First cohomology for finite groups of Lie type: simple modules with small dominant weights

Let $k$ be an algebraically closed field of characteristic $p>0$, and let $G$ be a simple, simply connected algebraic group defined over $\mathbb{F}_p$. Given $r \geq 1$, set $q=p^r$, and let $G(\mathbb{F}_q)$ be the corresponding finite Chevalley group. In this paper we investigate the structure of the first cohomology group $H^1(G(\mathbb{F}_q),L(\lambda))$ where $L(\lambda)$ is the simple $G$-module of highest weight $\lambda$. Under certain very mild conditions on $p$ and $q$, we are able to completely describe the first cohomology group when $\lambda$ is less than or equal to a fundamental dominant weight. In particular, in the cases we consider, we show that the first cohomology group has dimension at most one. Our calculations significantly extend, and provide new proofs for, earlier results of Cline, Parshall, Scott, and Jones, who considered the special case when $\lambda$ is a minimal nonzero dominant weight.

1. Introduction 1.1. Let k be an algebraically closed field of characteristic p > 0, and let G be a simple, simplyconnected algebraic group scheme defined over F p . Let F : G → G be the standard Frobenius map on G, and let F r be the r-th iterate of F . Set q = p r . The r-th Frobenius kernel G r of G is the scheme-theoretic kernel of F r , and the finite Chevalley group G(F q ) consists of the fixed points in G under F r . It is well-known that the representation theories of G, G r and G(F q ) are interrelated (cf. [Hum2,Nak]), and one can relate the cohomology theories via various spectral sequences and limiting techniques [CPSK, BNP]. Even with our current knowledge of these connections, our understanding of the dimensions of first cohomology groups for finite groups with non-trivial coefficients is limited.
In 1984, Guralnick [Gur] stated a conjecture for a universal upper bound on the dimension of first cohomology groups for finite groups with coefficients in a faithful simple module. Since that time counterexamples have been found to the strong form of Guralnick's conjecture. For example, Scott [Sco] and others have shown that there exist simple modules for finite Chevalley groups for which the first cohomology group has dimension at least 3. Guralnick in work with Aschbacher [AG] and Hoffman [GH] provided an upper bound on the the dimension of the aforementioned first cohomology group by one-half times the dimension of the given module. More recently, for semisimple algebraic groups, Cline, Parshall and Scott have demonstrated an upper bound (depending only on the associated root system of the group) for the first cohomology with coefficients in a simple module. Further work along these lines is provided in [PS2]. Results in the cross characteristic case for the dimension of the first cohomology group were proved by Guralnick and Tiep [GT].
In this paper we investigate the structure of the cohomology group H 1 (G(F q ), L(λ)) where L(λ) is a simple G(F q )-module. In order to understand how the corresponding algebraic group and Lie algebra cohomology are related to this computation we use the powerful filtration techniques developed by Lin and Nakano [LN] and extended more recently by Friedlander [Fri] using the idea of Weil restriction. This approach enables us to employ knowledge about the geometry of the flag variety G/B, in particular, Andersen's famous results [And] on the socle of the sheaf cohomology group H 1 (G/B, L(µ)) for µ an arbitrary weight.
With our machinery we are able to give a complete description of H 1 (G(F q ), L(λ)) when p > 2 for Φ = A n , D n , p > 3 for Φ = B n , C n , E 6 , E 7 , F 4 , G 2 , p > 5 for Φ = E 8 , q > 3, and λ is a fundamental dominant weight. Under certain additional mild restrictions we can improve these results to calculate this cohomology group when λ is less than or equal to a fundamental dominant weight. In particular, we show under these restrictions that if λ is less than or equal to a fundamental dominant weight, then dim H 1 (G(F q ), L(λ)) ≤ 1. 1 One can view our results in the framework of Guralnick's conjecture on the size of the first cohomology group when the coefficient module is taken in a certain subcollection of simple modules.
Our work extends the seminal results of Cline, Parshall and Scott [CPS1], of Jones [Jon], and of Jones and Parshall [JP], where they considered the special case that λ is a minimal nonzero dominant weight. The computations in [CPS1] were used in Wiles' proof of Fermat's Last Theorem [Wil] to show that certain deformation spaces of modular forms and elliptic curves have the same dimension. Our work uses completely different methods than those in [CPS1] and may have connections and uses for other number theoretic questions.
1.2. Main results. The main results of the paper are stated below. Let k be an algebraically closed field of characteristic p > 0, and let G be a simple, simply-connected algebraic group over k, which is defined over the field F p and with associated root system Φ. Let r ≥ 1 and set q = p r . For a complete explanation of notation, see Section 1.3.
Cline, Parshall, Scott and van der Kallen [CPSK,Theorem 7.4] proved that the restriction map H 1 (G, L(λ)) → H 1 (G(F q ), L(λ)) is injective when λ is a p r -restricted weight (i.e., λ ∈ X r (T )). Our first main result, proved in Section 4.1, is that the restriction map is an isomorphism when λ is less than or equal to a fundamental dominant weight, and when the prime p satisfies some fairly mild restrictions depending on the root system. Theorem 1.2.1. Assume that p > 2 when Φ = A n , D n , p > 3 when Φ = B n , C n , E 6 , E 7 , F 4 , G 2 , and p > 5 when Φ = E 8 . Suppose λ ≤ ω j for some j. If q > 3, then the restriction map is an isomorphism.
