Simple endotrivial modules for linear, unitary and exceptional groups

0.39 MB·English

by  Caroline Lassueur

SIMPLE ENDOTRIVIAL MODULES FOR LINEAR,
UNITARY AND EXCEPTIONAL GROUPS
CAROLINE LASSUEUR AND GUNTER MALLE
Abstract. Motivated by a recent result of Robinson showing that simple endotrivial
5 modules essentially come from quasi-simple groups we classify such modules for finite
1 speciallinearandunitarygroupsaswellasforexceptionalgroupsofLietype. Ourmain
0 tool is a lifting result for endotrivial modules obtained in a previous paper which allows
2
us to apply character theoretic methods. As one application we prove that the ℓ-rank

n of quasi-simple groups possessing a faithful simple endotrivial module is at most 2. As
a a second application we complete the proof that principal blocks of finite simple groups
J
cannot have Loewy length 4, thus answering a question of Koshitani, Ku¨lshammer and
2
Sambale. Our results also imply a vanishing result for irreducible characters of special

linear and unitary groups.
]
T
R
.
h
1. Introduction
t
a
m LetG beafinitegroupandk afieldofprimecharacteristic ℓ dividing |G|. AkG-module
[ V is called endotrivial if V ⊗V∗ ∼= k ⊕P, with a projective kG-module P. Endotrivial

1 modules have seen a considerable interest in the last fifteen years. Due to a recent result
v of Robinson, the focus has moved to simple endotrivial modules for quasi-simple groups.
0
In our predecessor paper [16] we classified such modules for several families of finite quasi-
0
4 simple groups. The present paper is a continuation of this project. We obtain a complete
0
classification for special linear and unitary groups and almost complete results for groups
0
. of exceptional Lie type.
1
We also give a necessary and sufficient condition for a trivial source module to be en-
0
5 dotrivial, depending only on the values of the associated ordinary character on ℓ-elements
1
of the group. This allows us to settle some cases left open in [16] for simple modules of
:
v
sporadic groups.
i
X It turned out in our previous work [16] that simple modules of quasi-simple groups G
r are rarely ever endotrivial, and if such modules exist at all, then this seems to severely
a
restrict the structure of Sylow ℓ-subgroups of G. Combining our new results with those
of [16] and [17] we can show Conjecture 1.1 in [16]:
Theorem 1.1. Let G be a finite quasi-simple group having a faithful simple endotrivial
module in characteristic ℓ. Then the Sylow ℓ-subgroups of G have rank at most 2.
Date: January 5, 2015.
2010 Mathematics Subject Classification. Primary 20C20; Secondary 20C30, 20C33, 20C34.
Key words and phrases. simple endotrivial modules, quasi-simple groups, special linear and unitary
groups, Loewy length, zeroes of characters.
The authors gratefully acknowledge financial support by ERC Advanced Grant 291512. The
first author also acknowledges financial support by SNF Fellowship for Prospective Researchers
PBELP2−143516.
1
2 CAROLINELASSUEURANDGUNTER MALLE
In the case of cyclic Sylow ℓ-subgroups, we calculated in [16, §3] the number of ℓ-blocks
of a group containing simple endotrivial modules and proved that they correspond to
non-exceptional liftable characters. In ℓ-rank 2, simple endotrivial modules only seldom
occur. We show that the following covering groups of linear, unitary and exceptional
groups do have faithful simple endotrivial modules in ℓ-rank 2: for ℓ = 3, the groups
2.L (4), 4 .L (4), 4 .L (4) and U (q) with q ≡ 2,5 (mod 9), for ℓ = 5, the groups F (2),
3 1 3 2 3 3 4
2.F (2), and for ℓ = 7, the group 2.F (2). We note that all these modules have trivial
4 4
source, apart from the ones for 4 .L (4).
1 3
As a further application of our results we are able to complete an investigation begun
by Koshitani, Ku¨lshammer and Sambale [12] and show:
Theorem 1.2. Let G be a finite non-abelian simple group and ℓ > 2 a prime such that
Sylow ℓ-subgroups of G are noncyclic. Then the principal ℓ-block of G cannot have Loewy
length 4.
Our proofs also yield a vanishing result for characters of linear and unitary groups of
large enough ℓ-rank:
Theorem 1.3. Let ℓ > 2 be a prime and G a finite quasi-simple covering group of L (q)
n
or U (q) of ℓ-rank at least 3. Then for any non-trivial χ ∈ Irr(G) there exists an ℓ-singular
n
element g ∈ G with χ(g) = 0, unless one of:
(1) ℓ|q is the defining characteristic of G;
(2) ℓ = 5, G = L (q) with 5||(q−1) and χ(1) = q2Φ ; or
5 5
(3) ℓ = 5, G = U (q) with 5||(q+1) and χ(1) = q2Φ .
