Adelic Openness for Drinfeld Modules in Special Characteristic

0.56 MB·

by  Anna Devic

Adelic Openness for Drinfeld Modules
in Special Characteristic
2 Anna Devic Richard Pink
∗ ∗∗
1
0
January 28, 2012
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n
a
J

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Abstract
2

] For any Drinfeld module of special characteristic p over a finitely
0
T
generated field, we study the associated adelic Galois representation at all
N
places differentfrom p and anddeterminetheimages of thearithmetic
0
. ∞
h and geometric Galois groups up to commensurability.
t
a
m
[

Mathematics Subject Classification: 11G09 (11R58)
3
v Keywords: Drinfeld modules, torsion points, Galois representations
8
9
3
3
.
3
0
1
1
:
v
i
X
r
a
∗Email: [email protected]
∗∗Dept. of Mathematics, ETH Zu¨rich, CH-8092 Zu¨rich, Switzerland, [email protected]
1
Contents
1 Introduction and overview 3
1.1 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Outline of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Structure of the article . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Subgroups of SL over a field 8
n
2.1 Root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Some algebraic relations . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Linear algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Finite groups of Lie type . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 Arbitrary finite groups . . . . . . . . . . . . . . . . . . . . . . . . 18
3 Subgroups of SL over a complete valuation ring 20
n
3.1 Adjoint representation . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Successive approximation . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 Successive approximation in the case p = n = 2 . . . . . . . . . . 25
3.4 Trace criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4 Preliminary results on Drinfeld modules 31
4.1 Endomorphisms rings . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2 Isogenies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.3 Tate modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.4 Non-singular model . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.5 Frobenius action . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.6 Good reduction and lattices . . . . . . . . . . . . . . . . . . . . . 38
4.7 Bad reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.8 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.9 Images of Galois groups . . . . . . . . . . . . . . . . . . . . . . . 42
4.10 Ring of traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5 Proof of the main result 46
5.1 Residual surjectivity at a single prime . . . . . . . . . . . . . . . . 46
5.2 Surjectivity at a single prime . . . . . . . . . . . . . . . . . . . . . 49
5.3 Residual surjectivity at several primes . . . . . . . . . . . . . . . 49
5.4 Adelic openness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.5 Absolute Galois group . . . . . . . . . . . . . . . . . . . . . . . . 53
6 Arbitrary endomorphism ring 55
6.1 The isotrivial case . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.2 The non-isotrivial case . . . . . . . . . . . . . . . . . . . . . . . . 58
2
1 Introduction and overview
1.1 Main result
Let K be a field that is finitely generated over a finite field κ of characteristic p.
Let Ksep be a fixed separable closure of K, and let κ be the algebraic closure
of κ in Ksep. Let G := Gal(Ksep/K) denote the absolute Galois group and
K
Ggeom := Gal(Ksep/Kκ) the geometric Galois group of K.
K
Let F bea finitely generated fieldoftranscendence degree 1over F . Let Abe
p
the ring of elements of F which are regular outside a fixed place of F. Let ϕ :
∞
A K τ be a Drinfeld A-module of rank r over K of special characteristic p .
0
→ { }
For any prime p = p of A let ρ : G GL (A ) denote the homomorphism
0 p K r p
6 →
describing the Galois action on the Tate module T (ϕ). We are interested in the
p
image of the associated adelic Galois representation
ρ := (ρ ) : G GL (A ).
ad p p K r p
−→
pY6=p0
By Anderson [And86], 4.2, it is known that the composite of ρ with the
ad
§
determinant map is the adelic Galois representation associated to some Drinfeld
module of rank 1 of the same characteristic p . Thus the image of ρ (Ggeom)
0 ad K
under the determinant is finite: see Proposition 6.3 below. Consequently, the
image of ρ (G ) under the determinant is an extension of a finite group and
ad K
a pro-cyclic group and therefore far from open. Also, the main problem in
determining ρ (G ) lies in determining ρ (Ggeom) SL (A ).
