arXiv:1701.00125v1 [] 31 Dec 2016

Irreducible representations of simple algebraic groups
in which a unipotent element is represented by a
matrix with single non-trivial Jordan block
Donna Testerman and A.E. Zalesski∗

1

Introduction

The representation theory of algebraic groups is based on the study of weight
spaces, which are nothing other than the homogeneous components with respect
to a maximal torus. As every semisimple element belongs to a maximal torus,
the knowledge of weights and their multiplicities, in a given representation, yields
rich information on eigenspaces of the element under consideration.
It is much harder to obtain information on properties of unipotent elements,
for example, their fixed point space, their minimal polynomial, or in the best
case, their Jordan block structure in a given representation. The situation is
better for certain classes of elements, such as root elements, but in general,
problems of this kind can be difficult.
One such question was raised several years ago by the second author, specifically:
Determine the irreducible representations φ of a simple algebraic group G
such that, for some unipotent element u, the Jordan normal form of φ(u) has
exactly one block of size greater than 1?
The main motivation for considering this question is to supply an additional
tool for the recognition of linear groups via properties of a single element. However, one can also view this question as a test of how well the general theory is
adapted for solving computational problems on unipotent elements.
The first contribution was made by I. Suprunenko in [17, Theorem 1.9], who
solved the problem in the case where φ(u) has exactly one Jordan block. Later
she obtained a solution to the above problem for classical groups [19, Theorem
3] (see [20] for the proof). The current manuscript grew out of our work on
overgroups of regular elements in simple algebraic groups (see [23, 24]). At
that time, Suprunenko had announced a result which can be used for solving
Acknowledgement. A part of this work was carried out with the generous support of the
Bernoulli Center, at the Swiss Federal Institute of Technology Lausanne, when the second
author participated in the workshop “Local representation theory and simple groups” (2016).
∗

1

the above question for elements of order p in the exceptional groups other than
G2 ; see Remark after Theorem 4.2 for details. We have recently learned that
David Craven is working on similar questions for finite simple groups and their
automorphism groups.
Our main result answers the above question by considering all unipotent
elements in all simple algebraic groups of exceptional type.
Theorem 1.1. Let G be a simply connected simple linear algebraic group of
exceptional Lie type over an algebraically closed field F of characteristic p ≥
0, and let u ∈ G be a nonidentity unipotent element. Let φ be a non-trivial
irreducible representation of G. Then the Jordan normal form of φ(u) contains
at most one non-trivial block if and only if G is of type G2 , u is a regular
unipotent element and dim φ ≤ 7.
Theorem 1.1 remains true when replacing G by a finite quasi-simple group
of Lie type, as every irreducible F -representation of such a group lifts to a
representation of an appropriate simple algebraic group.
Our method is different from those used in [17, 20] and in a sense is indirect.
We first consider the case where G is of type A1 , and for a representation ρ,
not necessarily irreducible, we prove that the condition that ρ(G) contains a
unipotent element with only one non-trivial Jordan block implies that all nonzero weights of the representation are of multiplicity 1. Then we consider a
special case where p = 0 or |u| = p, and use a result of [22, 13] saying that, with
the exception of one class of elements in G2 , when p = 3, u is contained in a
simple algebraic subgroup of G of type A1 . This implies that all non-zero weights
of φ are of multiplicity 1. The irreducible representations with this property are
determined in [24]; the list is very short for G of exceptional type. The Jordan
normal form of all classes of unipotent elements in these representations was
computed by Lawther [8]. This yields the result for p = 0 or |u| = p. If
|u| > p > 0 then there is a suitable parabolic subgroup P such that u ∈
/ Ru (P ),
the unipotent radical of P . So the projection u′ into a Levi subgroup L of P
is non-trivial. Then one can observe that Jor(τ (u′ )) has single non-trivial block
for every composition factor τ of the restriction of φ to L. This allows us to use
induction on the rank of G.
Notation Throughout the paper p denotes a prime or 0, and F an algebraically closed field of characteristic p. Unless otherwise stated, G is a simple
simply connected algebraic group over F . All representations of G and F Gmodules are rational. To say that a representation ρ of G or an F G-module M
is irreducible, we write ρ ∈ Irr G or M ∈ Irr G. We let {α1 , . . . , αn } be a base of
the root system of G. Our labelling of Dynkin diagrams is as in [3].
For an integer n > 0 we denote by Jn the Jordan block of size n, that is, the
(n × n)-matrix with 1 at the positions (i, i) and (i, i + 1) for i = 1, . . . , n, and
0 elsewhere. The Jordan block J1 is called trivial. For a matrix x we denote
2

by Jor(x) a Jordan normal form of x. If x is a linear transformation of a vector
space V we write JorV (x) for a Jordan normal form of x, especially when we
need to specify V . A diagonal matrix with diagonal entries x1 , . . . , xn is denoted
by diag(x1 , . . . , xn ). A similar notation is used for a block-diagonal matrix.