With this isomorphism we are able to compute the cohomology for G(F q ) with coefficients in a simple module having fundamental highest weight.
Theorem 1.2.2 is proved for the classical groups in Section 5, and for the exceptional groups in Section 6. Our techniques are also applicable to the case when the simple coefficient module has highest weight less than or equal to a fundamental dominant weight. To obtain complete results, we must enlarge the prime to p > 7 for the cases Φ = E 7 , E 8 .
Theorem 1.2.3. Let λ ∈ X(T ) + be such that λ ≤ ω j for some j. Assume that q > 3 and 1 Our techniques can also be employed to make calculations for larger dominant weights, provided that one either possesses more detailed information about the structure of the cohomology group H 1 (U1, L(λ)), or that one imposes stronger restrictions on p and q. p > 2 if Φ has type A n , D n ; p > 3 if Φ has type B n , C n , E 6 , F 4 , G 2 ; and p > 7 if Φ has type E 7 or E 8 .
Theorem 1.2.3 is proved in Section 6.
1.3. Definitions and notation. Much of the notation used here for algebraic groups is standard and can be found in [Jan2]. Let k be an algebraically closed field of characteristic p > 2, and let G be a simple, simply-connected algebraic group over k. Let T ⊂ G be a maximal torus, defined and split over F p , and let Φ be the root system of T in G. Let ∆ = {α 1 , . . . , α n } ⊂ Φ be a set of simple roots in Φ, and let Φ + and Φ − be the corresponding systems of positive and negative roots in Φ. In this paper we use the ordering of the simple roots given in [Hum1], following Bourbaki. Let B ⊂ G be the Borel subgroup of G containing T that corresponds to Φ − , and let U ⊂ B be the unipotent radical of B. Write W for the Weyl group of Φ, and let w 0 be the longest element in W .
Let E be the Euclidean space spanned by Φ. It possesses a W -invariant inner product, denoted by (·, ·). Given α ∈ Φ, write α ∨ = 2α/(α, α) for the corresponding coroot. Let α 0 be the highest short root in Φ, and set ρ = 1 2 α∈Φ + α. Then the Coxeter number associated to Φ is h = (ρ, α ∨ 0 )+1. The weight lattice X(T ) is the Z-span in E of the set of fundamental dominant weights {ω 1 , . . . , ω n }, which are defined by the equations (ω i , α ∨ j ) = δ i,j (Kronecker delta). Given λ ∈ X(T ) and w ∈ W , write λ → wλ for the usual action of W on X(T ), and write w · λ = w(λ + ρ) − ρ for the dot action of W on X(T ). The weight lattice is partially ordered by the relation µ ≤ λ if λ − µ is a nonnegative integral combination of simple roots. Write X(T ) + for the set of dominant weights in X(T ), and X r (T ) for the set of p r -restricted dominant weights in X(T ) + .
Let F : G → G be the Frobenius morphism of G. For r ≥ 1 and q = p r , set G(F q ) = G F r , the fixed-point subgroup of G under the r-th iterate F r : G → G, and set G r = ker F r , the schemetheoretic kernel of the map F r : G → G. For H ⊂ G a closed F -stable subgroup (scheme) of G, write H(F q ) = H F r and H r = ker(F r | H : H → H). Since T , B and U are closed F -stable subgroups of G, there are finite subgroups B(F q ), U (F q ), and T (F q ) of G(F q ), and the finite subgroup schemes B r , U r , and T r of G r .
Set u = Lie(U ), the Lie algebra of U . Then u is a p-restricted Lie algebra over k, and there exists a p-restricted Lie algebra u Fp over F p , obtained via reduction mod p from a Chevalley basis for u, such that u = u Fp ⊗ Fp k. Set u Fq = u Fp ⊗ Fp F q . Let u(u) be the restricted enveloping algebra of u. Then u(u) is isomorphic to Dist(U 1 ), the algebra of distributions on the finite group scheme U 1 [Jan2, I.9.6(4)]. The category of U 1 -modules is naturally equivalent to the category of Dist(U 1 ) ∼ = u(u)-modules [Jan2,I.8.6]. Henceforth, given a u(u)-module (equivalently, a U 1 -module) M , we often identify without further comment the spaces H • (u(u), M ) and H • (U 1 , M ).
Write mod(G) to denote the category of rational G-modules. Then any M ∈ mod(G) is by restriction also a module for G(F q ), G r , U r , etc. Given λ ∈ X(T ) + , let L(λ) be the simple rational G-module of highest weight λ. If λ ∈ X r (T ), then L(λ) remains simple upon restriction to G(F q ) and upon restriction to G r [Hum2, Theorems 2.5 and 2.11].
2. An analysis of 1-cohomology for algebraic groups, finite groups, and Lie algebras Let M be a finite dimensional rational G-module. In this section we relate the first cohomology group H 1 (G(F q ), M ) for G(F q ) to the corresponding cohomology groups for the algebraic groups G and U and the Frobenius kernel U 1 . When M = L(λ) with λ less than or equal to a fundamental dominant weight, we obtain a vanishing criterion for H 1 (G(F q ), M ) in terms of cohomology for U 1 . Throughout Section 2, we will assume that p is excellent for the root system Φ (cf. [LN,Section 1.4]), that is, p = 2 when Φ = B n , C n , F 4 , and p > 3 in type G 2 . Note that p is excellent whenever the requirements on p stated in the main results are satisfied.