ad K ad K ∩ p6=p0 r p
Recall that two subgroups of a group are called commensurable if their inter-
Q
section has finite index in both. We will show that ρ (Ggeom) is commensurable
ad K
to an explicit subgroup of SL (A ) whose definition depends only on infor-
p6=p0 r p
mation on certain endomorphism rings associated to ϕ. We will also determine
Q
ρ (G ) up to commensurability.
ad K
First, since the Galois representation commutes with the endomorphisms of
ϕ over K, the image of ρ must be contained in the centralizer of End (ϕ)
ad K
in GL (A ). Second, enlarging K does not change the image of Galois
p6=p0 r p
up to commensurability, but may increase the endomorphism ring. Since all
Q
endomorphisms of ϕ over any extension of K are defined over a finite separable
extension, the relevant endomorphism ring is therefore End (ϕ).
Ksep
For a Drinfeld module in generic characteristic it turns out that the image of
ρ up to commensurability, which was determined in [PR09a], indeed depends
ad
only on End (ϕ). But in special characteristic this cannot be so, due to a
Ksep
phenomenon described in [Pin06b]. The problem is that the endomorphism ring
of a Drinfeld module in special characteristic can be non-commutative. As a
consequence, it is possible that for some integrally closed infinite subring B A,
⊂
the endomorphism ring of the Drinfeld B-module ϕ B is larger than that of ϕ.
|
3
TheGaloisrepresentationassociatedtoϕmustthencommutewiththeadditional
operators coming from endomorphisms of ϕ B, forcing the image of ρ to be
ad
|
smaller. But using the results of [Pin06b] one can reduce the problem to the case
where this phenomenon of growing endomorphism rings does not occur.
For the following results let a be any element of A that generates a positive
0
power of p . View a as a scalar element of GL (A ) via the diagonal
0 0 p6=p0 r p
embedding A ֒ A , and let a denote the pro-cyclic subgroup that is
→ p6=p0 p h 0i Q
topologically generated by it.
Q
In the simplest case, where the endomorphism ring of ϕ over Ksep is A and
does not grow under restriction, our main result is the following:
Theorem 1.1.
Let ϕ be a Drinfeld A-module of rank r over a finitely generated field K of special
characteristic p . Assume that for every integrally closed infinite subring B A
0
⊂
we have End (ϕ B) = A. Then
Ksep
|
(a) ρ (Ggeom) is commensurable to SL (A ), and
ad K p6=p0 r p
(b) ρ (G ) is commensurable to aQ SL (A ).
ad K h 0i· p6=p0 r p
More generally, set R := EndKsep(ϕ)Qand F := Quot(A). Assume for the
moment that the center of R is A. Then R F is a central division algebra
A
⊗
over F of dimension d2 forsome d dividing r. Forany primep = p ofA, theTate
0
6
module T (ϕ) is a module over R := R A , which is an order in a semisimple
p p A p
⊗
algebra over F . Let D denote the commutant of R in End (T (ϕ)), which is
p p p Ap p
an order in another semisimple algebra over F . Let D1 denote the multiplicative
p p
group of elements of D of reduced norm 1. This is isomorphic to SL (A ) for
p r/d p
almost all p by Proposition 4.11, and equal to SL (A ) for all p if R = A.
r p
In this situation a version of our main result is the following:
Theorem 1.2.
Let ϕ be a Drinfeld A-module over a finitely generated field K of special char-
acteristic p . Assume that R := End (ϕ) has center A and that for every
0 Ksep
integrally closed infinite subring B A we have End (ϕ B) = R. Let D1 and
⊂ Ksep | p
a denote the subgroups defined above. Then
0
h i
(a) ρ (Ggeom) is commensurable to D1, and
ad K p6=p0 p
(b) ρ (G ) is commensurable to aQ D1.
ad K h 0i· p6=p0 p
Theorem 1.2 is the central result of thiQs article; its special case R = A is just
Theorem 1.1. Sections 2 to 5 are dedicated to proving Theorem 1.2. In Section
6 we deduce corresponding results without any assumptions on End (ϕ) that
Ksep
are somewhat more complicated to state.
