩ = p⟨σ, α^{∨}⟩ − 2⟨λ + ρ, α^{∨}⟩.  (4.6.2)

Since |⟨γ, α^{∨}⟩| ≤ h − 1 and λ is p-restricted, we see that ⟨σ, α^{∨}⟩ must equal 0, 1, or 2.

Case 1.1: ⟨σ, α^{∨}⟩ = 0. Then ⟨γ, α^{∨}⟩ = −2⟨λ + ρ, α^{∨}⟩ and therefore ⟨λ + γ + ρ, α^{∨}⟩ = ⟨λ + ρ, α^{∨}⟩ + ⟨γ, α^{∨}⟩ = −⟨λ + ρ, α^{∨}⟩. This is not possible because λ + γ is a dominant weight.

Case 1.2: ⟨σ, α^{∨}⟩ = 1. Then ⟨γ, α^{∨}⟩ = p − 2⟨λ + ρ, α^{∨}⟩ and therefore

⟨λ + ρ, α^{∨}⟩ = \frac{p}{2} − \frac{⟨γ, α^{∨}⟩}{2}.  (4.6.3)

Let us assume that G is not of type A_{1} or C_{n}. Then there exists a root β with ⟨α, β^{∨}⟩ = −1 (see also [Jan1, II 12.1]). Taking the inner product of equation (4.6.1) with β^{∨} gives

⟨γ, β^{∨}⟩ = −p⟨s_{α}(σ), β^{∨}⟩ + ⟨λ + ρ, α^{∨}⟩.  (4.6.4)

We substitute equation (4.6.3) into equation (4.6.4) and obtain

2⟨γ, β^{∨}⟩ + ⟨γ, α^{∨}⟩ = −2p⟨s_{α}(σ), β^{∨}⟩ + p.  (4.6.5)

The absolute value of the left hand side of the above equation is at most 3(h − 1) while the right hand side is a non-zero multiple of p. From p ≥ 3(h − 1) it follows that p = 3(h − 1). But that is impossible since p is prime and the rank of G is at least 2. Therefore ⟨σ, α^{∨}⟩ = 1 implies that G is of type A_{1} or C_{n}. Moreover, for any root β we have that ⟨α, β^{∨}⟩ is divisible by 2. This implies that α = α_{n}, the unique long (last) simple root, where the simple roots are denoted by {α_{1}, α_{2}, …, α_{n}}. We use (4.6.3) to substitute for ⟨λ + ρ, α_{n}^{∨}⟩ in (4.6.1) and take the inner product with an arbitrary simple root β to obtain

2⟨γ, β^{∨}⟩ − ⟨γ, α_{n}^{∨}⟩⟨α_{n}, β^{∨}⟩ = −2p⟨s_{α_{n}}(σ), β^{∨}⟩ − 2p⟨\frac{α_{n}}{2} , β^{∨}⟩.  (4.6.6)

The absolute value of the left hand side of the above equation is at most 4(h − 1) while the right hand side is a multiple of 2p. The size of p forces both sides to be zero. The right hand side yields s_{α_{n}}(σ) = −\frac{α_{n}}{2} , which forces σ = \frac{α_{n}}{2} . It follows from (4.6.1) that γ is a multiple of \frac{α_{n}}{2} . From equation (4.6.3) we obtain ⟨λ, α_{n}^{∨}⟩ = \frac{p−2−c}{2}, where c is odd and 0 < |c| ≤ h − 1. These are exactly the properties listed under (b) in this theorem.

Case 1.3: ⟨σ, α^{∨}⟩ = 2. Then

⟨λ + ρ, α^{∨}⟩ = p − \frac{⟨γ, α^{∨}⟩}{2} ≥ p − \frac{h − 1}{2}.

Substituting the above for ⟨λ + ρ, α^{∨}⟩ in equation (4.6.1) and taking the inner product with any simple root β yields

⟨γ, β^{∨}⟩ = −p⟨s_{α}(σ), β^{∨}⟩ − (p − \frac{⟨γ, α^{∨}⟩}{2}) ⟨α, β^{∨}⟩,

which gives

⟨γ, β^{∨}⟩ − \frac{1}{2}⟨γ, α^{∨}⟩⟨α, β^{∨}⟩ = p⟨−s_{α}(σ) − α, β^{∨}⟩.

