A note on Humphreys' conjecture on blocks

Humphreys' conjecture on blocks parametrises the blocks of reduced enveloping algebras $U_\chi({\mathfrak g})$, where ${\mathfrak g}$ is the Lie algebra of a reductive algebraic group over an algebraically closed field of characteristic $p>0$ and $\chi\in{\mathfrak g}^{*}$. It is well-known to hold under Jantzen's standard assumptions. We note here that it holds under slightly weaker assumptions, by utilising the full generality of certain results in the literature. We also provide a new approach to prove the result for ${\mathfrak g}$ of type $G_2$ in characteristic 3, a case in which the previously mentioned weaker assumptions do not hold. This approach requires some dimensional calculations for certain centralisers, which we conduct in the Appendix for all the exceptional Lie algebras in bad characteristic.


Introduction
One of the most powerful tools in representation theory is the notion of the block decomposition.Given a finite-dimensional K-algebra A, the block decomposition of A gives a partition of the set of irreducible A-modules.We may then study the representation theory of A block-by-block.In particular, if we wellunderstand one block (say, a block containing a trivial module) then we can often use translation functors to gain insight into the structure of other blocks.
If G is an algebraic group over an algebraically closed field K of characteristic p > 0 and g is its Lie algebra, then we may form, for each χ ∈ g * , the reduced enveloping algebra U χ (g).This is a finite-dimensional K-algebra which is important to the representation theory of g, and so we would like to understand its blocks.The leading result in this direction is Humphreys' conjecture on blocks.
Conjecture (Humphreys' conjecture on blocks).Suppose G is reductive and let χ ∈ g * be nilpotent.Then there exists a natural bijection between the blocks of U χ (g) and the set Λ χ /W • .In particular, Here, Λ χ is a certain finite subset of h * , where h is the Lie algebra of a maximal torus T of G, and W is the Weyl group of (G, T ), which acts on h * via the dot-action and thus induces an equivalence relation on Λ χ .The requirement that χ is nilpotent means that χ vanishes on the Lie algebra b of a Borel subgroup B of G (which we may assume contains T ).
This conjecture was proved by Humphreys [8] in 1971 for χ = 0, subject to the requirements that G be semisimple and that p > h, where h is the Coxeter number of (G, T ).Humphreys then extended the result further to χ in so-called standard Levi form in 1998 in [9] (the paper [9] doesn't explicitly state what assumptions are being made, but the argument holds for any connected reductive algebraic group whose derived group is simply-connected).Under three assumptions (which we will call Jantzen's standard assumptions [12,13] and denote (A), (B) and (C)), the conjecture was then proved by Brown and Gordon in [2] for all χ ∈ g * when p > 2, and then improved by Gordon in [7] to include the p = 2 case (so long as (A), (B) and (C) still hold).In fact, under assumptions (A), (B) and (C), Humphreys' conjecture on blocks allows us to count the number of blocks of U χ (g) for all χ ∈ g * , as these assumptions are sufficient to reduce the computation to the case of nilpotent χ (see [5], also Remark 3,infra).Furthermore, Braun [1] recently proved the conjecture for g = sl n with p|n, where assumptions (A) and (B) hold but (C) doesn't.In this case, however, the restriction to nilpotent χ is necessary, as the analogous result for semisimple χ was shown in [1] to fail when p = n = 3.
Let us now explain Jantzen's standard assumptions.These are: (A) that the derived group of G is simplyconnected; (B) that the prime p is good for G; and (C) that there exists a non-degenerate G-invariant bilinear form on g.The primes that are not good for a given G can be listed explicitly (and are all less that or equal to 5), and the existence of a non-degenerate G-invariant bilinear form on g holds whenever g is simple.
The question motivating this note is: what happens to Humphreys' conjecture on blocks for nilpotent p-characters if we remove assumptions (B) and/or (C)?We see in Section 3 that there is a natural surjection f : {Blocks of U χ (g)} → Λ χ /W • under only assumption (A).It turns out that this can be deduced from the literature [13,15].Furthermore, we show in Theorem 4.1 that the known proof of the injectivity of f works without assumption (B).This therefore confirms Humphreys' conjecture for the almost-simple groups over algebraically closed fields of the following bad characteristics: Corollary 1.1.Let G be an almost-simple group over an algebraically closed field K of bad characteristic p > 0. Then Humphreys' conjecture holds for G when p = 2 and G is of type E 6 , E 8 or G 2 , when p = 3 and G is of type E 7 , E 8 or F 4 , and when p = 5 and G is of type E 8 .