2

Preliminaries

In Lemma 2.1 below ρreg
S denotes the F S-module afforded by the regular representation of a finite group S.
Lemma 2.1. Let F be an algebraically closed field of characteristic p > 0,
let G be a finite group with Sylow p-subgroup S of order p, and let M be an
indecomposable F G-module. Suppose that NG (S)/S is abelian. Then there is
−dim L
· ρreg
an indecomposable F S-module L such that M |S = dim M|S|
S ⊕ L and
dim L < p.
Proof. If the restriction M |S is a projective F S-module then the statement is
obvious with L = 0. Suppose otherwise. Set N = NG (S). By [1, §19, Theorem
1], M |N = L ⊕ P where P is projective, and L is indecomposable. As S is cyclic,
−dim L
· ρreg
every projective F S-module is free, so P |S = dim M|S|
S . Recall that L is
uniserial ([5, Theorem VII.2.4]), that is, the submodule lattice of L is a chain.
Let S = hyi, and set x = 1−y in the group algebra F N , L0 = L and Li = xi L for
i = 1, . . . , d assuming Ld = 0, Ld−1 6= 0. So d ≤ p. Observe that L1 is an F N module. (Indeed, for n ∈ N we have nL1 = (1 − nyn−1 )L = (1 − y j )L for some
integer j > 0 and 1 − y j = (1 − y) + (1 − y)y + · · · + (1 − y)y j−1 .) Therefore, Li is
an F N -module for every i. As S acts trivially on every Li /Li+1 , it is completely
reducible as F N -module. Since L is uniserial, every Li /Li+1 is irreducible. Since
N/S is abelian, dim(Li /Li+1 ) = 1. So d = dim L. Here d < |S| as otherwise L|S
is free and hence so is M |S . This completes the proof.
Lemma 2.2. Let Jm ∈ GLm (F ), Jn ∈ GLn (F ), 1 < n < m be Jordan blocks of
size m, n respectively. Then the Jordan form of Jm ⊗ Jn contains at least two
blocks of size greater than 1 unless m = n = 2 and p 6= 2.
Proof. Let X be a cyclic p-group if F is a field of characteristic p > 0, otherwise
an infinite cyclic group. Let Vm , Vn be indecomposable F X-modules of dimensions m, n, respectively. Let Vi ⊂ Vm , Vj ⊂ Vm be submodules of dimensions
i, j, respectively. Then Vi ⊗ Vj is a submodule of Vm ⊗ Vn . The number of indecomposable summands of an F X-module M of dimension ≥ k is not less than
that on any submodule of M . It follows that the result follows by induction as
soon as one verifies this for (m, n) = (2, 2), (3, 2).
If m = n = 2 then V2 ⊗ V2 = W1 ⊕ W2 , where the pair (dim W1 , dim W2 ) is
(2, 2) if p = 2 and (3, 1) if p 6= 2. (This is well known. If p 6= 2, see [5, Ch, VII,
3

Theorem 2.7]. If p = 2 then V2 is free, and hence so is V2 ⊗ V2 .) By induction,
the lemma is true for p = 2.
Let m = 3, n = 2. If p = 3 then V3 is free, and hence so is V3 ⊗ V2 . If p 6= 2, 3
then the lemma again follows by [5, Ch, VII, Theorem 2.7].
Corollary 2.3. Let G be an algebraic group and u ∈ G unipotent. Let M be an
irreducible FG-module such that JorM (u) has a single non-trivial block. Then
either M is tensor-indecomposable or G = A1 and dim M = 4.
We will require the following generation result, due to Guralnick and Saxl.
Lemma 2.4. [6, Theorems 5.1 and 5.4] Let G be an exceptional finite group of
Lie type, of untwisted rank l, and x ∈ (G \ Z(G)). Then G can be generated
by l + 3 conjugates of x, except, possibly, for the case G = F4 , q even, x2 = 1,
where G can be generated by 8 conjugates of x.
Lemma 2.5. Let G be an irreducible subgroup of GLn (F ) and g ∈ G. For an
eigenvalue λ of g set d = dim(Id −λ−1 g)V . Suppose that G is generated by m
conjugates of g. Then n ≤ dm.
In addition, if G is an exceptional group of Lie type, of untwisted rank l, then
n ≤ d(l + 3), except, possibly, for G of type F4 , q even, x2 = 1, where n ≤ 8d.
Proof.PLet G = hg1 , . . . , gm i, where gi (1 ≤ i ≤ m) is conjugate to g in G. Set
−1
′
′
′
′
V′ = m
i=1 (Id −λ gi )V . Then dim V ≤ md and GV = V , whence V = V ,
and the first statement follows.
If G is a finite exceptional group of Lie type then the additional statement
follows from Lemma 2.4.

3

Some representations of groups SL2(p) and SL2(F )

Lemma 3.1. Let D = SL(2, p) ⊂ G = SL(2, F ), u ∈ D a unipotent element and
let K be a tensor-decomposable irreducible F G-module. Suppose that JorK (u)
contains a single non-trivial block. Then p > 2 and dim K = 4. In addition,
K|D has a composition factor of dimension 3, and u has a block of size 3.
Proof. Let K = K1 ⊗ K2 , where K1 is a tensor-indecomposable F G-module and
d := dim K1 > 1. By Lemma 2.2, JorK1 (u) and JorK2 (u) consist of blocks of size
at most 2. As K1 is irreducible and tensor-indecomposable, JorK1 (u) consists of
a single Jordan block. Therefore, dim K1 = 2 and p 6= 2. Obviously, JorK2 (u)
cannot have more than one block. It follows that K2 is tensor-indecomposable,
and again by Lemma 2.2, dim K2 = 2. As K1 |D ∼
= K2 |D , K|D contains as a
direct summand the adjoint F G-module, which is of dimension 3 for p > 2.
The following result is well known (see e.g. Humphreys [7, 12.4]):
4