5 10
One of the main tools for our investigations is our earlier result [16, Thm. 1.3] which
asserts that all endotrivial modules are liftable to characteristic 0. This allows us to
apply character theoretic methods in our investigation. More specifically, be can employ
the ordinary character theory of groups of Lie type as developed by Lusztig.
The paper is built up as follows. In Section 2 we derive Theorem 2.2 and apply it
first in Corollary 2.4 to rule out some examples in sporadic and exceptional type groups,
and in Corollary 2.9 to compute the torsion subgroup TT(G) of T(G) for several small
sporadic simple groups. In Section 3 we show that special linear groups in rank at least 3
have no faithful simple endotrivial modules (see Theorem 3.10). In Section 4 we classify
simple endotrivial modules for special unitary groups (see Theorem 4.5) and complete the
proof of Theorem 1.1 (in Section 4.3) and of Theorem 1.3 (in Section 4.5). We investigate
simple endotrivial modules for exceptional type groups in Section 5, leaving open only
one situation in groups of types E , 2E and E , respectively, see Theorem 5.2. Finally,
6 6 7
in Section 6 we complete the proof of Theorem 1.2.
Acknowledgement: We thank Frank Lu¨beck for information on the Brauer trees of
6.2E (2) for the prime 13, and Shigeo Koshitani for drawing our attention to [12].
6
2. Characters of trivial source endotrivial modules
Let G be a finite group, ℓ a prime number dividing |G|, k an algebraically closed field
of characteristic ℓ. Let (K,O,k) be a splitting ℓ-modular system for G and its subgroups.
Let 1 ∈ Irr(G) denote the trivial character of G.
G
SIMPLE ENDOTRIVIAL MODULES 3
If M is an indecomposable trivial source kG-module, then M lifts uniquely to an
indecomposable trivial source OG-lattice, denoted henceforth by Mˆ. Let χ denote
Mˆ
the ordinary character of G afforded by Mˆ. The following was proven by Landrock–Scott,
see [15, II, Lem. 12.6]:
Lemma 2.1. Let M be an indecomposable trivial source kG-module and x ∈ G be an
ℓ-element. Then:
(a) χ (x) ≥ 0 is an integer (corresponding to the multiplicity of the trivial khxi-
Mˆ
module as a direct summand of M| );
hxi
(b) χ (x) 6= 0 if and only if x belongs to a vertex of M.
Mˆ
From this we derive the following necessary and sufficient condition:
Theorem 2.2. Let M be an indecomposable trivial source kG-module. Then M is en-
dotrivial if and only if χ (x) = 1 for all non-trivial ℓ-elements x ∈ G.
Mˆ
Proof. If dim M = 1, then M is endotrivial and satisfies χ (x) = 1 for all ℓ-elements
k Mˆ
x ∈ G, therefore we may assume dim M > 1. We may also assume dim M is prime
k k
to ℓ. Indeed, on the one hand endotrivial modules have dimension prime to ℓ (see [16,
Lem. 2.1]), and on the other hand if the sufficient condition is assumed, then dim M ≡
k
χ (x) = 1 (mod ℓ) for any non-trivial ℓ-element x ∈ G. Thus, by [2, Thm. 2.1], the
Mˆ
trivial module occurs as a direct summand of M ⊗ M∗ with multiplicity 1. Since M has
k
trivial source we may write
M ⊗ M∗ ∼= k ⊕N ⊕...⊕N
k 1 r
where N ,...,N are non-trivial indecomposable trivial source modules. Clearly M is
1 r
endotrivial if and only if the module N has vertex {1} for every 1 ≤ i ≤ r. At the level
i
of characters we have
χ χ = 1 +χ +···+χ .
Mˆ Mˆ∗ G Nˆ1 Nˆr
Thus, by Lemma 2.1(b), M is endotrivial if and only if (χ χ )(x) = 1 for every non-
Mˆ Mˆ∗
trivial ℓ-element x ∈ G. Moreover since dim M is prime to ℓ, M has vertex a Sylow
k
ℓ-subgroup P, so that, by Lemma 2.1(a) and (b), χ (x) is a positive integer for every
Mˆ
non-trivial ℓ-element x ∈ G. The claim follows. (cid:3)
Henceforth we denote by T(G) the abelian group of isomorphisms classes of indecom-
posable endotrivial modules, with multiplication induced by the tensor product ⊗ . Then
k
T(G) is known to be finitely generated. For notation and background material on T(G)
we refer to the survey [4] and the references therein.
