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Non-G-completely reducible subgroups of the exceptional algebraic groups

Let G be an exceptional algebraic group defined over an algebraically closed field k of characteristic p>0 and let H be a subgroup of G. Then following Serre we say H is G-completely reducible or G-cr if, whenever H is contained in a parabolic subgroup P of G, then H is in a Levi subgroup of that parabolic. Building on work of Liebeck and Seitz, we find all triples (X,G,p) such that there exists a closed, connected, simple non-G-cr subgroup H


Introduction
Let G be an algebraic group defined over an algebraically closed field k of characteristic p > 0 and let H be a subgroup of G. Then following Serre [Ser98] we say H is G-completely reducible or G-cr if, whenever H is contained in a parabolic subgroup P of G, then H is in a Levi subgroup of that parabolic. This is a natural generalisation of the notion of a group acting completely reducibly on a module V : if we set G = GL(V ) then saying H is G-completely reducible is precisely the same as saying that H acts semisimply on V .
This notion is important in unifying some other pre-existing notions and results. For instance, in [BMR05], it was shown that a subgroup H is G-cr if and only if it satisfied Richardson's notion of being strongly reductive in G. It also allows one to state some previous results due to Liebeck-Seitz and Liebeck-Saxl-Testerman on the subgroup structure of the exceptional algebraic groups in a particularly satisfying form: Assume G is simple of one of the five exceptional types and let X be a simple root system. The result [LS96, Theorem 1] asserts a number N (X, G) such that if H is closed, connected and simple, with root system X, then H is G-cr whenever the characteristic p of k is bigger than N (X, G). In particular if p is bigger than 7 then they show that all closed, connected, reductive subgroups of G are G-cr. There is some overlap in that paper with the contemporaneous work of [LST96]. If H is a simple subgroup of rank greater than half the rank of G, then [Theorem 1, ibid.] finds all conjugacy classes of simple subgroups of G, the proofs indicate where these conjugacy classes are G-completely reducible. With essentially one class of exceptions, all subgroups, including the non-G-cr subgroups, can be located in 'nice' so-called subsystem subgroups of G. We shall mention these in greater detail later.
More recently, [Ste10a] and [Ste12] find all conjugacy classes of simple subgroups of exceptional groups of types G 2 and F 4 . One consequence of this is to show that the numbers N (X, G) found above can be made strict. (One need only change N (A 1 , G 2 ) from 3 to 2.) The main purpose of this article is to make all the N (X, G) strict. That is, for each of the five types of exceptional algebraic group G, for each prime p = char k and for each simple root system X, we give in a table of Theorem 1 an example H = E(X, G, p) of a connected, closed, simple non-G-cr subgroup H 1 with root system X, precisely when this is possible. In other words we classify the triples (X, G, p) where there exists a connected, closed, simple non-G-cr subgroup H with root system X. Moreover, in all but one case (where (X, G, p) = (G 2 , E 7 , 7)), we can locate E(X, G, p) in a subsystem subgroup.
Our main theorem can thus be viewed as the best possible improvement of the result [LS96, Theorem 1], in the spirit of that result. Before we state our main theorem in full, we need a definition: A subsystem subgroup of G is a simple, closed, connected subgroup Y which is normalised by a maximal torus T of G. Let Φ be the root system of G corresponding to a choice of Borel subgroup B ≥ T and for α ∈ Φ, let U α denote the T -root subgroup corresponding to α. Then Y = U α |α ∈ Φ 0 where either Φ 0 is a closed subsystem of Φ or (Φ, p) is (B n , 2), (C n , 2), (F 4 , 2) or (G 2 , 3) and Φ 0 lies in the dual of a closed subsystem. The subsystem subgroups of G are easily determined by the Borel-de Siebenthal algorithm. Most of our examples H = E(X, G, p) are described in terms of an embedding of H into a subsystem subgroup M . Here we describe M just by giving its root system. Theorem 1. Let G be an exceptional algebraic group defined over an algebraically closed field k of characteristic p > 0. Suppose there exists a non-G-cr closed, connected, simple subgroup H of G with root system X. Then (X, G, p) has an entry in Table 1.
