THE JACOBSON–MOROZOV THEOREM AND COMPLETE REDUCIBILITY
OF LIE SUBALGEBRAS

arXiv:1507.06234v4 [math.RT] 2 Oct 2017

DAVID I. STEWART AND ADAM R. THOMAS*

Abstract. In this paper we determine the precise extent to which the classical sl2 -theory of
complex semisimple finite-dimensional Lie algebras due to Jacobson–Morozov and Kostant can be
extended to positive characteristic. This builds on work of Pommerening and improves significantly
upon previous attempts due to Springer–Steinberg and Carter/Spaltenstein. Our main advance
arises by investigating quite fully the extent to which subalgebras of the Lie algebras of semisimple
algebraic groups over algebraically closed fields k are G-completely reducible, a notion essentially
due to Serre. For example, if G is exceptional and char k = p ≥ 5, we classify the triples (h, g, p) such
that there exists a non-G-completely reducible subalgebra of g = Lie(G) isomorphic to h. We do
this also under the restriction that h be a p-subalgebra of g. We find that the notion of subalgebras
being G-completely reducible effectively characterises when it is possible to find bijections between
the conjugacy classes of sl2 -subalgebras and nilpotent orbits and it is this which allows us to prove
our main theorems.
For absolute completeness, we also show that there is essentially only one occasion in which a
nilpotent element cannot be extended to an sl2 -triple when p ≥ 3: this happens for the exceptional
orbit in G2 when p = 3.

1. Introduction
The Jacobson–Morozov theorem is a fundamental result in the theory of complex semisimple Lie
algebras, due originally to Morozov, but with a corrected proof by Jacobson. One way to state it
is to say that for any complex semisimple Lie algebra g = Lie(G), there is a surjective map
(*)

{conjugacy classes of sl2 -triples} −→ {nilpotent orbits in g},

induced by sending the sl2 -triple (e, h, f ) to the nilpotent element e. That is, any nilpotent element
e can be embedded into some sl2 -triple. In [Kos59], Kostant showed that this can be done uniquely
up to conjugacy by the centraliser Ge of e; i.e. that the map (*) is actually a bijection. Much work
has been done on extending this important result to the modular case, that is where g = Lie(G) for
G a reductive algebraic group over an algebraically closed field k of characteristic p > 0. We mention
some critical contributions. In [Pom80], Pommerening showed that under the mild restriction that
p is a good prime, one can always find an sl2 -subalgebra containing a given nilpotent element, but
this may not be unique; in other words, the map (*) is still surjective, but not necessarily injective.
If h(G) denotes the Coxeter number1 of G, then in [SS70] Springer and Steinberg prove that the
uniqueness holds whenever p ≥ 4h(G) − 1 and in his book [Car93], Carter uses an argument due
to Spaltenstein to establish the result under the weaker condition p > 3h(G) − 3; both proofs
go essentially by an exponentiation argument. One major purpose of this article is to finish this
2010 Mathematics Subject Classification. 17B45.
*Supported by an LMS 150th Anniversary Postdoctoral Mobility Grant 2014-2015 Award.
1In case the root system of G is trivial, we define h(G) to be 0.
1

project and improve these bounds on the characteristic optimally, thus to state precisely when the
bijection (*) holds.
Theorem 1.1. Let G be a connected reductive group over an algebraically closed field k of characteristic p > 22 and let g be its Lie algebra. Then (*) is a bijection if and only if p > h(G).3
In fact, we will do even more than this, also determining when there is a bijection
(**)

{conjugacy classes of sl2 -subalgebras} −→ {nilpotent orbits in g},

and when a bijection exists, we will be able to realise it in a natural way. The equivalence of
bijections (*) and (**) is easily seen in large enough characteristics by exponentiation, but there
are quite a few characteristics where there exists a bijection (**), but not (*). To state our result,
we define for any reductive group the number b(G) as the largest prime p such that the Dynkin
diagram of G contains a subdiagram of type Ap−1 or p is a bad prime for G. Alternatively, b(G)
is the largest prime which is not very good for some Levi subgroup of G. If G is classical of type
(An , Bn , Cn , Dn ) we have b(G) is the largest prime which is no larger than (n + 1, n, n, n) and if G
is exceptional of type (G2 , F4 , E6 , E7 , E8 ) then b(G) = (3, 3, 5, 7, 7). If G is reductive then b(G) is
the maximum over all simple factors and is 0 in case the root system of G is trivial.
Theorem 1.2. Let G be a connected reductive group over an algebraically closed field k of characteristic p > 2 and let g be its Lie algebra. Then the number of conjugacy classes of sl2 -subalgebras
and nilpotent orbits is the same if and only if p > b(G). Moreover, when p > b(G), there is a natural
bijection (**) realised by sending an sl2 -subalgebra h to the nilpotent orbit of largest dimension that
intersects h non-trivially.
(To emphasise our improvement, [Car93, Thm. 5.5.11] gives the existence of such a bijection for E8
when p > 87, whereas we require just p > 7.)
For many applications, the Kempf–Rousseau theory of optimal cocharacters (whose consequences
were worked out in [Pre03]) is a sufficient replacement for much of the sl2 -theory one would typically
employ when working over C—indeed, this paper uses cocharacter theory quite extensively. But
it should not be a surprise that the unique smallest simple Lie algebra over k should continue to
play a role in modular Lie theory. We are aware of at least one example where our results are
likely to be used: on considering a maximal subgroup H of a finite group of Lie type G(q), one
frequently discovers the existence of a unique 3-dimensional submodule on the adjoint module that
must correspond to an sl2 -subalgebra of g. Then Theorem 1.2 promises certain useful properties of
this subalgebra which can be exploited to show that H lives in a positive-dimensional subgroup of
the ambient algebraic group G; typically this implies it is not maximal.
The question of the existence of bijections (*) and (**) turns out to be intimately connected to
J.-P. Serre’s notion of G-complete reducibility [Ser05]. Say a subgroup H of G is G-completely
reducible if whenever H is contained in a parabolic subgroup P of G, then H is contained in a
Levi subgroup L of P . The notion is inspired by a general philosophy of Serre, Tits and others to
generalise concepts of representation theory by replacing homomorphisms of groups H → GL(V )
2Note that if p = 2 then sl is non-simple and the question of finding subalgebras containing given nilpotent
2

elements becomes murky since one might consider it proper to consider the non-simple non-isomorphic subalgebras
pgl2 in addition.
3Interestingly, this theorem gives an optimal answer as to when the secondary demands of [DeB02, Hypothesis
4.2.5] are met; however it is known to the authors that [DeB02, Hypothesis 4.2.4] on which it is dependent holds for
every nilpotent orbit only under strictly worse conditions.
2

with homomorphisms of groups H → G, where G is any reductive algebraic group. Indeed, when
G = GL(V ), using the description of the parabolic subgroups and Levi subgroups of G as stabilisers
of flags of subspaces of V , the idea that a subgroup H is G-completely reducibly recovers the usual
idea of H acting completely reducibly on a representation V . There is a remarkably widespread web
of connections between G-complete reducibility and other areas of mathematics, such as geometric
invariant theory, the theory of buildings and the subgroup structure of algebraic groups, amongst
other things. In our proofs of Theorem 1.1 and 1.2 we will find yet another connection with Serre’s
notion, this time with the study of modular Lie algebras.
The natural extension of Serre’s idea to Lie algebras is due to McNinch, [McN07] and is developed
further in [BMRT13]. We say a subalgebra h of g is G-completely reducible (or G-cr) if whenever
h is contained in a parabolic subalgebra p of g, then h is in a Levi subalgebra of that parabolic.
(Recall that a parabolic subalgebra is by definition Lie(P ) for P a parabolic subgroup and a Levi
subalgebra is Lie(L) for L a Levi subgroup of P .) We will establish the following result, crucial for
our proof of Theorem 1.1.
Theorem 1.3. Let G be a connected reductive algebraic group over an algebraically closed field k
of characteristic p > 2. Then all semisimple subalgebras of g are G-completely reducible if and only
if p > h(G).
The proof of Theorem 1.3 reduces easily to the case where G is simple. Then work of S. Herpel and
the first author in [HS16b] on complete reducibility of representations of semisimple Lie algebras
can be adapted to prove the theorem when G is classical. The bulk of the work involved is showing
the result when G is exceptional. Let then G be an exceptional algebraic group. At least thanks
to [HS16a] and some work of A. Premet together with the first author, the isomorphism classes of
semisimple Lie subalgebras of the exceptional Lie algebras are known in all good characteristics.4
Our following theorem gives, for p ≥ 5 (in particular, for p a good prime) a full description of when
a simple subalgebra in one of those known isomorphism classes can be non-G-cr.
Theorem 1.4. Suppose h is a simple subalgebra of g = Lie(G) for G a simple algebraic group of
exceptional type over an algebraically closed field k of characteristic p ≥ 5. Then either h is G-cr
or one of the following holds:
(i) h is of type A1 and p < h(G);
(ii) h is of type W (1; 1), p = 7 and G is of type F4 ; or p = 5 or 7 and G is of type E6 , E7 or
E8 ;
(iii) Up to isomorphism we have (G, h, p) = (E7 , G2 , 7), (E8 , B2 , 5) or (E8 , G2 , 7).
Moreover, for each exception (G, h, p) above, there exists a non-G-cr subalgebra of the stated type.
Since we consider Lie algebras g = Lie(G) for G an algebraic group, g inherits a [p]-map arising
from the Frobenius morphism on the group. Then a subalgebra h of g is a p-subalgebra if and
only if it is closed under the [p]-map. Asking when p-subalgebras are G-cr gives a slightly different
answer, with an important connection to the existence of the bijection (**).
4Up to isomorphism, one gets only Lie algebras coming from algebraic groups and the first Witt algebra W (1; 1)

of dimension p, together with some semisimple subalgebras which are not the direct sum of simple Lie algebras,
existing only when p = 5 or 7 and g is of type E7 or E8 . An example of such a Lie algebra is the semidirect product
sl2 ⊗ (k[X]/hX p i) + 1 ⊗ W (1; 1) where the factor W (1; 1) commutes with the sl2 factor but acts by derivations on
the truncated polynomial ring k[X]/hX p i.
3

Theorem 1.5. Suppose h is a simple p-subalgebra of g = Lie(G) for G a simple algebraic group of
exceptional type over an algebraically closed field k of characteristic p ≥ 5. Then either h is G-cr
or one of the following holds:
(i) h is of type A1 and p ≤ b(G);
(ii) h is of type W (1; 1), p = 7 and G is of type F4 ; or p = 5 or 7 and G is of type E6 , E7 or
E8 ;
(iii) Up to isomorphism we have (G, h, p) = (E7 , G2 , 7), (E8 , B2 , 5) or (E8 , G2 , 7).
Moreover, for each exception, there exists a non-G-cr p-subalgebra of the stated type.
To appreciate fully the connection of Theorems 1.3 and 1.4 with Theorem 1.1 we will see that the
failure of the uniqueness part of the Jacobson–Morozov theorem to hold in characteristics less than
or equal to the Coxeter number h(G) comes exactly from the failure of some subalgebras isomorphic
to sl2 to be G-cr. (And this is precisely how we construct examples of extra conjugacy classes of
sl2 subalgebras when p ≤ h(G).) Moreover, so long as G contains neither a factor of type G2 when
p = 3 nor a factor of type Ap−1 , then the bijection (**) in Theorem 1.2 exists precisely when there
is an equivalence
H is G-completely reducible ⇐⇒ H is reductive
for all connected reductive subgroups H of G.
Another result concerns a connection between Seitz’s idea of subgroups of type A1 being good with
the study of modular Lie algebras. Recall from [Sei00] that a closed subgroup H of type A1 of an
algebraic group G is good if it has weights no bigger than 2p − 2 on the adjoint module. Again, this
idea forms part of the philosophy of generalising concepts of representation theory from GL(V ) to
other reductive groups. This time, Seitz’s notion gives us the correct generalisation of the notion
of a restricted representation of H := SL2 : If H acts with weights less than p on V , then it gives a
good A1 -subgroup of GL(V ), since H will have weights no more than 2p − 2 on gl(V )|H ∼
= V ⊗ V ∗.
In ibid. Seitz proves in particular that all unipotent elements of order p have a good A1 -overgroup
and that any two such overgroups are conjugate; this itself connects to questions raised in Serre’s
fundamental paper [Ser05] by providing a solution to finding overgroups of unipotent elements
which are so-called ‘saturated’. Our result is as follows.
Theorem 1.6. Let G be a connected reductive algebraic group over an algebraically closed field
of characteristic p > 2. Then every sl2 subalgebra of G is Lie(H) for H a good A1 if and only if
p > h(G).
Lastly, for completeness we have checked the following, improving the Jacobson–Morozov theorem
itself optimally, using the classification of nilpotent orbits in characteristic p ≥ 3.
Theorem 1.7. Let G be a simple algebraic group over an algebraically closed field of characteristic
p ≥ 3. Then any nilpotent element e ∈ g = Lie(G) belonging to the orbit O can be extended to an
(3)
sl2 -triple if and only if (G, p, O) 6= (G2 , 3, Ã1 ).
Acknowledgements. We thank Sasha Premet for some discussion and the referee for helpful
suggestions for improvement.
4

2. Preliminaries
2.1. Notation. In the following G will be a reductive algebraic group over an algebraically closed
field k of characteristic p > 2, and g will be its Lie algebra.
Throughout the paper we use the terms classical and exceptional when referring to both simple
algebraic groups and simple Lie algebras. When we say a simple Lie algebra is classical (or of
classical type) we mean that it is of type A–G. However, for a simple algebraic group, we use the
term classical to mean of type A–D, and exceptional otherwise.
All notation unless otherwise mentioned will be consistent with [Jan03]. In particular, all our
reductive groups are assumed to be connected. The root system R contains a simple system S
whose elements will be denoted αi , with corresponding fundamental dominant weight ̟i . We shall
denote roots in R by their coefficients in S labelled consistently with Bourbaki. For a dominant
weight λ = a1 ̟1 +a2 ̟2 +· · ·+an ̟n we write λ = (a1 , a2 , . . . , an ) and write L(λ) = L(a1 , a2 , . . . , an )
to denote the irreducible module of highest weight λ. Given modules M1 , . . . , Mk , the notation
V =
P
M1 | . . . |Mk denotes a module with a socle series as follows: Soc(V ) ∼
= Mk and Soc(V / j≥i Mj ) ∼
=
Mi−1 for k ≥ i > 1. We write M1 + M2 for M1 ⊕ M2 . We also write T (λ) for a tilting module of
high weight λ for an algebraic group G. In small cases, the structure of these is easy to write down.
For example, when V (λ) ∼
= L(λ)|L(µ), we have T (λ) ∼
= L(µ)|L(λ)|L(µ). The module VE6 (̟1 ) will
be denoted V27 and the module VE7 (̟7 ) denoted V56 .
When G is simple and simply-connected, we choose root vectors in g for a torus T ⊆ G and a
basis for t = Lie(T ) coming from a basis of subalgebras isomorphic to sl2 corresponding to each
of the simple roots. We write these elements as {eα : α ∈ R} and {hα : α ∈ S} respectively.
As g = Lie(G), we have that g inherits a [p]-map x 7→ x[p] , making it a restricted Lie algebra;
see [Jan03, I.7.10].
Recall also the first Witt algebra W (1; 1) := Derk (k[X]/X p ), henceforth denoted W1 . The Lie
algebra W1 is p-dimensional with basis {∂, X∂, . . . , X p−1 ∂} and commutator formula [X i ∂, X j ∂] =
(j − i)X i+j−1 ∂. In §4.3 we use a little of the representation theory of W1 . All that we need is
contained in [BNW09] for example. In particular, Derk (k[X]/X p ) is a module with structure S|k
where S is an irreducible module of dimension p − 1.
2.2. Parabolic subalgebras. Let P = LQ be a standard parabolic subgroup of an exceptional
algebraic group G with unipotent radical Q and Levi complement L, corresponding to a subset J
of S. In particular, letting RJ = R ∩ ZJ, we have P = hUα , T | α ∈ R+ ∪ RJ i. In this section
we discuss the structure of Q in terms of the action of L. Forgetting the Lie algebra structure on
q := Lie(Q), we obtain a module for l := Lie(L). We will see that if h is a subalgebra of l such that
q has no h-composition factors V with non-trivial first cohomology (which we will recall shortly)
then all complements to q in the semidirect product of Lie algebras h + q are conjugate to h by
elements of Q, hence all are G-cr.
The unipotent radical has (by [ABS90]) a well-known central filtration Q = Q1 ≥ Q2 ≥ . . . with
successive factors Qi /Qi+1 isomorphic to the direct product
of all root
P
P groups corresponding to the
set Φi of roots of level i, where the level of a root α = i∈S ci αi is i∈S\J ci , via the multiplication
Q
map π : Ga × · · · × Ga → G; (t1 , . . . , tn ) 7→ α∈Φi xα (ti ). The filtration {Qi } is L-stable and the
quotients have the structure of L-modules. That is, they are L-equivariantly isomorphic to the Lmodule Lie(Qi /Qi+1 ) = Lie(Qi )/ Lie(Qi+1 ), as is verified in [ABS90]. Moreover it is straightforward
to compute the L-composition factors of each subquotient; see [LS96, Lem. 3.1]. One observes all of
5

the high weights are restricted when p is good for G (and for p = 5 when G = E8 ). We may therefore
immediately conclude by differentiating the L-modules concerned that the same statement is true
of the l-composition factors of the l-module Lie(Qi )/ Lie(Qi+1 ). The following lemma records this.
Lemma 2.1. Let g be the Lie algebra of a simple exceptional algebraic group G in good characteristic
(or p = 5 when G = E8 ) and let p = l + q be a parabolic subalgebra of g. The l-composition factors
within q have the structure of high weight modules for l. If l0 = Lie(L0 ) for L0 a simple factor of
L, then the possible high weights λ of non-trivial l0 -composition factors are as follows:
(i) l0
if
(ii) l0
(iii) l0
(iv) l0
(v) l0

= An : λ = 2̟1 , 2̟n , 3̟1 , ̟j or ̟n+1−j (j = 1, 2, 3) (note that 2̟1 , 2̟n only occur
g = F4 and n ≤ 2 and 3̟1 only if g = G2 and n = 1);
= Bn or Cn (n = 2 or 3, g = F4 ): λ = ̟1 , ̟2 or ̟3 ;
= Dn : λ = ̟1 , ̟n−1 or ̟n ;
= E6 : λ = ̟1 or ̟6 ;
= E7 : λ = ̟7 .