4
1.2 Outline of the proof
In this outline we explain the key steps in the proof of Theorem 1.2 in the case
R = A; the general case follows the same principles. So we assume that for every
integrallyclosedinfinitesubringB AwehaveEnd (ϕ B) = A. Afterreplac-
Ksep
⊂ |
ing K by a finite extension, we may assume that ρ (Ggeom) SL (A ).
Let Γgeom denote its image in SL (A ) for any singleadprimKe p =⊂p op6=fpA0 , arnd lpet
p r p 0
∆geom denote its image in SL (k ) over the residue field k :=6A/pQ.
p r p p
A large part of the effort goes into proving that ∆geom = SL (k ) for almost
p r p
all p. The key arithmetic ingredients for this are the absolute irreducibility of
the residual representation from [PT06], the Zariski density of Γgeom in SL
p r,Fp
from [Pin06a], and the characterization of k by the traces of Frobenius elements
p
in the adjoint representation from [Pin06b].
In fact, the absolute irreducibility combined with a strong form of Jordan’s
theorem on finite subgroups of GL from [LP11] shows that ∆geom is essentially a
r p
finitegroupofLietypeincharacteristic p := char(F). LetH denotetheambient
p
¯
connected semisimple linear algebraic group over an algebraic closure k of k .
p p
If H is a proper subgroup of SL , the eigenvalues of any element of H must
p r,k¯p p
satisfy one of finitely many explicit multiplicative relations that depend only
on r. In this case we show that the eigenvalues of any Frobenius element in the
residual representation satisfy a similar relation. If this happens for infinitely
many p, the fact that the adelic Galois representation is a compatible system
implies that the eigenvalues of Frobenius over any single F satisfy the same
p
kind of relation. But that is impossible, because Γgeom is Zariski dense in SL .
p r,Fp
Therefore H is equal to SL for almost all p.
p r,k¯p
This means that∆geom isessentially thegroupofk′-rationalpointsof amodel
p p
of SL over a subfield k′ k¯ . To identify this subfield we observe that the
r,k¯p p ⊂ p
trace in the adjoint representation for any automorphism of the model is an
element of k′. We show that this holds in particular for the images of Frobenius
p
elements. But by [Pin06b] the images of the traces of all Frobenius elements in
the adjoint representation of SL generate k for almost all p. It follows that
r p
k k′ for almost all p, and then the inclusion ∆geom SL (k ) must be an
p ⊂ p p ⊂ r p
equality for cardinality reasons.
Wealsoneed toprove thatthehomomorphism Ggeom SL (k ) SL (k ) is
K → r p1 × r p2
surjectiveforanydistinctp ,p outsidesomefinitesetofprimes. Thisagainrelies
1 2
on traces of Frobenius elements. Indeed, if the homomorphism is not surjective,
the surjectivity to each factor and Goursat’s lemma imply that its image is
∼
essentially the graph of an isomorphism SL (k ) SL (k ). This isomorphism
r p1 → r p2
must come from an isomorphism of algebraic groups over an isomorphism of the
∼
residue fields σ : k k . Using this we show that the traces of Frobenius in
p1 → p2
the adjoint representation of SL map to the subring graph(σ) k k . But
r ⊂ p1 × p2
that again contradicts the result from [Pin06b] unless p or p is one of finitely
1 2
5
many exceptional primes.
Next we prove that Γgeom = SL (A ) for almost all p. For this we may already
p r p
assume that ∆geom = SL (k ). That alone does not imply much, because A is a
p r p p
local ring of equal characteristic, and so the Teichmu¨ller lift of the residue field
k ֒ A induces a lift SL (k ) ֒ SL (A ). But using successive approximation
p p r p r p
→ →
in SL (A ) we reduce the problem to showing that Γgeom surjects to SL (A/p2).
r p p r
This in turn we can guarantee for almost all p using traces of Frobenius elements
again.