Conversely, for each (X, G, p) given in Table 1, the last column guarantees an example of a closed, connected, simple, non-G-cr subgroup E(X, G, p) with root system X.
In particular we can improve on [LS96, Theorem 1]. In the table in Corollary 2 we have struck out the primes which were used in the hypotheses in [loc. cit.]. This is done partly to show where we have made improvements but mainly to facilitate reading the proof of the first part of Theorem 1. B 4 2 E ≤ A 8 ; V 9 ↓ E = 1000/0000 Table 1. Simple non-G-cr subgroups of type X in the exceptional groups a reductive algebraic group G, containing a maximal torus T of G. Recall that for each dominant weight λ ∈ X + (T ) for G, the space H 0 (λ) := H 0 (G/B, λ) = Ind G B (λ) is a G-module with highest weight λ and with socle Soc G H 0 (λ) = L(λ), the irreducible G-module of highest weight λ. The Weyl module of highest weight λ is V (λ) ∼ = H 0 (−w 0 λ) * where w 0 is the longest element in the Weyl group. We identify X(T ) with Z r for r the rank of G and for λ ∈ X(T ) + ∼ = Z r ≥0 ≤ X(T ), write λ = (a 1 , a 2 , . . . , a r ) = a 1 ω 1 + · · · + a r ω r where ω i are the fundamental domninant weights; a Z ≥0 -basis of X(T ) + . Put also L(λ) = L(a 1 , a 2 , . . . , a r ). When 0 ≤ a i < p for all i, we say that λ is a restricted weight and we write λ ∈ X 1 (T ). Recall that any module V has a Frobenius twist V [n] induced by raising entries of matrices in GL(V ) to the p n th power. Steinberg's tensor product theorem states that L(λ) = L(λ 0 ) ⊗ L(λ 1 ) [1] ⊗ · · · ⊗ L(λ n ) [n] where λ i ∈ X 1 (T ) and λ = λ 0 + pλ 1 + · · · + p n λ n is the p-adic expansion of λ ∈ Z r + . We refer to λ 0 as the restricted part of λ.
The right derived functors of Hom(V, * ) are denoted by Ext i G (V, * ) and when V = k, the trivial G-module, we have the identity Ext i G (k, * ) = H i (G, * ) giving the Hochschild cohomology groups. We recall some standard modules; when G is classical, there is a 'natural module' which we refer to by V nat ; or V m where m is the dimension of V nat . It is always the Weyl module V (ω 1 ), which is irreducible unless p = 2 and G is of type B n ; in the latter case it has a 1-dimensional radical. Certain properties of these modules is described in [Jan03,8.21]. Of importance to us is the fact that when G = SL n , r (L(ω 1 )) = L(ω r ) for r ≤ n − 1. We use this fact without further reference.
We will often want to consider restrictions of simple G-modules to reductive subgroups H of G. Where we write V 1 |V 2 | . . . |V n we list the composition factors V i of an H-module. For a direct sum of H-modules, we write V 1 + V 2 . Where a module is uniserial, we will write V 1 / . . . /V n to indicate the socle and radical series: here the head is V 1 and the socle V n . On rare occasions we use V /W to indicate a quotient. It will be clear from the context which is being discussed.
Recall also the notion of a tilting module as one having a filtration by modules V (µ) for various µ and also a filtration by modules H 0 (µ) for various µ (equiv. dual Weyl modules). Let us record in a lemma some key properties of tilting modules which we use: Lemma 2.1.