Since |⟨α, β^{∨}⟩| ≤ 3 the absolute value of the left hand side of the above equation is less than or equal to \frac{5}{2}(h − 1). But p ≥ 3(h − 1). Therefore, −s_{α}(σ) = α, which forces σ = α and γ = (p − ⟨λ + ρ, α^{∨}⟩)α. Moreover, λ + γ is just the reflection of the restricted weight λ across the (α, p)-wall. Recall that we also had the condition ⟨λ + γ, α^{∨}⟩ = ap^{n} − 1. This forces λ + γ to be on the (α, p)-wall and hence γ = 0.

Case 2: There exists a dominant weight η, a unique simple root α, and integers n > 0 and 0 < a < p, such that

λ − pσ = η − ap^{n}α.  (4.6.7)

Moreover, it follows that soc_{G} H^{1}(λ − pσ) ≅ L(η), which implies that η ↑ λ + γ.

We rewrite equation (4.6.7) in the form

η − λ = p(ap^{n−1}α − σ).  (4.6.8)

Sine η is dominant and λ is restricted, taking the inner product with any simple root β yields

−(p − 1) ≤ ⟨η − λ, β^{∨}⟩ = p⟨ap^{(n−1)}α − σ, β^{∨}⟩.

It follows that ⟨ap^{(n−1)}α − σ, β^{∨}⟩ ≥ 0 and therefore ap^{(n−1)}α − σ is a dominant weight. On the other hand, because of η ↑ λ + γ we have

h − 1 ≥ ⟨γ, α_{0}^{∨}⟩ = ⟨λ + γ − λ, α_{0}^{∨}⟩ ≥ ⟨η − λ, α_{0}^{∨}⟩ = p⟨ap^{(n−1)}α − σ, α_{0}^{∨}⟩ ≥ 0.

Now the size of p forces ap^{(n−1)}α − σ = 0 and therefore λ = η ↑ λ + γ. Finally it follows from [Jan1, II 5.16] that a = 1. _{□}

Corollary . Let p ≥ 3(h − 1), λ ∈ X_{1}(T), ν ∈ Γ, and γ a weight of L(ν). If Ext_{B_{1}}^{1}(V (λ + γ), λ) ≠ 0, then one of the following holds:

(a) λ ↑ λ + γ and λ ≠ λ + γ;

(b) the root system of G is of type A_{1} or C_{n}, and

(i) ⟨λ, α_{n}^{∨}⟩ = \frac{p−2−c}{2}, where α_{n} is the unique long simple root and c is odd with 0 < |c| ≤ h − 1,

(ii) γ = \frac{c}{2}α_{n}.

Proof. We now generalize Andersen’s argument that can be found in [Jan1, II 12.3-12.5]. If the B-module Ext_{B_{1}}^{1}(V (λ + γ), λ) ≠ 0, then there exists a weight σ such that

Hom_{B/B_{1}}(pσ, Ext_{B_{1}}^{1}(V (λ + γ), λ)) ≠ 0.

We apply the five term exact sequence corresponding to the spectral sequence

Ext_{B/B_{1}}^{i}(pσ, Ext_{B_{1}}^{j}(V (λ + γ), λ)) ⇒ Ext_{B}^{i+j}(V (λ + γ), λ − pσ).

We will distinguish two cases. First suppose that γ ≠ 0. It follows from Proposition 4.4a that the terms

E^{i,0} = Ext_{B/B_{1}}^{i}(pσ, Hom_{B_{1}}(V (λ + γ), λ)) = 0.

We obtain the isomorphism

Hom_{B/B_{1}}(pσ, Ext_{B_{1}}^{1}(V (λ + γ), λ)) ≅ Ext_{B}^{1}(V (λ + γ), λ − pσ).

Now the assertion follows from the previous theorem.