We also provide a different approach to the proof of the injectivity in Proposition 4.2, which demonstrates that injectivity in fact holds whenever there exists a collection of irreducible modules of a certain nice form (namely, which are so-called baby Verma modules).Premet's theorem [17] shows the existence of such irreducible modules under assumptions (A), (B) and (C), and we observe in Corollary 4.3 that the existence also holds for the almost-simple algebraic group of type G 2 in characteristic 3 (where assumption (C) fails).This thus proves Humphreys' conjecture on blocks for G 2 in characteristic 3, which could not be deduced using the previous approach.
In the Appendix, we conduct some calculations with a view to finding other examples where these irreducible modules exist.Unfortunately, the calculations do not lead to further examples, but we hope the calculations are interesting in their own right, as they demonstrate divisibility bounds for irreducible modules for certain nice χ and small primes.
Statements and Declarations: The author was supported during this research by the Engineering and Physical Sciences Research Council, grant EP/R018952/1, and later by a research fellowship from the Royal Commission for the Exhibition of 1851.
Acknowledgments: The author would like to thank Ami Braun and Dmitriy Rumynin for suggesting this question, and Simon Goodwin for engaging in many useful discussions regarding this subject and comments on earlier versions of this paper.

Preliminaries on Lie algebras
Throughout this note we work with a connected algebraic group G over an algebraically closed field K of characteristic p > 0.More precise assumptions on G are given section-by-section, but it is always at least a reductive algebraic group with simply-connected derived subgroup.Inside G, we fix a maximal torus T and a Borel subgroup B of G containing T .Write X(T ) for the character group of T , Y (T ) for the cocharacter group of T , and •, • : X(T ) × Y (T ) → Z for the natural pairing.We write g for the Lie algebra of G, b for the Lie algebra of B and h for the Lie algebra of T .As Lie algebras of algebraic groups these are all restricted, so come equipped with p-th power maps g → g (resp.b → b, h → h) written x → x [p] .
Set Φ to be the root system of G with respect to T , Φ + to be the positive roots corresponding to B and Π to be the simple roots.For α ∈ Φ we set α ∨ ∈ Y (T ) to be the corresponding coroot, and we write g α for the root space of α in g.We then define n + = α∈Φ + g α and n − = α∈Φ + g −α , so g = n − ⊕ h ⊕ n + .For α ∈ Φ we define h α := dα ∨ (1) ∈ h, and we choose e α ∈ g α and e −α ∈ g −α so that [e α , e −α ] = h α (see, for example, [14] for more details on this procedure).We also choose a basis h 1 , . . ., h d of h with the property that h Set W to be the Weyl group of Φ, which acts naturally on X(T ) and h * .We fix ρ ∈ X(T ) ⊗ Z Q to be the half-sum of positive roots in Φ.This then allows us to define the dot-action of W on X(T ) as w • λ = w(λ + ρ) − ρ (noting that this action makes sense even if ρ / ∈ X(T )).When ρ ∈ X(T ), dρ(h α ) = 1 for all α ∈ Π.If ρ / ∈ X(T ), we may still define dρ ∈ h * such that dρ(h α ) = 1 for all α ∈ Π, since the derived subgroup being simply-connected implies that these h α are linearly independent in h.We may therefore define the dot action on h * similarly to how it was defined on X(T ).When we wish to specify that W is acting through the dot-action, we may write W • instead of W .
We write U (g) for the universal enveloping algebra of g.We write Z p for the central subalgebra of U (g) generated by all x p − x [p] with x ∈ g, which we call the p-centre of U (g).Given χ ∈ g * , we write U χ (g) for the reduced enveloping algebra U χ (g) := U (g)/ x p − x [p] − χ(x) p | x ∈ g .Each irreducible g-module is finitedimensional [12,Theorem A.4] and so, by Schur's lemma, each irreducible g-module is a U χ (g)-module for some χ ∈ g * .For χ ∈ g * , we recall that the centraliser of χ in g is defined as c g (χ The adjoint action of G on g induces the coadjoint action of G on g * , and if χ, µ ∈ g * lie in the same coadjoint G-orbit then U χ (g) ∼ = U µ (g).The derived group of G being simply-connected implies (see [13,15]) that any µ ∈ g * lies in the same G-orbit as some χ ∈ g * with χ(n + ) = 0. Putting these two observations together, we always assume χ(n + ) = 0 throughout this paper.