We therefore find the following restrictions on the dimensions of l-composition factors of q (hence
also on the h-composition factors of q).
Corollary 2.2. With the hypothesis of the lemma, let V be an l-composition factor of q. Then
dim V ≤ 64.
Proof. This follows from the lemma if l′ is simple. Moreover, if g 6= E8 then the number of positive
roots is at most 56 and the result follows. So suppose g = E8 . The product of the dimensions
of the possible simple factors is at most 64 in all cases, except for l′ of type A1 A6 for which a
module L(1) ⊗ L(̟3 ) has dimension 2 × 35 = 70. However, an easy calculation shows the actual l′ composition factors are L(1)⊗L(̟2 ), L(̟4 ), L(1)⊗L(̟6 ) and L(̟1 ). Hence the largest dimension
of any l′ -composition factor is 42.

We recall a concrete description of the 1-cohomology of Lie algebras; see [Wei94, §7.4]. Let h be a
Lie algebra and V an h-module. A 1-cocycle is a map ϕ : h → V such that
(1)

ϕ([x, y]) = x(ϕ(y)) − y(ϕ(x)).

Let Z 1 (h, V ) denote the set of 1-cocycles. For v ∈ V the map h → V : x 7→ x(v) is a 1-cocycle
called a 1-coboundary; denote these by B 1 (h, V ). Two 1-cocycles are equivalent if they differ by a
1-coboundary; explicitly ϕ ∼ ψ if there is some v ∈ V such that ϕ(x) = ψ(x) + x(v) for all x ∈ h.
In this case we say ϕ and ψ are conjugate by v. One then has H1 (h, V ) = Z 1 (h, V )/B 1 (h, V ).
Note that V can be considered as a Lie algebra with trivial bracket. Then one may form the
semidirect product h + V . A complement to V in the semidirect product h + V is a subalgebra h′
such that h′ ∩ V = 0 and h′ + V = h + V . Just as for groups, one has a vector space isomorphism
Z 1 (h, V ) ←→ {complements to V in h + V } ,
by ϕ 7→ {x + ϕ(x) : x ∈ h}.
We wish to realise the conjugacy action of V on h + V in terms of a group action. Suppose
dim V = n. For our purposes it will do no harm to identify h with its image in gln . Furthermore,
it will be convenient to embed V into gln+1 as strictly upper-triangular matrices with non-zero
6

entries only in the last column, viz.





V := 



0


∗
0
∗

..  .
..
.
.

0 ∗
0

Then the action of GLn on V is realised as conjugation of the block diagonal embedding of GLn
in GLn+1 via x 7→ diag(x, 1). Clearly adding the identity endomorphism of V commutes with the
action of GLn . Hence the group Q := 1 + V is GLn -equivariant to its Lie algebra V . Now suppose
h′ is a complement to h in h + V given as h′ = {x + ϕ(x) : x ∈ h} ⊂ gln+1 . If q = 1 + v ∈ Q with
v ∈ V we have
(x + ϕ(x))q = xq + ϕ(x) = (1 − v)x(1 + v) + ϕ(x).
Then since x normalises V and any two endomorphisms v1 , v2 ∈ V ⊆ End(V ) satisfy v1 v2 = 0,
we see easily that (x + ϕ(x))q = x + [x, v] + ϕ(x) = x + x(v) + ϕ(x), showing that two cocycles
in Z 1 (h, V ) are equivalent if and only if they are conjugate under the action ϕq (x) = x(v) + ϕ(x),
where q = 1 + v.
We have proven the following proposition:
Proposition 2.3. Let h ⊆ gl(V ) be a subalgebra with dim V = n. Then realised as a subalgebra of
gln+1 as above, all complements to V in h + V are conjugate to h via elements of 1 + V if and only
if H1 (h, V ) = 0.
We also require the following crucial result, generalising the above proposition to the case where q
is non-abelian. This is one of our main tools in proving Theorem 1.4:
Proposition 2.4. Suppose P = LQ is a parabolic subgroup of a reductive algebraic group G
with unipotent radical Q and Levi factor L. Let h be a subalgebra of Lie(L) and suppose that
H1 (h, Lie(Qi /Qi+1 )) = 0 for each i ≥ 1. Then all complements to q in h + q are Q-conjugate to h.
Proof. Let Q = Q1 ≥ Q2 ≥ . . . denote the filtration of Q by L-modules described at the beginning
of §2.2 and let qi := Lie(Qi ). We prove inductively that all complements to q/qi in h + q/qi
are Q/Qi -conjugate to h. If i = 1 this is trivial, so assume all complements to q in h + q/qi−1 are
Q/Qi−1 -conjugate to h. Take a complement h′ to q/qi in h+q/qi . Then by the inductive hypothesis,
we may replace h′ by a Q/Qi -conjugate so that h + qi−1 /qi = (h + qi−1 )/qi = (h′ + qi−1 )/qi . Thus h′
is a complement to qi−1 /qi in the semidirect product h + qi−1 /qi . We have h′ = x + γ(x) for some
cocycle γ : h → qi−1 /qi . But since H1 (h, qi−1 /qi ) = 0, we may write γ(x) = [x, v] for some v ∈ q.
Now γ|ch (q) is identically zero. Thus γ factorises through h → gl(qi−1 /qi ), so it suffices to consider
the image of h in gl(qi−1 /qi ). Now by Proposition 2.3 the image of h′ is Qi−1 /Qi -conjugate to h.
It follows that h′ is Qi−1 /Qi -conjugate to h.

2.3. Cohomology for G, G1 and g. In this section, G will be simple and simply-connected. In
the subsequent analysis, it will be necessary to know the ordinary Lie algebra cohomology groups
H1 (g, V ) for V a simple restricted g-module of small dimension. We will first reduce our considerations to the restricted Lie algebra cohomology. In this section, G1 denotes the first Frobenius
kernel of G—we warn the reader that we sometimes use subscripts to index groups, but it will
7

G p
V
dim V H1 (G1 , V )[−1]
A1 p < 64 L(p − 2)
p − 1 L(1)
A2 p = 5, 7 L(p − 2, 1)
18, 33 L(1, 0)
p=5
L(3, 3)
63
k
B2 p = 5, 7 L(p − 3, 0)
13, 54 k
p=5
L(1, 3)
52
L(0, 1)
G2 p = 7
L(2, 0)
26
k
A3 p = 5
L(3, 1, 0)
52
L(1, 0, 0)
A4 p = 5
L(1, 0, 0, 1)
23
k
A6 p = 7
L(1, 0, 0, 0, 0, 1) 47
k
Table 1. Non-trivial G1 -cohomology of irreducible modules of dimension at most 64.

be clear from the context our intent. We may identify restricted cohomology for g with the usual
Hochschild cohomology for G1 . Recall the exact sequence [Jan03, I.9.19(1)]:
0 → H1 (G1 , V ) → H1 (g, V ) → Homs (g, V g )
(2)

→ H2 (G1 , V ) → H2 (g, V ) → Homs (g, H1 (g, V )).

From this it follows that if V g = 0 (i.e. g has no fixed points on V ) then H1 (G1 , V ) → H1 (g, V )
is an isomorphism. This happens particularly in the case that V is simple and non-trivial. The
main result of this section is Proposition 2.6 below, which gives every instance of a simple restricted
G-module V of dimension at most 64 such that the groups H1 (G1 , V ) ∼
= H1 (g, V ) are non-trivial.
In order to prove this proposition, we will need some auxiliary results. The following useful result
relates to weights in the closure of the lowest alcove, C̄Z = {λ ∈ X(T ) | 0 ≤ hλ + ρ, β ∨ i ≤
p for all β ∈ R+ }. It is immediate from [BNP02, Cor. 5.4 B(i)].
Lemma 2.5. Let G be simple and simply connected and suppose L = L(µ) with µ ∈ C̄Z and p ≥ 3.
Then we have H1 (G1 , L) = 0 unless G is of type A1 and L = L(p − 2).
In the next proposition and elsewhere, we use repeatedly the linkage principle for G1 , [Jan03,
II.9.19]. This states that if ExtiG1 (L(λ), L(µ)) 6= 0 for any i ≥ 0 then λ ∈ Wp · µ + pX(T ), where ·
denotes the usual action of the affine Weyl group Wp on X(T ), shifted by ρ.
If V is a G-module, then as G1 is a normal subgroup scheme of G, the cohomology group H1 (G1 , V )
inherits a G-module structure [Jan03, I.6.6]; since G1 acts trivially on this, such an action must
factor through the Frobenius morphism, hence can be untwisted to yield a G-module H1 (G1 , V )[−1] .
Of course, any simple module for G1 can always be given a G-structure in a unique way and is
associated to a unique highest weight λ ∈ X1 (T ), where X1 (T ) denotes the p-restricted dominant
weights; see [Jan03, II.3.15].
Proposition 2.6. Let G be a simple algebraic group of rank no more than 7, let p ≥ 5 and suppose
V is a restricted irreducible G-module of dimension at most 64. Then either H1 (G1 , V ) = 0 or
(G, V, p) is listed in Table 1 (up to taking duals).
Proof. The values of H1 (G1 , V ) where G is of type A1 are well-known and can be found, for example
in [Ste10, Prop. 2.2]. The statement in the cases where G is of type A2 or B2 , or type G2 when p ≥ 7
is immediate from, respectively, [Yeh82] (see [Ste12, Prop. 2.3]), [Ye90, Table 2] and [LY93, Table 2].
8

Consider the remaining cases. A list of all non-trivial restricted modules of dimension at most 64 is
available from [Lüb01]. We then use the G1 -linkage principle to remove any modules L(λ) such that
λ is not G1 -linked to µ = 0. Explicitly, one may calculate w · 0 for any w ∈ W (a finite list) and add
an appopriate (uniquely defined) element of pX(T ) to produce a collection of p-restricted weights
in X1 (T ); since we assume H1 (G1 , V ) 6= 0 we know that λ is one in this collection. Furthermore,
we remove any modules L(λ) such that λ is in the lowest alcove, since H1 (G1 , L(λ)) = 0 by Lemma
2.5. This reduces the list of possibilities considerably. For any cases still remaining, we appeal
to [BNP04, Thm. 3A] (case r = 1), recalling H 0 (λ) := IndG
B (λ). This implies that
(
H 0 (ωα )[1] if λ = pωα − α for α ∈ S
(3)
H1 (G1 , H 0 (λ)) =
.
0
otherwise
Let us take G = A3 and explicitly give the details. Using [Lüb01, 6.7], we see that the following
is a complete list of the high weights of all non-trivial restricted modules V = L(λ) such that
dim V ≤ 64 (up to taking duals): (1, 0, 0), (0, 1, 0), (2, 0, 0), (1, 0, 1), (1, 1, 0), (0, 2, 0), (3, 0, 0),
(2, 0, 1), (1, 1, 1), (0, 3, 0), (4, 0, 0), (5, 0, 0), (1, 2, 0) and when p = 5 the weight (3, 1, 0). A weight
λ = (a, b, c) is in C̄Z if and only if a + b + c ≤ p − 3. So if p ≥ 11, all weights are in C̄Z , if p = 7
only (5, 0, 0) is not in C̄Z and if p = 5 the first six weights in the list are in C̄Z .
We may reduce this list using the linkage principle for G1 . In our case, this implies that the only
restricted weights G1 -linked to (0, 0, 0) up to duals are (p−2, p−2, p−2), (p−2, 1, 0), (p−2, p−3, 0),
(0, p − 3, 2), (p − 4, 1, p − 2), (p − 4, 0, 0), (p − 2, 2, p − 2), (0, p − 4, 0), (1, p − 4, 1), (1, p − 2, 1),
(p − 3, 0, p − 3), (p − 3, 0, 1), (p − 3, p − 2, 1), (p − 2, p − 2, 2), (p − 3, 2, p − 3). By comparison with
the list above, we may discount the possibility that p = 7 and the list of possible modules V with
H1 (G1 , V ) 6= 0 to just L(1, 1, 1), L(2, 0, 1) and L(3, 1, 0) when p = 5.
We now use (3) to find that H1 (G1 , H 0 (λ)) = 0 for λ = (1, 1, 1), (2, 0, 1) and H1 (G1 , H 0 (λ)) =
L(1, 0, 0)[1] for λ = (3, 1, 0). Now, the structure of the induced modules H 0 (λ) can be deduced:
each is the indecomposable reduction modulo p of a certain lattice in the simple module LC (λ); by
comparing the weight multiplicities in [Lüb01], one finds there are just two composition factors, in
each case. Since L(λ) is the socle of H 0 (λ) one gets H 0 (1, 1, 1) = L(0, 1, 0)|L(1, 1, 1), H 0 (2, 0, 1) =
L(1, 0, 0)|L(2, 0, 1) and H 0 (3, 1, 0) = L(2, 0, 1)|L(3, 1, 0). Consider the following exact sequence for
a 2-step indecomposable G1 -module M = M1 |M2 .
0 → H0 (G1 , M2 ) → H0 (G1 , M ) → H0 (G1 , M1 )
→ H1 (G1 , M2 ) → H1 (G1 , M ) → H1 (G1 , M1 ) → . . .
Applying this to H 0 (λ) for λ = (1, 1, 1), (2, 0, 1) yields that H1 (G1 , L(1, 1, 1)) = H1 (G1 , L(2, 0, 1)) =
0. Moreover, applying the sequence to H 0 (3, 1, 0) and using the fact H1 (G1 , L(2, 0, 1)) = 0, it follows
that H1 (G1 , L(3, 1, 0)) ∼