Lemma 3.2. Let E be an indecomposable rational module of composition length
2 for a simple algebraic group. Let µ, µ′ be the highest weights of E/L, L, resp.,
where L is the maximal submodule of E. Then either µ < µ′ or µ > µ′ , and in
the latter case E is of shape Wµ /M , where Wµ is the Weyl module of highest
weight µ and M is a submodule of Wµ .
Corollary 3.3. Let G = A1 and let V be an F G-module. Suppose that W is
either a Weyl module or indecomposable of composition length at most 2. Then
all weights of W are of multiplicity 1.
Proof. If p = 0 then all weights of an irreducible F G-module are well known to
be of multiplicity 1, and hence so are the weights of any Weyl module of G for
any p > 0. If W is an indecomposable of composition length 2 then, by Lemma
3.2, either W or the dual of W is a quotient of a Weyl module, whence the
claim.
Lemma 3.4. [2, Corollary 3.9] Let G = A1 and Vaω1 , Vbω1 P
be irreducible F Gi
modules of highest weights aω1 , bω1 , respectively. Let a =
i≥0 ai p and b =
P
i
i≥0 bi p be the p-adic expansions of a and b, respectively. Let vp (a + 1) denote
the maximum r such that pr |a + 1. Suppose that there exists an indecomposable
F G-module of composition length 2 with factors Vaω1 and Vbω1 .Then there exists
a natural number k ≥ vp (a+1) such that ai = bi for i 6= k, k+1, and ak = p−bk −2
and ak+1 = bk+1 ± 1. In particular, either a ≥ p or b ≥ p.
Corollary 3.5. Let p > 3 and let G, a, b be as in Lemma 3.4. Let E be an
F G-module with composition factors Vaω1 and Vbω1 . Suppose that a = pi + pj
and b = pr + pt where i < j, r < t. Then E is completely reducible.
Proof. Suppose the contrary. Note that a + 1 is coprime to p as p > 2. So
vp (a + 1) = 0. We can assume (by swapping the modules) that i ≤ r. Suppose
that i < r. Then bi = 0 and ai = 1; by Lemma 3.4, ai = p − 2, which is false as
p > 3. So i = r. Then j 6= t, and we can assume j < t. Then aj = 1, bj = 0,
and, by Lemma 3.4, 1 = aj = p − 2, a contradiction.
Remark.The assumption on a, b in Corollary 3.5 is equivalent to saying that
Vaω1 and Vbω1 are tensor-decomposable and dim Vaω1 = dim Vbω1 = 4.
Lemma 3.6. Let D = SL(2, p) and let S be an indecomposable F D-module. Let
u ∈ D, o(u) = p and suppose that JorS (u) contains a single non-trivial block.
Then dim S ≤ p + 1, and one of the following holds (where l is the composition
length of S):
(1) l = 1 and dim S ≤ p;
(2) l = 2, p > 2 and dim S ∈ {p − 1, p + 1} or p = 2 and dim S = 2;
(3) l = 3, p > 3, dim S = p + 1 and the dimensions of the composition factors
of S are 2, p − 3, 2;
5

(4) S has a composition factor of dimension p − 2 and all other factors are
trivial.
In addition, if dim S ≥ p then JorS (u) contains a block of size p.
Proof. The first claim is well known if S is irreducible. Suppose S is reducible,
and set U = hui. By Lemma 2.1, the Jordan form of u is (m · Jp , Jd ) for some
d < p. By assumption, 0 ≤ m ≤ 1, and m = 1 implies d ≤ 1. Therefore,
dim S ≤ p + 1. The additional claim (after item (4)) is obvious.
Consider the options for l. If l = 1 then the dimension of S is well known
to be at most p. Suppose that l > 1 and p = 2. Then D ∼
= SL2 (2), and the
non-trivial composition factors of S are projective D-modules. So either S is
irreducible or trivial on the subgroup of D of order 3. Then dim S = 2. Let
p > 2. If l = 2 then dim S = p − 1 or p + 1, see [2, p. 49] or [7, p.111]. If l ≥ 3
then (3) and (4) follow by applying (2) to the factors of S of composition length
2. Indeed, let T be an indecomposable submodule of S of composition length
2. By (2), dim T = p ± 1, and dim S ≤ p + 1 by the above. So dim T = p − 1,
and m := dim S/T ≤ 2. Let d, e be the dimensions of the composition factors
of T , where d + e = p − 1. As S is indecomposable, there is an indecomposable
quotient of S of dimension d + m or e + m. By (2), d + m or e + m equals p − 1.
We may assume that d + m = p − 1 (by reordering d, e). If m = 2 then d = p − 3
and e = 2 so (3) holds. Here p 6= 3 as d 6= 0. If m = 1 then d = p − 2 and e = 1,
that is (4) holds.
The following fact is trivial but it is convenient to state it explicitly as this
is frequently used.
Lemma 3.7. Let M be an F G-module, and u ∈ G unipotent. Suppose that
JorM (u) contains a single non-trivial block. Then the Jordan form of u on any
submodule or quotient module of M contains at most one non-trivial block. The
same is true for every quotient M2 /M1 , where M1 ⊂ M2 are F G-submodules of
M.
Proof. Indeed, u has a single block of size k > 1 on M if and only if the module
(u − 1)M is uniserial as an F hui-module. This property is inherited by submodules. Applying this to the dual of M , we get the result for quotient modules.
These also imply the result for M2 /M1 .
Lemma 3.8. Let G be of type A1 , and let u ∈ G be a unipotent element. Let
M be an F G-module and M0 the maximal trivial submodule of M . Suppose
that JorM (u) contains a single non-trivial block. Then the composition series of
M contains at most two non-trivial terms. More precisely, one of the following
holds:
(A) the composition series of M contains at most one non-trivial term; or
6