Indeed, suppose first that (p,r) = (2,2). Then the result from [Pin06b]
6
implies that the images of the traces of all Frobenius elements in the adjoint
representation of SL generate A/p2 for almost all p. In particular these traces
r
do not all lie in the Teichmu¨ller lift k A/p2, and so the images of Frobenius
p
⊂
elements in GL (A ) cannot all lie in the lift of GL (k ). The desired surjectivity
r p r p
Γgeom ։ SL (A/p2) follows from this using some group theory.
p r
In the remaining case p = r = 2 it may happen that the traces of Frobenius in
the adjoint representation do not generate the field F, but the subfield of squares
F2 := x2 x F , of which F is an inseparable extension of degree 2. This
{ | ∈ }
phenomenon stems from the fact that the adjoint representation of SL on psl
2 2
in characteristic 2 factors through the Frobenius Frob : x x2. In that case,
2
7→
the result from [Pin06b] implies that the images of the traces of all Frobenius
elements in the adjoint representation of SL generate the subring k p2/p3
r p
⊕
of A/p3 for almost all p, where k denotes the canonical Teichmu¨ller lift of the
p
residue field k of p. By digging deeper into the structure of SL (A/p3), and
p 2
replacing K by a finite extension at a crucial step in the argument, we can again
show that Γgeom surjects to SL (A/p2).
p r
Finally, using group theory alone the above results about SL (k ) SL (k )
r p1 × r p2
and SL (A ) imply that the homomorphism Ggeom SL (A ) is sur-
r p K → p6∈P3 r p
jective for some finite set of primes P . On the other hand, the homomor-
3
phism Ggeom SL (A ) has open image by the mQain result of [Pin06b].
K → p∈P3 r p
While this does not directly imply that the image of the product homomorphism
Ggeom SQL (A ) is open, because the image of a product map may be
K → p6=p0 r p
smaller than the product of the images, some variant of the argument can be
Q
made to work, thereby finishing the proof of Theorem 1.2 (a).
Theorem 1.2 (b) is deduced from this as follows. Since ρ (Ggeom) is already
ad K
open in SL (A ), it suffices to show that detρ (G ) is commensurable to
p6=p0 r p ad K
a within A×. As the determinant of ρ is the adelic Galois representa-
h 0i Q p6=p0 p ad
tion associated to some Drinfeld module of rank 1 of the same characteristic p ,
0
Q
this reduces the problem to the case that r = 1 and that ϕ is defined over a finite
field, say over κ itself. Then Frob acts through multiplication by an element
κ
a A which is a unit at all primes p = p but not at p . It follows that (a) = pi
∈ 6 0 0 0
for some positive integer i. The same properties of a show that (a ) = pj for
0 0 0
some positive integer j. Together it follows that (aj) = pij = (ai), and so aj/ai
0 0 0
6
is a unit in A×. As the group of units is finite, we deduce that ajℓ = aiℓ for
0
some positive integer ℓ. In particular ρ (G ) = a is commensurable to a ,
ad K 0
h i h i
as desired. This finishes the proof of Theorem 1.2 (b).
1.3 Structure of the article
Section 1 is the present introduction and overview. Sections 2 and 3 deal with
subgroups of SL and GL . They are independent of Drinfeld modules, of the
n n
rest of the article, and of each other.
Section 2 deals with subgroups of SL and GL over a field and establishes
n n
suitable conditions for such a subgroup to be equal to SL . It is based on some
n
calculations in root systems, on known results on finite groups of Lie type, and
on a strong form of Jordan’s theorem from [LP11].
Section3dealswithclosedsubgroupsofSL andGL overacompletediscrete
n n
valuation ring R of equal characteristic p with finite residue field, and establishes
suitable conditions for such a subgroup to be equal to SL (R). The method uses
n
successive approximationoverthecongruencefiltrationofSL (R), respectively of
n
GL (R), whosesubquotients arerelated totheadjoint representation. Curiously,
n
the case p = n = 2 presents special subtleties here, too, because the Lie bracket
on sl in characteristic 2 is not surjective.
2
In Section 4 we list known results about Drinfeld modules in special charac-
teristic or adapt them slightly to the situation at hand. This includes properties
of endomorphism rings, Galois representations on Tate modules, characteristic
polynomials of Frobenius, and bad reduction. We also create the setup in which
the proof of Theorem 1.2 takes place, and list the main arithmetic ingredients
from [Pin06a], [Pin06b], and [PT06] with their immediate consequences.