= H1 (G1 , H 0 (3, 1, 0)) ∼
= L(1, 0, 0)[1] .
Finally, we record the following result, which is presumably well-known. We were unable to locate
a proof in the literature, so we give one.
Lemma 2.7. Let G be a simple connected algebraic group of type A1 , let p > 2 and 0 ≤ a, b ≤ p − 1.
Then Ext1G1 (L(a), L(b)) ∼
= Ext1g (L(a), L(b)) unless a = b = p − 1. Moreover, Ext1G1 (L(a), L(b)) 6= 0
if and only if a = p − 2 − b and a, b 6= p − 1, and Ext1g (L(p − 1), L(p − 1)) ∼
= (g∗ )[1] .
Proof. We prove the second statement about Ext1G1 (L(a), L(b)) first. Since w·a is either a or −a−2,
the linkage principle for G1 implies that Ext1G1 (L(a), L(b)) = 0 unless b = a or b = p − 2 − a. If
9

b = a then Ext1G1 (L(a), L(b)) = 0 by [Jan03, II.12.9] so we may now assume b = p − 2 − a. But now
Ext1G1 (L(a), L(p−2−a)) ∼
= Ext1G1 (k, L(a)⊗L(p−2−a)) ∼
= Ext1G1 (k, L(p−2)⊕L(p−4)⊕· · ·⊕L(0)).
Then the only term which survives in this expression is Ext1G1 (k, L(p − 2)) = H1 (G1 , L(p − 2)) =
H1 (G1 , H 0 (p − 2)) whose structure as a G-module can be read off from [Jan03, II.12.15] or [BNP04,
Thms. 3(A-C)].
Now, in sequence (2), put V = L(b) ⊗ L(a)∗ . Then we have an isomorphism Ext1G1 (L(a), L(b)) ∼
=
1
s
g
g
Extg (L(a), L(b)) if we can show that Hom (g, V ) = 0. But if V were non-zero then we must have
L(a) ∼
= k. Thus we now assume a = b.
= L(b) and V g ∼
The assignation of V to the sequence (2) is functorial, thus, associated to the G-map k → V , there
is a commutative diagram
∼
=
0 −−−−→ H1 (g, k) = 0 −−−−→ Homs (g, k g ) ∼
= (g∗ )[1] −−−−→ H2 (G1 , k)






∼
,
=y
θy
y

0 −−−−→

H1 (g, V )

ζ
−−−−→ Homs (g, V g ) ∼
= (g∗ )[1] −−−−→ H2 (G1 , V )

where the natural isomorphism kg → V g induces the middle isomorphism. We want to show that ζ
is injective, since then it would follow that H1 (g, V ) = 0. To do this it suffices to show that θ is an
injection (g∗ )[1] → H2 (G1 , V ) and for this, it suffices to show that the simple G-module (g∗ )[1] does
not appear as a submodule of H1 (G1 , V /k). If (g∗ )[1] did appear as a submodule, there must be a
composition factor L(ν) of V /k such that H1 (G1 , L(ν)) ∼
= (g∗ )[1] . Writing L(ν) = L(ν0 ) ⊗ L(ν1 )[1]
using Steinberg’s tensor product theorem, we have H1 (G1 , L(ν)) ∼
= H1 (G1 , L(ν0 )) ⊗ L(ν1 )[1] and the
latter is non-zero only when ν0 = p − 2. Since the weights of V are all even and bounded above by
2p − 2, with equality if and only if a = b = p − 1, we must have ν0 = p − 2, ν1 = 1 and indeed,
a = b = p − 1. But then H2 (G1 , V ) ∼
= Ext2G1 (L(p − 1), L(p − 1)) which is zero since L(p − 1) is the
projective Steinberg module. Thus from the bottom line of the diagram we have an isomorphism
H1 (g, V ) ∼

= (g∗ )[1] as required.
2.4. Nilpotent orbits. At various points we use the theory of nilpotent orbits, particularly the
results of [Pre95a]. Everything we need can be found in [Jan04, §1-5]. We particularly use the fact
that a nilpotent element e has an associated cocharacter; that is a homomorphism τ : Gm → G
such that under the adjoint action, we have τ (t) · e = t2 e and τ evaluates in the derived subgroup
of the Levi subgroup in which e is distinguished. Recall that an sl2 -triple is a triple of elements
(e, h, f ) ∈ g × g × g such that [h, e] = 2e, [h, f ] = −2f and [e, f ] = h. In the case that a nilpotent
element e is included in an sl2 -triple in g, the theory of associated cocharacters can be used to
prove the following useful result.
Proposition 2.8 (cf. [HS16a, Prop. 3.3(iii)]). Suppose the nilpotent element e is not in an orbit
containing a factor of type Ap−1 , and that h is a toral element in the image of ad e with [h, e] = 2e.
Then there is a cocharacter τ associated to e with Lie(τ (Gm )) = hhi.
We also need to use the Jordan block structure of nilpotent elements on the adjoint and minimal
modules in good characteristic. For the adjoint modules we may use [Law95], and see [PS16] which
provides the validity of these tables for the nilpotent analogues of the unipotent elements considered
there. For the minimal modules we may use [Ste16, Thm 1.1].
10

When referring to nilpotent orbits of Lie(G) for a simple algebraic group G, we use the labels defined
in [LS12]. In particular, when G is of exceptional type these labels are described in Chapter 9 of
[loc. cit].
At certain points we make use of the notion of a reachable nilpotent element. A (nilpotent) element
e ∈ g is said to be reachable if it is contained in the derived subalgebra of its centraliser. That is
e ∈ [ge , ge ]. The reachable elements of g have all been classified in [PS16]. Then the main point
is that long root elements in simple subalgebras are almost always reachable in those subalgebras,
hence are reachable elements of g. We will need the following result.
Lemma 2.9. Let h = Lie(H) for H a simple algebraic group, not of type A1 . Then any long root
element e is reachable, except possibly if H is of type Cn and p = 2.
Proof. Since all long root elements are in a single H-orbit, it suffices to prove the lemma in the case
e = eα̃ for α̃ the highest root. Then it suffices to find two root elements α and β with α + β = α̃
such that [eα , eβ ] 6= 0.
This is a simple case-by-case check of the root systems. In all cases except Cn , one can take α and
β to be long roots. For a Chevalley basis, we have [eα , eβ ] = ±eα̃ and so we are done. In case Cn ,
one may take two short roots α and β, and one has [eα , eβ ] = ±2 · eα̃ , which is non-zero provided
p 6= 2.

3. Irreducible and completely reducible subalgebras of classical Lie algebras.
The proof of Theorem 1.3.
For the time being, assume p > 2. In this section we show that Theorem 1.3 holds in the case G is
simple and classical; this is Proposition 3.4 below. Let G be a simple, simply-connected algebraic
group of classical type and let h be a subalgebra of g = Lie(G). We first give a condition for h to
be G-irreducible. That is, that h is in no proper parabolic subalgebra of g. This is given in terms
of the action of h on the natural module V for G, as it is in the group case—see [LT04, Lem. 2.2].
Proposition 3.1. The algebra h is G-irreducible if and only if one of the following holds:
(i) g = sl(V ) and h acts irreducibly on V ;
(ii) g = sp(V ) or so(V ) and h stabilises a decomposition V ∼
= V1 ⊥ V2 ⊥ · · · ⊥ Vn , where
the Vi are a set of mutually orthogonal, non-degenerate and non-isomorphic h-irreducible
submodules of V .
Proof. By [MT11, Prop 12.13], the (proper) parabolic subgroups of G are precisely the stabilisers
of (non-trivial) flags F • of totally isotropic subspaces of V (where G = SL(V ) preserves the 0form on V ). Let F • be a flag of subspaces such that the k-points of its stabiliser is a parabolic
subgroup P . We claim that Stab(F • ) is smooth. We certainly have Lie(Stab(F • )) contains Lie(P )
so that Lie(Stab(F • )) is maximal rank, thus corresponds to a subsystem of the root system. Any
root space heα i contained in Lie(Stab(F • )) gives rise to a root subgroup of Stab(F • )(k) = P , via
t 7→ exp(t.eα ). Thus Lie(Stab(F • )(k)) = Lie(Stab(F • )), as required.
The case G = SL(V ) is now clear since h fixes a non-trivial subspace of V if and only if it is
contained in a parabolic subalgebra.
Now suppose G = Sp(V ) or SO(V ). Firstly, let h be a G-irreducible subalgebra of g and suppose
V1 is a minimal non-zero h-invariant subspace of V , so V1 is an h-irreducible submodule of V . Then
11

V1 must be non-degenerate or else h would stabilise a non-zero totally isotropic subspace and hence
be contained in a proper parabolic subalgebra. We then use an inductive argument applied to V1⊥
to see that h stabilises a decomposition V ∼
= V1 ⊥ V2 ⊥ · · · ⊥ Vn of non-degenerate, mutually
orthogonal, h-irreducible submodules. If Vi |h ∼
= Vj |h by an isometry φ : Vi → Vj for i 6= j then h
preserves the totally isotropic subspace {vi + iφ(vj )} ⊂ Vi ⊕ Vj (where i2 = −1). Thus the Vi are
pairwise non-isomorphic. Finally, it remains to note that that any subalgebra h preserving such a
decomposition as in (ii) is G-irreducible since it stabilises no totally-isotropic subspaces of V by
definition.

Since the Levi subalgebras of classical groups are themselves classical one gets the following, using
precisely the same argument as in [Ser05, Ex. 3.2.2(ii)]. (We remind the reader of our assumption
that p > 2.)
Lemma 3.2. The subalgebra h of g is G-cr if and only if it acts completely reducibly on the natural
module V for g.
To prove the next proposition, we use Lemma 3.2 together with the following non-trivial result.
Theorem 3.3 ( [HS16b, Cor. 8.12]). Let G be a semisimple algebraic group and let V be a g-module
with p > dim V . Assume that g = [g, g]. Then V is semisimple.
Proposition 3.4. Let G be a simple algebraic group of classical type with h(G) its Coxeter number
and g its Lie algebra. If p > h(G) then any semisimple subalgebra h of g is G-cr.
Proof. Let G be a simple algebraic group of classical type with p > h(G). Now assume, looking for a
contradiction, that h is a non-G-cr subalgebra of g. Thus h is in a non-trivial parabolic subalgebra,
projecting isomorphically to some proper Levi subalgebra l with Coxeter number h1 < h(G). One
checks that the condition p > h(G) implies that p > dim V for V a minimal-dimensional module
for any simple factor of l. Now [Str73, Main Thm.] implies that the projection of h to the simple
factors of l are all direct products of classical-type Lie algebras, hence h itself is a direct product of
classical-type Lie algebras, isomorphic to Lie(H) for H some semisimple algebraic group. Thus by
Theorem 3.3 we have that h acts completely reducibly on V . Hence by Lemma 3.2 we have that h
is G-cr.

4. G-complete reducibility in exceptional Lie algebras. Proof of Theorem 1.4.
Let G be reductive and P be a parabolic subalgebra of G with Levi decomposition P = LQ. We
begin with some general results on G-complete reducibility of subalgebras. For our purposes, they
will be used in order to generate examples of non-G-completely reducible subalgebras.
Lemma 4.1 ( [McN07, Lem. 4]). Let G be a reductive algebraic group and let L be a Levi subgroup
of G. Suppose h ⊆ Lie(L) is a Lie subalgebra. Then h is G-cr if and only if h is L-cr.
Lemma 4.2 ( [BMRT13, Thm. 5.26(i)]). Let G be a reductive algebraic group and let P = LQ be
a parabolic subgroup of G. Suppose h is a subalgebra of g contained in p = Lie(P ). Then if h is not
Q-conjugate to a subalgebra of l = Lie(L) then h is non-G-cr.
The following lemma provides a strong connection between the structure of modular Lie algebras
and the notion of G-complete reducibility and will be used very often.
12

Lemma 4.3. Let G be a reductive algebraic group and suppose p is a good prime for G. Suppose
further that h is a simple G-cr subalgebra of g which is restrictable as a Lie algebra. Then either h
is a p-subalgebra of g or h is L-irreducible in a Levi subalgebra l = Lie(L) of g with a factor of type
Arp−1 for some r ∈ N.
Proof. If h is not a p-subalgebra of g then its p-closure, hp , is a p-envelope of h strictly containing
h. Since h is restrictable, by [SF88, Thm. 2.5.8] hp has structure hp = h ⊕ J for J an ideal of hp
centralised by h. Now suppose L is chosen minimally such that h ⊆ l; as h is G-cr, it follows that
h is L-irreducible. If p is a very good prime for l, then l = l′ ⊕ z(l) where l′ is the direct sum of
simple Lie algebras and a central torus z(l), both of which are p-subalgebras. Since h is simple, it
has trivial projection to z(l) and so J ⊆ l′ . But since p is good, the centraliser of an element of
the semisimple Lie algebra l′ is in a proper parabolic of l′ and h is G-cr; thus h is in a proper Levi
subalgebra of l′ and so L was not minimal, a contradiction. It follows that p is not a very good
prime for l′ , but this precisely means that L contains a factor of type Arp−1 .

Corollary 4.4. Let g = Lie(G) be an exceptional Lie algebra in good characteristic. Suppose
h = Lie(H) for H a simple algebraic group not of type A1 and that h is a non-G-cr subalgebra of g.
Choose p = l + q minimal subject to containing h. Then the projection h̄ of h to l is a p-subalgebra.
Proof. By the proof of Lemma 4.3, if h̄ is not a p-subalgebra then l is of type Ap−1 and h̄ projects
to a subalgebra of slp acting irreducibly on the natural module. Since h is not of type A1 , the KacWeisfeiler conjecture (which is true for this situation by [Pre95b]) implies the only non-restricted
representations of h have dimension at least p2 . This is a contradiction.

Unless otherwise mentioned, for the rest of this section G will denote an exceptional algebraic group
over an algebraically closed field k of characteristic p ≥ 5 and we will let g = Lie(G). Our strategy
to prove Theorem 1.4 is as follows—we use the opportunity to fix notation for the remainder of the
paper as we explain this:
Suppose there exists a non-G-cr simple subalgebra h of g of a given type. Then h must be contained
in a parabolic subalgebra p = l + q = Lie(P ) = Lie(LQ), which from now on, we assume is minimal
subject to containing h. It follows that the projection h̄ of h to l is L-irreducible. Since h is not
conjugate to a subalgebra of l, Proposition 2.4 implies that q contains an h-composition factor V
for which H1 (h, V ) 6= 0. If h is not isomorphic to pslp or W1 then h ∼
= Lie(H) for some simple
simply-connected algebraic group H of the same type and the remarks at the beginning of §2.3
imply that H1 (H1 , V ) 6= 0 for H1 the first Frobenius kernel of H. In fact the same isomorphism
holds with H = SLp and h = pslp as any pslp -module can be lifted to a module for slp by allowing
the centre to act trivially; then one may apply the exact sequence of [Wei94, 7.5.3]. By Lemma 2.2
we must have that the dimension of V is less than 64. By Proposition 2.6 it now follows that:
(4)
Either h ∼
= W1 or V appears in Table 1.
By analysing the structure of q closely, we will find that in most cases no such V appears as a
composition factor of q so that these cases are ruled out: see Lemmas 4.5 and 4.6 below.
One set of cases requires more work: If l is a Levi subalgebra of g of type E6 or E7 , then we investigate all possible actions of h on the smallest dimensional modules for E6 and E7 (of dimensions
27 and 56, respectively). We will see that in any such action, a regular nilpotent element of h does
not act consistently with the Jordan block sizes of nilpotent elements on the relevant modules, as
described in [Ste16]. Having reduced the possible cases as above, we show that the remaining cases
13

of (G, h, p) do indeed give rise to non-G-cr subalgebras, recorded in Lemmas 4.7, 4.11, 4.12 and
4.14.
Lemma 4.5. Suppose h is a simple non-G-cr subalgebra not of type A1 or W1 and that p = l + q
is a minimal parabolic containing it. Then (G, h, p) is (E8 , B2 , 5), (E7 , G2 , 7) or (E8 , G2 , 7); or l′
has a simple factor of type E6 or E7 .
Proof. Without loss of generality we may assume G is simply-connected. Suppose P = LQ with
Lie(P ) = p and let L′ = L1 . . . Lr be an expression for the derived subgroup of L as a central
product of simple factors not containing any exceptional factors. As each Li is simply connected,
we may write l′ = l1 ⊕ · · · ⊕ lr with li = Lie(Li ). Since P was minimal, we must have that the
projection of h to each li is Li -irreducible; call this hi .
Since h is not of type A1 , all of the Li factors have rank at least 2. As Li is classical by assumption,
it has a natural module Vi and it follows from Lemma 3.1 and Corollary 4.4 that there exists an
Li -irreducible restricted subgroup Hi whose action on Vi differentiates to that of hi . The action of
hi on Vi determines it up to Li -conjugacy, except if Li is of type Dn and there are two conjugacy
classes interchanged by a graph automorphism (or up to three if Li is of type D4 ). Thus we may
write h = Lie(H) for H a diagonal subgroup of H1 . . . Hr . One may now compute a list of possible
h-factors of q by differentiating the H-factors on Q; the latter are available from [LS96, Lem. 3.4].
By (4) this implies (H, p, λ) = (B2 , 5, (2, 0)), (B2 , 5, (1, 3)) or (G2 , 7, (2, 0)).
Suppose H is a subgroup of type B2 . The only Levi factors for which L(2, 0) or L(1, 3) occur as an
H-composition factor are A3 A4 and D7 . Hence G is of type E8 . Similarly, if H is a subgroup G2
then the only Levi factors for which L(2, 0) occurs as an H-composition factor are A6 and D7 , and
so G is of type E7 or E8 .