2.1. Reduction to Sylow p-subgroups. The first step in establishing the relationship between the cohomology groups H 1 (G(F q ), M ) and H 1 (U 1 , M ) is to consider a suitable subspace of the cohomology for the finite subgroup The torus T (F q ) acts on the groups G(F q ) and U (F q ) by conjugation, and the conjugation actions together with the defining action of T (F q ) on M induce actions of T (F q ) on H • (G(F q ), M ) and H • (U (F q ), M ). The restriction map in cohomology is then a homomorphism of T (F q )-modules. For any group G ′ and any kG ′ -module N , the inner automorphisms of G ′ all induce the identity map on H • (G ′ , N ) [Eve, Proposition 4.1.1]. Then T (F q ) acts trivally on H • (G(F q ), M ), and the restriction homomorphism defines for each n ≥ 0 an injective map 2.2. Weil restriction. The next step in establishing the relationship between G(F q )-cohomology and U 1 -cohomology is to relate cohomology for the finite group U (F q ) to cohomology for a suitable restricted Lie algebra. For this we need the Weil restriction functor constructed by Friedlander.
Definition 2.2.1. [Fri,Definition 1.4] Let r ≥ 1 and set q = p r . Then the Weil restriction R Fq/Fp (u Fq ) of the F q -Lie algebra u Fq is the F p -Lie algebra obtained by viewing the underlying F q -vector space of u Fq as an F p -vector space, and by viewing the F q -bilinear bracket on u Fq as an F p -bilinear map on the underlying F p -vector space. The Weil restriction R Fq/Fp (u Fq ) is made a p-restricted Lie algebra by considering the p-restriction operator on u Fq as a p-restriction operator on the underlying F p -vector space.
Proposition 2.2.2. [Fri,Proposition 1.7] Let r ≥ 1 and set q = p r . Then R Fq/Fp (u Fq ) ⊗ Fp k ∼ = u ⊕r as p-restricted Lie algebras over k.
. This identification is an isomorphism of p-Lie algebras, where the Lie bracket and p-operation as p-restricted Lie algebras over F q . Extending scalars to k, one gets R Fq/Fp (u Fq ) ⊗ Fp k ∼ = u ⊕r .
There exists a natural embedding ι : u Fp ֒→ R Fq/Fp (u Fq ) of F p -Lie algebras corresponding to the fact that u Fp is naturally an F p -vector subspace of u Fq . Explicitly, u Fp identifies with the subspace u Fp ⊗ Fp 1 of u Fp ⊗ Fp F q ∼ = R Fq/Fp (u Fq ). Extending scalars to k, one obtains an embedding which is just the diagonal embedding of u into u ⊕r . To see this, observe that an element x ∈ u Fp ⊂ u maps under ι⊗ Fp k to x⊗ Fp (1⊗ Fp 1) ∈ u Fp ⊗ Fp (F q ⊗ Fp k) ∼ = R Fq/Fp (u Fq )⊗ Fp k, and 1⊗ Fp 1 ∈ F q ⊗ Fp k is the sum of the r primitive orthogonal idempotents that yield the decomposition F q ⊗ Fp k ∼ = k ×r . There also exists a surjection R Fq/Fp (u Fq )⊗ Fp k ∼ = u Fp ⊗ Fp (F q ⊗ Fp k) ։ u Fp ⊗ Fp k ∼ = u induced by the natural multiplication map F q ⊗ Fp k → k. This surjection then identifies with the r-fold addition map u ⊕r ։ u. From now on, it will be convenient to denote the Weil restriction R Fq/Fp (u Fq ) simply by u Fq . Then u(u Fq ⊗ Fp k) ∼ = u(u ⊕r ).
The groups T (F p ) and T (F q ) act on u Fp and u Fq , respectively, by the adjoint action, and the embedding u Fp ֒→ u Fq is a homomorphism of T (F p )-modules. Upon scalar extension to k, the T (F p )module homomorphism u Fp ֒→ u Fq lifts to a T -module homomorphism u Fp ⊗ Fp k ֒→ u Fq ⊗ Fp k, which under the identifications u Fp ⊗ Fp k ∼ = u and u Fq ⊗ Fp k ∼ = u ⊕r is just the usual adjoint action of T .
2.3. The gr operation. Set A = kU (F q ), the group algebra over k of U (F q ), and let I ⊂ A be the augmentation ideal of A. Then the powers of I form a multiplicative filtration of A. Set gr A = i≥0 (I i /I i+1 ), the associated graded ring. By [LN,Theorem 2.3], gr A is isomorphic as a Hopf algebra to u(u Fq ⊗ Fp k). The isomorphism is a map of T (F q )-modules, where T (F q ) acts on U (F q ) by conjugation, and the action of Then gr M is naturally a graded module for the graded algebra gr A ∼ = u(u Fq ⊗ Fp k). We now follow the discussion in [PS1,Section 2]. Let N and Q be kU (F q )-modules, and let , which means that Im σ ⊆ I.Q. Since gr takes surjections to surjections, we have by [PS1, Section 2] an extension The resulting extension is non-split because Im σ ′ ⊆ i>0 Q i . Therefore, we get an injective map If N is also a B(F q )-module then gr N is a gr A ⋊ T (F q )-module and this map also induces an injection on the space of T (F q ) fixed points: The associated graded module gr M is a gr A-module. Note that gr M might not coincide with gr M . However, in the cases we consider the two filtrations will give rise to the same module.