Section 5 then contains (what remains of) the proofof Theorem 1.2, following
the outline expained above.
In Section 6 we determine ρ (Ggeom) and ρ (G ) up to commensurability
ad K ad K
for arbitrary Drinfeld modules in special characteristic. The main ingredients for
this are the special case of Theorem 1.2 and some reduction steps from [Pin06b].
This article is based on the doctoral thesis of the first author [Dev10]; its
results are roughly the same as the results there. We are grateful to Florian Pop
for pointing out Theorem 4.13.
7
2 Subgroups of SL over a field
n
In a nutshell, the main goal of this section is to establish suitable conditions for
subgroups of SL over a field to be equal to SL . We first give conditions for
n n
root systems to be simple of type A , and then deal with the case of connected
ℓ
semisimple linear algebraic groups over a field. Based on this we treat the case of
finite groups of Lie type, which must also take inner forms of SL into account.
n
The main results are Theorems 2.14, 2.20, and 2.21. We also recall a strong form
of Jordan’s theorem from [LP11].
2.1 Root systems
Let Φ be a non-trivial root system generating a euclidean vector space E. Let
W be the associated Weyl group, and let S be a W-orbit in E. We are interested
in the conditions:
(a) S generates E as a vector space.
(b) There are no distinct elements λ ,...,λ S such that λ +λ = λ +λ .
1 4 1 2 3 4
∈
(c) There are no distinct elements λ ,...,λ S such that λ + λ + λ =
1 6 1 2 3
∈
λ +λ +λ .
4 5 6
Theorem 2.1.
Assume (a) and (b). Then Φ is simple of type A for some ℓ 1. Moreover, if
ℓ
≥
Φ = (e e ) 0 i < j ℓ E = Rℓ+1/diag(R)
i j
{± − | ≤ ≤ } ⊂
in standard notation, and if ℓ = 2 or in addition (c) is satisfied, then
6
S = ce 0 i ℓ
i
{ | ≤ ≤ }
for some constant c = 0.
6
The proof of this result extends over the rest of this subsection. Throughout
we assume conditions (a) and (b). Note that (a) implies that 0 S.
6∈
Lemma 2.2.
Let λ S and α , α be two orthogonal roots in Φ. Then λ α or λ α .
1 2 1 2
∈ ⊥ ⊥
Proof. Let s W denote the simple reflection associated to α . The fact that
αi ∈ i
α α implies that
1 2
⊥
2(λ,α )
i
s (λ) = λ α , and
αi − (α ,α ) · i
i i
2(λ,α ) 2(λ,α )
1 2
s s (λ) = λ α α ,
α1 α2 − (α ,α ) · 1 − (α ,α ) · 2
1 1 2 2
8
and hence
λ+s s (λ) = s (λ)+s (λ).
α1 α2 α1 α2
But if λ is not orthogonal to α or α , these are four distinct elements of S,
1 2
contradicting condition (b).
Lemma 2.3.
The root system Φ is simple.
Proof. Assume that Φ = Ψ + Ψ is decomposable and let λ S. Since Φ
1 2
∈
generates E, there exists an α Φ which is not orthogonal to λ. Suppose
∈
without loss of generality that α Ψ . Then, by Lemma 2.2, the vector λ is
2
∈
orthogonal to all roots that are orthogonal to α; in particular λ Ψ . Then
1
⊥
w(λ) Ψ for all w W and therefore S Ψ . However, this contradicts
1 1
⊥ ∈ ⊥
condition (a).
Lemma 2.4.
The root system Φ does not contain a root subsystem of type B .
2
Proof. Assume that Ψ Φ is a root subsystem of type B . Then the subspace
2
⊂
RΨ possesses a basis e ,e such that Ψ consists of eight roots e , e , and
1 2 1 2
{ } ± ±
e e , and where e e and e +e e e . Thus for any λ S, Lemma
1 2 1 2 1 2 1 2
± ± ⊥ ⊥ − ∈
2.2 implies that λ e for some i = 1, 2, and that λ e e for some choice
i 1 2
⊥ ⊥ ±
of sign. Together this gives four cases, in each of which we deduce that λ e .