Under the assumption that the normal ℓ-rank of the group G is greater than one, trivial
source endotrivial modules coincide with torsion endotrivial modules. Then Theorem 2.2
says that the torsion subgroup TT(G) of T(G) is a function of the character table of G.
Corollary 2.3. Assume the Sylow ℓ-subgroups of G are neither cyclic, nor semi-dihedral,
nor generalised quaternion. If V is a self-dual endotrivial kG-module, then χ (1) ≡ 1
Vˆ
(mod |G| ) and χ (x) = 1 for all non-trivial ℓ-elements x ∈ G.
ℓ Vˆ
Proof. If V is self-dual, then the class of V is a torsion element of the group of endotrivial
modules T(G). If P ∈ Syl (G), then restriction to P induces a group homomorphism
ℓ
4 CAROLINELASSUEURANDGUNTER MALLE
ResG : T(G) −→ T(P) : [M] 7→ [M| ], where under the assumptions on P the group
P P
∼
T(P) is torsion-free (see [23, §7 and §8]). It follows that V| = k ⊕(proj), that is, V is
P
a trivial source kG-module. The congruence modulo |G| is immediate and the second
ℓ
(cid:3)
claim is given by Theorem 2.2.
This allows us to settle some of the cases left open in [16] corresponding to self-dual
modules not satisfying the conclusion of Corollary 2.3:
Corollary 2.4. The candidate characters for the sporadic groups Fi′ and M from [16,
24
Table 6] are not endotrivial. Similarly neither are the unipotent characters φ for E (q)
80,7 6
nor φ for 2E (q) from [16, Table 4].
16,5 6
The following examples show how Theorem 2.2 enables us to find torsion endotrivial
modules using induction and restriction of characters and to settle some other cases for
covering groups of sporadic simple groups left open in [16, Table 6] or which could so
far be discarded via Magma computations only. Below we denote ordinary irreducible
characters by their degrees, and the labelling of characters and blocks is that of the GAP
character table library [22].
Example 2.5 (The character of degree 55 of M in characteristic ℓ = 3).
22
Let G = M and let P ∈ Syl (G). Let V denote the simple kG-module affording the
22 3 55
character 55 ∈ Irr(G). Let e denote the block idempotent corresponding to the principal
1 0
3-block of kG, which is the unique block with full defect. The group G has a maximal
subgroup H ∼= A .2 such that H ≥ N (P), and inducing the non-trivial linear character
6 3 G
1 ∈ Irr(H) to G yields e ·IndG(1 ) = 55 , so that 55 is the character of a trivial source
2 0 H 2 1 1
module. Thus V is endotrivial by Theorem 2.2.
55
Example 2.6 (Thetwo faithfulcharactersofdegree154of2.M incharacteristic ℓ = 3).
22
Let G = 2.M , let P ∈ Syl (G). Let V ,V∗ denote the dual simple kG-modules
22 3 154 154
of dimension 154 affording the characters 154 ,154 ∈ Irr(G). They both belong to
2 3
the faithful block B . The group G has a subgroup H ∼= (2 × A ).2 ≥ N (P), and
6 6 3 G
inducing the linear characters 1 ,1 ∈ Irr(H) to G we have e · IndG(1 ) = 154 and
3 4 6 H 3 2
e · IndG(1 ) = 154 , where e is the block idempotent corresponding to B . Therefore
6 H 4 3 6 6
V , V∗ are trivial source modules, thus by Theorem 2.2 it follows from the values of
154 154
154 ,154 that they are endotrivial. We note that the two faithful simple endotrivial
2 3
modules V ,V∗ affording the characters 10 ,10 ∈ Irr(G) from [16, Thm. 7.1] are also
10 10 1 2
trivial source modules. Indeed 10 ⊗10 = 45 +55 where 45 has defect zero. Whence
1 1 2 1 2
V ⊗ V ∼= V ⊕(proj) and V ,V∗ must be trivial source since V is.
10 k 10 55 10 10 55
Example 2.7 (The three characters of degree 154 of HS in characteristic ℓ = 3).
Let G = HS, and let V ,V ,V denote the three simple self-dual kG-modules of di-
1 2 3
mension 154, affording the characters 154 ,154 ,154 ∈ Irr(G) respectively. Let H be a
1 2 3
maximal subgroup of G isomorphic to M . Then 154 | = 55 + 99 , where 55 is the
22 1 H 1 1 1
character afforded by the simple endotrivial kM -module V of Example 2.5, and 99
22 55 1
∼
has defect zero. Thus V | = V ⊕(proj) is endotrivial, and so is V (see [16, Lem. 2.2]).