Now suppose that γ = 0. We have

E^{i,0} = Ext_{B/B_{1}}^{i}(pσ, Hom_{B_{1}}(V (λ), λ)) ≅ H^{i}(B, −pσ).

As pointed out in the proof of Proposition 4.4b, there exists a G_{1}B^{+}-projection of Z_{1}(λ) onto V (λ). We substitute the exact sequence

0 → M → Z_{1}(λ) → V (λ) → 0

for [Jan1, II 12.3(3)] and apply [Jan1, II 12.4-12.5] to obtain E^{2,0} ≅ H^{2}(B, −pσ) = 0. Now assume that Ext_{B}^{1}(V (λ), λ − pσ) ≠ 0. Applying the preceding theorem, conclusion (a) must hold since γ ≠ 0, and so σ = p^{m}α, for some simple root α. Moreover, by [Jan1, II 5.20] we obtain

E^{1} ≅ Ext_{B}^{1}(V (λ), λ − pσ) ≅ k ≅ H^{1}(B, −pσ) ≅ E^{1,0}.

This forces E^{0,1} = 0 which contradicts our assumption. _{□}

4.7 Returning to the reduction criterion in Corollary 4.5, if Hom_{B/B_{1}}(pσ, Ext_{B_{1}}^{1}(V (λ), λ ⊗ L(ν))) ≠ 0, the results in Section 4.6 can be used to find necessary conditions on σ and λ. These conditions are given in the following result.

Proposition . Let p ≥ 3(h − 1), λ ∈ X_{1}(T), and ν ∈ Γ. If there exists a weight σ such that Hom_{B/B_{1}}(pσ, Ext_{B_{1}}^{1}(V (λ), λ ⊗ L(ν))) ≠ 0, and either

(i) the underlying root system of G is not of type A_{1} or C_{n} or (ii) ⟨λ, α_{n}^{∨}⟩ ≠ \frac{p−2−c}{2}, where α_{n} is the unique long simple root and c is odd with 0 < |c| ≤

h − 1, then the following hold:

(a) σ is equal to a simple root α.

(b) ⟨λ + ρ, α^{∨}⟩ ≥ p − \frac{(h−1)}{2} and the weight λ̄, which denotes the image of λ after reflection across the (α, p)-wall, is dominant.

Proof. Since ν ∈ Γ, L(ν) ≅ H^{0}(ν) ≅ V (−w_{0}ν)^{∗}. Therefore,

Ext_{B_{1}}^{1}(V (λ), λ ⊗ L(ν)) ≅ Ext_{B_{1}}^{1}(V (λ) ⊗ V (−w_{0}ν), λ).

The module V (λ) ⊗ V (−w_{0}ν) has a Weyl filtration [Mat] with factors whose weights are contained in the set J = {μ ∈ X(T)_{+}| Hom_{G}(V (λ) ⊗ V (−w_{0}ν), H^{0}(μ)) ≠ 0}. Choose a total ordering μ_{1}, μ_{1}, ..., μ_{n} on J such that μ_{i} > μ_{j} implies i < j. By dualizing the “good filtration” argument in [Jan1, II 4.16], we construct a Weyl filtration 0 = V_{0} ⊂ V_{1} ⊂ V_{2} ... ⊂ V_{n} = V of V = V (λ) ⊗ V (−w_{0}ν) by setting V_{i} equal to the kernel of the projection of V_{i+1} onto V (μ_{i+1}). Set m = max ({i | μ_{i} > λ} ∪ {0}). Then the submodule M = V_{m} has a Weyl filtration whose factors V (μ_{i}) satisfy μ_{i} ≰ λ. The quotient Q = V/V_{m} has a Weyl filtration whose factors V (μ_{i}) satisfy μ_{i} ≯ λ.

The short exact sequence 0 → M → V → Q → 0 induces the the long exact sequence

... → Ext_{B_{1}}^{1}(Q, λ) → Ext_{B_{1}}^{1}(V, λ) → Ext_{B_{1}}^{1}(M, λ) → ... .