We can define, for each λ ∈ h * , a one-dimensional b-module K λ on which n + acts as zero and h acts via λ.The assumption that χ(n + ) = 0 means that K λ is a U χ (b)-module if and only if λ ∈ Λ χ , where and that all irreducible U χ (b)-modules are of this form.We therefore may define the baby Verma module Z χ (λ) = U χ (g) ⊗ Uχ(b) K λ , a U χ (g)-module of dimension p N , where N = |Φ + |.Every irreducible U χ (g)module is the quotient of some baby Verma module (see [12,Lem. B.4]).
Since W • acts on h * , we may define an equivalence relation on Λ χ by setting λ ∼ µ if and only if there exists w ∈ W with w • λ = µ.We write Λ χ /W • for the set of equivalence classes of Λ χ under this relation. If In this case, W • in fact acts on Λ χ , so Λ χ /W • is the set of W • -orbits for this action.The condition that χ(b) = 0 is sufficiently important in this paper that we make the definition b If I = Π we say that χ is regular nilpotent in standard Levi form.In general, we say χ ∈ g * is regular nilpotent if it is in the same G-orbit as the µ ∈ g * which is regular nilpotent in standard Levi form.

Preliminaries on Blocks
Let us briefly recall the definition of the blocks of a finite-dimensional K-algebra A (one can find more details in [3, I.16, III.9], for example).We say that one irreducible A-module M is linked to another irreducible A-module N if Ext 1 (M, N ) = 0.This is not an equivalence relation, but we may refine it to one.The equivalence classes under the resulting equivalence relation are then called the blocks of A.
In this note, we are concerned with the case of A = U χ (g) with χ ∈ b ⊥ .Under assumptions (A), (B) and (C) the results in this section are well-known -for example, they are contained within the proof of Proposition C.5 in [12].Nonetheless, we recall them to highlight when assumptions (A), (B) and (C) are or are not necessary.Remember from Section 2 that each irreducible U χ (g)-module is a quotient of a baby Verma module Z χ (λ), and thus all irreducible U χ (g)-modules appear as composition factors of baby Verma modules.Recall also that the Grothendieck group G (U χ (g)) of the category of finite-dimensional U χ (g)-modules is the abelian group generated by symbols [M ], for M running over the collection of all finite-dimensional U χ (g)-modules, subject to the relation that where Irr(U χ (g)) is the set of isomorphism classes of irreducible U χ (g)-modules and [Z χ (λ) : L] indicates the composition multiplicity of L in Z χ (λ).
We wish to define the map as follows.Let B be a block of U χ (g), and let E be an irreducible module in this block.There must exist For this to be well-defined, it is necessary to see that it does not depend on our choice of E ∈ B or on our choice of Z χ (λ) ։ E. For this, we note that U (g) G ⊆ Z(U (g)) acts on the baby Verma module Z χ (λ) via scalar multiplication as follows.Under the assumption that the derived group of G is simply-connected (assumption (A)), the argument of Kac and Weisfeiler in [15, Th. 1] (c.f.[13, Th. 9.3]) shows that there exists an isomorphism π : U (g) G → S(h) W• , where the dot-action on S(h) is obtained by identifying S(h) with the algebra P (h * ) of polynomial functions on h * and then defining (w • F )(λ) = F (w −1 • λ) for w ∈ W , F ∈ P (h * ) and λ ∈ h * .This isomorphism allows us, as in [13], to define a homomorphism cen λ : U (g) G → K which sends u ∈ U (g) G to π(u)(λ), viewing π(u) as an element of P (h * ).Then U (g) G acts on Z χ (λ) via the character cen λ , for λ ∈ Λ 0 .
If E and E ′ lie in the same block then it is easy to see that U (g) G must act the same on both modules, and if G acts on E via cen λE and on E ′ by cen λ E ′ .Thus, cen λE = cen λ E ′ and so, as in [13, Cor.9.4] (see also [15,Th. 2]), we have . This shows that f is well-defined.Furthermore, f is clearly surjective (just take the block containing an irreducible quotient of the desired Z χ (λ)).
The above discussion also shows that Thus, there is a bijection In particular, we get the following proposition (which also may more-or-less be found in [12, C.5]), observing that at no point thus far have we required assumptions (B) or (C).