1 H 55 1
The characters 154 ,154 lie in the non-principal block of full defect B . Let e be the
2 3 2 2
∼
corresponding block idempotent. Then G has a maximal subgroup L = U (5) : 2 with a
3
non-trivial linear character 1 such that e · IndG(1 ) = 154 . Reducing modulo 3, this
2 2 L 2 2
SIMPLE ENDOTRIVIAL MODULES 5
proves that V is a trivial source module, hence endotrivial by Theorem 2.2. So is V , the
2 3
image of V under the outer automorphism of G.
2
Example 2.8 (The four characters of degree 61776 of 6.Fi in characteristic ℓ = 5).
22
Let G = 6.Fi . The characters 61776 ,61776 belong to block 107 and 61776 ,61776
22 1 3 2 4
to block 108. Let e ,e denote the corresponding block idempotents respectively.
107 108
Then G has two non-conjugate subgroups H and L isomorphic to O+(2).3.2, such that
8
e · IndG(1 ) = 61776 , e · IndG(1 ) = 61776 , e · IndG(1 ) = 61776 and
107 H H 1 108 H H 2 107 L L 3
e ·IndG(1 ) = 61776 . Therefore the simple reductions V , V , V , V
108 L L 4 617761 617762 617763 617764
modulo5ofthesecharactersaretrivialsourcemodulesandareendotrivialbyTheorem2.2.
We recall from [16, Rem. 7.2] that Fi also has a simple trivial source endotrivial mod-
22
ule V affording 1001 ∈ Irr(Fi ), and 3.Fi has two faithful simple trivial source
1001 1 22 22
endotrivial modules V , V∗ affording 351 ,351 ∈ Irr(3.Fi ).
351 351 1 2 22
The structure of the group T(G) is not known for the sporadic groups and their covers.
The above examples enable us to describe explicitly the elements of the torsion subgroup
TT(G) in these cases.
Corollary 2.9.
(a) In characteristic 3, with notation as in Examples 2.5 and 2.6 and with M an
indecomposable endotrivial kM -module affording the character 154 ∈ Irr(M )
22 1 22
we have
TT(M ) = h[V ],[M]i ∼= Z/2⊕Z/2,
22 55
TT(2.M ) = h[V ],[V ]i ∼= Z/2⊕Z/4.
22 10 154
(b) In characteristic 3, with notation as in Example 2.7 we have
TT(HS) = h[V ],[V ]i ∼= Z/2⊕Z/2.
2 3
(c) In characteristic 5, with notation as in Example 2.8 we have
TT(Fi ) = h[V ]i ∼= Z/2,
22 1001
TT(3.Fi ) = h[V ],[V ]i ∼= Z/2⊕Z/3,
22 1001 351
TT(6.Fi ) = h[V ],[V ]i ∼= Z/2⊕Z/6.
22 1001 617761
Proof. For G ∈ {M ,2.M ,HS,Fi ,3.Fi ,6.Fi }, let ℓ be as in the statement, P ∈
22 22 22 22 22
Syl (G) and set N := N (P). In all cases the group TT(G) injects via restriction into the
ℓ G
group X(N) of one-dimensional kN-modules, so that its elements are isomorphism classes
of endotrivial trivial source modules, which are the Green correspondents of modules
in X(N). (See [4, §4].)
(a) If G = 2.M , then P ∼= 32 and X(N) ∼= 2 × 4. The group 2.M has six simple
22 22
trivial source endotrivial modules: k, the modules V ,V∗ , V , V∗ of Example 2.6, as
154 154 10 10
well as the module V of Example 2.5 (seen as a k[2.M ]-module). As TT(G) injects
55 22
in X(N), it must have 8 elements, hence TT(G) ∼= Z/2 ⊕Z/4. The set of generators is
obvious since both V and V are not self-dual, therefore of order 4.
10 154
The claim that TT(G) ∼= (Z/2)2 for G = M follows from the above, since TT(G)
22
is a subgroup of TT(2.M ) via inflation and X(N) ∼= 22 in this case. Using the 3-
22
decomposition matrix of G we conclude that with 154 ,10 ∈ Irr(2.M ) the tensor prod-
2 1 22
uct V ⊗ V equals M ⊕ Q, where M is an indecomposable endotrivial kG-module
154 k 10
6 CAROLINELASSUEURANDGUNTER MALLE
affording the character 154 ∈ Irr(G) and Q is a projective kG-module. Whence the set
1
of generators.