The next lemma reduces the proof of Theorem 1.4 to considering all simple subalgebras of rank at
most 2.
Lemma 4.6. Suppose h is a simple subalgebra of rank at least 3. Then h is G-cr.
Proof. Statement (4) above implies that we are done unless h is of type A3 , A4 or A6 . Firstly,
suppose h is of type A3 and is a non-G-cr subalgebra of g. Then (4) tells us p = 5 and either
V := L(3, 1, 0) or its dual is an h̄-composition factor of q. Now dim V = 52 and so we are forced
to conclude that G is of type E8 . By Lemma 4.5 and dimensions again, we must have l′ of type
E7 . But the only non-trivial factor of q has dimension 56. If V were a composition factor of the
self-dual module V56 , then so would be its dual; a contradiction by dimensions.
Now suppose h is of type A4 . Statement (4) tells us p = 5 and q contains an h̄-composition
factor V := L(1, 0, 0, 1). By Lemma 4.5 we may also assume that G is of type E7 or E8 with l′
chosen minimally of type E6 or E7 . If l′ is of type E6 , the non-trivial l′ -composition factors of q
are either V27 or its dual. Since V appears amongst q ↓ h̄, the h̄-composition factors of V27 are
L(1, 0, 0, 1)/k4 . Since the restriction of a natural module L(1, 0, 0, 0) for h to a Levi sl2 -subalgebra
has composition factors L(1)/k3 , and L(1, 0, 0, 1) is a composition factor of L(1, 0, 0, 0)⊗L(0, 0, 0, 1),
we may calculate that the restriction to a Levi sl2 -subalgebra of V27 is a completely reducible module
with composition factors L(2)/L(1)6 /k12 . A non-zero nilpotent element of this subalgebra acts with
Jordan blocks 3 + 26 + 112 , though this is impossible by [Ste16, Table 4]. In case l′ is of type E7 ,
we see that V56 contains a composition factor L(1, 0, 0, 1) and the remaining composition factors
must have dimension 33 or less, and if not self-dual, must have dimension 16 or less. Up to duals,
14

the possible composition factors together with their restrictions to a Levi subalgebra s of type sl2
are in the following table.
λ
(1, 0, 0, 1)
(1, 0, 0, 0)
(0, 1, 0, 0)

L(λ) ↓ s
L(2) + L(1)6 + k8
L(1) + k3
L(1)3 + k4

The restriction of any resulting module to s is completely reducible, and so the Jordan blocks
of a non-zero nilpotent element e ∈ s are determined by the h̄-composition factors on V56 . It is
easily checked that there is no way of combining these composition factors compatibly with the
possibilities in [Ste16, Tables 2, 3].
Finally, suppose that h is of type A6 . Arguing in a similar fashion, we find that G is of type E8 with
p = 7; the type of l′ is E7 ; and the h̄-composition factors on V56 are L(̟1 + ̟6 )/k9 . Restricting to
a Levi sl2 -subalgebra and comparing once again with [Ste16, Tables 2, 3] yields a contradiction. 
4.1. Subalgebras of type A1 . In this section we show that Theorem 1.4 holds in case h is of type
A1 . The following also deals with one direction of Theorem 1.3.
Lemma 4.7. Let G be a reductive algebraic group and let p > 2. Then whenever p ≤ h(G), there
exists a non-G-cr sl2 -subalgebra containing a regular nilpotent element e, a toral element h and an
element f regular in a proper Levi subalgebra of g.
Proof. It suffices to tackle the case where G is simple. Let gZ be a lattice defined via a Chevalley
basis in the simple complex Lie algebra gC of the same type as g and let e be the regular element
given by taking a sum of all simple root vectors in gZ . Then there is an element f ∈ gZ which is a
sum of negative root vectors such that (e, h, f ) is an sl2 -triple. These are easily constructed in the
case G is classical and are given explicitly by [Tes92, Lem. 4] in the case G is exceptional:
(i)
(ii)
(iii)
(iv)

g = G2 , f = 6e−α1 + 10e−α2 ;
g = F4 , f = 22e−α1 + 42e−α2 + 30e−α3 + 16e−α4 ;
g = E7 , f = 34e−α1 + 49e−α2 + 66e−α3 + 96e−α4 + 75e−α5 + 52e−α6 + 27e−α7 ;
g = E8 , f = 92e−α1 + 136e−α2 + 182e−α3 + 270e−α4 + 220e−α5 + 168e−α6 + 114e−α7 + 58e−α8 .

Reducing everything modulo p > 2 gives an sl2 -subalgebra h of g. Moreover, since p < h(G) we have
that e[p] 6= 0, which follows from the description of the Jordan blocks of regular nilpotent elements
in [Jan04, §2] for G classical, and is immediate from [Law95, Table D] in case G is exceptional.
Therefore, in each case h is a non-p-subalgebra of type A1 (noting that there is only one p-structure
on h by [SF88, Cor. 2.2.2(1)]). Since E6 contains F4 as a p-subalgebra, the non-p-subalgebra h
contained in F4 is also a non-p-subalgebra of E6 .
Suppose G is not of type Ap−1 . Then since h contains a regular nilpotent element of g, it is certainly
not contained in a Levi subalgebra of type Arp−1 , hence being non-p-subalgebras, these subalgebras
are non-G-cr, by Lemma 4.3. In particular each is in a proper parabolic subalgebra of g. Thus f is
no longer regular in g. This can be seen explicitly in the case g is exceptional as p < h(G) implies
that p divides at least one of the coefficients of the root vectors of f . In the classical case, e acts as
a single Jordan block on the natural module in types A, B and C acting on standard (orthonormal)
basis vectors as e(ei ) = ei−1 whereas f (ei ) = λi ei+1 for λi the coefficient of the simple root vector
15

e−αi . Since h is in a proper parabolic subalgebra, it stabilises a subspace, meaning that some
non-zero collection of the λi must be congruent to 0 modulo p. The remainder determine a regular
element in some proper Levi subalgebra obtained by removing the appropriate nodes of the Dynkin
diagram. In type Dn , the regular nilpotent element e acts with Jordan blocks of size 2n − 1 and 1
on the natural module, and only regular elements act in such a way. Since V |so2n−1 ∼
= V ′ ⊕ k for V ′
the natural module for so2n−1 , and a regular element of the latter acts as a full Jordan block on V ′ ,
we must have that e is contained in subalgebra of type so2n−1 . Hence we may embed e in a regular
sl2 -subalgebra h in the so2n−1 subalgebra such that h is toral and f is regular in a Levi subalgebra
of so2n−1 . This implies that f is regular in a Levi subalgebra of g = so2n . Finally to check the
theorem in case G of type Ap−1 , simply observe from Proposition 2.6 that sl2 has indecomposable
representations of dimension p of the form k|L(p − 2), which can be used to embed h in g such that
h does not act completely reducibly on the natural module for G with e being regular but f not.
The subalgebra h is then non-G-cr by Lemma 3.2.
(Specifically, we may take







e := 





0 1 0 0 ...
.
0 1 0 ..
.
0 1 ..
.
0 ..
..
.

0
0
0
0
1
0





0

0



 λ1 0





λ2


 , f := 









0

0

0

0

0
..
.

0
0
λp−2

...
..
.
..
.
..
.
..
.
0

1
0
0
0
0
0

where λi = −i(i + 1) mod p. Thus f is a regular nilpotent element of type Ap−2 .)















Lemma 4.8. Theorem 1.4 holds in the case h is of type A1 .
Proof. The case p ≤ h(G) is supplied by Lemma 4.7, so suppose p > h(G) and h is a non-G-cr
subalgebra of type A1 . By statement (4), there is an h̄-composition factor of q isomorphic to
L(p − 2), of dimension p − 1. In each case we will show this is impossible.
When g is of type G2 , we have h(G) = 6 and the largest dimension l′ -composition factor occurring
is 4-dimensional. Hence g is not of type G2 .
When g is of type F4 , we have h(G) = 12. Only a Levi subalgebra of type C3 has an l′ -composition
factor of dimension 12 or more; the composition factor is L(0, 0, 1), which is 14-dimensional. Using
Proposition 3.1 we see that a Lie algebra of type C3 contains two subalgebras of type A1 not
contained in parabolics when p > 12, acting either as L(5) or L(3) + L(1) on the natural module.
V
Since 3 (L(1, 0, 0)) ∼
= L(0, 0, 1) + L(1, 0, 0) one calculates that the composition factors of such sl2
subalgebras on L(0, 0, 1) are L(9)/L(3) and L(5)/L(3)2 , respectively. In particular, neither has a
composition factor L(p − 2) for p > 12 and so g is not of type F4 .
When g is of type E6 , we also have h(G) = 12. Since all of the Levi subalgebras of g are of classical
type, we use Proposition 3.1 to find the L-irreducible subalgebras of type A1 . As in the F4 case, it
is straightforward to check that none of them has a composition factor L(p − 2) when p > 12. (To
find restrictions of the spin modules for Levi subalgebras of type D, one uses [LS96, Prop. 2.13].)
When g is of type E7 , we have h(G) = 18. The same approach as used above rules out h̄ ⊆ l for l
consisting of classical Levi factors. Suppose l′ is of type E6 and that h̄ is a subalgebra of type A1
16

with a composition factor L(p−2) on V27 . Then the action of a regular nilpotent element of h̄ on V27
has a Jordan block of size at least p − 1 ≥ h(G) = 18. This is a contradiction, since [Ste16, Table 4]
shows that the Jordan blocks of the action of any nilpotent element of E6 have size at most 17.
For g of type E8 , we have h(G) = 30. As above, one similarly rules out the cases where L is classical
or type E6 . So suppose L is of type A1 E6 . Restricting to the first factor, the only composition
factors are trivial or isomorphic to L(1), while the composition factors for the second factor on
q are trivial, or isomorphic to V27 or its dual. Since h̄ has a composition factor of high weight
p − 2 on q it follows that h̄ acts on V27 with a composition factor of high weight p − 3 or p − 2.
Therefore, the action of a regular nilpotent element e of h̄ on V27 has a Jordan block of size at least
p − 2 ≥ h(G) − 1 = 29. This contradicts [Ste16, Table 4], as before. Finally, if l′ is of type E7 , then
V56 has an h̄-composition factor L(p − 2). The action of a regular nilpotent element e of h̄ has a
Jordan block of size at least p − 1 ≥ h(G) = 30 on V56 . Using [Ste16, Tables 2, 3], we see this is a
contradiction since the largest Jordan block of a nilpotent element acting on V56 is 28.

4.2. Subalgebras of rank 2. In this section we prove Theorem 1.4 holds for all simple subalgebras
of g of rank 2 in a series of lemmas, taking each isomorphism class of h in turn. First of all we turn
to a very special case.
Lemma 4.9. Let p = 5, g = E7 and h ∼
= sl3 be a p-subalgebra of g. Suppose the highest root
element e = eα̃ ∈ h is nilpotent of type A4 + A1 . Then h = Lie(H) for H a maximal closed
connected subgroup of type A2 in G.
Proof. One has in h the relation [eα , eβ ] = e holds, with eα , eβ ∈ he ⊂ ge ⊆ g(≥ 0). Now one
sees from [LT11] that ge (0) is toral, hence the nilpotence of eα and eβ imply that they are in
fact contained in ge (> 0). The projections eα , eβ to ge (1) are hence non-trivial and we must have
[eα , eβ ] = e. Recall e is contained in an sl2 -triple (e, hα̃ , fα̃ ) ∈ h × h × h. Now [hα̃ , e] = 2e and hα̃
is in the image of ad e. Since e has a factor of type Ap−1 in g there is more than one element hα̃
satisfying these conditions. Indeed if h1 ∈ Lie(τ (Gm )) has this property, then so does h′ := h1 +λh0
for h0 ∈ z(sl5 ), where the projection of e to its A4 factor is regular in sl5 (for more details, see
the argument in [HS16a, §A.2]). The cases where λ ∈ F5 give all such instances where h′ is also a
toral element, therefore we may restrict to these five cases. Furthermore, the cases where λ is 1 or
2 respectively, are conjugate to the cases where λ is 4 or 3 respectively, via a representative of an
element of the Weyl group of E7 inducing a graph automorphism on the A4 factor, and centralising
the A1 factor ([Car72, Table 10]). Now it is an easy direct calculation using the elements given
′
in 
[HS16a, Table 4] that
 if λ 6= 0 then a basis for the elements of ge (1) on which h has weight 1
is

e−000001 , e000011 . One checks that the commutator of these two is eα6 . The latter is not of
0

0

type A4 + A1 . Hence we conclude λ = 0.

As hα̃ ∈ τ (Gm ), there is a standard sl2 -triple (e, hα̃ , f ), that is a regular sl2 -subalgebra of a Levi
subalgebra of type A4 A1 . We have fα̃ − f ∈ ge ⊆ ge (≥ 0) so that fα̃ projects to f in g(−2). Now
τ −1 is associated to f and hence ad f is injective on ge (1 + rp) for each r ≥ 1. It follows that eα and
eβ can have no non-zero component in ge (1 + rp), in other words they are homogeneous in ge (1).
Looking at ge (1) in [LT11], we conclude that eα and eβ are both of the form
λ1 · e000011 +4 · λ2 · e111100 + 3 · λ2 · e111110 + 4 · λ2 · e012100 + λ2 · e011110 + λ3 · e−000001
0

0

1

1

0

1

+ 2 · λ4 · e−111100 + 4 · λ4 · e−011100 + 4 · λ4 · e−001110 + λ4 · e−011110 ,
0

1

17

1

0

with λi ∈ k. If eα arises from the coefficients (λ1 , λ2 , λ3 , λ4 ) and eβ from (µ1 , µ2 , µ3 , µ4 ) then
calculating the commutator and insisting that the answer be e one sees that the equations
(*)

λ1 µ 2 = λ2 µ 1 ,

λ4 µ3 = λ3 µ4 ,

λ4 µ2 − λ2 µ4 = 1,

λ1 µ3 − λ3 µ1 = 1,

must be satisfied. If µ1 6= 0 then by replacing eα by eα − νeβ for suitable ν, we may assume that
λ1 = 0. Otherwise we may swap eα and eβ to assume λ1 = 0. Then using the equations of (∗)
we have λ3 µ1 = −1, thus µ1 6= 0 and so λ2 = 0. Subsequently λ4 µ2 = 1. Now replacing eα by
a multiple we can arrange λ3 = 1, thus µ1 = −1. Additionally, (using [LT11]) one checks that
the element h1 (t6 )h2 (t9 )h3 (t12 )h4 (t18 )h5 (t15 )h6 (t10 )h7 (t5 ) centralises e and the element of ge (1)
corresponding to coordinates (0, 0, 1, 0), while acting as a non-trivial scalar on (0, 0, 0, 1). It follows
that we may replace eα with a conjugate such that its coordinates are (0, 0, 1, 1), that is, λ1 = λ2 = 0
and λ3 = λ4 = 1. Thus µ2 = 1 and µ3 = µ4 . Replacing now eβ by eβ − µ3 eα , we may assume
µ3 = µ4 = 0 so that the coordinates of eβ are (−1, 1, 0, 0). Hence eα and eβ are completely
determined.
Now we show that fα̃ is unique up to conjugacy by Ge ∩ Ghα̃ ∩ Geα ∩ Geβ . Again if (e, hα̃ , f )
is
La standard sl2 -triple, then fα̃ − f ∈ ge (0) and as hα̃ has weight −2 on it, in fact, fα̃ − f ∈
r>0 ge (−2 + rp) = ge (3) ⊕ ge (8). Checking [LT11], we have fα̃ is of the form
λ1 ·e111000 + λ2 · e111110 + λ2 · e012110 − λ3 · e123211 + λ3 · e123221 + e−100000 + 2 · e−000000
1

0

1

2

1

1

1

+ 2 · e−001000 + e−000010 − λ4 · e−001100 + λ4 · e−011100 + λ5 · e−112211 + λ5 · e−012221
1

1

0

1

0

0

In h we have the relation [[fα̃ , eα ], eα ] = 0. This implies λ2 = λ3 = λ4 = 0. Additionally, the
relation [[fα̃ , eβ ], eβ ] = 0 implies λ5 = 0. Lastly, [[eα , fα̃ ], fα̃ ] = 0 implies λ1 = 0. Thus fα̃ is fully
determined.
We obtain in addition e−α = [fα̃ , eβ ] and e−β = [eα , fα̃ ] giving in total,
eα := e−000001 + 2 · e−111100 + 4 · e−011100 + 4 · e−001110 + e−011110
0

1

1

0

0

eβ := −1 · e000011 + 4 · e111100 + 3 · e111110 + 4 · e012100 + e011110
1

1

0

1

0

e−α := e000001 + 2 · e111100 + 4 · e011100 + 4 · e001110 + 4 · e011110
0

1

1

0

0

e−β := e−000011 + 4 · e−111100 + 2 · e−111110 + e−012100 + 4 · e−011110 .
0

0

1

1

1

It is automatic from the Serre relations that these elements generate a subalgebra isomorphic to
A2 . However if one chases through the proof of [LS04, Lem. 4.1.3], we discover that the conjugacy
class of maximal A2 subgroups of E7 have Lie algebras whose root elements are of type A4 + A1 .
It follows that h = Lie(H) for one of these subgroups.