Proposition 3.1.Let G be a connected reductive algebraic group over an algebraically closed field K of characteristic p > 0, with simply-connected derived subgroup, and let χ ∈ b ⊥ .Then there exists a natural surjection between the set of blocks of U χ (g) and the set Remark 1.We have used in the above argument the fact that, when assumption (A) holds, there exists an isomorphism U (g) G ∼ − → S(h) W• .This result dates back to Kac and Weisfeiler [15], who proved it for connected almost-simple algebraic groups under the assumption that G = SO 2n+1 (K) when p = 2. 1 According to Janzten [13, Rem.9.3], the argument of Kac and Weisfeiler holds for reductive g whenever assumption (A) holds.Jantzen further gives an argument [13, 9.6] using reduction mod p techniques which holds under his standard assumptions.In fact, slightly weaker assumptions are sufficient: assumption (B) is only needed to ensure p is not a so-called torsion prime of Φ ∨ (in the sense of [4,Prop. 8]), which is also satisfied for the bad prime 3 in case G 2 , while assumption (C) is only needed to ensure that the (derivatives of the) simple roots are linearly independent in h * , which is also satisfied for p = 2 in type F 4 and p = 3 in type G 2 .In particular, the argument of Kac-Weisfeiler is unnecessary for our later result (Corollary 4.3) that Humphreys' conjecture on blocks holds for the almost-simple algebraic group of type G 2 in characteristic 3.

Upper bound
Humphreys' conjecture on blocks claims that the map f defined in the previous section is, in fact, a bijection.What remains, therefore, is to show that Gordon [7] has shown that this inequality holds under assumptions (A), (B) and (C), and a similar argument is reproduced in [12, C.5].We give a version of this argument here in order to observe that it does not require assumption (B), and to highlight where assumption (C) is necessary: The discussion in Section 3 shows that U χ (g) has |Λ 0 /W • | blocks if, for each λ ∈ Λ 0 , all composition factors of the baby Verma module Z χ (λ) lie in the same block.This property holds for the µ ∈ g * which is regular nilpotent in standard Levi form, since the corresponding baby Verma module has a unique maximal submodule and so is indecomposable, and it is well-known that all composition factors of an indecomposable module lie in the same block.Therefore U χ (g) has |Λ 0 /W • | blocks for all χ in the G-orbit of µ.
Suppose now that the intersection of b ⊥ with the G-orbit of µ is dense in b ⊥ .By [6, Prop.2.7], Humphreys' conjecture on blocks would follow.
When can we say that (G • µ) ∩ b ⊥ is dense in b ⊥ ?Well, if there exists a G-equivariant isomorphism Θ : g ∼ − → g * , we can set y := Θ −1 (µ).Then [11, 6.3, 6.7] (which make no assumptions on p) establish that the G-orbit of y is dense in the nilpotent cone N of g, and so the G-orbit of µ is dense in Θ(N ).Thus, (G • µ) ∩ b ⊥ is dense in b ⊥ , and so (cf.[7,Th. 3.6]) under assumptions (A) and (C) we get Humphreys' conjecture of blocks: Theorem 4.1.Let G be a connected reductive algebraic group over an algebraically closed field K of characteristic p > 0, with simply-connected derived subgroup.Suppose that there exists a G-module isomorphism Θ : Remark 2. It is straightforward to see that this theorem implies Corollary 1.1 from the Introduction.
Remark 3. Suppose χ ∈ g * with χ(n + ) = 0.Under assumption (C), there exists a G-module isomorphism Θ : g → g * , so we may fix x ∈ g such that Θ(x) = χ.In g it is well-known that x has a (unique) Jordan decomposition x = x s + x n , where x s is semisimple, x n is nilpotent, and [x s , x n ] = 0, and thus we may define the Jordan decomposition χ = χ s + χ n where χ s = Θ(x s ) and χ n = Θ(x n ).In fact, Kac and Weisfeiler [15,Th. 4] show that a Jordan decomposition of χ may be defined even when assumption (C) does not hold, so long as assumption (A) does instead: we say that χ = χ s + χ n is a Jordan decomposition if there exists Under assumptions (A) and (B), Friedlander and Parshall [5] show that there is an equivalence of categories between {U χ (g) − modules} and {U χ (c g (χ s )) − modules} (the categories of finite-dimensional modules).It can then further be shown under those assumptions (see, for example, [12, B.9]) that there is an equivalence of categories between {U χ (c g (χ s )) − modules} and {U χn (c g (χ s )) − modules}.Under assumptions (A) and (B), this then often allows us to reduce representation-theoretic questions to the case of nilpotent χ.