We have an isomorphism of algebras gr A ∼ = u(u Fq ⊗ Fp k) ∼ = u(u ⊕r ). If M is a rational B-module, then the linear isomorphism M → gr M is an isomorphism of u(u ⊕r )-modules, where the action on gr M is given by gr A, and the action of u(u ⊕r ) on M is the k-linear extension of the restriction of the rational action of U to u Fq regarded as a Lie algebra over F p (cf. [LN,Proposition 2.4] and [Fri,Theorem 4.3]). Put another way, u(u ⊕r ) acts on M via the surjection u(u ⊕r ) ։ u(u) discussed in Section 2.2 composed with the natural action of u(u) on M . In particular, the normal subalgebra of u(u ⊕r ) that is isomorphic to u(u) and that corresponds to the first component of the direct sum u ⊕r acts on M via the natural action of u(u) on M .
We can now give via Lie algebra cohomology an upper bound for the dimension of H 1 (G(F q ), M ) when M is in mod(G). Fq ) . Also, recall that gr A is isomorphic as a Hopf algebra to u(u Fq ⊗ Fp k). The isomorphism is a map of T (F q )-modules, where T (F q ) acts on U (F q ) by conjugation, and the action of T (F q ) on u(u Fq ⊗ Fp k) is described in Section 2.2.
From [LN,Theorem 3.2] we get the May spectral sequence .
2.5. Let N be a G-module, and let Γ denote the following composition of maps: For the last map we are identifying gr A with u(u ⊕r ), and considering the map in cohomology induced by the inclusion of u(u) into the first component of u(u ⊕r ) (i.e., induced by the inclusion of u into the first component of u ⊕r ). We have also identified the cohomology groups for u(u) with those for U 1 [Jan2, I.8.6, I.9.6]. We will next prove that if N = L(λ) * with λ ∈ X 1 (T ), then the composition of maps (2.5.1) fits into a commutative square of first cohomology groups.
Proof. First observe that if M is a rational B-module such that M is generated as a u(u)-module by a highest weight vector, then gr M = gr M as a u(u ⊕r )-module. This can be seen by analyzing the action of root subgroups (as in the proof of [LN,Proposition 2.4]) to show that the (weight) filtration coincides with the radical filtration on M . In particular, this applies when M = L(λ) with λ ∈ X 1 (T ).
The commutativity of the diagram reduces to proving that if is a nonsplit extension of rational B-modules, then the extension of gr A-modules is equivalent to the extension obtained via restriction (to u Fq ) of u(u ⊕r )-modules. Let φ : Q → L(λ) be the map of B-modules given above. Let φ 1 : Q → L(λ) denote the restriction of φ by considering Q and L(λ) as u(u ⊕r )-modules, and let φ 2 : gr Q → gr L(λ) be the induced map of gr A-modules, which is also a surjection. By the preceding paragraph we have gr Q = gr Q (because Q is generated by a highest weight vector, namely, the inverse image under φ of a highest weight vector in L(λ), or else the sequence splits) and gr L(λ) = gr L(λ). Let δ 1 : Q → gr Q (resp. δ 2 : L(λ) → gr L(λ)) be the isomorphism described in the preceding section. By checking on weight spaces (cf. [LN,Proposition 2.4]) one can show that we have a commutative diagram of gr A-modules: This proves the equivalence of the extensions.
Recall that Γ is a composition of maps. All the maps are injective except possibly the restriction map: res : . We shall prove that this map is injective under the assumption that λ ≤ ω j and q > 3. The statement clearly holds for r = 1, in which case res is the identity, so we can assume that r > 1.
Consider now the Lyndon-Hochschild-Serre (LHS) spectral sequence for u(u Fq ⊗ Fp k) ∼ = u(u ⊕r ) and its normal subalgebra u(u) (i.e., the subalgebra corresponding to the first component of u ⊕r ): . Then the 5-term exact sequence of the new spectral sequence has initial terms [ML,Theorem X.7.4]. In particular, 2 ) T (Fq) = 0, and consequently Γ is injective. 2.6. The injectivity of the map Γ allows us to state the following vanishing result that will be used throughout the paper. Jan2,II.4.11]. By Lemma 3.3.1 below, the weights of H 1 (U 1 , k) are simple roots, and none of these is T (F q )-invariant if q > 3. (The only T (F 3 )-invariant simple roots occur when Φ is of type A 1 or B 2 .) The vanishing of H 1 (G(F q ), k) now follows from Corollary 2.6.1(b). The vanishing of H 1 (G(F q ), k) can also be proved directly by an argument using the Frattini subgroup.
Because of the above remark, we may henceforth restrict our attention to λ = 0.