(b) For G = HS we have P ∼= 32, N ∼= 2 × (32.SD ) and X(N) ∼= 23. If e denotes
16 0
the principal block idempotent of kG, then e ·IndG(1 ) = 175 with L ∼= U (5) : 2 as in
0 L L 1 3
Example 2.7. It follows that the kG-Green correspondent of a one-dimensional module
in X(N) affords the character 175 ∈ Irr(G) but is not endotrivial by Theorem 2.2. This
1
together with Example 2.7 forces TT(G) = h[V ],[V ]i ∼= (Z/2)2.
2 3
∼
(c) For G = Fi we have X(N) = 4. The module V is endotrivial and trivial
22 1001
source, thus must be the Green correspondent of a module in X(N), whereas the Green
correspondents of the two other non-trivial modules in X(N) are not endotrivial: this
follows easily by inducing the corresponding linear characters of N to G and checking
that their characters cannot satisfy the criterion of Theorem 2.2. Whence TT(G) =
h[V ]i ∼= Z/2.
1001
∼
For G = 3.Fi , X(N) = 3 × 4 and TT(Fi ) ≤ TT(G) via inflation. Thus in view
22 22
of Example 2.8, we must have TT(G) = h[V ],[V ]i ∼= Z/3 ⊕ Z/2. For G = 6.Fi ,
1001 351 22
∼
X(N) = 6 × 4 and TT(3.Fi ) ≤ TT(G). Now the four faithful simple modules of di-
22
∼
mension 61776 of Example 2.8 are not self-dual. This forces TT(G) = h[V ],[V ]i =
617761 1001
Z/6⊕Z/2. (cid:3)
3. Special linear groups
In this section we investigate simple endotrivial kSL (q)-modules for n ≥ 3, where k
n
is a field of characteristic ℓ not dividing q. (The case of n = 2 or that ℓ|q was already
considered in [16, Prop. 3.8 and Thm. 5.2].) Furthermore, we may assume that ℓ 6= 2 by
[16, Thm. 6.7]. Also, we exclude the case of cyclic Sylow ℓ-subgroups for the moment,
partial results for that situation will be given in Section 3.6.
Our argument will proceed in three steps. First we treat unipotent characters, then we
deal with the case when ℓ divides q−1, and finally we consider the general case. But first
we need to collect some auxiliary information.
3.1. Regular elements and maximal tori. Let F : GL → GL be the standard
n n
Frobenius map on the linear algebraic group GL over a field of characteristic p corre-
n
sponding to an F -rational structure. The conjugacy classes of F-stable maximal tori of
q
GL and of SL are parametrized by conjugacy classes of the symmetric group S (and
n n n
thus by partitions of n) in such a way that, if T ≤ GL corresponds to w ∈ S with
n n
cycle shape λ = (λ ≥ λ ≥ ...), then |TF| = (qλi − 1), while for T ≤ SL we have
1 2 i n
(q −1)|TF| = i(qλi −1). In both groups T hQas automizer NGF(T)/TF isomorphic to
CSn(w) (see e.gQ. [19, Prop. 25.3]). For a prime ℓ > 2 not dividing q we write dℓ(q) for the
multiplicative order of q modulo ℓ.
Lemma 3.1. Let λ ⊢ n be a partition, and T a corresponding F-stable maximal torus of
SL . Assume that either all parts of λ are distinct, or q ≥ 3 and at most two parts of λ
n
are equal. Then:
(a) TF contains regular elements.
Now let ℓ be a prime such that some part of λ is divisible by d := d (q).
ℓ
SIMPLE ENDOTRIVIAL MODULES 7
(b) If either d > 1, or λ has at least two parts then TF contains ℓ-singular regular
elements.
(c) If either d > 1, or λ has at least three parts, or ℓ divides (q − 1)/gcd(n,q − 1)
and λ has at least two parts, then TF contains ℓ-singular regular elements with
non-central ℓ-part.
Proof. Write λ = (λ ≥ ... ≥ λ ). First consider the corresponding torus T˜ = TZ(GL )
1 s n
in GL . It is naturally contained in an F-stable Levi subgroup GL of GL such that
n i λi n
T := GL ∩T˜ is a Coxeter torus of GL . In particular, TF ∼= F× is a Singer cycle. Let
i λi λi i Qqλi
x ∈ TF be such that its eigenvalues are generators of F× , and x˜ = (x ,...,x ) ∈ T˜F. If
i i qλi 1 s
all λ are distinct, then clearly all eigenvalues of x˜ are distinct, so x˜ is regular. Multiplying
i
x by the inverse of detx˜ yields a regular element x in TF = T˜ ∩SL (q). If q ≥ 3 then
1 n
there are at least two orbits of generators of F× under the action of Gal(F /F ), so we
qλi qλi q
may again arrange for an element with distinct eigenvalues, proving (a). Clearly, if λ is
i
divisible by d then o(x ) is divisible by ℓ. If d > 1 or λ has at least two parts, this also
i
holds for our modified element x, so we get (b).