(i) For each λ ∈ X(T ) + there is a unique indecomposable tilting module T (λ) of high weight λ; (ii) A direct summand of a tilting module is a tilting module; (iii) The tensor product of two tilting modules is a tilting module; As we are considering very low weight representations in general, it is possible to spot that a module is a T (λ); for instance when p = 2, the natural Weyl module for B n has a 1-dimensional radical, so its structure is W (λ 1 ) = L(λ 1 )/k. It is then the case that giving the Loewy series for a module k/L(λ 1 )/k uniquely characterises it as a tilting module T (λ 1 ).
Recall that a parabolic subgroup P of G has a Levi decomposition, where Q is the unipotent radical of the P . Recall also L = L ′ Z(L) with L ′ being semisimple.

Outline
Theorem 1 has two facets. The first proves that if p ∈ N (X, G) for N (X, G) as defined in Corollary 2, then X is G-cr. The second proves the existence of the examples given in Table 1 and proves that they are non-G-cr.
The proof of the first part runs along the same lines as that of [ From [ABS90], Q has a filtration Q = Q 1 ≥ Q 2 ≥ Q 3 . . . with successive quotients being known (usually semisimple) L-modules.
Now, for an exceptional algebraic group G over k of characteristic p and a simple root system X we consider possible embeddingsH ≤ L ′ whereH is an L ′ -irreducible subgroup (which can be determined using 4.8 and/or by working down through 4.9). The composition factors V of the restrictions of the L-modules Q i /Q i+1 are investigated, and then conditions for the vanishing of For the proof of the second part of Theorem 1, we must show that for each of the remaining cases (where some composition factor V of Q [1] has H 1 (H, V ) = 0), we exhibit a non-G-cr subgroup H with the required root system over the required characteristic. In almost all cases we can give an example in a classical subgroup of G. Here is it easy to see when it is in a parabolic subgroup using 4.8. In two cases this is not possible, yet we can assert the existence of such a group using a cohomological argument.

Preliminaries
One needs to be careful about the notion of complements in semidirect products of algebraic groups. These are treated systematically in [McN10]. We recall some of the main facts. A closed subgroup H ′ of G is a complement to Q if it satisfies the following equivalent conditions: for a discussion. Note that [LS96] uses item (iv) above as its definition of a complement, without the last condition on Lie algebras.
. We write Z 1 (H, Q) for the set of 1-cocycles.
We say γ ∼ δ if there is an element q ∈ Q(k) with q −h γ(h)q = δ(h) for each h ∈ H(k). We write H 1 (H, Q) for the set of equivalence classes of 1-cocycles Z 1 (H, Q)/ ∼. 2 We use the data on Frank Lübeck's website which accompanies [Lüb01].
We recall some results from [Ste10b]. In almost all cases the cohomology group H 1 (G, V ) for a semisimple algebraic group G satisfies ). This fact allows us to reduce our considerations to simple modules with non-trivial restricted parts.
Lemma 4.5. Let G be a simple algebraic group and V a simple G-module. There are many papers finding the values Ext n H (L, M ) with H of low rank and L, M simple. Taking L = k, the trivial module, one gets the following result, where we have included more data than necessary for our purposes for completion's sake.
Lemma 4.6. Let V be a simple module for a simple algebraic group H where H is one of SL 2 , SL 3 , Sp 4 over an algebraically closed field of any characteristic p; G 2 for p = 2, 3 or p ≥ 13; or SL 4 , Sp 6 or Sp 8 when p = 2. Then H 1 (H, V ) is at most one-dimensional, and is non-zero if and only if V is a Frobenius twist of one of the modules in the following table.
In the table we also give some useful dimension data, often in only specific characteristic.

Proof of Theorem 1
In [Ste10a] and [Ste12] we find all semisimple non-G-cr subgroups of G where G is G 2 and F 4 respectively. So the result follows for these cases. It remains to deal with the cases G = E 6 , E 7 and E 8 . We start by honing the Liebeck and Seitz result to show that if H is a closed, connected, simple subgroup of G with root system X and p is not in our list N (X, G) then H is G-cr. Then we check that the examples given in Table 1 are indeed non-G-cr.