Cohomology for the Frobenius kernel U 1
In this section we study the cohomology group H 1 (U 1 , L(λ)) of Theorem 2.5.1. Eventually we will specialize to the case where λ is less than or equal to a fundamental dominant weight. Throughout this section, we maintain the standing assumption that p > 2.
3.1. Weight spaces in the socle of U 1 cohomology. Given λ ∈ X(T ), set λ * = −w 0 λ. Observe that the involution λ → λ * restricts to involutions on ∆ and X 1 (T ). In particular, if λ ∈ X 1 (T ) is less than or equal to a fundamental dominant weight, then so is λ * . Also, Ext 1 U 1 (L(λ), k) ∼ = Ext 1 U 1 (k, L(λ * )) = H 1 (U 1 , L(λ * )), so understanding the T (F q )-invariants in Ext 1 U 1 (L(λ), k) will enable us to apply Theorem 2.5.1 to study cohomology for G(F q ). Our first step is to analyze the socle of Ext 1 U 1 (L(λ), k) as a B/U 1 ∼ = (U/U 1 ) ⋊ T -module. Every simple rational B/U 1 -module is one-dimensional of T -weight −µ − pν for some µ ∈ X 1 (T ) and some ν ∈ X(T ). The dimension of the (−µ − pν)-isotypic component in the socle of a rational B/U 1 -module M is equal to dim Hom B/U 1 (−µ − pν, M ). Then to compute the socle of Ext 1 U 1 (L(λ), k) as a B/U 1 -module, it suffices to consider, for µ ∈ X 1 (T ) and ν ∈ X(T ), the dimensions of the Hom-spaces (3.1.1) Here we have used the fact that T 1 ∼ = B 1 /U 1 is a normal subgroup scheme in B/U 1 with quotient B/B 1 . The last isomorphism follows by applying the LHS spectral sequence for the group extension 1 → U 1 → B 1 → T 1 → 1 and using the fact that modules over T 1 ∼ = B 1 /U 1 are completely reducible.
It is interesting to note that the contribution to the socle of Ext 1 U 1 (L(λ), k) comes from two sources. The factors in the first direct summand seem to come from Kostant's classical theorem for the cohomology of complex semisimple Lie algebras, while for p large, the multiplicities dim Ext 1 G (L(λ), H 0 (σ)) of the other factors arise as coefficients of Kazhdan-Lusztig polynomials. In the special case when the Weyl module V (λ) is simple, we obtain the following corollary.
3.3. Constraints on weights. Next we examine the structure of M := Ext 1 U 1 (L(λ), k) as a B/U 1module. We are interested in cases for which the socle of M is equal to the entire module.
Lemma 3.3.1. Let V be a finite dimensional rational B-module. Let µ be a weight of T in Ext 1 U 1 (V, k) ∼ = H 1 (U 1 , V * ). Then µ = β − ν for some β ∈ ∆ and some weight ν of V .
Proof. First, if µ is a weight of T in H 1 (U 1 , V * ), then µ is also a weight of T in H 1 (U 1 , k) ⊗ V * by the argument in [UGA,§2.5]. Also, the weights of V * are precisely {−ν : V ν = 0}. Next, write Dist(U 1 ) + for the augmentation ideal of the algebra Dist(U 1 ). By inspecting the low degree terms in the cobar resolution computing H The T -weights of the latter space are precisely the simple roots in ∆.
For any µ ∈ X(T ), consider the injective hull I(µ) of µ in the category of rational B/U 1 -modules. Set Q = α∈∆ I(−s α · λ) ⊕ σ↑λ I(−σ) ⊕mσ . We have soc B/U 1 M = soc B/U 1 Q by Corollary 3.2.2, so there exists an injection M ֒→ Q. To show that soc B/U 1 M = M , it suffices to show that no weight from the second socle layer of Q can be a weight of M . (Recall that the second socle layer of Q is defined as soc B/U 1 (Q/ soc B/U 1 Q).) For µ ∈ X, one has I(µ) ∼ = k[U/U 1 ] ⊗ µ as a B/U 1 -module by [Jan2,I.3.11], where k[U/U 1 ] denotes the coordinate ring of the unipotent group U/U 1 . Lemma 3.3.2. The second socle layer of the B/U 1 -module I(µ) consists of one-dimensional modules of the form µ + p m γ with γ ∈ ∆ and m > 0.