Since |Z(SL (q))| = (n,q − 1), (c) is clear when d > 1. So now assume that d = 1,
n
that is, ℓ|(q −1). If λ has at least three parts, then we may arrange so that the ℓ-parts
of the various x are not all equal, even inside SL (q), so that we obtain an element with
i n
non-central ℓ-part. The same is possible if (q−1)/(n,q−1) is divisible by ℓ and λ has at
(cid:3)
least two parts.
We will also need some information on centralizers of semisimple elements in PGL =
n
GL /Z(GL ) containing F-stable maximal tori from given classes. For this, let H denote
n n
an F-stable reductive subgroup of PGL . If H contains a maximal torus of type w, then
n
the Weyl group W of H will have to contain a conjugate of w. Now W is a parabolic
H H
subgroup of the Weyl group S of PGL . Then F permutes the factors of this Young
n n
subgroup, and thus WF is a product of various symmetric groups, some of them in an
H
imprimitive action. Thus we conclude the following:
Lemma 3.2. Let H ≤ PGL be a reductive subgroup containing F-stable maximal tori
n
corresponding to cycle shapes λ ,...,λ . If no intransitive or imprimitive subgroup of S
1 r n
contains elements of all these cycle shapes, then H = PGL .
n
3.2. Unipotent characters of SL (q). We first investigate possible endotrivial simple
n
unipotent modules of SL (q). Recall that these are naturally labelled by partitions λ ⊢ n
n
(see e.g. [6, §13]), and we write χ for the complex unipotent character with label λ. We
λ
first describe their values on regular semisimple elements.
Proposition 3.3. Let G = SL (q). Let t ∈ G be a regular semisimple element, and let
n
w ∈ W = S be the label of the unique F-stable maximal torus T ≤ SL containing t. Let
n n
λ ⊢ n be a partition, χ ∈ Irr(G) the corresponding unipotent character, and ϕ ∈ Irr(W)
λ λ
the corresponding irreducible character of W. Then ρ (t) = ϕ (w).
λ λ
8 CAROLINELASSUEURANDGUNTER MALLE
Proof. If t is contained in a unique F-stable maximal torus T of SL , then the character
n
formula in [6, Prop. 7.5.3] for the Deligne–Lusztig character RT,1 simplifies to
1
RT,1(t) = 1 = |NG(T) : T| = |CW(w)|
|T|
g∈G
tX∈Tg
(where T = TF), and RT′,1(t) = 0 for any F-stable maximal torus T′ not G-conjugate
to T. The unipotent characters in type A coincide with the almost characters (see [6,
§12.3]), so
1 1
χλ(t) =|W| ϕλ(x)RTx,1(t) = |W| ϕλ(x)RT,1(t)
x∈W x∈[w]
X X
|[w]|
= ϕ (w)|C (w)| = ϕ (w),
λ W λ
|W|
where T denotes an F-stable maximal torus parametrized by x ∈ W, and [w] is the
x
conjugacy class of w in W. (cid:3)
We will also need two closely related statements in our later treatment of exceptional
groups of Lie type. For this, assume that G is connected reductive with Steinberg endo-
morphism F : G → G, and set G := GF. We consider s ∈ G∗ semisimple. The almost
characters in the Lusztig series E(G,s) are indexed by extensions to W .F of F-invariant
s
irreducible characters of the Weyl group W of C (s), and we denote them by R , for ϕ˜
s G∗ ϕ˜
a fixed extension of ϕ ∈ Irr(W )F.
s
Proposition 3.4. In the above setting, let t ∈ G be regular semisimple and ϕ ∈ Irr(W )F.
s
Then Rϕ˜(t) = 0 unless t lies in an F-stable maximal torus T such that T∗ ≤ CG∗(s) and
T∗ is indexed by wF ∈ W F such that ϕ˜(wF) 6= 0.