When assumption (C) holds, we may do this for Humphreys' conjecture on blocks (we assume here that χ is chosen so that g may be taken as 1 in the definition of the Jordan decomposition, recalling that reduced enveloping algebras are unchanged by the coadjoint G-action on their corresponding p-character).The equivalence of categories between {U χ (g) − modules} and {U χn (l) − modules} (where l := c g (χ s )) clearly preserves the number of blocks of the respective algebras.Thus, Humphreys' conjecture on blocks for (l, χ n ) will imply it for (g, χ) if and only if , where W ′ is the Weyl group corresponding to l.What is W ′ ?Well, the root system for l is {α ∈ Φ | χ s (h α ) = 0} so it is easy to see that W ′ lies inside W (Λ χ ), the set of w ∈ W which fix Λ χ setwise (it is straightforward to see under our assumptions that it doesn't matter in defining this subgroup whether we consider the usual action or the dot-action of W , since ρ ∈ Λ 0 ).When assumption (C) holds, W (Λ χ ) is parabolic (see [16,Lem. 7], [10, Prop.1.15]), and so one can easily check that W ′ = W (Λ χ ) in this case (see [2, Rem.3.12(3)]).This then obviously implies that • as required.Braun [1, Th. 6.23, Ex. 6.25] has shown that when assumption (C) fails to hold, it can be the case that Humphreys' conjecture on blocks holds for nilpotent χ but fails for general χ.Specifically, set g = sl 3 , p = 3, and choose χ ∈ sl * 3 such that χ(e 11 − e 22 ) = χ(e 22 − e 33 ) = 0 (using e ij for the usual basis elements of gl 3 ).Recalling that the Weyl group for sl 3 is the symmetric group S 3 , one can check that W (Λ χ ) = {Id, (1, 2, 3), (1, 3, 2)} and so is not a parabolic subgroup of W . Thus, W ′ = W (Λ χ ) and so there can be linkages under W which do not exist under W ′ .In particular, choosing suitable χ, one can use this to show that Braun's argument then shows that the latter value is the number of blocks of U χn (l) and so the number of blocks of U χ (g).We note that this argument highlights that [16,Lem. 7] requires the assumption that p be very good for the root system.
The argument above highlights one approach to proving Humphreys' conjecture on blocks; namely, to obtain the desired result it suffices to find a dense subset of b ⊥ lying inside D |Λχ/W•| .Note that b ⊥ = K N , where N = |Φ + |, and recall that any non-empty open subset is dense in K N when it is equipped with the Zariski topology.For each λ ∈ Λ 0 , define Finding the desired C λ therefore provides an approach to proving Humphreys' conjecture on blocks, and in the rest of this section we explore one particular way of obtaining such C λ .
Let us consider a bit further the condition that m ∩ c g (χ) = 0. Let x ∈ m ∩ c g (χ).We can then write The fact that x ∈ c g (χ) means that χ([x, g]) = 0.This is equivalent to the requirement that χ([x, e β ]) = 0 for all β ∈ Φ and χ([x, h]) = 0 for all h ∈ h.Let ∆ be the subset of Φ − such that χ(e α ) = 0 for α ∈ ∆.We then have, for Showing that m ∩ c g (χ) = 0 then involves showing that there is no non-zero solution to these equations in c γ .We now turn to the application of these propositions.In each case, we take χ to be regular nilpotent in standard Levi form and we apply one of the propositions or its corollaries to determine a divisibility bound for the dimensions of U χ (g)-modules.We do this for Φ of exceptional type.Principally, we compute the centraliser c g (χ) and use its description to determine the bound.For Φ = G 2 we give the explicit computations, but for the larger rank examples the results were obtained using Sage [A7].Because of this, when there is a choice we take the structure coefficients to be as used in the Sage class LieAlgebraChevalleyBasis with category.However, we use the labelling of the simple roots as given in [A8].