If Ext_{B_{1}}^{1}(Q, λ) ≠ 0 then there exists a V (μ_{i}) in the filtration of Q with Ext_{B_{1}}^{1}(V (μ_{i}), λ) ≠ 0. By Corollary 4.6, μ_{i} > λ which contradicts how Q was constructed. Therefore, the exact sequence becomes

0 → Ext_{B_{1}}^{1}(V, λ) → Ext_{B_{1}}^{1}(M, λ) → ... .

Using left exactness of the functor Hom_{B/B_{1}}(pσ, −), there is a injection:

0 → Hom_{B/B_{1}}(pσ, Ext_{B_{1}}^{1}(V, λ)) ↪ Hom_{B/B_{1}}(pσ, Ext_{B_{1}}^{1}(M, λ)).

Hence, Hom_{B/B_{1}}(pσ, Ext_{B_{1}}^{1}(M, λ)) ≠ 0.

For i = 0, 1, …, m − 1, we have V_{i+1}/V_{i} ≅ V (μ_{i+1}). Consider the short exact sequences

0 → V (μ_{i+1}) → M/V_{i} → M/V_{i+1} → 0.

The construction of M allows only factors V (μ) with μ ≠ λ. Hence, it follows from Proposition 4.4a that the long exact sequence

... → Hom_{B_{1}}(V (μ_{i+1}), λ) → Ext_{B_{1}}^{1}(M/V_{i+1}, λ) → Ext_{B_{1}}^{1}(M/V_{i}, λ) → Ext_{B_{1}}^{1}(V (μ_{i+1}), λ) → ...

becomes

0 → Ext_{B_{1}}^{1}(M/V_{i+1}, λ) → Ext_{B_{1}}^{1}(M/V_{i}, λ) → Ext_{B_{1}}^{1}(V (μ_{i+1}), λ) → ... .

Again by using the left exactness of the functor Hom_{B/B_{1}}(pσ, −), we obtain exact sequences for all i = 0, …, m − 1:

0 → Hom_{B/B_{1}}(pσ, Ext_{B_{1}}^{1}(M/V_{i+1}, λ)) → Hom_{B/B_{1}}(pσ, Ext_{B_{1}}^{1}(M/V_{i}, λ)) →
Hom_{B/B_{1}}(pσ, Ext_{B_{1}}^{1}(V (μ_{i+1}), λ)) → ... .

Since

0 ≠ Hom_{B/B_{1}}(pσ, Ext_{B_{1}}^{1}(M, λ)) = Hom_{B/B_{1}}(pσ, Ext_{B_{1}}^{1}(M/V_{0}, λ))

while clearly Hom_{B/B_{1}}(pσ, Ext_{B_{1}}^{1}(M/V_{m}, λ)) = 0 there exists a μ ≠ λ with

Hom_{B/B_{1}}(pσ, Ext_{B_{1}}^{1}(V (μ), λ)) ≠ 0.

We repeat the argument of Case 1 in the proof of Corollary 4.6 to conclude that

Ext_{B}^{1}(V (μ), λ − pσ) ≠ 0.

Now Theorem 4.6 says that σ = p^{m}α where α denotes a simple root.

To see that m = and σ = α, we once more look at the exact sequence (see the proof of Proposition 4.4b)

0 → N → Z_{1}(λ) → V (λ) → 0

and the argument in [Jan1, II 12.4]. Z_{1}(λ) is projective as a B_{1}-module and we obtain a surjection Hom_{B_{1}}(N, λ⊗L(ν)) → Ext_{B_{1}}^{1}(V (λ), λ⊗L(ν)). If pσ is a weight of Ext_{B_{1}}^{1}(V (λ), λ⊗ L(ν)) then it is also a weight of Hom_{B_{1}}(N, λ ⊗ L(ν)). Now there has to exist a weight γ of L(ν) such that λ + γ − pσ is a weight of N and hence of Z_{1}(λ). Now γ − pσ + (p − 1)ρ has to be a weight of the Steinberg module Z_{1}((p − 1)ρ) ≅ Z_{1}(λ) ⊗ ((p − 1)ρ − λ). For any simple root β we have therefore |⟨γ − pσ + (p − 1)ρ, β^{∨}⟩| ≤ (p − 1)(h − 1). Thus

⟨γ + (p − 1)ρ, β^{∨}⟩ − (p − 1)(h − 1) ≤ ⟨pσ, β^{∨}⟩ ≤ ⟨γ + (p − 1)ρ, β^{∨}⟩ + (p − 1)(h − 1).