s
Proof. Since t is assumed to be regular, it is contained in a unique F-stable maximal
torus T of G. The character formula in [6, Prop. 7.5.3] for the Deligne–Lusztig characters
then shows that RT,θ(t) = 0 for all θ ∈ Irr(TF) unless s ∈ T∗ up to conjugation. So
now assume that T∗ ≤ CG∗(s) is indexed by the F-conjugacy class of w ∈ Ws. Then by
definition the almost character for ϕ is given by
1 1
Rϕ˜(t) = |W | ϕ˜(xF)RTx,θ(t) = |W | ϕ˜(wF)RT,θ(t) = 0
s s
xX∈Ws xX∼w
unless ϕ˜(wF) 6= 0, as claimed, where the sum runs over the F-conjugacy class of w. (cid:3)
Proposition 3.5. In the above setting, let t ∈ G be regular semisimple and assume that
the orders of t and s are coprime. Then R (t) is divisible by |W(T)/W(T,θ)| for all
ϕ˜
ϕ ∈ Irr(W )F, where W(T) is the Weyl group of the unique maximal torus T containing
s
t, and W(T,θ) is the stabilizer of θ ∈ Irr(T) corresponding to s.
Proof. Let T be the unique F-stable maximal torus of G containing x, and T = TF. Let
(T,θ) correspond to s ∈ G∗. By [6, Prop. 7.5.3] we have
1 |N (T)|
|RT,θ(t)| = θ(tg) = G = |W(T)|,
|T| |T|
g∈XNG(T)
SIMPLE ENDOTRIVIAL MODULES 9
since the order of the linear character θ is coprime to that of t by assumption, and so θ
takes value 1 on any conjugate of t. Thus
1 |W(T)||W /W(T,θ)| |W(T)|
s
Rϕ˜(t) = ϕ˜(wF)RT,θ(t) = ± ϕ˜(wF) = ± ϕ˜(wF)
|W | |W | |W(T,θ)|
s s
x∼w
X
(cid:3)
as claimed.
Lemma 3.6. Let n = 2d+r with d ≥ 2 and 1 ≤ r ≤ d−1. There exists a semisimple
element t ∈ SL (q) of order (qd −1)(qr −1) with centralizer GL (qd)(qr −1) in GL (q)
n 2 n
such that the unipotent character ρ with λ = (d+r,r +1,1d−r−1) takes value ±qd on t.
λ
Proof. Let f,g ∈ F [X] be irreducible of degrees d, r respectively with f(0)2g(0) = (−1)n,
q
and t ∈ SL (q) be a semisimple element with minimal polynomial fg and characteristic
n
polynomial f2g. Then clearly t has centralizer C ∼= GL (qd)(qr−1) in GL (q), and thus is
2 n
as in the statement. Since the unipotent characters of SL (q) are the restrictions of those
n
of GL (q), we may now arguein the latter group. Now t is only contained in the two types
n
of maximal tori T ,T of C, of orders (qd−1)2(qr−1) and (q2d−1)(qr−1), parametrized
1 2
by the partitions µ = (d,d,r) and µ = (2d,r) respectively. Thus the Deligne–Lusztig
1 2
characters take values
R (t) = ±(qd +1), R (t) = ∓(qd −1),
T1,1 T2,1
where the signs are opposed (since ǫ = −ǫ in the notation of [6, Prop. 7.5.3]). Fur-
T1 T2
thermore, the Murnaghan–Nakayama formula gives χ (µ ) = (−1)d+r+1 = −χ (µ ) for
λ 1 λ 2
the values of the corresponding characters of S . The value ρ (t) can now be computed
n λ
(cid:3)
as in the proof of Proposition 3.3.
We will also need a q-analogue of Babbage’s congruence, which was first shown by
Andrews [1] under stronger assumptions; here for integers 0 ≤ k ≤ n we set
n k xn−k+i −1
:= ∈ Z[x].
k xi −1
x
h i Yi=1
Lemma 3.7. Let x be an indeterminate and d,h ≥ 2. Then in Z[x] we have
hd−1
≡ x(h−1)(d2) (mod Φ (x)2).
d
d−1
(cid:20) (cid:21)x
Proof. When d is a prime, then Φ (x) = (xd −1)/(x−1), and in that case the claim is
d
proved in [1, Thm. 2]. But inspection shows that the argument given there is valid for all
d ≥ 2 if congruences are considered just modulo Φ (x)2 instead of (xd−1)2/(x−1)2. (cid:3)
d
Proposition 3.8. Let χ be a non-trivial unipotent character of G = SL (q), n ≥ 3, and
n
2 6= ℓ6 |q a prime such that n ≥ 2d where d := d (q). Then χ is not the character of a
ℓ
simple endotrivial module for any central factor group of G.