Corollary 5.2. With the hypotheses of the lemma, let V be an L ′ -composition factor of Q and suppose L ′ does not contain a component of type A 1 . Then either dim V ≤ 60 or G = E 8 , L ′ = D 7 and V is a spin module for L ′ of dimension 64.
Proof. If L ′ is itself simple, this follows from the lemma. Also, if G = E 6 or E 7 then the number of positive roots is less than 56, so the result is clear. So we may assume G = E 8 . The possibilities for L are A 2 A 2 , A 2 A 3 , A 2 A 4 , A 3 A 3 , A 3 A 4 , A 2 D 4 and A 2 D 5 . Since V is simple, it must be a tensor product of simple modules for the two factors, with the simple modules occurring in the lemma. One checks that the highest dimension possible for this is when L = A 3 A 4 , V = L(λ 2 ) ⊗ L(λ 2 ) with dim V = 6 × 10 = 60.
For the second part, if G = E 7 and L ′ is simple this follows from Lemma 5.1, the largest case occurring when L ′ = A 6 . If L ′ is not simple, then it is A 4 A 2 , A 3 A 2 or A 2 A 2 . Then the largest possible dimension comes from the first option and is at most 10 × 3 = 30 ≤ 35-dimensional.
p ∈ N (X, G) implies that H is G-cr. Since we are building on [LS96, Theorem 1], we need only deal with the struck out numbers in the table in Corollary 2.

Proof of the first statement of Theorem 1:
Looking for a contradiction, we will assume H is non-G-cr; then we can make the following assumption, using 4.4: We Suppose H is of type B 2 and p = 3. SinceH is D 7 -irreducible, it must have act on the natural module V 14 for L ′ as specified in 4.8. Checking [Lüb01], one finds the simple untwisted representations of dimension no more than 14 are L(0, 1), L(1, 0), L(0, 2), L(2, 0) with dimensions 4, 5, 10 and 14, respectively. But L(0, 1) is the natural representation for Sp 4 , thus carries a symplectic structure, which cannot be non-degenerate. Hence V 14 ↓H = L(2, 0); moreover, as L(2, 0) is an irreducible Weyl module when p = 3, the embeddingH ֒→ L ′ can be seen as the reduction mod p of an embeddingH Z ֒→ L ′ Z . Now [LS96, Proposition 2.12] gives that V Z ↓H Z is the irreducible Weyl module V (1, 3). Using [Lüb01] one can calculate the composition factors of a reduction mod 3 of this module; one sees that V ↓H has composition factors L(1, 3)|L(2, 1)|L(0, 1). Since none of these modules appears in 4.6, this rules out (X, G, p) = (B 2 , •, 3).
By 5.2 the largest possibility for the dimension V when G = E 7 is 35; when G = E 6 it is 16.
For (A 2 , E 8 , 5), the fact that V has dimension at least 54 forces L ′ = E 7 , D 7 or A 7 but simple E 7and D 7 -modules are self-dual, so the possibilities for V coming from 4.6 are discounted as they are not self-dual. Thus we may assume that L ′ = A 7 and V = L(ω 3 ) = 3 (L(ω 1 )). SinceH is L ′ -ir, L ′ must act irreducibly on the natural 8-dimensional module V 8 for L ′ . A check of [Lüb01] forces V 8 |L ′ = L(1, 1). But 3 L(1, 1) has highest weights (2, 2) and (0, 3). But the weights appearing in 4.6 are all higher than these (in the dominance order). This rules out (A 2 , E 8 , 5).
Since there are no embeddings of a subgroup of type C 4 into any proper Levi of E 6 , this case is ruled out too.
This completes the proof of the first statement of Theorem 1.
p ∈ N (X, G) implies the existence of a non-G-cr subgroup H with root system X. The examples when G = G 2 and F 4 were shown already in [Ste10a, Theorem 1] and [Ste12, Theorem 1(A)(B)] to be non-G-cr, so we need only deal with the cases G = E 6 , E 7 and E 8 .