This forces p|⟨σ, β^{∨}⟩| ≤ p − 1 + h − 1 + (p − 1)(h − 1) = ph − 1 and hence |⟨σ, β^{∨}⟩| ≤ h − 1. As σ = p^{m}α for some simple root α, the size of p and the above inequality force m = 0. Hence σ = α.

Recall that Ext_{B}^{1}(V (μ), λ − pα) ≅ Hom_{G}(V (μ), H^{1}(λ − pα)). By [Jan1, II 5.17] the Gsocle of H^{1}(λ − pα) is isomorphic to L(λ). Moreover, by [Jan1, II 5.15], the maximal weight of H^{1}(λ − pα) is λ̄ = s_{α} · (λ − pα), if λ̄ is dominant, otherwise the maximal weight is λ. Hence, λ ↑ μ ↑ λ̄. Now μ ≠ λ and λ̄ is the reflection of λ across the (α, p)-wall. It follows that μ = λ̄ and λ̄ is dominant. Moreover, μ is of the form λ + γ with γ a weight of L(ν) with 0 < ⟨γ, α^{∨}⟩ = ⟨λ̄ − λ, α^{∨}⟩ ≤ (h − 1). Hence, ⟨λ + ρ, α^{∨}⟩ ≥ p − \frac{(h−1)}{2}. _{□}

4.8 In the previous proposition, under the hypotheses, it was shown that ⟨λ + ρ, α^{∨}⟩ ≥ p− \frac{(h−1)}{2} for some simple root α. This condition allows us to prove an isomorphism between Hom and Ext^{1} groups. This result will be crucial in our later analysis of self-extensions.

Proposition . Let p ≥ 3(h − 1), λ ∈ X_{1}(T), ν ∈ Γ, and α ∈ Δ. If ⟨λ + ρ, α^{∨}⟩ ≥ p − \frac{(h−1)}{2} and the weight λ̄, which denotes the image of λ after reflection across the (α, p)-wall, is dominant, then

Ext_{B}^{1}(V (λ), (λ − pα) ⊗ L(ν)) ≅ Hom_{B}(V (λ), λ ⊗ L(ν)).

Proof. Let P (α) and L(α) denote parabolic and Levi subgroups that correspond to the simple root α. We apply the spectral sequence

E_{2}^{i,j} = Ext_{P (α)}^{i}(V (λ), R^{j} ind_{B}^{P (α)}(λ − pα) ⊗ L(ν)) ⇒ Ext_{B}^{i+j}(V (λ), (λ − pα) ⊗ L(ν)).

Since λ − pα is not dominant, we obtain the following isomorphism:

Hom_{P (α)}(V (λ), R^{1} ind_{B}^{P (α)}(λ − pα) ⊗ L(ν)) ≅ Ext_{B}^{1}(V (λ), (λ − pα) ⊗ L(ν)). (4.8.1)

As an SL_{2}-module R^{1} ind_{B}^{P (α)}(λ − pα) is just the Weyl module with highest weight λ̄. It has a composition series of length two. The simple module with highest weight λ, L_{α}(λ), is its socle and the simple module with highest weight λ̄, L_{α}(λ̄), is its head. For any simple root β ≠ α we have 0 ≤ ⟨λ̄, β^{∨}⟩ ≤ ⟨λ, β^{∨}⟩ ≤ p − 1. Therefore we can write λ̄ = λ̄_{0} + pω_{α}, where λ̄_{0} ∈ X_{1}(T) and ω_{α} denotes the fundamental weight corresponding to α. By Steinberg’s tensor product theorem, L_{α}(λ̄) ≅ L_{α}(λ̄_{0}) ⊗ L_{α}(ω_{α})^{(1)}. Each of the weights λ, λ̄_{0}, and ω_{α} is p-restricted. Therefore, L_{α}(λ) ≅ ind_{B}^{P (α)}(λ) and L_{α}(λ̄) ≅ ind_{B}^{P (α)}(λ̄_{0}) ⊗ (ind_{B}^{P (α)}(ω_{α}))^{(1)}.