Lemma 4.10. Theorem 1.4 holds when h is of type A2 .
Proof. Suppose, for a contradiction, that h is a non-G-cr subalgebra of type A2 . By (4) we have
that there is an h̄-composition factor V of q with V = L(λ) or L(λ)∗ where λ is (3, 1) or (3, 3) when
p = 5, or (5, 1) when p = 7. Furthermore, by Lemma 4.5 we may assume that l′ is of type E6 or
E7 and that h̄ is a p-subalgebra by Corollary 4.4.
18

λ
(2, 2)
(3, 1)
(3, 0)
(1, 1)
(2, 0)
(1, 0)

dim L(λ)
19
18
10
8
6
3

L(λ) ↓ sα̃
L(4) + L(3)2 + L(2)2
L(4) + L(3)2 + L(2) + L(1)
L(3) + L(2) + L(1) + k
L(2) + L(1)2 + k
L(2) + L(1) + k
L(1) + k

weights
43 /35 /25 /13 /03
44 /34 /24 /14 /02
42 /32 /22 /12 /02
42 /3/2/12 /02
4/3/2/1/02
4/1/0

Table 2. The restrictions of various sl3 -modules

Suppose p = 5. Since dim L(3, 3) = 63, it cannot occur as an h̄-composition factor of q. We will
require more work to show that L(3, 1) or L(1, 3) cannot occur as an h̄-composition factor of q.
Since the argument is the same for each, we assume V = L(3, 1).
By Lemma 2.9, any root vector of h̄, say eα̃ corresponding to the highest root α̃, has the property
that it is reachable. Then the possibilities are given by [PS16]. Let also hα̃ be an element in the
Cartan subalgebra of h̄ for which there is a nilpotent element fα̃ with sα̃ = heα̃ , hα̃ , fα̃ i ∼
= sl2 .
We need certain data about the restrictions to sα̃ of all irreducible restricted non-self-dual h̄-modules
of dimension at most 10 and all self-dual irreducible h̄-modules of dimension at most 20. We also
note that eα̃ has a Jordan block of size 5 on V .
Suppose first that l′ is of type E6 . Then V occurs as a h̄-composition factor of V27 or its dual.
One finds that in E6 there is just one reachable element with a Jordan block of size at least 5 on
V27 , namely that with label 2A2 + A1 . Using Table 2, we will now compare the weights of hα̃ on
V27 with those on various modules of dimension 27 containing V as a composition factor. It will
be convenient to do this by considering the action of elements of E6 on elements of q/[q, q] where
q is the nilradical of an E6 parabolic subalgebra of E7 . The tables in [LT11] give a cocharacter τ
associated to 2A2 + A1 in E7 . As eα̃ does not contain a factor of type Ap−1 then by Proposition 2.8,
we may assume hα̃ ∈ Lie(τ (Gm )) hence it suffices to compute the weights of τ on q/[q, q], which is
an easy computation in the root system of E7 (and which we perform in GAP). We find the weights
of τ on V27 are −4/ − 32 / − 24 / − 14 /05 /14 /24 /32 /4. Reduction modulo 5 then gives the weights of
hα̃ on V27 , namely 45 /36 /26 /15 /05 . Since one composition factor is V = L(3, 1) itself, we must find
4/32 /22 /13 /03 from the remaining composition factors of V27 . It is then a straightforward matter
to compare this with the weights of the tables above to see that this is impossible.
Thus we must have l′ of type E7 . Then V and its dual occur on the self-dual module V56 ↓ h̄. It
follows that a subset of the composition factors of V56 ↓ sα̃ is L(4)2 /L(3)4 /L(2)2 /L(1)2 . Hence a
root element eα̃ of sα̃ is represented on V56 by a matrix whose 4th power has rank at least two
and 3rd power has rank at least 6. Since it is also reachable, by comparison with [PS16] there
are just three options for the nilpotent orbit containing eα̃ , namely A3 + A2 + A1 , 2A2 + A1 and
A4 + A1 . The case A3 + A2 + A1 can be ruled out: in both sl3 and g there is a unique toral element
h ∈ im ad eα̃ which has weight 2 on a highest root element, and has weight 1 on the simple root
elements. As geα̃ ⊆ geα̃ (≥ 0), it follows that geα̃ (1) must be non-zero. But [LT11] reveals that for
eα̃ in E7 of type A3 + A2 + A1 the space geα̃ (1) is zero. (The element eα̃ can in fact be found in
[geα̃ (0), geα̃ (2)].) This is a contradiction. Hence eα̃ is not of type A3 + A2 + A1 .
Suppose eα̃ is of type 2A2 + A1 . Since eα̃ is not regular in a Levi with a factor of type Ap−1 ,
Proposition 2.8 implies that there is a unique hα̃ ∈ h such that hα̃ has weight 2 on eα̃ and hα̃ ∈
19

im ad eα̃ ; we take hα̃ ∈ Lie(τ (Gm )) for τ an associated cocharacter to eα̃ . The precise description
of such a τ comes from [LT11]. In an E7 -parabolic subalgebra of E8 , the space q/[q, q] affords a
representation V56 for the Levi. Applying τ to the roots of q/[q, q] gives the following multiplicities
for τ :
wt -4 -3 -2 -1 0 1 2 3 4
dim 2 4 8 8 12 8 8 4 2
Since ad eα̃ kills the L(4) weight space, on which hα̃ has weight 4, we must have two composition
factors isomorphic to L(4). These afford weights (4, 2, 0, −2, −4) on q/[q, q]. Removing the weights
of these composition factors and considering the remaining weights, we see inductively that the
composition factors of sα̃ are L(4)2 /L(3)4 /L(2)6 /L(1)4 /k4 .
Both V and its dual occur as composition factors of h̄ on V56 , so using Table 2, we find that the restriction to sα̃ of the remaining h̄-composition factors of V56 has composition factors L(2)4 /L(1)2 /k4 .
Again using Table 2, we find that either L(1, 1) occurs or both L(2, 0) and L(0, 2) occur as h̄composition factors of V56 . In the first case, the remaining sα̃ -composition factors are L(2)3 /k3 .
But no combination of modules in Table 2 have such restriction, a contradiction. So L(2, 0) and
L(0, 2) both occur, and the remaining sα̃ -composition factors are L(2)2 /k2 . Again we see that this
is impossible. This rules out the case eα̃ of type 2A2 + A1 .
If eα̃ is of type A4 + A1 , then Lemma 4.9 implies that h = Lie(H) for H a maximal A2 subgroup
of G. However from [LS04, Lem. 4.1.3] we have found the Lie algebra of the maximal connected
subgroup of type A2 in E7 . But this does not act on V56 with a composition factor of high weight
(3, 1) or (1, 3). This contradiction completes this case.
Finally, we consider the possibility that V = L(5, 1) when p = 7. Since dim L(5, 1) = 33 > 27, we
cannot have l′ of type E6 ; also the self-duality of V56 implies that we would need L(5, 1) and L(1, 5)
as composition factors of V56 , but 66 > 56. This is a contradiction.

Lemma 4.11. Theorem 1.4 holds when h is of type B2 .
Proof. Assume h is of type B2 and non-G-cr. By (4) we have at least one of L(2, 0) and L(1, 3)
occurs as an h̄-composition factor of q when p = 5 or L(4, 0) occurs as an h̄-composition factor of
q when p = 7. Moreover, by Lemma 4.5 we have g is of type E7 or E8 .
First suppose p = 5. When g is of type E8 we construct an example of a non-G-cr subalgebra
of type B2 . Let m be a maximal subsystem subalgebra of type D8 . We embed a subalgebra h of
type B2 into m via the representation T (2, 0) + k = k|L(2, 0)|k + k. (Note that T (2, 0) is self-dual
and odd-dimensional, hence is an orthogonal representation for h.) Therefore h is contained in a
D7 -parabolic subalgebra of m: by Lemma 3.1, it is in some parabolic, and it stabilises a singular
vector, the stabiliser of which is a D7 -parabolic. Then h is thus in a D7 -parabolic of g. We know
that a Levi subalgebra of type D7 acts as L(̟1 )2 + L(̟2 ) + L(̟6 ) + L(̟7 ) + 0 on g. In particular,
the largest summand is 91-dimensional.
Now consider H, a subgroup B2 of G = E8 embedded into
V
D8 via T (2, 0)+k. Then 2 (T (2, 0)+k) occurs as direct summand of g ↓ H. For any p > 2, we have
V
V
that 2 W is a direct summand of W ⊗2 , hence if W is tilting, 2 W is also. In our case, this implies
V2
(T (2, 0) + k) ∼
= T (2, 0) + L(0, 2) + T (2, 2). Therefore, H has a 95-dimensional indecomposable
summand M ∼
= L(2, 0)|L(2, 2)|L(2, 0) on g. We may identify T (2, 2) with the H1 -injective
= T (2, 2) ∼
hull of L(2, 0), which restricts to H1 indecomposably. As the category of representations for H1 is
equivalent to the category of p-representations for h, this shows h has an indecomposable summand
20

M ↓ h of dimension at least 95 on g; thus h cannot live in a Levi subalgebra of type D7 , proving it
is non-G-cr.
Now suppose g is of type E7 . Then by Lemma 4.5, we have l′ is of type E6 and V27 ↓ h̄ has a
composition factor L(2, 0) (we rule out L(1, 3) since it is 52-dimensional). Using [Lüb01, 6.22], the
only irreducible h̄-modules of dimension at most 14 are L(2, 0), L(1, 1), L(0, 2), L(1, 0), L(0, 1) and
k. Let s denote a long root sl2 -subalgebra of h̄. The following table lists the composition factors of
the restrictions of the irreducible h̄-modules above to s.
λ
(0, 0)
(0, 1)
(1, 0)
(0, 2)
(1, 1)
(2, 0)

dim L(λ)
1
4
5
10
12
13

L(λ) ↓ s
k
L(1)/k 2
L(1)2 /k
L(2)/L(1)2 /k3
L(2)2 /L(1)3
L(2)3 /L(1)2

A non-zero nilpotent element e of s satisfies e ∈ [h̄e , h̄e ] i.e. it is reachable in g. Thus from [PS16]
it is of type A1 , 2A1 , 3A1 , A2 + A1 , A2 + 2A1 or 2A2 + A1 . As in Lemma 4.10 we establish that
the composition factors of s on V27 must be as follows:
O
A1
2A1
3A1
A2 + A1
A2 + 2A1
2A2 + A1

V27 ↓ s
L(1)6 /k15
L(2)/L(1)8 /k8
L(2)3 /L(1)6 /k6
L(3)/L(2)4 /L(1)4 /k3
L(3)2 /L(2)3 /L(1)4 /k2
L(4)/L(3)2 /L(2)3 /L(1)2 /k

Comparing the above two tables, we find that there is just one possibility: O is of type 3A1 and
the h̄-composition factors of V27 are L(2, 0)/L(1, 0)2 /k4 . Let he, h, f i = s ⊂ h̄ be an sl2 -triple. By
Proposition 2.8, up to conjugacy by Ge we have that h ∈ Lie(τ (Gm )) for τ an associated cocharacter
to e. Up to conjugacy then, we may assume e = eα1 + eα3 + eα6 , h = hα1 + hα3 + hα6 and there is a
nilpotent element e′ in an sl2 -subalgebra s′ of h̄ which commutes with e and h. The subalgebra s′
must also be a long root sl2 -subalgebra of h̄, and so e′ is also reachable of type A31 . Now, from [LT11]
we see that ge (0) = Ce for Ce a reductive group of type A2 A1 . We have that Ce has six nilpotent
orbits on ge (0) corresponding to partitions of (3, 2); viz. {3, 2 + 1, 1 + 1 + 1} × {2, 1 + 1}. These can
be computed in GAP as having orbit types in E6 with labels 2A2 A1 , 2A2 , 3A1 , 2A1 , A1 , 0. Up to
conjugacy by Ce then, there is just one possibility for e′ , which may be taken as eα4 +α5 + eα5 +α6 .
The element es := e + e′ is then a subregular nilpotent element of h̄, which in g is checked to have
type A2 + 2A1 . A corresponding sl2 subalgebra, ss say has composition factors on L(2, 0) which
are L(4)/L(2)2 /k4 . But the existence of the L(4) composition factor is already incompatible with
the action of an ss on V27 from the table above. This is a contradiction.
Now suppose p = 7. Then by Proposition 2.6, we have L(4, 0) occurring as a h̄-composition factor of
q. Thus the h̄-composition factors on V56 are L(4, 0)/k 2 . One computes that the restriction of V56 to
a Levi subalgebra sl2 is completely reducible with composition factors L(4)5 /L(3)4 /L(2)3 /L(1)2 /k3 .
This action is inconsistent with any of the Jordan blocks in [Ste16, Tables 2, 3].

21

Lemma 4.12. Theorem 1.4 holds when h is of type G2 .
Proof. Assume that h is a non-G-cr subalgebra of type G2 . By (4), we have p = 7 and g is of type
E7 or E8 . In both cases, we will construct a non-G-cr subalgebra of type G2 .
Let g be of type E7 and p = 7. Then g contains a parabolic subalgebra p = l + q, with l′ of type
E6 and the l′ -composition factor of q is V27 . Now, l′ contains a maximal subalgebra h̄ of type G2
and V27 ↓ h̄ = L(2, 0). By Proposition 2.6, we have H1 (h̄, q) ∼
= k. Thus, by Lemma 4.2, there must
exist a non-G-cr subalgebra h of type G2 .
Now let g be of type E8 and p = 7. Then g has a Levi factor l′ of type E7 and by Lemma 4.1, the
non-L′ -cr subalgebra of E7 constructed above is therefore a non-G-cr subalgebra of g.