L(1, 1) + k 6 + L(1, 0) 4 + L(0, 1) 4 , so V 27 ↓ H = L(1) ⊗ L(1) + L(1) 8 + k 7 = T (2) + L(1) 8 + k 7 . In [Ste12] it is shown that H is in a long A 1 -parabolic of F 4 , hence H is in an A 1 -parabolic of E 6 (so that L ′ of type A 1 ). But V 27 ↓ L ′ = L(1) 9 + k 9 and so H is not GL(V 27 )-conjugate to (a subgroup of) L ′ , let alone E 6 -conjugate. Now To show it is also non-E 8 -cr, note that L(E 8 ) ↓ E 7 = L(E 7 ) + L(T 1 ) + L(Q) 2 where Q is the unipotent radical of an E 7 -parabolic of E 8 , with L(Q) ↓ E 7 = V 56 + k. Thus L(E 8 ) ↓ H contains at least two submodules isomorphic to T (2) (contained in the two V 56 s). On the other hand L(E 8 ) ↓ L ′ = L(A 1 T 1 ) + k 6 + M 2 where M is the restriction to L ′ of the Lie algebra of the unipotent radical of an A 1 -parabolic. Using 5.1, M has composition factors with high weights 1 or 0, which must be semisimple since Ext 1 A 1 (L(1), L(1)) = Ext 1 A 1 (L(1), L(0)) = 0. In particular, while the direct summand L(A 1 T 1 ) is an indecomposable module T (2) for L ′ , it is the only one in L(E 8 ); for H there are at least two such (in L(Q)). Thus H is also non-E 8 -cr.
(G, X, p) = (E 6 , A 2 , 3) Let τ denote a graph automorphism of G with induced action on the Dynkin diagram for G. If G τ denotes the fixed points of τ in G, we have G τ ∼ = F 4 such that the root groups corresponding to simple short roots are contained in the subsystem (of type A 2 A 2 ) determined by the nodes in the Dynkin diagram of G on which τ acts non trivially. Thus H is contained in A 2Ã2 ≤ F 4 by x → (x, x). It is shown in [Ste12,4.4.1,4.4.2] that this subgroup is in a B 3 -parabolic of F 4 with V 7 ↓H = L A 2 (11).
In [Ste12,5.1] the restrictions of the F 4 -module V 26 = V (0001) ∼ = 0001/0000 to H andH is calculated. Using this together with V 27 ↓ F 4 = T (0001) = 0000/0001/0000 we see that V 27 ↓H cannot be the same as V 27 ↓ H: the former is an extension by the trivial module of V 26 ↓H = 11 3 + 00 5 where the resulting module is self-dual, so must be 11 3 + 00 6 whereas the latter is forced to be T (11) 3 . By a similar argument as before, we also get that this subgroup is non-E 7 -cr and non-E 8 -cr.
We give an example of a subgroup not arising from a non-F 4 -cr subgroup (these being found in [Ste12]): (G, X, p) = (E 6 , A 1 , 5).
The remaining cases where X = A 1 are similar.
Let us now vouch for the existence of the subgroup asserted in case (G, X, p) = (E 8 , C 3 , 3).
There is one further case where we could not give a nice embedding as we have done above. Let H = (E 7 , G 2 , 7).
Suppose H had a proper reductive overgroup in G. Then by 4.9 it would have to lie in a subsystem subgroup of type A 7 . Also it cannot lie in any parabolic subgroup of A 7 since thenH would not be E 6 -irreducible. Checking [Lüb01] one sees that there are no irreducible 8-dimensional representations of H ∼ = G 2 . This is a contradiction. Thus H has no proper reductive overgroup in G as required.
The remaining cases are all similar and easier. This completes the proof of Theorem 1.