From the short exact sequence of P (α)- modules

0 → ind_{B}^{P (α)}(λ) → R^{1} ind_{B}^{P (α)}(λ − pα) → ind_{B}^{P (α)}(λ̄_{0}) ⊗ (ind_{B}^{P (α)}(ω_{α}))^{(1)} → 0

one obtains the exact sequence

0 → Hom_{P (α)}(V (λ), ind_{B}^{P (α)}(λ) ⊗ L(ν)) → Hom_{P (α)}(V (λ), R^{1} ind_{B}^{P (α)}(λ − pα) ⊗ L(ν))

→ Hom_{P (α)}(V (λ), ind_{B}^{P (α)}(λ̄_{0}) ⊗ (ind_{B}^{P (α)}(ω_{α}))^{(1)} ⊗ L(ν)) → ....

Now it follows from the isomorphism (4.8.1) that it suffices to show that

Hom_{P (α)}(V (λ), ind_{B}^{P (α)}(λ̄_{0}) ⊗ (ind_{B}^{P (α)}(ω_{α}))^{(1)} ⊗ L(ν)) = 0.  (4.8.2)

The module (ind_{B}^{P (α)}(ω_{α}))^{(1)} is self-dual as a P (α)-module. Therefore (4.8.2) is equivalent to

Hom_{P (α)}(V (λ) ⊗ (ind_{B}^{P (α)}(ω_{α}))^{(1)}, ind_{B}^{P (α)}(λ̄_{0}) ⊗ L(ν)) = 0.

The evaluation map

ε : L(ω_{α}) ≅ ind_{B}^{G}(ω_{α}) ≅ ind_{P (α)}^{G}(ind_{B}^{P (α)}(ω_{α})) → ind_{B}^{P (α)}(ω_{α})

is a non-zero homomorphism of P (α)-modules and ind_{B}^{P (α)}(ω_{α}) is simple as a P (α)-module. Thus, ε is onto and we obtain a projection of V (λ) ⊗ L(ω_{α})^{(1)} onto V (λ) ⊗ (ind_{B}^{P (α)}(ω_{α}))^{(1)}.

Furthermore, there is a projection of G-modules from V (λ + pω_{α}) onto V (λ) ⊗ L(ω_{α})^{(1)}. From these maps, we obtain the following sequence of embeddings:

Hom_{P (α)}(V (λ) ⊗ (ind_{B}^{P (α)}(ω_{α}))^{(1)}, ind_{B}^{P (α)}(λ̄_{0}) ⊗ L(ν))
↪ Hom_{P (α)}(V (λ) ⊗ L(ω_{α})^{(1)}, ind_{B}^{P (α)}(λ̄_{0}) ⊗ L(ν))
↪ Hom_{P (α)}(V (λ + pω_{α}), ind_{B}^{P (α)}(λ̄_{0}) ⊗ L(ν))
≅ Hom_{G}(V (λ + pω_{α}), H^{0}(λ̄_{0}) ⊗ L(ν)).

The last term is zero unless there exists a weight γ of H^{0}(ν) = L(ν) such that

λ + pω_{α} = λ̄_{0} + γ = s_{α} · (λ − pα) − pω_{α} + γ,

which implies that 2pω_{α} = (p − ⟨λ + ρ, α^{∨}⟩)α + γ. Taking the inner product with α yields 2p ≤ h − 1 + ⟨γ, α^{∨}⟩ ≤ 2(h − 1), which is impossible. The assertion follows. _{□}

4.9 Vanishing of B_{1}-cohomology The following theorem serves the same purpose as [And1, Theorem 3.4] in Andersen’s proof of the vanishing of self-extensions for the first Frobenius kernel.