4.3. Subalgebras of type W1 . Our strategy in this section is slightly different. When p = 7 and
g is of type F4 or p = 5, 7 and g is of type E6 , E7 or E8 we construct an example of a non-G-cr
subalgebra of type W1 . In all other cases, we use calculations in GAP to show that each subalgebra
of type W1 is G-cr. To do this we rely on the following result:
Lemma 4.13 (cf. [HS16a, Thm. 1.1]). Let g be a simple classical Lie algebra of exceptional type.
Suppose h ∼
= W1 is a p-subalgebra of g and p is a good prime for g. Let ∂ ∈ h be represented by the
nilpotent element e ∈ g. Then the following hold:
(i) e is a regular element in a Levi subalgebra l of g and the root system associated to l is
irreducible.
(ii) For h(L) the Coxeter number of l, we have either p = h(L) + 1 or l is of type An and
p = h(L).
This will allow us to construct in GAP a generic subalgebra h of type W1 for a representative of
each possible nilpotent element representing ∂ and then show that X p−1 ∂ is also contained in l,
hence h = h∂, X p−1 ∂i is contained in l.
Lemma 4.14. Theorem 1.4 holds when h is of type W1 .
Proof. By Corollary 4.4, we may assume that h is non-G-cr and the projection of h to l′ is a
p-subalgebra unless p = 5, 7 and g has a Levi subalgebra of type Ap−1 .
When g is of type G2 all Levi factors are of type A1 , so this case is immediately discounted.
Now suppose g is of type F4 . When p = 5 we show that all subalgebras of type W1 are G-cr but
we postpone doing this here and give a general method below. When p = 7 we claim the following
subalgebra is non-G-cr:
h = he0100 + e0010 + e0001 , e−0122 + e−1222 + 4 · e−1231 i.
By checking the commutator relations hold, we see that h is isomorphic to W1 (with the first
generator mapped to ∂ and the second mapped to X p−1 ∂). Moreover, h is evidently contained in
a C3 -parabolic subalgebra. Now, using the MeatAxe in GAP, we calculate that the socle of the
adjoint module g ↓ h is 15-dimensional. On the other hand, any subalgebra of type W1 contained
in a Levi subalgebra of type C3 is conjugate to h̄ = he0100 + e0010 + e0001 , e−0122 i but using the
MeatAxe, we calculate that the socle of g ↓ h̄ is 24-dimensional. Therefore h is not contained in a
Levi subalgebra of C3 and is thus non-G-cr. When p ≥ 11, all subalgebras of g of type W1 are G-cr
by Lemma 4.13, since the largest Coxeter number of a proper Levi subalgebra of g is 6.
22

Now let g be of type E6 . We construct an example of a non-G-cr subalgebra of type W1 when
p = 5, 7. For p = 5 the subalgebra h ∼
= W1 embedded in a Levi subalgebra of type A4 via the
representation k[X]/X p ∼
= L(4)|k is non-A4 -cr and hence non-G-cr by Lemma 4.1. For p = 7,
consider the following subalgebra.
h = he10000 + e01000 + e00100 + e00010 + e00001 ,
0

0

0

0

0

e−11111 − 2 · e−11211 + e−12210 + e−01221 i
0

1

1

1

Again, one checks that h is isomorphic to W1 and is evidently contained in an A5 -parabolic subalgebra. We then use the MeatAxe to calculate that the socle of g ↓ h is 21-dimensional, whereas any
subalgebra of type W1 contained in a Levi subalgebra of type A5 acts on g with a 43-dimensional
socle. Therefore h is non-G-cr. When p ≥ 11, all subalgebras of type W1 are G-cr by Lemma 4.13,
since the largest Coxeter number of a proper Levi subalgebra of g is 8.
Finally, suppose g is of type E7 or E8 . Both contain an E6 -Levi subalgebra and therefore contain
a non-G-cr subalgebra of type W1 when p = 5, 7 by Lemma 4.1. We now consider the case p ≥ 11.
Therefore we have that h̄ is a p-subalgebra. By Lemma 4.13, it follows that (g, l′ , p) = (E7 , D6 , 11),
(E7 , E6 , 13), (E8 , D6 , 11), (E8 , E6 , 13), (E8 , D7 , 13) or (E8 , E7 , 19) and that ∂ = e is regular in l.
We rule out each possibility, as well as (F4 , B2 , 5), using calculation in GAP. All of the cases are
similar and we give the general method.
Let e be the regular nilpotent element of l. Then following the proof of [HS16a, Lem. 3.11] we
have an associated cocharacter τ with Lie(τ (Gm )) = hX∂i. This cocharacter is explicitly given
in [LT11]. Now suppose X p−1 ∂ is represented by the nilpotent element f . Then as [X∂, X p−1 ∂] =
(p − 1)X p−1 ∂, one calculates that f is in the direct sum of the τ weight spaces congruent to
−2p + 4 modulo p. As in [HS16a, §A.2] we use GAP to construct a generic nilpotent element f1
in such weight spaces. Using the commutator relations in W1 , for example ad(e)p−1 f = −e and
[f, ad(e)i (f )] = 0 for all 1 ≤ i ≤ p − 3, we then find that f1 , and hence f , is contained in l and thus
h = he, f i is contained in l. Thus all subalgebras of type W1 are G-cr in each possibility.

4.4. Proofs of Theorems 1.3 and 1.4. Of importance to us will be the following theorem:
Theorem 4.15 ( [HS16a, Thm. 1.3]). Let g be a simple classical Lie algebra of exceptional type.
Suppose p is a good prime for g and let h be a simple subalgebra of g. Then h is either isomorphic
to W1 or it is of classical type.
Proof of Theorem 1.4. As h must project to an isomorphic subalgebra of a proper Levi subalgebra
in good characteristic, the theorem now follows from Theorem 4.15 and Lemmas 4.6, 4.7, 4.8, 4.10,
4.11, 4.12 and 4.14 above.

Proof of Theorem 1.3. Suppose G is connected reductive with g its Lie algebra and h some semisimple subalgebra. Lemma 4.7 provides the forward implication and so it remains to prove the reverse
one. Since we are assuming that p > h(G), it is in particular a very good prime, and so we have
g∼
= g1 × g2 × · · · × gr × z where each gi is simple and z is a central torus of g. The parabolic subalgebras of g are the direct products of parabolic subalgebras of the simple factors, and similarly for
the corresponding Levi factors. Hence if h is in a parabolic subalgebra p of g, then it is in a Levi
subalgebra l of p if and only if the projection of h to each simple factor gi of g also has this property.
Thus we reduce the proof of the theorem to the case G is simple. Now if G is classical, the result
is supplied by Proposition 3.4. Thus we may assume that G is exceptional. It will be shown in
23

a forthcoming paper [PS] by A. Premet and the first author, that all semisimple subalgebras are
direct sums of simple Lie algebras when p > h(G). Putting this together with Theorem 4.15 and
Lemma 4.13, we have that h = h1 × · · · × hr for each hi a simple classical Lie algebra.
Assume h is a subalgebra of a parabolic subalgebra p = l + q of g. We will be done by Proposition
2.4 if we can show that any simple h-composition factor V ∼
= V1 ⊗· · ·⊗Vr with Vi a simple hi -module
satisfies H1 (h, V ) = 0. By the Künneth formula, we are done if we can show that H1 (hi , Vi ) = 0 for
each i. This has already been shown to be impossible in the proof of Theorem 1.4: in the context
of statement (4) we assumed the existence of such a module Vi , the contradiction of which proved

Theorem 1.4.
5. Unique embeddings of nilpotent elements into sl2 -subalgebras. Proofs of
Theorems 1.1 and 1.6
Proof of Theorem 1.1. First suppose p > h(G). We wish to show that the bijection (*) from the
introduction holds. To start with, [Pom80] provides the surjectivity. It remains to prove that the
map is injective.
Let (e, h, f ) be an sl2 -triple of g. By Theorem 1.3 we have that h = he, h, f i is G-cr and by Lemma
4.3, we have that h is a p-subalgebra. In particular, the element e is nilpotent with e[p] = 0. We
will show, under our hypotheses, that h is L-irreducible in a Levi subalgebra l = Lie(L) of G if and
only if the element e of h is distinguished in l. One way round is easy: If h is contained in a Levi
subalgebra l and e is a distinguished element, then h cannot be in a proper parabolic subalgebra
of l, since if it did, then by Theorem 1.3, h, hence also e, would be in a proper Levi subalgebra of
l. This is a contradiction as e is assumed distinguished.
For the other direction, assume h is L-irreducible in some Levi subalgebra l = Lie(L) of g and
assume, looking for a contradiction, that e is not distinguished. Let us see that this implies h is in a
proper Levi subalgebra of l. To do this, note first that h is a subalgebra of the Lie algebra l′ = [l, l];
we have l′ is semisimple, due to our hypothesis that p > h(G). We wish to show that h centralises a
vector w ∈ l′ , since then h ⊆ (l′ )w will be in a proper parabolic subalgebra of l′ , hence by Theorem
1.3, in a proper Levi subalgebra of l′ . To see that h does indeed fix a vector, let us start by noting
that e has at least one Jordan block of size 1 on the adjoint module of l′ . And for this, let e ∈ k ⊆ l
be a Levi subalgebra of l in which e is distinguished and note that p > h(G) implies that k contains
no factors of type Ap−1 so that k = k′ ⊕ z(k). That z(k) 6= 0 provides the existence of the requisite
Jordan block. Moreover, [HS16a, Prop. 3.3] implies that h ∈ k also, so that z(k) is also centralised by
h. Let 0 6= v ∈ z(k). Then [e, v] = 0 and v 6∈ im ad e. If [f, v] = 0 then h centralises v, so we are done.
Otherwise, consider the h-submodule W := hv, ad(f )v, . . . , ad(f )p−1 vi. (This is a submodule since
each ad(f )i (v) is an ad h-eigenvector and so W is ad e-stable; additionally it is ad(f ) stable since
the fact that h is a p-subalgebra implies that ad(f )p = 0.) Since W ′ := had(f )v, . . . , ad(f )p−1 vi is
both ad e- and ad f -stable, we have that the h-submodule W is a non-trivial extension of W ′ by k.
But dim W ′ ≤ p − 1 and the only simple h-module which extends the trivial is the module L(p − 2)
f be an indecomposable summand
of dimension p − 1. It follows that W ∼
= k|L(p − 2). Let W
f
of l containing W as a submodule. We cannot have W projective since then its restriction to
the subalgebra ke ⊂ h would give Jordan blocks of size p, which is not possible by the choice
f is indecomposable and reducible. The structure of such modules was determined
of v. Hence W
in [Pre91] (see [Far09, §4.1] for a more recent account): they have Loewy length 2 with isotypic socle
f consists of trivial modules. But then the socle of W
f ∗ ⊆ (l′ )∗ ∼
and head. Thus the head of W
= (l′ )
24

consists of trivial submodules. This implies that h fixes a 1-space on l′ . This justifies the claim
that h is in a proper Levi subalgebra of l, which is the contradiction sought.
We now wish to show that if h is an L-irreducible subalgebra containing the nilpotent element e
distinguished in l, then it is unique up to conjugacy in L. For this, recall that for any nilpotent
element
L e there is, by [Pre95a], an associated cocharacter τ : Gm → L which gives a grading
l =
i∈Z l(τ ; i). Moreover, the images of any two such cocharacters are conjugate by Le . By
Proposition 2.8, we may assume that if (e, h, f ) is an sl2 -triple, that the element h is contained in
Lie(τ (Gm )) and thus is unique up to conjugacy by Le , contained in the graded piece l(τ ; 0). Now
′
suppose e is distinguished and (e, h, f ) and (e, h, f ′ ) are two sl2 -triples. Then
P f − f ∈ le = le (≥ 0).
′
′
But the weight of h on f and f is −2, so that f − f is an element of i>0 le (−2 + ip). Since
p > h(G) and
P the largest j such that l(j) 6= 0 is 2h − 2 (this follows from [McN05, Prop. 30]), we
have that i>0 le (−2 + ip) = le (−2 + p). But since e is distinguished in l we have that le (i) = 0
for all odd i, hence that f − f ′ = 0 as required. This proves Theorem 1.1 for p > h(G).
For p ≤ h(G) we appeal to Lemma 4.7. If (e, h, f ) is an sl2 -triple as described in the statement
of the lemma, then f is a regular nilpotent in a proper Levi subalgebra l, say. Thus by [Pom80]
(or just another application of Lemma 4.7) it can be embedded into an sl2 -triple (e′ , h′ , f ) inside
l. Since (e, h, f ) is not contained in l we have that (f, −h, e) and (f, −h′ , e′ ) are non-conjugate
sl2 -triples containing the common nilpotent element f as required.

Proof of Theorem 1.6. We wish to see that h = Lie(H) for H a good A1 -subgroup of G. Fix a pair
e ∈ h. The assumption p > h(G) implies that all unipotent elements of G are of order p. Let u be
one corresponding to e under a Springer isomorphism. Now [Sei00, Props. 4.1 & 4.2] furnish us with
a good A1 -overgroup of any unipotent element u. Since all unipotent elements of H are conjugate,
the Lie algebra of a root group of H will contain e. Thus e ∈ h = Lie(H) as required.

6. Complete reducibility of p-subalgebras and bijections of conjugacy classes of
sl2 -subalgebras with nilpotent orbits. Proofs of Theorems 1.2 and 1.5
In this section we prove that there is a bijection
{conjugacy classes of sl2 subalgebras} → {nilpotent orbits}
if and only if p > b(G), where b(G) is defined in the introduction. Here, the bijection is realised by
sending a conjugacy class of sl2 -subalgebras to the nilpotent orbit of largest dimension meeting it.
It is not a priori clear that this would be well defined (as one conjugacy class of sl2 subalgebras
could contain two non-conjugate nilpotent orbits of the same dimension) but we show that this
never happens.
We will need two lemmas.
Lemma 6.1. If p > b(G) then for any sl2 -triple (e, h, f ) the elements e and f are nilpotent and
the element h is toral.
Proof. Suppose the statement is false. Then we may assume that h = he, h, f i ∼
= sl2 is a nonp-subalgebra. As there are no Levi subalgebras of type Arp−1 , we have that h is non-G-cr by
Lemma 4.3. Thus h lives in a parabolic subalgebra p = l + q which may be chosen minimally
subject to containing h such that the projection h̄ of h to a Levi subalgebra l is G-irreducible. By
Lemma 4.3 again, it follows that h̄ is a p-subalgebra. In particular, the images ē and f¯ of e and
25

f respectively are p-nilpotent. But as e and f are contained in the p-nilpotent spaces hēi + q and
hf¯i + q, respectively, they are also p-nilpotent.
Now h is a complement to q in the semidirect product h̄ + q and q has a filtration by restricted
l-modules, thus a filtration by restricted h̄-modules. By (4), it follows that one of the h̄-composition
factors of q is isomorphic to the h̄-module L(p − 2). We have H1 (sl2 , L(p − 2)) ∼
= k2 . Let us describe
a set of cocycle classes explicitly. Define γa,b : sl2 → L(p − 2) on a basis e, h, f ∈ sl2 via γa,b (h) = 0,
γa,b (e) = av−p+2 , γa,b (f ) = bvp−2 where vp−2 and v−p+2 are a chosen pair of highest and lowest
weight vectors in L(p − 2). Then one checks that the γa,b satisfy the cocycle condition (1), so for
instance
0 = γa,b (h) = γa,b ([e, f ]) = eγa,b (f ) − f γa,b (e) = 0 − 0.
Further, if for some v ∈ L(p − 2), we have γa,b (x) = γa′ ,b′ (x) + x(v) for all x ∈ sl2 , then applying to
h, we see that h(v) = 0, so that 0 is a weight of L(p − 2). This happens if and only if p = 2, which is
excluded from our analysis. Now, since H1 (sl2 , L(p−2)) ∼
= k2 , we see that the classes [γa,b ] are a basis
1
for H (sl2 , L(p − 2)). In particular, in any equivalence class of cocycles in H1 (sl2 , L(p − 2)), there is
a cocycle which vanishes on h. In the present situation, this means, following the argument in 2.4,
(or simply by observing that the restriction map H1 (h̄, Lie(Qi /Qi+1 )) → H1 (π(h), Lie(Qi /Qi+1 )) is
zero) that h can be replaced by a conjugate in which the element h ∈ h satisfies π(h) = h, so that
in particular, h is toral.