Remark 7. Our computations of dim c g (χ) can be compared with the computations of dim c g (e) for e = α∈Π e α which can be deduced from [A8, Cor.2.5, Thm.2.6].The results are listed in Table A. When g is simple, χ and e are identified through the G-equivariant isomorphism g ∼ − → g * , and thus c g (χ) = c g (e).In the subsections below, we nonetheless include calculations of c g (χ) for the bad primes for which g is simple, since we give explicit bases for the centralisers in these case and in some instances we use such bases to show the reducibility of the corresponding baby Verma modules.In the other cases (which we label with an asterisk (*) in Table ??), however, we find that the dimensions of c g (e) and c g (χ) differ from each other.Note also that we give in Table ?? the dimension of c g (χ) for G 2 in characteristic 3, even though we do not give it in Subsection A.2 below, because it is easy to compute.Remark 8.In our discussion of g so far, the Lie algebra g of G has been obtained as g = g Z ⊗ Z K, where g Z is a Z-form of the complex simple Lie algebra g C .In particular, g Z is the Z-form coming from the chosen Chevalley basis of g C , which is what gives our Chevalley basis of g.We may then also define g Fp = g Z ⊗ Z F p , so that g = g Fp ⊗ Fp K. Therefore, if χ Fp : g Fp → F p is a linear form, we may define χ : g → K by linear extension.It is clear that any χ in standard Levi form may be obtained in this way.Our calculations in Sage are calculations with g Fp and χ Fp rather than g.However, when χ is obtained through scalar extension from an F p -linear form, the above discussion shows that determining the elements of g which lie in c g (χ) comes down to finding solutions to certain linear equations with coefficients in F p .This in particular shows that c g Fp (χ Fp ) ⊗ Fp K = c g (χ), so our calculations over F p also lead to the results over K.
A.1.G 2 in characteristic 2. Suppose Φ = G 2 and p = 2. Since p is non-special in this case, we may apply Proposition A.1.Let us therefore compute c g (χ).Set x ∈ g be written as x = γ∈Φ c γ e γ + γ∈Π d γ h γ , with the c γ , d γ lying in K. Then the relations required for x ∈ c g (χ) are as follows: 2 (14 − 4) = 5, and so by Proposition A.1 we conclude that every finite-dimensional U χ (g)-module has dimension divisible by 2 5 .
We furthermore note that a U χ (g)-module of dimension 2 5 does indeed exist in this case.Let λ ∈ Λ 0 be such that λ(h β ) = 0, and let us write ω 1 ∈ Λ 0 for the map with ω 1 (h α ) = 1 and ω 1 (h β ) = 0. We may then define a U χ (g)-module homomorphism This has a kernel of dimension 2 5 and so both the kernel and image of this homomorphism are U χ (g)-modules of dimension 2 5 .
A.2. G 2 in characteristic 3. Suppose Φ = G 2 and p = 3.Note that this Lie algebra is not simple, since it has an ideal generated by the short roots.In this case p is not non-special for Φ so we cannot apply Proposition A.1.Instead, we want to apply Proposition A.2, and so we need to find an appropriate m.Take m = n − .In this case, Ψ = Φ + is closed and χ(e −γ−δ ) = 0 for all γ, δ ∈ Ψ.In the notation of the previous discussion, we have ∆ = {−α, −β}.Let x = γ∈Φ + c α e −α .Then the relations required for x ∈ c g (χ) are as follows: It is easy to see that these relations force x = 0, so m ∩ c g (χ) = 0. Hence, Proposition A.2 shows that every finite-dimensional U χ (g)-module has dimension divisible by 3 6 , which is 3 dim n − .So in this case each baby Verma module Z χ (λ) is irreducible.A.3.F 4 in characteristic 2. Set Φ = F 4 and p = 2. Since p is not non-special in this case we need to use Proposition A.2; in fact, we use Corollary A.4. Set m to be the subspace of n − with basis given by the elements e −α for α ∈ Ψ := Φ + \ {α 2 + 2α 3 }.It is straightforward to see that Ψ is 2-closed.We want to see that m ∩ c g (χ) = 0. We do this by giving a basis of c g (χ) as follows:  In particular, dim(c g (χ)) = 16 and so d(χ) = 116 = |Φ + | − 4. Proposition A.1 then says that all finitedimensional U χ (g)-modules have dimension divisible by 2 116 .A.10. E 8 in characteristic 3. Suppose Φ = E 8 and p = 3.Since p is non-special in this case, we may apply Proposition A.1.We must therefore give c g (χ), and Sage computations show that c g (χ) is the K-subspace of g with the following basis: (    In particular we see that dim c g (χ) = 10, and so d(χ) = 119 = |Φ + | − 1.Hence, every finite-dimensional U χ (g)-module has dimension divisible by 5 119 .
are in the same block of U χ (g) }, and define C := λ∈Λ0 C λ .It is straightforward from the arguments in Section 3 to see that C ⊆ D |Λχ/W•| .Furthermore, if for each λ ∈ Λ 0 we can find a dense open subset C λ of b ⊥ with C λ ⊆ C λ , then C := λ∈Λ0 C λ would be a dense open subset of b ⊥ contained in C ⊆ D |Λχ/W•| .

Remark 4 .
From the discussion in Section 3, it is sufficient to check the condition of Proposition 4.2 for representatives λ ∈ Λ 0 /W • .