(B) p > 2, the composition length of M/M0 is 2 and p + 1 ≤ dim M/M0 ≤
p + 2.
Moreover, in case (B), we have:
(C) If aω1 , bω1 are the highest weights of the composition factors of M/M0 ,
with a ≥ b, then a ≥ p.
Proof. For p = 0 the lemma is trivial. So we assume p > 0. Obviously, we may
assume that M is indecomposable. Let D denote the subgroup of G isomorphic
to SL2 (p). Then M |D = S ⊕ T , where T is a trivial D-module and S is an
indecomposable one. We first prove (A) and (B), in a sequence of steps (1) to
(11).
(1) Every submodule of M/M0 is indecomposable. In particular, the socle of
M/M0 is irreducible.
Indeed, if L is a submodule of M/M0 and L = L1 ⊕ L2 , where L1 , L2 are nonzero F G-modules, then one of them is trivial by Lemma 3.7, which contradicts
Lemma 3.4 and the definition of M0 .
(2) Let M1 ⊂ M2 be F G-submodules of M . Suppose that JorM2 /M1 (u) has a
block of size p. Then M1 ⊆ M0 and M/M2 is a trivial F G-module.
Indeed, M2 /M1 has an indecomposable F hui-submodule X of dimension p.
Hence X is projective and injective, so M/M1 |hui = X ⊕ Y , where Y is an F huimodule. By Lemma 3.7, Y is a trivial F hui-module. As X ⊂ M2 /M1 , it follows
that u is trivial on M/M2 , and hence M/M2 is a trivial F G-module. Applying
this to the dual of M , we observe that u acts trivially on M1 . So the claim
follows.
(3) Let K be a composition factor of M . If JorK (u) has a block of size p then
statement (A) holds.
This follows from (2).
(4) If p = 2 then the statement (A) holds.
Indeed, in this case |u| = 2 and M has a non-trivial composition factor K,
say. Then u must have a block of size 2 on K, so the result is true by (3).
From now on we assume p > 2.
(5) If M has a composition factor K of dimension p then (A) holds.
Since dim K = p, K is tensor-indecomposable, and hence K|D is irreducible.
Then it is a projective D-module and JorK (u) consists of a single block of size
p. So the claim follows from (3).
(6) Let M1 ⊂ M2 ⊂ M3 be F G-submodules of M such that M2 /M1 is irreducible
and M3 /M1 is indecomposable. If M3 /M2 is trivial then (A) holds.
7

Set K = M2 /M1 and L = M3 /M1 . Suppose the contrary. Then it suffices
to handle the case where dim(L/K) = 1. If dim K ≤ p − 1 then dim L ≤ p.
By [11, Theorem 2], L is completely reducible, contrary to the assumption.
So dim K > p by (5). Therefore, K is tensor-decomposable. By Lemma 3.1,
dim K = 4 and p > 3 by Lemma 3.1 and (3). Then dim L = 5 ≤ p, so L is
decomposable by [11, Theorem 2]. This is a contradiction.
(7) Either statement (A) holds or all composition factors of M/M0 are non-trivial
and M/M0 is uniserial.
If M/M0 has a trivial composition factor then the claim follows from (6).
Otherwise, this follows from Lemma 3.7 and (1).
(8) Suppose that M has a tensor-decomposable composition factor K, say. Then
either (A) holds and dim K = 4 or p > 3 and (B) holds.
By Lemmas 3.7 and 3.1, dim K = 4, the composition factors of K|D are of
dimensions 1, 3 and JorK (u) has a block of size 3. Suppose that (A) is false.
Then p > 3 by (3). Furthermore, K can be included in a subquotient L, say, of
composition length 2. Let K ′ be the second factor of L. By (7), K ′ is non-trivial
and L is indecomposable.
Suppose first that K ′ is tensor-decomposable. Then dim K ′ = 4 and, by
Lemma 3.5, L is completely reducible, which is false.
So K ′ is tensor indecomposable, and hence K ′ |D is irreducible. Set m =
dim K ′ , so dim L = 4 + m. Then 1 < m ≤ p, and m < p by (5). Then L|D has
composition factors of dimensions 3, 1, m, and hence is decomposable (otherwise
contradicts Lemma 3.6(3)). As JorL (u) has a single non-trivial block, it follows
that L|D contains an indecomposable submodule X, say, with composition factors of dimensions 3, m. By Lemma 3.6, we have 3 + m = p − 1 or p + 1. In the
former case dim L = p, which is false in view of [11, Theorem 2]. So 3+m = p+1,
and hence dim L = p + 2. Furthermore, by Lemma 3.6, JorX (u) contains a block
of size p. Let L = M2 /M1 for some F G-modules M1 ⊂ M2 . Then, by (2), M/M2
and M1 are trivial F G-modules. So we deduce that M1 = M0 and M = M2 , i.e.
L = M/M0 , so (B) follows.
(9) If the restriction (M/M0 )|D has a trivial composition factor then (A) or (B)
holds.
Suppose the contrary. Then by (7), all composition factors of M/M0 are nontrivial, and tensor-indecomposable factors remain irreducible upon restriction
to D. So one of the composition factors of M is tensor-decomposable, which
contradicts (8).
(10) Either (A) or (B) holds, or the restriction (M/M0 )|D is indecomposable
and has no trivial composition factor.
Suppose that neither (A) nor (B) holds. Then, by (9), (M/M0 )|D has no
trivial composition factor. By Lemma 3.1, every composition factor of M/M0 is
8

