Strongly and Weyl transitive group actions on buildings arising from Chevalley groups

0.18 MB·English

by  Peter Abramenko

STRONGLY AND WEYL TRANSITIVE GROUP ACTIONS ON
BUILDINGS ARISING FROM CHEVALLEY GROUPS
1
PETERABRAMENKOANDMATTHEWC.B.ZAREMSKY
1
0
2
n Abstract. LetKbeafieldandg(K)aChevalleygroup(scheme)overK. Let
a (B,N)bethestandardsphericalBN-pairing(K),withT =B∩N andWeyl
J groupW =N/T. Weprovethatthereexistnon-trivialelements w∈W such
that all representatives ofw inN have finiteorder. This allowsus toexhibit
5
examples of subgroups of g(Qp) that act Weyl transitively but not strongly
] transitively on the affine building ∆ associated with g(Qp). Such examples
R werepreviouslyknownonlyinthecasewheng(Qp)=SL2(Qp)and∆isatree
(see[AB1]).
G
.
h
t
a
m 1. Introduction
[
In building theory, there are two important concepts of transitivity that are

1
strongerthanchambertransitiveactions,namelystronglytransitiveandWeyltran-
v
3 sitive actions (for the precise definitions, we refer to Section 2 below). Strongly
1
transitive group actions on (thick) buildings are equivalent to BN-pairs, which is
1
1 reasonenoughfor studying them. However,if one considers buildings from the W-
1. metricpointofview,thereisanothernaturalnotionoftransitivity,whichwascalled
0 “Weyl transitivity” in [AB1]. Strong transitivity is always defined with respect to
1
a chosen (not necessarily complete) system of apartments of the building in ques-
1
: tion, whereas the definition of Weyl transitivity does not refer to apartments. It is
v
i wellknownthat strongtransitivity (with respect to any apartmentsystem)always
X
impliesWeyltransitivity,andthatthese twonotionsareequivalentforbuildings of
r
a sphericaltype. Itwasalsoexpectedthatfornon-sphericalbuildingsWeyltransitiv-
ity is strictly weaker than strong transitivity. However, the first explicit examples
ofthistypeweredocumentedonlyafewyearsagoin[AB1],followingsomesugges-
tions made by J. Tits in [T, Section 3.1, Example (b)]. Tits suggested to analyze
the actions of anisotropic groups over global fields on suitable Bruhat-Tits build-
ings,andin [AB1]this wasdone in the simplestcase,namely for the norm1 group
of a quaternion division algebra D over Q acting on the Bruhat-Tits tree T of
p
SL (Q ) for suitable p. Providedthat -1 is not a square in D, this yielded the first
2 p
explicit examples of Weyl transitive actions which are not strongly transitive with
respect to any apartment system. However, trees are rather special buildings, and
the question remained whether one can produce examples of groups acting Weyl
transitively but not strongly transitively on buildings of arbitrary (affine) type.
1
2 PETERABRAMENKOANDMATTHEWC.B.ZAREMSKY
It is the goalof this paper to presentsome examples of this kind. Our approach
here will not use anisotropic algebraic groups but will rather generalize a second
type of counter-example discussed in Section 6.10.2 of [AB2]. Here, dense sub-
groups of SL (Q ) are exhibited that do not act strongly transitively on T . By a
2 p p
general principle, formulated and proved as Proposition 3.4 in [AB1] and restated
as Lemma 2.5 below, the density immediately implies that these subgroups act
Weyltransitively. Theargumentgivenin[AB2,Section6.10.2]thatshowsthatthe
actions are not strongly transitive appears to be rather special and is restricted to
the tree case. Just based on this argument, it is not clear how to show for higher
dimensionsordifferenttypesthatagivenWeyltransitiveactionthatisnotstrongly
transitive with respect to a certain apartment system is in fact not strongly tran-
sitive with respect to any apartment system. In this paper we will generalize the
group SL (Q ) to an arbitrary p-adic Chevalley group g(Q ). The generalization
2 p p
of the argument given in [AB2, Section 6.10.2] that does the job is the following