Proof. It suffices to describe the second socle layer of the B/U 1 -module be a minimal injective resolution of the B/U 1 -module k. Then soc I 1 ∼ = soc(I(0)/ soc I(0)). Also, for all ν ∈ X(T ) and all i ≥ 0, one has Hom B/U 1 (ν, In particular, the weight ν occurs in the second socle layer of I(0) with multiplicity dim Hom B/U 1 (ν, I 1 ) = dim H 1 (B/U 1 , −ν). Now consider the LHS spectral sequence . It gives rise to the 5-term exact sequence Set Q Kos = α∈∆ I(−s α · λ), a submodule of Q. By Lemmas 3.3.1 and 3.3.2, if the second socle layer of Q Kos contains a vector of the same weight as a vector in M , then −s α · λ + p m γ = β − ν for some α, β, γ ∈ ∆, some m > 0, and some weight ν of L(λ). Equivalently, Since ν ≤ λ, the right-hand-side of (3.3.1) must be an element of NΦ + . Since α, β, γ are simple roots, this implies that β ∈ {α, γ}. Given J ⊆ ∆, let H 0 J (λ) be the induced module of highest weight λ for the standard Levi subgroup L J of G; see [Jan2,II.5.21]. Then [Jan2,II.5.21]. Suppose equation (3.3.1) holds. Then Since ν is a weight of L(λ), and hence also of H 0 (λ), we conclude from (3.3.2) that ν must be a weight of H 0 J (λ) for some J ⊆ ∆ with |J| ≤ 2. Now set Q KL = σ↑λ I(−σ) ⊕mσ . As before, if the second socle layer of Q KL contains a vector of the same weight as a vector in M , then for some dominant weight σ ∈ X(T ) + with σ ↑ λ, some γ, β ∈ ∆, some integer m > 0, and some weight ν of L(λ). Taking the inner product with the dual root γ ∨ , we get 3.4. Semisimplicity of U 1 cohomology. We can now give conditions under which the Ext-group Ext 1 U 1 (L(λ), k) is semisimple as a B/U 1 -module, that is, Ext 1 U 1 (L(λ), k) = soc B/U 1 Ext 1 U 1 (L(λ), k). Theorem 3.4.1. Let λ ∈ X(T ) + with λ ≤ ω j for some fundamental weight ω j . Assume that p > 2 if Φ has type A n , D n ; p > 3 if Φ has type B n , C n , E 6 , E 7 , F 4 ; p > 5 if Φ has type E 8 or G 2 . Then as a B/U 1 -module, where m σ = dim Ext 1 G (L(λ), H 0 (σ)). Proof. We show that no weight in the second socle layer of Q = Q Kos ⊕ Q KL can be a weight of M . First suppose that a weight from the second socle layer of Q Kos is a weight of M . Then by the discussion in Section 3.3, there exists a subset J = {α, β, γ} ⊆ ∆ with |J| ≤ 2, an integer m > 0, and a weight ν of H 0 J (λ) such that λ − ν = −β + (λ + ρ, α ∨ )α + p m γ. In Section 7.2 we consider the restriction of λ to all root subsystems of Φ of rank ≤ 2, and in each case compute all possible values for λ − ν. An elementary case-by-case analysis shows that, under the stated restrictions on p, the equation λ − ν = −β + (λ + ρ, α ∨ )α + p m γ has no solutions, so no weight in the second socle layer of Q Kos can be a weight of M . Now suppose that a weight from the second socle layer of Q KL is a weight of M . Note that Q KL = {0} in types A n and D n by Corollary 3.2.3, because in these types λ ≤ ω j implies λ = ω i for some i, and L(ω i ) = V (ω i ) (cf. Section 5.1). So we may assume that Φ is not type A n or D n . Then by (3.3.4), there exist simple roots β, γ ∈ ∆, a dominant weight σ ∈ X(T ) + with σ ↑ λ, an integer m > 0, and a weight ν of L(λ) such that −σ + p m γ = β − ν. Assume for the moment that Φ is not of type E 8 . Then by (3.3.6) and the stated assumption on p, Since σ ↑ λ implies σ ≤ λ, one gets (σ, γ ∨ ) ∈ {0, 1, 2, 3} from the list of possible values for σ in the Appendix (Section 7.1). Then necessarily (−ν, γ ∨ ) ≥ 5. Choose w ∈ W such that α := wγ is dominant (so α will be either the highest short root or the highest long root). Then, recalling that λ * = −w 0 λ and −w 0 ω j = ω i for some i, which is strictly less than 5, and leads to a contradiction. Finally, suppose Φ is of type E 8 with p > 5. Then the previous argument leads to a contradiction, because (3.4.1) becomes (−ν, γ ∨ ) + (σ, γ ∨ ) ≥ 12, whereas we still have (σ, γ ∨ ) ≤ 3, and (3.4.2) demonstrates that (−ν, γ ∨ ) ≤ 6.

4.
1. An isomorphism with G cohomology. The results of Section 3 enable us to give conditions under which the restriction map H 1 (G, L(λ)) → H 1 (G(F q ), L(λ)) is an isomorphism, and hence allow us to prove Theorem 1.2.1. For the sake of smoothness of exposition, we handle the case of Φ of type G 2 first as a separate result, because some subtleties arise there when treating the case p = 5. Our proof of Theorem 1.2.1 for type G 2 also establishes Theorems 1.2.2 and 1.2.3 for G 2 .
We now prove Theorem 1.2.1 for the remaining Lie types.