tensor-indecomposable, and hence irreducible for D. Then (M/M0 )|D is indecomposable in view of Lemma 3.7.
(11) Statement (A) or (B) holds.
Suppose the contrary. Then, by (8), the composition factors of M are tensorindecomposable and hence are irreducible for D. By (10), (M/M0 )|D is indecomposable with no trivial composition factor. Then, by Lemma 3.6, p > 2 and
dim(M/M0 ) ≤ p + 1. If dim(M/M0 ) ≤ p then M/M0 is completely reducible by
[11], which contradicts (1).
So we have p > 2 and dim(M/M0 ) = p + 1, and all composition factors are
irreducible for D. If case (3) of Lemma 3.6 holds, then M/M0 has composition
length 3 with tensor indecomposable factors of dimension 2, p − 3, 2, contradicting Lemma 3.4. So Lemma 3.6(2) must hold and M/M0 has composition length
2 as in (B).
Finally, statement (C) follows from (B) and Lemma 3.4.
Lemma 3.9. Let G ∼
= A1 and let M be an F G-module. Let u ∈ G be a unipotent
element. Suppose that JorM (u) contains a single non-trivial block. Then all nonzero weights of M , with respect to a fixed maximal torus of G, are of multiplicity
1.
Proof. For p = 0 the lemma is trivial, for p > 0 this follows from Lemma 3.8
and Corollary 3.3.
Lemma 3.10. Let G be a simple algebraic group and X ∼
= A1 a subgroup of
G. Let u ∈ X be a unipotent element and M an F G-module. Suppose that
JorM (u) contains a single non-trivial block. Then all non-zero weights of M ,
with respect to a fixed maximal torus of G, are of multiplicity 1. Moreover M is
tensor-indecomposable, unless p 6= 2, G = A1 and dim M = 4.
Proof. Suppose the contrary, and fix a maximal torus T of G, and a maximal
torus T1 of X with T1 ⊂ T . Let Mλ be a T -weight space of weight λ 6= 0
such that dim Mλ > 1. Then dim Mw(λ) > 1 for every w ∈ W , where W is the
Weyl group of G. By Lemma 3.9, T1 acts trivially on Mw(λ) for every w ∈ W .
Recall that the weights of G are elements of Hom(T, GL1 (F )), which is a Zlattice of rank r equal to the rank of G. The Weyl group acts on T and hence on
Hom(T, GL1 (F )), so W is realized as a subgroup of GLr (Z). Let R be the vector
space over the rational number field Q spanned by the weights, and this yields
an embedding of W into GLr (Q). It is well known that W is an irreducible
subgroup of GLr (Q). The subspace of R spanned by {w(λ) : w ∈ W } is W stable,
P and hence coincides with R. Therefore, every weight µ can be written
as w∈W aw w(λ) with aw ∈PQ. Let m be an integer such that maw ∈ Z for
every w ∈ W . Then mµ = w∈W (maw )w(λ), where the coefficients maw are
integers. This implies that (mµ)(T1 ) = 1, whence µ(T1m ) = 1. Note that for
9

every t1 ∈ T1 there is an element t ∈ T1 such that tm = t1 , in other words
T1 = T1m . Therefore, µ(T1 ) = 1. This is true for every weight µ of T . This
implies that T1 acts trivially on M , which is a contradiction.
For the second assertion in the lemma see Corollary 2.3.
Theorem 3.11. Let G be a simple algebraic group, u ∈ G a unipotent element
and M an F G-module. Suppose that JorM (u) contains a single non-trivial block.
If p > 0 and up = 1, or if p = 0, then all non-zero weights of M , with respect to
a fixed maximal torus of G, are of multiplicity 1, unless possibly G = G2 , p = 3
(3)
and u lies in the class A1 as in [13].
Proof. By the main results of [22, 13], every element of order p in a simple
algebraic group in defining characteristic p is contained in a simple algebraic
(3)
subgroup of type A1 , with the exception of the class of elements labelled A1 in
G = G2 when p = 3. If p = 0, every unipotent element is well known to lie in a
subgroup of type A1 . So the statement follows from Lemma 3.10.

4

Representations of groups of exceptional type

In view of Theorem 3.11, it is useful to know which irreducible representations of
exceptional algebraic groups have all non-zero weights of multiplicity 1. Moreover, for our application to the question about the Jordan block structure of
unipotent elements in the representation space, Corollary 2.3 shows that we can
restrict our attention to tensor-indecomposable representations. We have the
following result taken from [24].
Lemma 4.1. Let G be a simple algebraic group of exceptional type and let M
be a tensor-indecomposable irreducible F G-module with highest weight ω 6= 0.
Suppose that all non-zero weights of M are of multiplicity one. Then (G, ω) ∈
{(E6 , pa ωi ), i = 1, 2, 6, (E7 , pa ωj ), j = 1, 7, (E8 , pa ω8 ), (F4 , pa ωk ), k = 1, 4, (G2 , pa ωl ), l =
1, 2}, for some a ≥ 0.
Theorem 4.2. Let G be a simple algebraic group of exceptional type and let
1 6= u ∈ G be a unipotent element of order p. Let M be an irreducible F Gmodule. Then one of the following holds:
(a) JorM (u) contains at least two non-trivial blocks;
(b) G = G2 , ω = pa ω1 , p ≥ 7, u is regular and JorM (u) has precisely one block;
(3)

(c) G = G2 , p = 3 and u lies in the class labelled A1 .
Proof. By Corollary 2.3, M is tensor indecomposable. As |u| = p, Theorem 3.11
leaves us with inspection of the unipotent block structure for the cases listed
in Lemma 4.1, unless we are in the situation of (c) above. The Jordan block
10