Proof. Unipotent characters have the centre of G in their kernel, hence we may see χ as a
character of any central factor group S of G. Let λ ⊢ n denote the label for χ. First note
that we have λ 6= (n), since (n) parametrizes the trivial character of G. The Steinberg
character, parameterized by (1n), can never be the character of a simple endotrivial kS-
(n)
module, forithasdegreeq 2 , socanbecongruentto±1modulo|S| onlywhenn = ℓ = 3,
ℓ
10 CAROLINELASSUEURANDGUNTER MALLE
q ≡ 4,7 (mod 9). But by [14, Tab. 4] the Steinberg character is reducible modulo 3 in
this case.
If χ is endotrivial, then in particular its values on ℓ-singular elements have to be of
absolute value one (see [16, Cor. 2.3]). We start with the case that d = 1, so ℓ|(q − 1)
(and hence q ≥ 4 as ℓ 6= 2). First assume that moreover (q −1)/(n,q−1) is divisible by
ℓ, so |Z(G)| = (n,q −1) is not divisible by the full ℓ-part of q −1. Let T be a maximal
torus of G corresponding to an element w ∈ S of cycle shape (n−1)(1). By Lemma 3.1,
n
T contains a regular element x of G with non-central ℓ-part. Then χ(x) = ϕ (w) by
λ
Proposition 3.3. By the Murnaghan–Nakayama formula the latter is zero unless λ has an
n−1-hook. Similarly, with a maximal torus corresponding to the cycle shape (n−2)(2)
we see that λ has to possess an n−2-hook. When n ≥ 6 we may also argue that λ has
an n−3-hook, using the cycle shape (n−3)(3). The only partitions possessing all three
types of hooks are (n) and (1n), which we excluded above. When n ≤ 5, there are also the
possibilities λ = (2,2),(3,2) and (22,1). From the known values of unipotent characters
in SL (q) and SL (q) provided in Chevie [9] it follows that all three characters vanish on
4 5
the product of a Jordan block of size n−1 with a commuting non-central ℓ-element.
Next assume that d = 1, but now (q−1)/(n,q−1) is prime to ℓ, so that in particular ℓ
divides n. Thus if n ≤ 6, we only need to consider the pairs (n,ℓ) ∈ {(3,3),(5,5),(6,3)}.
If ℓ divides |Z(S)|, then χ cannot take absolute value one values on central ℓ-elements.
Otherwise, it can be checked from the known character tables of unipotent characters [9]
thattherearenoexamplesofdegreecongruentto±1modulo|S| . Forn ≥ 7,arguingwith
ℓ
maximal toricorresponding tocycle shapes(n−2)(1)2, (n−3)(2)(1), we see by Lemma 3.1
that λ must possess n−2- and n− 3-hooks, whence λ or its conjugate partition is one
of (n),(n−1,1),(n−3,3),(n−4,22) (recall that q ≥ 4). Now removing an n−3-hook
from the second and third of these partitions or their conjugate partitions leaves a 2-core,
so the unipotent characters labelled by these vanish on regular semisimple elements of a
torus of type (n − 3)(2)(1). Finally, the character labelled by the partition (n − 4,22)
vanishes on regular elements in a torus of type (n−4)(3)(1), so is not endotrivial. This
completes the discussion for d = 1.
So now let us assume that d ≥ 2. Write n = ad+r with 0 ≤ r < d. Let T be a maximal
torus of G corresponding to an element w ∈ S of cycle shape (n−r)(r). As before by
n
Lemma 3.1 and Proposition 3.3 we conclude that there is an ℓ-singular regular element
x ∈ T such that χ(x) = ϕ (w). By the Murnaghan–Nakayama formula the latter is zero
λ
unless λ has an n − r-hook such that the partition obtained by removing that hook is
an r-hook. Now first assume that r ≥ 3. Then similarly, with a torus corresponding to
an element w ∈ S of cycle shape (n−r)(r −1)(1) we see that the remaining partition
n
has to possess an r −1-hook. The only partitions with that property are (up to taking
the conjugate partition), λ = (n − r − s + 1,2s,1r−s−1) for some 0 ≤ s ≤ r − 1, and
λ = (n−r−s,r+1,1s−1) for some 1 ≤ s ≤ n−2r−1. Next, consider tori corresponding
to the cycle shape (n−d)(d), which again contain ℓ-singular regular elements. This forces
λ to possess an n− d-hook such that removing it leaves a d-hook. For the first type of
partition, this is only possible when s = 0, so λ = (n − r + 1,1r−1) is a hook. For the
second type, it forces s = d − r, whence λ = (n − d,r + 1,1d−r−1). So, up to taking
conjugates, λ is one of (n−r +1,1r−1) or (n−d,r+1,1d−r−1).