Proof of Theorem 1.2.1. The case when Φ is of type G 2 is handled by Theorem 4.1.1, so we assume for the remainder of the proof that Φ is not of type G 2 . By Remark 2.6.2 we may also assume λ = 0. By Theorem 3.4.1, where m σ = dim Ext 1 G (L(λ * ), H 0 (σ)). Note that λ * = −w 0 λ is again dominant and less than or equal to a fundamental dominant weight. Consider −s α · λ * = −λ * + (λ * + ρ, α ∨ )α. Consulting the lists, provided in Section 7.1, of dominant weights less than or equal to a fundamental dominant weight, and using the Cartan matrix to rewrite α as a sum of fundamental dominant weights, one can check that the coefficients of −s α · λ * are not all divisible by q − 1 when q > 3, and hence that the weight −s α · λ * does not contribute to the T (F q )-invariants in H 1 (U 1 , L(λ)). Consequently, Now observe that since λ * is less than or equal to a fundamental dominant weight, the only weight σ ∈ X(T ) + satisfying σ < λ * that gives a T (F q )-invariant in H 1 (U 1 , L(λ)) is the zero weight. Then dim H 1 (U 1 , L(λ)) T (Fq ) = m 0 = dim H 1 (G, L(λ)).
Theorem 4.2.1. Assume that p > 2 when Φ is of type A n or D n , p > 5 when Φ is of type E 8 , and p > 3 in all other cases. Assume that q > 3. Let λ ∈ X(T ) + with λ ≤ ω j for some fundamental dominant weight ω j . Then H 1 (G(F q ), L(λ)) = 0 if either L(λ) = H 0 (λ) or if λ is not linked to zero under the dot action of the affine Weyl group W p .

Type Minuscule Weights
Corollary 4.3.1. Assume that p > 2 when Φ is of type A n or D n , p > 5 when Φ is of type E 8 , and p > 3 in all other cases. Assume also that q > 3. Let ω j be a minuscule dominant weight. Then Proof. As ω j is not an element of the root lattice ZΦ, it cannot be linked to zero under the dot action of the affine Weyl group W p . Now apply Theorem 4.2.1.

Results for Classical Groups
5.1. Types A n , B n , D n . For the classical groups, the condition λ ≤ ω j implies that λ = 0 or that λ = ω i for some 1 ≤ i ≤ j. We proceed to verify Theorems 1.2.2 and 1.2.3 for the classical groups. For types A n , B n , and D n , if p > 2 then L(λ) = H 0 (λ) = V (λ) by [Jan2,II.8.21]. Then under the hypotheses of Theorems 1.2.2 and 1.2.3, H 1 (G(F q ), L(λ)) = 0 by Theorem 4.2.1.

5.2.
Non-vanishing in type C n . Assume that Φ has type C n with n ≥ 3. As noted in the Appendix, λ ≤ ω i implies that λ = ω j for some 0 ≤ j ≤ i (where ω 0 := 0). Kleshchev and Sheth [KS1,KS2], using results of Adamovitch on the submodule structure of the Weyl modules V (ω j ), completely determine the structure H 1 (G, L(ω j )). We formulate their result as follows.
Theorem 5.2.1 (Kleshchev-Sheth). Let Φ be of type C n with n ≥ 3 and p > 2. Write n + 1 = b 0 + b 1 p + · · · + b t p t with 0 ≤ b i < p and b t = 0. Then Proof. By [KS2,Corollary 3.6(ii)], H 1 (G, L(ω j )) is isomorphic to either 0 or k, and is isomorphic to k under precisely the following conditions: n + 1 − j = a 0 + a 1 p + · · · + a s p s where 0 ≤ a i < p, and j = 2(p − a i )p i for some i such that a i > 0 and either a i+1 < p − 1 or j < 2p i+1 . But if j = 2(p − a i )p i and a i > 0, then j = 2p i+1 − 2a i p i < 2p i+1 , so the "either. . . or. . . " condition is always true and thus superfluous. Writing j = (p − 2a i )p i + p i+1 , we have n + 1 = a 0 + . . . + a i p i + a i+1 p i+1 + · · · + a s p s This has the form given in the statement of the theorem, with 0 < b i = p − a i < p, j = 2b i p i , and at least one nonzero term beyond b i p i in the p-adic expansion of n + 1.
Combining Theorem 5.2.1 with Theorem 1.2.1 we obtain: Corollary 5.2.2. Let Φ be of type C n with n ≥ 3 and p > 3. Write n + 1 = b 0 + b 1 p + · · · + b t p t with 0 ≤ b i < p and b t = 0. Let λ ∈ X(T ) + with λ ≤ ω j for some j. Then The vanishing of H 1 (G(F q ), L(ω j )) for j odd and p > 3 can also be seen from Theorem 4.2.1, since for odd j the fundamental weight ω j is not in the root lattice, hence is not linked to zero under the dot action of the affine Weyl group W p . 6. Results for Exceptional Groups 6.1. Large prime vanishing. The results of Section 4 completely compute H 1 (G(F q ), L(λ)) when the underlying root system Φ is of classical type, when p > 3 (or p > 2 and q > 3 for types A n and D n ), and when λ is less than or equal to a fundamental dominant weight. For the exceptional types a number of open cases remain. The following lemma narrows down the list of remaining open cases to only finitely many values of p.