structure of all unipotent elements in the representations listed in Lemma 4.1
has been computed by Lawther [8]. We see that either (a) or (b) holds. (Note
that the case |u| = 2 can be deduced from classification of irreducible linear
groups generated by transvections, see for instance [12].)
Remark. Under the assumptions of Theorem 4.2 and assuming in addition
that G 6= G2 , a result of Suprunenko [18, Theorem 1] allows one to reduce the
problem under discussion to an analysis of modules M of dimension at most
4(l + 3) and a small list of further exceptions, most of which can be handled
using the Tables of [8]. However, the proof of Suprunenko’s result announced in
2005 has not been yet published, so we prefer to avoid using it. In addition, the
method based on Theorem 3.11 can also lead to an alternative proof of a similar
theorem for classical algebraic groups.
We wish to extend Theorem 4.2 to the case |u| > p. For this we use induction.
From Lemma 3.7 we get the following
Lemma 4.3. Let P be a parabolic subgroup in G with Levi factor L and unipotent
radical U. Let u ∈ P be unipotent and let u′ denote the projection of u into L.
Let V be an irreducible F G-module and 0 = V0 ⊂ V1 ⊂ V2 ⊂ · · · ⊂ Vk = V be
a socle series of V |P (that is, U acts trivially on the factors Vi /Vi−1 ). Suppose
that the Jordan form of u on V has at most one block of size greater than 1.
Then the same holds for the Jordan form of u′ on every factor Vi /Vi−1 .
Lemma 4.4. Let G be a simple algebraic group in characteristic p > 0, P a
parabolic subgroup and U = Ru (P ) the unipotent radical of P . If U is abelian
then the exponent of U equals p. If U is nilpotent of class 2 then U is of exponent
p or 4.
Proof. It is well known that U is generated by root subgroups Uα for some sets
of roots α, and each Uα is an abelian group of exponent p. This implies the
statement if U is abelian. Otherwise, let U ′ be the derived subgroup of U . Then
for every x ∈ U the mapping u → [x, u] (u ∈ U ) yields a group homomorphism
U → U ′ . Now U ′ is of exponent p as so is U/U ′ . If p = 2 then U is of exponent
4. If p > 2 then (xu)p = xp up [u, x]p(p−1)/2 = xp up as [u, x]p(p−1)/2 = 1. This
easily implies the lemma.
Remark. Below we will apply Lemma 4.4 to groups G of type E6 or F4 , and
to the maximal parabolic subgroups P corresponding to nodes 1, 6 for E6 and
1, 4 for F4 . Then U is of nilpotency class 2. This follows from the description
of U in [4, 4.4] for node 1 of E6 , and for node 6 this follows too as the graph
automorphism of E6 permutes the nodes 1,6. For G = F4 this similarly follows
from [4, 4.5] for node 1, and for node 4 from data at [4, p. 19]. (More precisely,
U is generated by the root subgroups Uα , where α runs over positive roots whose
11

expression in terms of simple roots contains α4 , and U contains a normal subgroup R generated by Uα for α such that the root α4 occurs in such expressions
with coefficient 2. As no positive root of F4 has α4 -coefficient greater than 2,
the claim follows from [4, 4.8(i)].)
Lemma 4.5. Let G be a simple exceptional algebraic group of rank l in defining
characteristic p, and u ∈ G unipotent. Let M be an irreducible G-module such
that JorM (u) contains a single non-trivial block. Let k ≥ 0 be an integer such
k
that |up | = p. Then dim M ≤ (p − 1)pk (l + 3), unless possibly G is of type F4 ,
p = 2, where dim M ≤ 2k+3 .
k

Proof. As JorM (u) contains a single non-trivial block, it follows that JorM (up )
contains at most pk non-trivial blocks, each of size at most p. Therefore,
k
dim(Id −up )M ≤ (p−1)pk . There is a finite group G1 ⊂ G of Lie type such that
u ∈ G1 and M is an irreducible F G1 -module. So the result follows by applying
k
Lemma 2.5 to up .
Example. If |u| ≤ 4 then dim M ≤ 2(l + 3) or 16 for F4 . If |u| = 9 then
dim M ≤ 6(l + 3).
Lemma 4.6. Let G be of type E6 , E7 , E8 or F4 , and let 1 6= u ∈ G be a unipotent
element. Let V be an irreducible G-module such that dim V > 1. Then JorV (u)
has at least two non-trivial blocks.
Proof. By Lemma 3.1, we can assume that V = Vω is tensor-indecomposable,
and hence without loss of generality the highest weight ω of V is p-restricted.
In view of Theorem 4.2 we can assume that |u| > p. If |u| = 4 then dim M ≤ 22
(see Example following Lemma 4.5); however G is well known to have no nontrivial irreducible representation of degree less than 25. So we can assume that
|u| > 2p. Suppose the contrary, that JorV (u) has a single non-trivial block.
Suppose first that G is of type E6 . Let Pi , i = 1, 6, be a maximal parabolic
of G corresponding to nodes 1, respectively 6, of the Dynkin diagram of G. Let
Li be a Levi subgroup of Pi and L′i the derived subgroup of Li . Then L′i is a
simple group of type D5 . By Lemma 4.4, u ∈
/ Ru (P ). Let u′ be the projection
of u into Li . Then 1 6= u′ ∈ L′i . By Lemma 4.3, the Jordan form of u′ has at
most one non-trivial block on every composition factor of the restriction of V to
L′i . Let λ be the highest weight of a non-trivial composition factor.
Recall that L′1 is generated by the root subgroups U±αi with i ∈ {2, 3, 4, 5, 6},
and P
L′2 is generated by the root subgroups U±αi with i ∈ {1, 2, 3, 4, 5}. Let
ω=
ai ωi , where ω1 , . . . , ω6 are the fundamental weights of E6 and 0 ≤ ai < p
for i = 1, . . . , 6. Let λ1 , . . . , λ5 be the fundamental weights of D5 . By Smith’s
theoremP[15], the restriction of V to L′1 contains a composition factor of highest
5
′
weight
i=1 ai λi , and the restriction of V to L2 has a composition factor of
highest weight a6 λ1 + a5 λ2 + a4 λ3 + a3 λ4 + a2 λ5 . (The root ordering for L′2 ∼
= D5
′
is inverse of that in L1 .) By a result of Suprunenko [20, Theorem 3] for p > 2
12