Lemma 6.2. Up to conjugacy by G = SO(V ) or Sp(V ), there is precisely one self-dual representation V of h := he, h, f i ∼
= sl2 of dimension r with p < r < 2p. We may construct V in such a
way that e ∈ h acts with a single Jordan block of size r and such that f acts with Jordan blocks
of sizes (p − 1 − i), (i + 1)2 . Moreover, V is uniserial with structure L(i)|L(p − 2 − i)|L(i) where
r = p + i + 1. Conversely, if V is uniserial with structure L(i)|L(p − 2 − i)|L(i) then up to swapping
e and f , we may assume e acts with a single Jordan block and V is self-dual.
Proof. Concerning the first sentence of the lemma, the existence of a representation V is implied
by [Pom80] applied to the regular nilpotent orbits in type Bn and Cn depending on the parity
of r. Suppose V did not consist of restricted composition factors. Then as p < r < 2p, there
would be precisely one which is not restricted, with at least one restricted factor. Thus V would
be decomposable and e would certainly not act with a single Jordan block. Hence the composition
factors of V are restricted. If the socle were not simple, then there would be two linearly independent
vectors u, v for which e · u = e · v = 0. But then the rank of e cannot be r − 1, contradicting the
hypothesis that it acts with a single Jordan block.
Thus the socle is simple, isomorphic to L(i) say. Then all the composition factors of V are L(i) or
L(p − 2− i) since otherwise the vanishing of Ext1 between either of these and any other composition
factor (by Lemma 2.7) would force a non-trivial direct summand. Also V / Soc V must contain a
submodule L(p − 2 − i), or the socle would split off as a direct summand. Since V contains the
submodule L(p − 2 − i)|L(i), self-duality forces it to contain a quotient L(i)|L(p − 2 − i). As the
simple socle is isomorphic to L(i), the head is also simple and isomorphic to L(i). Now, if there
were two composition factors isomorphic to L(p − 2 − i) then the dimension of V would be at least
2p, a contradiction. Thus the structure of V is precisely L(i)|L(p − 2 − i)|L(i), which has dimension
p + i + 1. Thus r = p + i + 1, as required.
We must now show that there is just one such representation up to conjugacy. As e has rank r − 1,
and h is toral by 6.1, there must be a vector of weight −i, say w, generating V under h. We have
her−i−1 · w, . . . , er−1 · wi spans Soc V , and if r − p ≤ j ≤ r − 1 then f · ej · w ∈ hej−1 · wi, by
26

uniqueness of the weights in the submodule L(p − 2 − i)|L(i). Also by uniqueness of the weights of
the quotient L(p − 2 − i)|L(i) we must have f · ei · w ∈ hei−1 · w, ei+p−1 · wi. As the endomorphism ep
commutes with e, f and h, the automorphism v 7→ v + ep · v of V is an sl2 -module homomorphism.
Moreover, as e preserves the form on V , so does ep and thus conjugating by an element 1 + ep of
G we may assume that f · ej · w ∈ hej−1 · wi for some 0 ≤ j ≤ i. It is easy to check that this
determines the action of f completely, hence the action of h.
Finally suppose V is of the form L(i)|L(p − 2 − i)|L(i). Consider the submodule U with two
composition factors. Then it is clear that either e or f has a Jordan block of size at least p and by
applying an automorphism if necessary, we may assume the former. Thus there is a vector v−ı̄ of
h-weight −ı̄ where ı̄ = p − 2 − i under which e generates U . Let v−ı̄+2s := es · v−ı̄ for 0 ≤ s ≤ p − 1
so that U is spanned by the vj . Inductively this determines the action of f except for the action of
f on v−ı̄ itself. Since f sends this vector into the −ı̄ − 2 = ith weight space of U which is spanned
by vi , we have f · v−ı̄ = λvi , for some λ ∈ k. Now take an −ith weight vector w−i ∈ V \ U . Define
w−i+2s = es · w−i for 0 ≤ s ≤ i. Then V is spanned by the wj and vj . Furthermore the action of f
is determined except on w−i itself. Since the ı̄ = p − 2 − i weight space is 1-dimensional we must
have f · w−i = µvı̄ . But calculating h · w−i = e · f · w−i − f · e · w−i we get −iw−i = µv−i − iw−i .
Thus µ = 0. Now the wj do not generate a submodule, so we must have e · wi = νv−ı̄ for some
ν 6= 0. But then h · wi = e · f · wi − f · e · wi gives us iwi = iwi − νλv−ı̄ so that λ = 0. Replacing the
vj by their division by ν we see that the action of e, f and h are now completely determined. It is
easy to check directly that the module is self-dual and the Jordan blocks of f are as claimed. 
Lemma 6.3. An indecomposable representation V of h := he, h, f i ∼
= sl2 of dimension r with
p < r < 2p is either a quotient or submodule of a projective restricted module, or self-dual of the
type described in Lemma 6.2.
Proof. A similar argument as used in the previous proof reduces us to the case that V consists of
restricted composition factors. We may assume V contains at least one composition factor each
of types L(i), L(ı̄) = L(p − 2 − i). We will show that the multiplicity of L(i) is 1 or 2. Assume
V has three composition factors isomorphic to L(i). It suffices to show there is no representation
V with composition series L(i)|L(ı̄)|(L(i) + L(i)). Let U be a module with composition series
L(ı̄)|(L(i) + L(i)). Then it is easy to see that e and f satisfy ep = f p = 0 as endomorphisms of
U . Thus U is a restricted representation. As it has simple head it is a quotient of the projective
cover P (ı̄) of L(ı̄) with composition series L(ı̄)|(L(i) + L(i))|L(ı̄). Let vı̄ , . . . , v−ı̄ be a set of weight
vectors for the 1-dimensional weight spaces of U coming from the composition factor L(ı̄). Then
one can check that an action of h on U is given (up to isomorphism) by e · vı̄ = u−i and f · v−ı̄ = wi
where {ui , . . . , u−i } and {wi , . . . , w−i } are a basis of weight vectors for the socle of U .
Now if V exists of this form, we may take a weight vector y−i of weight −i in V \ U . Then
y−i+2s := es · yi for 0 ≤ s ≤ i − 1 together with u, v and w spans V . If e · yi 6= 0 then it is a non-zero
multiple of vı̄ . Now h · yi = e · f · yi − f · e · yi leads to a contradiction. Similarly, we see that
f · y−i = 0. But this implies that hyi , . . . , y−i i is a submodule, which is a contradiction.
Thus V contains at most two composition factors isomorphic to L(i). If it contains two, then it
contains at most one of type L(ı̄). It is now clear that there are only three possible structures for
V , namely L(ı̄)|(L(i) + L(i)) or its dual, or L(i)|L(ı̄)|L(i), which is unique up to isomorphism by

Lemma 6.2.
Proof of Theorem 1.2. The reduction to the case where G is simple is easy. If G is classical, we
may argue using the natural module V for G. If G is of type An , then whenever p > b(An ) = n + 1
27

we may appeal to Theorem 1.1. On the other hand, whenever p ≤ b(An ) there is an Ap−1 -Levi
subgroup L of G, for which h acts on the natural module for L′ as an indecomposable module
k|L(p − 2). Now there is a bijection between nilpotent orbits of Ap−1 and isomorphism classes of
completely reducible sl2 -representations via partitions of p, so that h is in an extra conjugacy class
of sl2 -subalgebras, showing that there is no bijection between the sets in (**). It remains to consider
the cases where G is of type Bn , Cn or Dn . Note that p > b(G) implies 2p > dim V for V the natural
module for G. First note that completely reducible restricted actions of h are in 1-1 correspondence
with nilpotent orbits of G, which have associated partitions of dim V of size at most p, and these
account for all completely reducible actions by Lemma 4.3. Moreover, this bijection is realised by
sending h to any of the (G-conjugate) nilpotent elements it contains. If V ↓ h is not completely
reducible, we have that V ↓ h contains an indecomposable summand W which is not irreducible.
Suppose L(i) is a submodule of W . We have 0 ≤ i < p − 1 since L(p − 1) is in its own block
and we can have at most one factor of this type in V by dimensions. Then U ∼
= L(p − 2 − i)|L(i)
∗
∼
must be a submodule of W also. If U were a direct summand then U = L(i)|L(p − 2 − i) is
another submodule of V . But there can be no intersection between U and U ∗ . This implies that
dim V ≥ 2p, a contradiction. Hence U is not a direct summand. By dimensions there is at most one
indecomposable summand W , taking one of the forms discussed in Lemma 6.3. If it is the quotient
or submodule of a projective then a similar argument shows it has no intersection with its dual,
contradicting the dimension of W . Hence U lies in an indecomposable direct summand of the type
discussed in Lemma 6.2. Being the unique such in V , it must be non-degenerate. Thus h lives in
Lie(X × Y ) where X is the stabiliser of W ⊥ in G, of type Sp(W ⊥ ), SO(W ⊥ ) or O(W ⊥ ) (as the case
may be) with Y a similar stabiliser of W in G. Now, the necessarily restricted completely reducible
image of h in Lie(X) is determined up to X-conjugacy by the image of any nilpotent element in h.
By Lemma 6.2 the projection of h to Lie(Y ) is determined uniquely up to conjugacy in Y , with an
element of largest orbit dimension acting with a full Jordan block on W . In particular, a nilpotent
element of h of largest orbit dimension always determines h up to conjugacy.
In case G is exceptional, the analysis here is very case-by-case. Firstly, let e be any nilpotent
element of h. By Lemma 6.1, there is a toral element h ∈ h such that [h, e] = 2e and h ∈ im ad e.
By Proposition 2.8 we have h ∈ Lie(τ (Gm )) for τ a cocharacter associated to e. Now in the
grading of g associated with τ , we have e ∈ g(2), h ∈ g(0) and since [e, f ] = h, projecting f to its
¯
¯
component f¯ ∈ g(−2), we must have (e, h,
= ge (≥ 0). As also
Lf ) an sl2 -triple, with f − f ∈ ge L
[h, f − f¯] = −2(f − f¯), we have f − f¯ ∈ r>0 ge (−2 + rp). If the subspace r>0 ge (−2 + rp) is
trivial, we are automatically done. Looking at the tables in [LT11]5, this already rules out 111 of
the 152 orbits.
The strategy employed in the remaining cases is more subtle. The idea is to work inductively
through the remaining nilpotent orbits from largest dimension downwards, proving that for a given
nilpotent element e of orbit dimension d there is just one conjugacy class of nilpotent elements f
whose orbit dimension is d or lower and such that (e, h, f ) is an sl2 -triple. That is, we will show
that whenever f is not conjugate to f¯ by an element in Ge ∩ Gh , then f has higher orbit dimension
than that of e. To show this, we will effectively find all possible f such that (e, h, f ) is an sl2 -triple
and check each case.
To progress further, recall that Ge is the semidirect product Ce Re of its reductive part Ce and
unipotent radical Re . Since p is a good prime
L at this stage, one has Lie(Ce ) = ge (0) and Lie(Re ) =
ge (> 0). We also have Lie(Ge ∩ Gh ) ⊆
r≥0 ge (rp). We present two tools, which together deal
5In [LT11], the values of r such that g (r) 6= 0 are listed in the columns marked m.
e
28

with the remaining cases.
L For the first, henceforth Tool (a), suppose Ce acts with finitely many
orbits on the subspace r>0 ge (−2 + rp). In these cases one can write down all possibilities for the
element f up to conjugacy by Ge ∩ Gh . Then it is a simple matter to check the Jordan blocks of
the element f on the adjoint module for g in GAP and observe, by comparing with [Law95], that
the orbit dimension is larger than that of e. For example, if (g, O, p) is (E7 , (A5 )′ , 11) then we may
take
e = e100000 + e010000 + e001000 + e000100 + e000010 ,
0

0

0

0

0

h = 5 · hα1 + 8 · hα3 + 9 · hα4 + 8 · hα5 + 5 · hα6 ,
f¯ = 5 · e−100000 + 8 · e−010000 + 9 · e−001000 + 8 · e−000100 + 5 · e−000010 .
0

0

0

0

0

Ce◦

A21

and ge (−2 + p) is a module for C of high weight ̟1 for the first
of type
We have C :=
factor say. Thus C has two orbits on ge (−2 + p), namely the zero orbit and the non-zero orbit.
The element f¯ itself corresponds to the zero orbit, whereas if f = f¯ + f1 for 0 6= f1 ∈ ge (−2 + p)
then one checks that the Jordan blocks of the action of f on g are 23 + 173 + 15 + 11 + 93 + 3 + 13 ,
whereas those of f¯ are 11 + 102 + 93 + 7 + 66 + 53 + 42 + 3 + 16 . Comparing with [Law95] one sees
that f is in the orbit E6 whereas f¯ is in the orbit (A5 )′ .
The remaining cases all have the property that ge (p) 6= 0 with ge (p) having a basis of commuting
sums of root vectors. To describe Tool (b), suppose x ∈ ge (p) is a sum of commuting root vectors.
Then as p 6= 2, one may form the endomorphism δx := 1 + ad x + 21 (ad x)2 . Since x ∈ ge (p),
it follows that x commutes with both e and h, hence δx ∈ Ge ∩ Gh . Thus he, h, δx (f¯)i is an
sl2 -triple. For any y ∈ ge (p), we have [δx (f¯), δy (f¯)] = 0 modulo ge (> −2 + p) and so we get a
linear map δ• (f¯) : ge (p) → ge (−2 + p) by x 7→ δx (f¯). Now if z ∈ ge (0) then as f¯ ∈ ge (−2), we
have [f¯, z] ∈ g(−2) and since [e, [f¯, z]] = [h, z] = 0, we have [f¯, z] ∈ ge (−2) = 0. Thus f¯ is in the
centraliser of ge (0) so that Ce◦ also commutes with δ• (f¯); this means that δ• (f¯) is a Ce◦ -module map
from ge (p) → ge (−2 + p). Thus one may assume, replacing f by a conjugate, that the projection
of f to the image of δ• (f ) in ge (−p + 2) is zero. In particular, if this map is an isomorphism, one
concludes that any sl2 -tripleL(e, h, f ) is conjugate to another (e, h, f ′ ) such that the projection of
f ′ to ge (−2 + p) is zero. If r>0 ge (−2 + rp) = ge (−2 + p) (which it almost always is) this shows
that f is unique up to conjugacy. When using the fact that δ• (f¯) is a Ce◦ -module map, to check
the isomorphism, one finds that it always suffices to check that δ• (f¯) is non-zero on restriction to
high weights of the Ce◦ -modules ge (p), since these modules are always semisimple.
For example, suppose (g, O, p) is (E6 , D5 (a1 ), 7). Checking [LT11], one may choose
e = e10000 + e00000 + e01000 + e00010 + e00100 + e00110
0

1

0

0

1

0

and
h = 6 · hα1 + 7 · hα2 + 10 · hα3 + 12 · hα4 + 7 · hα5 .
Under these circumstances, one may calculate that the component of f in g(−2) is
f¯ := 6 · e−10000 + e−00000 + 10 · e−01000 + e−00010 + 6 · e−00100 + 6 · e−00110 .
0

L

0

1

0

1

0

From [LT11], one has r>0 ge (−2 + rp) = ge (5) of dimension 2, while ge (p) is generated over k by
the (commuting) root vectors x1 := e−00001 and x2 := e12321 . One checks that the images of x1 and
2

0

x2 under δ• (f ) are linearly independent. Thus δ• (f ) induces an isomorphism ge (p) → ge (−2 + p)
as required, showing the uniqueness of f in this case.
29

Lastly, using Tool (b) to assume that f projects trivially to the image of δ• (f ) and then applying
Tool (a) finishes the analysis in any remaining cases where ge (−2 + p) 6= 0. For example, if
(g, O, p) = (E8 , E7 (a4 ), 11), then Ce◦ is of type A1 and we may take
e = e1000000 + e0010000 + e0000010 + e0000110 + e0110000 + e0011000 + e0111000 + e0011100 .
0

0

0

1

0

1

0

Then δ• (f ) acts non-trivially on the element x = e2465431 ∈ ge (p) inducing a
3

1

Ce◦ -isomorphism

ge (p) → kCe◦ · (e2465421 + e2465321 ) ⊆ ge (−2 + p) and so one may assume
3

3

f ∈ f¯ + kCe◦ · (e2465431 ),
3

· (e2465431 ) ∼
with
= L(1) as a Ce◦ -module. Now, we use Tool (a). As Ce◦ has just one non-zero
3
orbit on the representation L(1), we may assume that f = f¯+ e2465431 . But computing the Jordan
kCe◦

3

blocks of f on the adjoint representation, one finds that f has a higher orbit dimension than that
of f¯.
There remain some cases where ge (−2 + 2p) 6= 0. These are (E8 , E8 (a5 ), 11), (E8 , E8 (b4 ), 11) and
(F4 , F4 (a2 ), 5). Since these cases are distinguished orbits, ge (−2 + p) = 0. Precisely the same
analysis as used in Tool (b) will work here, replacing (ge (−2 + p), ge (p)) with (ge (−2 + 2p), ge (2p)).
It remains to show that no bijection exists when 2 < p ≤ b(G). It is well known that the number of nilpotent orbits of g is finite and so it suffices to show there are infinitely
Pp−1 many classes
of sl2 -subalgebras. Suppose l is a Lie subalgebra of type Ap−1 and let e =
i=1 eαi and f =
Pp−1 2
he, f i is an sl2 -subalgebra with [e, f ] = diag(p − 1, p − 3, . . . ,
i=0 −i e−αi . Then one checks thatP
p
−p + 3, −p + 1). Further, let f0 = p−1
i=0 ie−αi and λ ∈ k with λ 6= λ. Then again one checks
that he, (f + λf0 )i is an sl2 -subalgebra, this time with [e, f ] = h + λI. We therefore have infinitely
many sl2 -subalgebras with pairwise non-isomorphic representations on the restriction of the natural
representation of l. The condition 2 < p ≤ b(G) implies that if G is not of type G2 then g has
a Levi subalgebra of type Ap−1 and when G is of type G2 then p = 3 and g has a pseudo-Levi
subalgebra of type A2 . In all cases we therefore have a subalgebra l of type Ap−1 and moreover, the
restriction of the adjoint representation of g to l contains a copy of the natural representation of
l. Thus we have infinitely many GL(g)-conjugacy classes of sl2 -subalgebras of g (all with pairwise
non-isomorphic representations) and so we certainly have infinitely many G-conjugacy classes of
sl2 -subalgebras.