Theorem . Let G be a simple algebraic group with p ≥ 3(h − 1), λ ∈ X_{1}(T), and ν ∈ Γ. If either

(a) G does not have underlying root system of type A_{1} or C_{n} or

(b) ⟨λ, α_{n}^{∨}⟩ ≠ \frac{p−2−c}{2}, where α_{n} is the unique long simple root and c is odd with 0 < |c| ≤ h − 1,

then Ext_{B_{1}}^{1}(V (λ), λ ⊗ L(ν)) = 0.

Proof. Suppose that Ext_{B_{1}}^{1}(V (λ), λ ⊗ L(ν)) ≠ 0. Then there exists a weight σ such that Hom_{B/B_{1}}(pσ, Ext_{B_{1}}^{1}(V (λ), λ ⊗ L(ν))) ≠ 0. According to Proposition 4.7, σ = α, a simple root.

Again we will use a modification of Andersen’s argument given in [Jan1, II 12.3-12.5]. There exists a spectral sequence

Ext_{B/B_{1}}^{i}(pα, Ext_{B_{1}}^{j}(V (λ), λ ⊗ L(ν))) ⇒ Ext_{B}^{i+j}(V (λ), (λ − pα) ⊗ L(ν)).

The beginning of the corresponding five-term exact sequence is

0 → Ext_{B/B_{1}}^{1}(pα, Hom_{B_{1}}(V (λ), λ ⊗ L(ν))) → Ext_{B}^{1}(V (λ), (λ − pα) ⊗ L(ν))
→ Hom_{B/B_{1}}(pα, Ext_{B_{1}}^{1}(V (λ), λ ⊗ L(ν))) → Ext_{B/B_{1}}^{2}(pα, Hom_{B_{1}}(V (λ), λ ⊗ L(ν))).

From Proposition 4.4b, Hom_{B_{1}}(V (λ), λ ⊗ L(ν)) is a direct sum of trivial modules. Then

Ext_{B/B_{1}}^{1}(pα, Hom_{B_{1}}(V (λ), λ ⊗ L(ν)) ≅ H^{1}(B/B_{1}, −pα) ⊗ Hom_{B_{1}}(V (λ), λ ⊗ L(ν))
Ext_{B/B_{1}}^{2}(pα, Hom_{B_{1}}(V (λ), λ ⊗ L(ν)) ≅ H^{2}(B/B_{1}, −pα) ⊗ Hom_{B_{1}}(V (λ), λ ⊗ L(ν)).

By [Jan1, II 5.20], H^{1}(B/B_{1}, −pα) ≅ k and by [Jan1, II 12.5], H^{2}(B/B_{1}, −pα) = 0. Therefore, the exact sequence becomes

0 → Hom_{B_{1}}(V (λ), λ ⊗ L(ν)) → Ext_{B}^{1}(V (λ), (λ − pα) ⊗ L(ν))
→ Hom_{B/B_{1}}(pα, Ext_{B_{1}}^{1}(V (λ), λ ⊗ L(ν))) → 0.

By applying part (b) of Proposition 4.7 together with Proposition 4.8, it follows that the first map is an isomorphism, thus Hom_{B/B_{1}}(pα, Ext_{B_{1}}^{1}(V (λ), λ ⊗ L(ν))) = 0 which is a contradiction. _{□}

The following Corollary follows immediately from Lemma 4.5.

Corollary . Let G be a simple algebraic group with p ≥ 3(h−1), λ ∈ X_{1}(T), and ν_{1}, ν_{2} ∈ Γ with ν_{1} ≠ 0. If either

(a) G does not have underlying root system of type A_{1} or C_{n} or

(b) ⟨λ, α_{n}^{∨}⟩ ≠ \frac{p−2−c}{2}, where α_{n} is the unique long simple root and c is odd with 0 < |c| ≤ h − 1,

then Ext_{G}^{1}(L(λ) ⊗ L(ν_{1})^{(1)}, L(λ) ⊗ L(ν_{2})) = 0.