and [19, Theorem 3] for p = 2, λ = pm ω1 for some integer m > 0. Applying this
to L′1 , we obtain a1 ≤ 1, a2 = · · · = a5 = 0, applying to L′2 , we get a6 ≤ 1. So
we are left to examine the cases where ω ∈ {ω1 , ω6 , ω1 + ω6 }. By Lawther [8, p.
4136], JorV (u) has at least two non-trivial blocks for ω = ω1 and ω6 .
Let ω = ω1 + ω6 . Then ω − α1 , respectively ω − α6 , affords the highest weight
of an F Li -composition factor for i = 1, resp. i = 2, with highest weight λ1 + λ4 .
This again contradicts [20, Theorem 3] for p > 2 and [19, Theorem 3] for p = 2.
If G is of type E7 then G has a parabolic subgroup P whose Levi factor
L′ is of type E6 , and Ru (P ) is abelian ([4, 4.4]), and hence of exponent p. As
above we deduce that u 6∈ Ru (P ). If ω = ω1 , ω affords an L′ -composition factor
of highest weight ω1 for L′ = E6 , and if ω = ω7 , the weight ω − α7 affords
an L′ -composition factor which is again one of the 27-dimensional irreducible
E6 -modules. But this contradicts the conclusion of the previous paragraph. In
an entirely similar way, and using [4, 4.4], the case G = E8 follows from that for
E7 .
P
Let G = F4 , and let ω =
ai ωi , where ω1 , . . . , ω4 are the fundamental
weights of F4 . Let Pi , i = 1, 4, be a maximal parabolic of G corresponding to
nodes 1 or 4 of the Dynkin diagram of G. Let Li be a Levi subgroup of Pi and
L′i the derived subgroup of Li . Then L′4 is simple of type B3 , and L′1 is simple
of type C3 . Let λ1 , λ2 , λ3 be the fundamental weights of L′4 and µ1 , µ2 , µ3 the
fundamental weights of L′1 . As above, by Smith’s theorem, the restriction of
V to L′4 contains a composition factor with highest weight a1 λ1 + a2 λ2 + a3 λ3 ,
and the restriction of V to L′1 has a composition factor with highest weight
a4 µ1 + a3 µ2 + a2 µ3 . Applying [20, Theorem 3] to L′4 , we get a1 = 1, a2 = a3 = 0;
applying to L′2 [20, Theorem 3] for p > 2 and [19, Theorem 3] for p = 2, we get
a4 ≤ 1. So we have to examine the cases ω ∈ {ω1 , ω4 , ω1 + ω4 }. By Lawther [8,
p. 4134, 4135], JorV (u) has at least two non-trivial blocks for ω = ω1 and ω4 .
If p = 2 and ω = ω1 + ω4 then V is tensor-decomposable (see [16, Corollary of
Theorem 41]), so JorV (u) has at most two non-trivial blocks by Lemma 2.3.
Let ω = ω1 + ω4 and p > 2. By Lübeck [9], dim Vω = 1053. As |u| > p > 2,
we have |u| ≤ 33 , 52 , 72 , 112 for p = 3, 5, 7, 11 respectively, see [8]. By Lemma
4.5, we have dim M ≤ 7 · 10 · 11 = 770. This is a contradiction.
Now we consider the remaining cases for the group G = G2 . So from now
on we assume G = G2 , that is, the unipotent elements which are either of order
greater than p or the one class of elements of order 3 for p = 3 which do not
lie in any A1 -type subgroup [13, Theorem 5.1]. We fix a maximal torus T of G
and root subgroups with respect to T . For all roots α, let xα : Ga → G be an
isomorphism whose image is the T -root subgroup Uα corresponding to α. By
[8], for example, we are left to consider the following:
a) u is regular and hence conjugate to x−α1 (1)x−α2 (1), p ≤ 5, and u has order
p2 or 8.
13

b) u is in the class G2 (a1 ), p = 2, u is conjugate to xα2 (1)x3α1 +α2 (1) and has
order 4.
c) u has order 3 and is conjugate to x2α1 +α2 (1)x3α1 +2α2 (1).
We note that the Jordan block structure of all unipotent elements acting
on the irreducible modules with highest weight ω1 , or ω2 for p 6= 3, is given in
Lawther [8]. We use this to show:
Lemma 4.7. Let 1 6= u ∈ G = G2 be unipotent and let V be one of the two
irreducible F G-modules with highest weight ω1 or ω2 . Then JorV (u) has a single
non-trivial block if and only if one of the following holds:
(1) u is regular and ω = ω1 .
(2) u is regular, p = 3 and ω = ω2 .
Proof. For the weight ω = ω2 , p 6= 3 and ω = ω1 the statement follows directly
from [8, Table 1]. So it remains to consider the case of the irreducible module
VG (ω2 ), when p = 3. We apply the exceptional graph automorphism of G
and see that any element acting with only one non-trivial Jordan block on Vω2
must have image an element acting with only one non-trivial Jordan block on
Vω1 . Then, by the above remarks, the image of the element under the graph
automorphism must be regular, which means the element itself is regular. The
result follows.
Proposition 4.8. Let G = G2 and let u ∈ G be unipotent. Let V be an irreducible G-module with highest weight ω. Then JorV (u) has a single non-trivial
block if and only if ω = pk ω1 or p = 3 and ω = pk ω2 for some integer k ≥ 0.
Proof. Suppose the contrary. By Corollary 2.3, V is tensor-indecomposable,
so we may then assume that V is p-restricted. Note that u is conjugate to
a unipotent element of G2 (p), and the restriction of a p-restricted irreducible
representation to G2 (p) remains irreducible. So it suffices to deal with G =
G2 (p), which we assume in some cases below. By Lemma 4.7, we can assume
that ω 6= ω1 , ω2 .
Let p = 2. Then |u| ≤ 8. Let u2 be a power of u such that |u2 | = 2. By
Lemma 4.5, we get dim V ≤ 20. However, as ω is 2-restricted and ω 6= ω1 , ω2 ,
we have ω = ω1 + ω2 . By [9], dim V = 64 in this case, which is a contradiction.
For elements of order p for p = 3, as in c) above, we similarly obtain the
bound dim V ≤ 10, whereas the minimal dimension of an irreducible representation of G with highest weight ω 6= 0, ω1 , ω2 exceeds 26 [10]. So we are left with
the case |u| = 9 for p = 3 and |u| = 25 for p = 5. In these cases u is a regular
unipotent element of G, say u = x−α1 (1)x−α2 (1).
Let Pi ≤ G be the parabolic subgroup with Pi ⊇ B − , the Borel subgroup
generated by the maximal torus T and the root subgroups corresponding to
negative roots, and whose Levi factor Li satisfies L′i = hU±αi i. Set Qi = Ru (Pi )
14