1
⊥
As λ was arbitrary, this shows that S e , contradicting condition (a).
1
⊥
Lemma 2.5.
The root system Φ is not of type G .
2
Proof. Choose simple roots α , α of Φ such that α is the shorter one. Then
1 2 1
Φ contains the root 2α +α which is orthogonal to α , and the root 3α +2α
1 2 2 1 2
which is orthogonal to α . Thus for any λ S, Lemma 2.2 implies that λ α
1 2
∈ ⊥
or λ 2α + α , and that λ α or λ 3α + 2α . By a simple calculation,
1 2 1 1 2
⊥ ⊥ ⊥
each of these four cases implies that λ = 0, contradicting condition (a).
Lemma 2.6.
The root system Φ does not contain a root subsystem of type D .
4
Proof. Assume that Ψ Φ is a root subsystem of type D . Then, up to scal-
4
⊂
ing the inner product on E, the subspace RΨ possesses an orthonormal basis
e ,e ,e ,e such that Ψ consists of the roots e e for all 1 i < j 4 and
1 2 3 4 i j
{ } ± ± ≤ ≤
all choices of signs. In particular, the roots e + e and e e are orthogonal
1 i 1 i
−
9
for every 2 i 4. Thus for any λ S, Lemma 2.2 implies that λ e +ε e
1 i i
≤ ≤ ∈ ⊥
for some ε = 1. Since the roots e ε e and ε e +ε e are also orthogonal,
i 1 2 2 3 3 4 4
± −
Lemma 2.2 implies that λ e ε e or λ ε e +ε e . Since
1 2 2 3 3 4 4
⊥ − ⊥
(e +ε e )+(e ε e ) = (e +ε e )+(e +ε e ) (ε e +ε e ) = 2e ,
1 2 2 1 2 2 1 3 3 1 4 4 3 3 4 4 1
− −
in both cases we deduce that λ 2e . As λ was arbitrary, this shows that
1
⊥
S 2e , contradicting condition (a).
1
⊥
Combining Lemmas 2.3 through 2.6, it follows that Φ is a simple root system
of type A for some ℓ 1. Using standard notation we may identify E with the
ℓ
≥
vector space Rℓ+1/diag(R), let e ,...,e E denote the images of the standard
0 ℓ
∈
basis vectors of Rℓ+1, and assume that Φ consists of the roots e e for all
i j
−
distinct 0 i,j ℓ. Then its Weyl group is the symmetric group S on l +1
ℓ+1
≤ ≤
letters, acting on E by permuting the coefficients.
Consider any λ S and write λ = (a ,...,a ) modulo diag(R). Since λ = 0
0 ℓ
∈ 6
in E, the coefficients a are not all equal.
i
Lemma 2.7.
Suppose that ℓ 3, and consider indices i and j satisfying a = a . Then for all
i j
≥ 6
indices i′ and j′ that are distinct from i and j we have ai′ = aj′.
Proof. The assumption implies that i = j, and the assertion is trivial unless also
6
i′ = j′. Then ei ej and ei′ ej′ are orthogonal roots, and so Lemma 2.2 implies
6 − −
that λ ei ej or λ ei′ ej′. This means that ai = aj or ai′ = aj′; but by
⊥ − ⊥ −
assumption only the second case is possible.
Lemma 2.8.
If ℓ 3, there exists an index i such that the a for all j = i are equal.
j
≥ 6
Proof. Since ℓ 3 and the a are not all equal, Lemma 2.7 implies that the a
i i
≥
are also not all distinct. Therefore there exist distinct indices i,j,j′ satisfying
ai = aj = aj′. Then for any i′ = i,j, Lemma 2.7 shows that ai′ = aj′. Thus i has
6 6
the desired property.
Lemma 2.9.
If ℓ = 2 and in addition condition (c) is satisfied, then the a are not all distinct.
i
Proof. Being an orbit under the Weyl group, the set S consists of the vectors
(a ,a ,a ), (a ,a ,a ), (a ,a ,a ),
0 1 2 1 2 0 2 0 1
(a ,a ,a ), (a ,a ,a ), (a ,a ,a )
0 2 1 1 0 2 2 1 0
10