Set u = Lie(U ). By [FP,Proposition 1.1], there exists a spectral sequence of B-modules satisfying with E i,j 2 = 0 if i is odd. Then H 1 (U 1 , L(λ)) is a T -module subquotient of H 1 (u, L(λ)). We claim that H 1 (u, L(λ)) T = 0. The assumption p ≥ h + h λ − 1 implies for all β ∈ Φ + that (λ + ρ, β ∨ ) ≤ p. Then by [UGA,Theorem 4.2.1], the weights of T in H 1 (u, L(λ)) are precisely {−s α · λ * : α ∈ ∆}. 2 2 The introduction to the paper [UGA] states that the Borel subgroup B and its unipotent radical U should correspond to the set of negative roots in Φ. However, for the theorems to be correctly stated these groups and the Lie algebra u = Lie(U ) should correspond the set of positive roots in Φ. The weights we have listed here for H 1 (u, L(λ)) are the correct weights when the Lie algebra u corresponds to the set of negative roots in Φ.
Proof. If 3 < p ≤ 31 and p does not equal one of the primes listed next to λ in the Hasse diagram for Φ, then λ is not linked to zero under the dot action of the affine Weyl group W p . If p = 7, Φ = E 8 , and λ = ω 3 , then L(λ) = H 0 (λ) by [Jan1,§4.6]. In any case, H 1 (G(F q ), L(λ)) = 0 by Theorem 4.2.1.
When Φ is of exceptional type, the largest possible value for h + h λ − 1 in Lemma 6.1.1 is 35, which occurs for type E 8 when 3ω 8 ≤ λ ≤ ω 4 . Thus Lemma 6.1.1 and Proposition 6.2.1 show that the only cases of Theorems 1.2.2 and 1.2.3 which we have not thus far explicitly calculated are H 1 (G(F q ), L(λ)) when the underlying root system is of exceptional type and when the prime p is one of those appearing next to the weight λ in the Hasse diagram for Φ in Section 7.1.
6.3. Non-zero cohomology groups for types F 4 , E 7 , and E 8 . Let E be the Euclidean space spanned by Φ. Recall that the affine Weyl group W p is generated by the set of simple reflections {s α : α ∈ ∆} ⊂ GL(E) together with the affine reflection s 0 := s α 0 ,p , which is defined for λ ∈ E by Theorem 6.3.1. Suppose that one of the following conditions is satisfied: (1) Φ has type F 4 and p = 13, so s 0 · 0 = 2ω 4 ; (2) Φ has type E 7 and p = 19, so s 0 · 0 = 2ω 1 ; or (3) Φ has type E 8 and p = 31, so s 0 · 0 = 2ω 8 .
The following additional non-vanishing result holds for type E 7 .
In summary, for p > 3 (and p > 5 when Φ = E 8 ), and for λ less than or equal to a fundamental dominant weight, we have computed all cohomology groups H 1 (G(F q ), L(λ)) except for 2 cases in type E 7 , λ = 2ω 7 , p = 5, and λ = ω 2 + ω 7 , p = 7 indicated in Figure 2, and the 3 cases λ ∈ {2ω 7 , ω 1 + ω 7 , ω 2 + ω 8 } for p = 7 in type E 8 indicated in Figure 3. In all the cases we have computed, we have found dim H 1 (G(F q ), L(λ)) ≤ 1. 7. Appendix 7.1. Hasse diagrams for fundamental weights. In this section we describe the restriction of the partial ordering ≤ on X(T ) to the dominant weights λ ∈ X(T ) + satisfying λ ≤ ω j for some fundamental dominant weight ω j . If Φ has classical type A n , B n , C n or D n , then all such λ are themselves fundamental dominant weights (or 0). Type A n . Each fundamental dominant weight ω j is minimal with respect to ≤.
Exceptional types. The Hasse diagrams for the exceptional types are given in Figures 1-5. If the weight µ appears below and is connected to the weight λ by a line, then µ < λ. White boxes with black borders indicate that the given weight is conjugate (i.e., linked) to 0 under the dot action of the affine Weyl group W p for the primes shown. Linkage relations were verified for 5 ≤ p ≤ 31 using the computer program GAP [GAP].   7.2. Weights of induced modules. In this section we analyze the weights ν of induced modules H 0 (τ ), when the root system has rank ≤ 2. Given a dominant weight τ , and given a weight ν of H 0 (τ ), one has τ − ν = θ for some θ ∈ NΦ + . For each possible restriction τ of a weight λ occurring in Section 7.1 to a rank one or rank two root subsystem, we list the corresponding possible values for ν and θ. When Φ has rank 2, write ∆ = {α 1 , α 2 }, and let ω 1 and ω 2 denote the corresponding fundamental dominant weights.
Type A 1 . In this case there is only a single fundamental dominant weight ω. Write ∆ = {α}.
Type A 1 × A 1 . The weights of H 0 (ω 1 + ω 2 ) and H 0 (2ω 1 + ω 2 ) are given in Table 3. The weights of H 0 (ω 1 ), H 0 (2ω 1 ), and H 0 (3ω 1 ) can be deduced from the data for Type A 1 in Table 2. Since the situation is symmetric with respect to the ordering of the fundamental dominant weights, the weights of H 0 (ω 2 ), H 0 (2ω 2 ), and H 0 (3ω 2 ) can also be deduced from the data in Table 2.
Type B 2 . Assume that α 1 is long and that α 2 is short. The weights of H 0 (ω 1 ) and H 0 (ω 2 ) are given in Table 5.
Type G 2 . Assume that α 1 is short and that α 2 is long. The weights of H 0 (ω 1 ) and H 0 (ω 2 ) are given in Table 6.