and let πi : Pi → Li be the canonical projection. Set ui = πi (u) ∈ L′i . As u ∈ Pi ,
u stabilizes the commutator series V ⊃ [V, Qi ] ⊃ [[V, Qi ], Qi ] ⊃ · · · , and acts on
the quotients, via the element ui . Then by Lemma 3.7, the matrix of u in its
action on every subquotient has at most one non-trivial Jordan block and, by
Lemma 3.8(C), each of these subquotients can have at most one non-trivial prestricted irreducible constituent. Note that we will abuse notation and write ωi
for the restriction of ωi to T ∩ L′i . Setting [V, Q0i ] = V , let [V, Qdi ] = [[V, Qd−1
]Qi ]
i
for d ≥ 1; we will use the following result from [14, 2.3]:
If p = 3, assume ω = rω1 , for some r. Fix an integer d ≥ 0. Then the quotient
[V, Qdi ]/[V, Qd+1
] is isomorphic to the direct sum of those weight spaces of V of
i
the form ω − dαj − mαi , for some m ≥ 0 and where {i, j} = {1, 2}.
Case p = 3.
Note that V is tensor-indecomposable if and only if ω = 2ω1 or 2ω2 (we
ignore the cases ω = ω1 , ω2 by the above), see [16, Corollary of Theorem 41].
The class of regular elements is invariant under the graph automorphism of
G and so the Jordan block structure of a regular element on Vaω1 is the same as
the Jordan block structure of this element on Vaω2 . So it suffices to deal with
ω = 2ω1 . Using the above quoted result [14, 2.3], we see that [V, Q11 ]/[V, Q21 ] has
three F L′1 -composition factors, afforded by ω − α1 − α2 and ω − 2α1 − α2 , (the
latter weight has multiplicity 2, see [10], and affords two composition factors)
with highest weights 3ω1 , respectively ω1 . This contradicts Lemma 3.8(G).
Case p = 5.
Throughout, we will refer to [10] for weight multiplicities, without further
reference.
Consider first the modules V = Vaω1 , a = 2, 3, 4. Here the F L′1 -module
[V, Q1 ]/[V, Q21 ] has F L1 -composition factors of highest weights (a + 1)ω1 and
(a − 1)ω1 , afforded by ω − α1 − α2 , respectively ω1 − 2α1 − α2 . Then Lemma 3.8
implies that a = 4. But in this case the second weight has multiplicity 2 and
affords a third non-trivial composition factor, contradicting Lemma 3.8(C).
Now turn to the modules whose highest weight is of the form bω2 . For V2ω2 ,
the F L′2 -module [V, Q32 ]/[V, Q42 ] has composition factors of highest weights 3ω2 ,
ω2 and ω2 , afforded by ω − 3α1 − α2 , respectively ω − 3α1 − 2α2 , the latter having
multiplicity 3 in V . This contradicts Lemma 3.8(C).
For V3ω2 , the F L′2 -module [V, Q32 ]/[V, Q42 ] has composition factors of highest
weights 4ω2 and 2ω2 , afforded by ω − 3α1 − α2 , respectively ω − 3α1 − 2α2 ,
contradicting Lemma 3.8(C).
Finally, for the F G-module V4ω2 , we consider the action of the parabolic
subgroup P1 . The F L′1 -module [V, Q31 ]/[V, Q41 ] has a composition factor R of
dimension 10 whose highest weight is 9ω1 (afforded by the weight ω − 3α2 ).
Then R is a tensor product of modules of dimensions 2 and 5, which contradicts
Lemma 2.2.
15

We now turn to modules Vaω1 +bω2 , where 0 < a, b < 5. By [21, 1.35], the
weight ω − α1 − α2 has multiplicity 2 in V if and only if (3b + a + 3) 6≡ 0 mod 5.
If ω − α1 − α2 has multiplicity 2, the F Li -module [V, Qi ]/[V, Q2i ] has composition
factors of highest weights a+3 and a+1, or b+1 and b−1, for i = 1, respectively 2,
afforded by ω − αj and ω − αj − αi , where {i, j} = {1, 2}. Now using repeatedly
Lemma 3.8 and Lemma 2.2, we deduce that ω ∈ {ω1 + 2ω2 , 3ω1 + bω2 (b =
1, 3, 4), 4ω1 +ω2 }. If b > 1, the weight ω −2α2 affords an F L1 -composition factor
of [V, Q21 ]/[V, Q31 ] which is tensor decomposable and contradicts Lemma 2.2. If
ω = aω1 +ω2 , for a = 3, 4, then ω −3α1 and ω −3α1 −α2 afford F L2 -composition
factors of [V, Q32 ]/[V, Q42 ], of highest weights 4ω2 , respectively 2ω2 , contradicting
Lemma 3.8(C).
This completes the consideration of the remaining cases for the group G = G2
and together with Theorem 4.2 completes the proof of Theorem 1.1.

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