With Theorem 1.2 in hand, we turn our attention to p-subalgebras and prove the last result,
Theorem 1.5.
Proof of Theorem 1.5. In light of Theorem 1.4, it suffices to prove the following three claims: All
p-subalgebras of type A1 are G-cr when b(G) < p ≤ h(G); there exists a non-G-cr p-subalgebra of
type A1 when 5 ≤ p ≤ b(G); and the examples of non-G-cr subalgebras of type B2 , G2 and W1
given in Section 4 are all p-subalgebras.
First, let b(G) < p ≤ h(G) and h = he, h, f i be a p-subalgebra of g, with e belonging to the largest
nilpotent orbit meeting h. Since h is a p-subalgebra we have e[p] = 0 and the restriction on p implies
that p is a good prime for G. As in the proof of Theorem 1.6, we are therefore furnished with a
good A1 -subgroup H of G such that e ∈ Lie(H), by [Sei00, Props. 4.1 & 4.2]. But now Theorem 1.4
implies that Lie(H) is conjugate to h, since both contain e and all nilpotent elements of Lie(H) are
30

conjugate (since all unipotent elements of H are conjugate) so e belongs to the largest dimensional
nilpotent orbit meeting Lie(H). By [Sei00, Prop. 7.2], good A1 -subgroups are G-cr. Therefore
Lie(H) is G-cr by [McN07, Thm. 1] and hence h is G-cr, as required.
Now suppose 5 ≤ p ≤ b(G). This implies that either G is of type E6 and p = 5 or G is of type
E7 , E8 and p = 5, 7. In each case we present a non-G-cr p-subalgebra of type A1 : By definition of
b(G), we have a Levi subgroup of type Ap−1 . We know l = Lie(L) has a p-subalgebra of type sl2
acting as L(p − 2)|k on the natural module for l (see the proof of Lemma 4.7). This subalgebra is
therefore non-L-cr and hence non-G-cr by Lemma 4.1.
Finally, we consider the subalgebras of type B2 , G2 and W1 from Section 4. The claim is clear for
the subalgebras of type B2 and G2 , since the given examples are Lie(H) for H a subgroup of G,
hence p-subalgebras. So it remains to consider the subalgebras of type W1 constructed in Section
4.3. Firstly, the subalgebra of type W1 contained in A4 when p = 5 is a p-subalgebra since it acts
via its canonical representation k[X]/X p ∼
= L(4)|k. For p = 7 we have two explicit constructions
of subalgebras of type W1 , one contained in a C3 -parabolic of F4 and the other contained in an
A5 -parabolic of E6 (and E7 , E8 ). One checks in GAP that the elements representing ∂ and X p−1 ∂
are both sent to 0 by the [p]-map in both cases and hence the subalgebras are p-subalgebras. 
Remark 6.4. Suppose G is simple and of exceptional type with Lie algebra g. Then we can extend
Theorem 1.5 to the case p = 3 when h is of type A1 . We have checked computationally that all
p-subalgebras of type A1 are G-cr when G is of type G2 . If G is not of type G2 then it has a Levi
subgroup L of type A2 . We know l = Lie(L) has a p-subalgebra of type sl2 acting as L(1)|k on
the natural 3-dimensional module. This subalgebra is therefore non-L-cr and hence non-G-cr by
Lemma 4.1.
Remark 6.5. The proofs and statements of the main theorems show that whenever p > b(G), there
is in fact a bijection of the form (*) between conjugacy classes of p-subalgebras of type sl2 and
nilpotent elements e ∈ g = Lie(G) such that e[p] = 0.
Finally, we turn our attention to the proof of Theorem 1.7.
Proof of Theorem 1.7. If the nilpotent element e belongs to an orbit whose label is defined over C
then we will see that e is contained in a Z-defined sl2 -subalgebra of gC such that the image of e in
gZ ⊗Z Fp = gFp is of the same type as e. In particular e is always contained in an sl2 -subalgebra
of g. If G is of classical type, this is straightforward as the natural representation is defined over
Z such that e is a sum of Jordan blocks with 1 on the super-diagonal, h is diagonal and f is
determined by a combinatorial formula given in [Car93, Proof of Prop. 5.3.1]. More carefully, for
sufficiently large p, there is a 1-1 correspondence between partitions of n of an appropriate sort,
depending on the root system of G, and direct sums of irreducible sl2 -representations of dimensions
corresponding to the partition which give an embedding into G. The irreducible representations
satisfy the condition of being defined over Z, with e represented by a sum of Jordan blocks with
1s on the superdiagonal. Thus we may assume that G is exceptional. Now [Tes95, Lem. 2.1, 2.4]
gives a Z-defined sl2 -triple containing e in the cases that e is not contained in a maximal rank
subalgebra. (That the representatives for the distinguished nilpotent elements in [Tes95] have the
same label over all primes follows from the results of [LS12].) Of course, if e is in a maximal rank
subalgebra then we are done by induction.
We are left with the distinguished elements that belong to orbits whose labels are not defined over
C. There are just two of these for p ≥ 3: one is the exceptional nilpotent orbit in G2 with label
31

(3)

(3)

A1 . The other is the exceptional nilpotent orbit in E8 with label A7 . For the latter we simply
exhibit an example cooked up with GAP. From [LS12] one may take
e := e0001110 + e0000111 + e1110000 + e1111000 + e0011100 + e0111100 + e0121000 + e0011111
0

0

1

0

1

0

0

1

and this is filled out to an sl2 -triple with
h = hα4 + hα5 + 2 · hα6 + hα8
f = 2 · e−0001110 + e−0011100 + e−0111100 + e−1111100 + 2 · e−0121000
0

1

0

1

0

+ 2 · e−0011110 + e−0111110 + e−0011111 + 2 · e−1121000 + 2 · e−0121100 .
1

0

0

1

1

(3)

Finally, if e is of type A1 in G2 then e can be taken to be e2α1 +α2 + e3α1 +2α2 . Now it is straightforward to check that the image of (ad e)2 does not contain e. (One can even do this by hand.) 

References
[ABS90]

H. Azad, M. Barry, and G. Seitz, On the structure of parabolic subgroups, Comm. Algebra 18 (1990),
no. 2, 551–562. MR MR1047327 (91d:20048)
[BMRT13] M. Bate, B. Martin, G. Röhrle, and R. Tange, Closed orbits and uniform S-instability in geometric
invariant theory, Trans. Amer. Math. Soc. 365 (2013), no. 7, 3643–3673. MR 3042598
[BNP02] C. P. Bendel, D. K. Nakano, and C. Pillen, Extensions for finite Chevalley groups. II, Trans. Amer. Math.
Soc. 354 (2002), no. 11, 4421–4454 (electronic). MR 1926882 (2003k:20063)
[BNP04]
, Extensions for Frobenius kernels, J. Algebra 272 (2004), no. 2, 476–511. MR 2028069
(2004m:20089)
[BNW09] B. D. Boe, D. K. Nakano, and E. Wiesner, Ext1 -quivers for the Witt algebra W (1, 1), J. Algebra 322
(2009), no. 5, 1548–1564. MR 2543622 (2011b:17027)
[Car72]
R. W. Carter, Conjugacy classes in the Weyl group, Compositio Math. 25 (1972), 1–59. MR 0318337 (47
#6884)
[Car93]
Roger W. Carter, Finite groups of Lie type, Wiley Classics Library, John Wiley & Sons Ltd., Chichester, 1993, Conjugacy classes and complex characters, Reprint of the 1985 original, A Wiley-Interscience
Publication. MR MR1266626 (94k:20020)
[DeB02]
Stephen DeBacker, Parametrizing nilpotent orbits via Bruhat-Tits theory, Ann. of Math. (2) 156 (2002),
no. 1, 295–332. MR 1935848
[Far09]
R. Farnsteiner, Group-graded algebras, extensions of infinitesimal groups, and applications, Transform.
Groups 14 (2009), no. 1, 127–162. MR 2480855
[HS16a]
Sebastian Herpel and David I. Stewart, Maximal subalgebras of Cartan type in the exceptional Lie algebras,
Selecta Math. (N.S.) 22 (2016), no. 2, 765–799. MR 3477335
[HS16b]
Sebastian Herpel and David I. Stewart, On the smoothness of normalisers, the subalgebra structure of
modular Lie algebras and the cohomology of small representations, Doc. Math. (2016), to appear.
[Jan03]
J. C. Jantzen, Representations of algebraic groups, second ed., Mathematical Surveys and Monographs,
vol. 107, American Mathematical Society, Providence, RI, 2003. MR MR2015057 (2004h:20061)
[Jan04]
, Nilpotent orbits in representation theory, Lie theory, Progr. Math., vol. 228, Birkhäuser Boston,
Boston, MA, 2004, pp. 1–211. MR 2042689 (2005c:14055)
[Kos59]
Bertram Kostant, The principal three-dimensional subgroup and the Betti numbers of a complex simple
Lie group, Amer. J. Math. 81 (1959), 973–1032. MR 0114875 (22 #5693)
[Law95]
R. Lawther, Jordan block sizes of unipotent elements in exceptional algebraic groups, Comm. Algebra 23
(1995), no. 11, 4125–4156. MR MR1351124 (96h:20084)
[LS96]
M. W. Liebeck and G. M. Seitz, Reductive subgroups of exceptional algebraic groups, Mem. Amer. Math.
Soc. 121 (1996), no. 580, vi+111. MR MR1329942 (96i:20059)
[LS04]
, The maximal subgroups of positive dimension in exceptional algebraic groups, Mem. Amer. Math.
Soc. 169 (2004), no. 802, vi+227. MR MR2044850 (2005b:20082)
32

[LS12]

[LT04]
[LT11]
[Lüb01]
[LY93]
[McN05]
[McN07]
[MT11]

[Pom80]
[Pre91]
[Pre95a]
[Pre95b]
[Pre03]
[PS]
[PS16]
[Sei00]
[Ser05]
[SF88]
[SS70]

[Ste10]
[Ste12]
[Ste16]
[Str73]
[Tes92]
[Tes95]
[Wei94]
[Ye90]

Martin W. Liebeck and Gary M. Seitz, Unipotent and nilpotent classes in simple algebraic groups and Lie
algebras, Mathematical Surveys and Monographs, vol. 180, American Mathematical Society, Providence,
RI, 2012. MR 2883501
Martin W. Liebeck and Donna M. Testerman, Irreducible subgroups of algebraic groups, Q. J. Math. 55
(2004), no. 1, 47–55. MR 2043006 (2005b:20087)
R. Lawther and D. M. Testerman, Centres of centralizers of unipotent elements in simple algebraic groups,
Mem. Amer. Math. Soc. 210 (2011), no. 988, vi+188. MR 2780340 (2012c:20127)
Frank Lübeck, Small degree representations of finite Chevalley groups in defining characteristic, LMS J.
Comput. Math. 4 (2001), 135–169 (electronic). MR 1901354 (2003e:20013)
Jia Chun Liu and Jia Chen Ye, Extensions of simple modules for the algebraic group of type G2 , Comm.
Algebra 21 (1993), no. 6, 1909–1946. MR 1215553 (94h:20051)
G. J. McNinch, Optimal SL(2)-homomorphisms, Comment. Math. Helv. 80 (2005), no. 2, 391–426.
MR 2142248 (2006f:20055)
George McNinch, Completely reducible Lie subalgebras, Transformation Groups 12 (2007), no. 1, 127–135.
Gunter Malle and Donna Testerman, Linear algebraic groups and finite groups of Lie type, Cambridge
Studies in Advanced Mathematics, vol. 133, Cambridge University Press, Cambridge, 2011. MR 2850737
(2012i:20058)
Klaus Pommerening, Über die unipotenten Klassen reduktiver Gruppen. II, J. Algebra 65 (1980), no. 2,
373–398. MR 585729 (83d:20031)
A. A. Premet, The Green ring of a simple three-dimensional Lie p-algebra, Izv. Vyssh. Uchebn. Zaved.
Mat. (1991), no. 10, 56–67. MR 1179217
Alexander Premet, An analogue of the Jacobson-Morozov theorem for Lie algebras of reductive groups of
good characteristics, Trans. Amer. Math. Soc. 347 (1995), no. 8, 2961–2988. MR 1290730 (95k:17012)
, Irreducible representations of Lie algebras of reductive groups and the Kac-Weisfeiler conjecture,
Invent. Math. 121 (1995), no. 1, 79–117. MR 1345285 (96g:17007)
, Nilpotent orbits in good characteristic and the Kempf-Rousseau theory, J. Algebra 260 (2003),
no. 1, 338–366, Special issue celebrating the 80th birthday of Robert Steinberg. MR 1976699
Alexander Premet and David I. Stewart, Classification of the maximal subalgebras of exceptional lie
algebras over fields of good characteristic, in preparation.
, Rigid orbits and sheets in reductive lie algebras over fields of prime characteristic, J. Inst. Math.
Jussieu (2016), to appear.
Gary M. Seitz, Unipotent elements, tilting modules, and saturation, Invent. Math. 141 (2000), no. 3,
467–502. MR 1779618 (2001j:20074)
J-P. Serre, Complète réductibilité, Astérisque (2005), no. 299, Exp. No. 932, viii, 195–217, Séminaire
Bourbaki. Vol. 2003/2004. MR 2167207 (2006d:20084)
H. Strade and R. Farnsteiner, Modular Lie algebras and their representations, Monographs and Textbooks
in Pure and Applied Mathematics, vol. 116, Marcel Dekker Inc., New York, 1988. MR 929682 (89h:17021)
T. A. Springer and R. Steinberg, Conjugacy classes, Seminar on Algebraic Groups and Related Finite
Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69), Lecture Notes in Mathematics,
Vol. 131, Springer, Berlin, 1970, pp. 167–266. MR 0268192 (42 #3091)
David I. Stewart, The second cohomology of simple SL2 -modules, Proc. Amer. Math. Soc. 138 (2010),
no. 2, 427–434. MR 2557160 (2011b:20134)
D. I. Stewart, The second cohomology of simple SL3 -modules, Comm. Algebra 40 (2012), no. 12, 4702–
4716. MR 2989676
David I. Stewart, On the minimal modules for exceptional Lie algebras: Jordan blocks and stabilizers,
LMS Journal of Computation and Mathematics 19 (2016), no. 1, 235–258.
H. Strade, Lie algebra representations of dimension p − 1, Proc. Amer. Math. Soc. 41 (1973), 419–424.
MR 0330247 (48 #8585)
D. M. Testerman, The construction of the maximal A1 ’s in the exceptional algebraic groups, Proc. Amer.
Math. Soc. 116 (1992), no. 3, 635–644. MR 1100666 (93a:20073)
Donna M. Testerman, A1 -type overgroups of elements of order p in semisimple algebraic groups and the
associated finite groups, J. Algebra 177 (1995), no. 1, 34–76. MR 1356359 (96j:20067)
Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics,
vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324 (95f:18001)
Jia Chen Ye, Extensions of simple modules for the group Sp(4, K), J. London Math. Soc. (2) 41 (1990),
no. 1, 51–62. MR 1063542 (91j:20105a)
33

[Yeh82]

S. el B. Yehia, Extensions of simple modules for the Chevalley groups and its parabolic subgroups, Ph.D.
thesis, University of Warwick, 1982.

University of Newcastle, UK
E-mail address: dis20@cantab.net (Stewart)
School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK, and the Heilbronn Institute
for Mathematical Research, Bristol, UK.
E-mail address: adamthomas22@gmail.com (Thomas)

34