4.10 Self-extensions For r = 1, we can now demonstrate that self-extensions between simple modules do not exist for groups G(𝔽_{p}) whose underlying root system is not of type A_{1} or C_{n}. And “most” of the time in the latter cases.

Theorem . Let G be a simple algebraic group with p ≥ 3(h − 1), and λ ∈ X_{1}(T). If either (a) G does not have underlying root system of type A_{1} or C_{n} or

(b) ⟨λ, α_{n}^{∨}⟩ ≠ \frac{p−2−c}{2}, where α_{n} is the unique long simple root and c is odd with 0 < |c| ≤ h − 1,

then Ext_{G(𝔽_{p})}^{1}(L(λ), L(λ)) = 0.

Proof. The claim follows from Corollary 4.5 and Theorem 4.9. _{□}

5 Completely reducible modules

5.1 In 1985, S.D. Smith [S] studied the reducibility of certain G(𝔽_{q})-modules and posed the following question.

(5.1.1) ([S, Question 5]) If a finite-dimensional G(𝔽_{q})-module M has a composition series with all composition factors being isomorphic must M be completely reducible?

Some remarks must be made at this point. First, throughout this paper a G(𝔽_{q})-module has properly designated a kG(𝔽_{q})-module. In [S], Smith considers 𝔽_{q}G(𝔽_{q})-modules. However, an 𝔽_{q}G(𝔽_{q})-module V is completely reducible if and only if V ⊗_{𝔽_{q}} k is completely reducible over kG(𝔽_{q}). Hence, for consistency, the results of this section will be stated over the field k. Secondly, Smith originally posed the question only for certain “high weight” modules. This assumption is not necessary for the succeeding results.

Smith answers (5.1.1) affirmatively ([S, Proposition 4]) under a further assumption that the high weight of the “unique” composition factor is minimal. On the other hand, he notes that the answer is no in general as there exists an indecomposable SL_{2}(𝔽_{p})-module with two isomorphic composition factors. Indeed, this is related to our exclusion of certain weights in Theorem 4.10. We can use the above results on self-extensions to show that the answer to (5.1.1) is yes in most cases when the prime is sufficiently large.

Theorem (A). Let G be a semisimple simply connected algebraic group with p ≥ 3(h − 1). Assume that r ≥ 2. If M is a finite-dimensional G(𝔽_{q})-module with all composition factors being isomorphic, then M is completely reducible.

Theorem (B). Let G be a simple algebraic group with p ≥ 3(h−1). Suppose M is a finitedimensional G(𝔽_{p})-module with all composition factors being isomorphic to L(λ) for some λ ∈ X_{1}(T). If the underlying root system of G is of type A_{1} or C_{n}, assume further that, ⟨λ, α_{n}^{∨}⟩ ≠ \frac{p−2−c}{2}, where α_{n} is the unique long simple root and c is odd with 0 < |c| ≤ h − 1. Then M is completely reducible.

5.2 The vanishing of extensions between simple modules for a ring R is equivalent to the ring being semisimple. Furthermore, for a ring R and R-module M, the condition that Ext_{R}^{1}(L, L′) = 0 for all (not necessarily distinct) composition factors L and L′ of M implies that M is completely reducible. Corollary 3.3 along with Corollary 3.4 can be used to deduce a somewhat more general result about complete reducibility of G(𝔽_{q})-modules when r ≥ 2.

Theorem . Let G be a semisimple simply connected algebraic group with p ≥ 3(h − 1) and r ≥ 2. Suppose that M is a finite-dimensional G(𝔽_{q})-module such that Ext_{G}^{1}(L(λ), L(μ)) = 0 for all composition factors L(λ) and L(μ) of M and for λ ≠ μ either

(i) λ_{r−1} = μ_{r−1} or (ii) there exists a root α such that |⟨λ̂ − μ̂, α^{∨}⟩| > h − 1. Then M is completely reducible.

References

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Department of Mathematics, 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