fact, which turns out to be true for all p-adic Chevalley groups g(Q ):
p
Proposition 1.1. If a subgroup of g(Q ) acts strongly transitively on the cor-
p
responding Bruhat-Tits building ∆, then it contains nontrivial elements of finite
order.
Anexplanationofhow∆arisesfromg(Q )willbegiveninSection3,andaproof
p
of this proposition will be given in Section 4. Then, using arguments in Section 5,
one obtains the following:
Proposition 1.2. g(Q ) has (many) dense torsionfree subgroups. The action of
p
any such subgroup on ∆ is Weyl transitive but not strongly transitive with respect
to any apartment system of ∆.
2. Some facts about strongly and Weyl transitive actions
Let ∆ be a thick building with associated Coxeter system (W,S) of finite rank
|S|. Denote by C the set of chambers of ∆ and by δ : C×C −→ W the associated
Weyl distance function. Recall that a group H is said to act Weyl transitively on
∆ if it acts on C preserving δ and such that for any given w ∈ W, the action of
H on {(C,C′) ∈C×C | δ(C,C′)= w} is transitive. In particular, Weyl transitive
actions are type-preserving and chamber transitive. It can be shown that a group
H acts Weyl transitively on some thick building if and only if H admits a “Tits
subgroup;” see [AB2, Proposition 6.34 and Definitions 6.39 and 6.45].
LetAbethecompletesystemofapartmentsof∆. Bydefinition,asubsetA⊆A
is a system of apartments of ∆ if for any two C,C′ ∈ ∆, there is a Σ ∈ A which
contains C and C′. A type-preserving action of a group H on ∆ is called strongly
transitivewithrespecttoAifitistransitiveontheset{(C,Σ)∈C×A|C ∈Σ}. The
TRANSITIVITY OF GROUP ACTIONS ON BUILDINGS FROM CHEVALLEY GROUPS 3
actionofH on∆iscalledstrongly transitive ifitis stronglytransitivewithrespect
to some system of apartments A of ∆. It is clear that H acts strongly transitively
on ∆ with respect to a system of apartments A if it acts chamber transitively, and
for some (and hence any) chamber C the stabilizer Stab (C) acts transitively on
H
{Σ∈A|C ∈Σ}. Equivalently, H acts transitively on A, and for some (and hence
any) Σ ∈ A the stabilizer Stab (Σ) acts transitively on the set of chambers of
H
Σ. It is well known that a group H acts strongly transitively on a thick building
if and only if H admits a BN-pair. If ∆ is spherical, it is also well known that
A is the only system of apartments of ∆, and that the following three statements
for the action of a group H on ∆ are equivalent: (i) H acts Weyl transitively;
(ii) H acts transitively on pairs of opposite chambers of ∆; (iii) H acts strongly
transitively. In the following lemma we collect some further (easy) results which
relatestrongandWeyltransitivity. The proofs canbe foundin[AB1, Section3]or
in [AB2, Section 6.1.3].
Lemma 2.1. Let H be a group acting on the building ∆.
(1) If H acts strongly transitively, then it also acts Weyl transitively.
(2) IfH actsWeyltransitively, thenHΣ={hΣ|h∈H}isasystemofapartments
of ∆ for any Σ∈A.
(3) H acts stronglytransitively if andonly it acts Weyl transitively and there exists
an apartment Σ∈A such that Stab (Σ) acts chamber transitively on Σ.
H
In view of (3) we make the following definition:
Definition 2.2. A type-preserving action of a group H on a building ∆ is called
weakly transitive if there exists an apartment Σ ∈ A such that Stab (Σ) acts
H
chamber transitively on Σ.
It is clear that an action is strongly transitive if and only if it is both Weyl and
weakly transitive. In order to find Weyl transitive actions that are not strongly
transitive,oneneedsagroup-theoreticcriterionequivalenttoweaktransitivity. We
willformulatesucha criterionin the followingsituation, whichwe shallassumefor
the rest of this section:
Let G be a group acting strongly transitively on ∆ with respect to A. Fix an
apartmentΣ inA,andsetN =Stab (Σ ), T ={t∈N |tC =C forallchambers
0 G 0
C in Σ }. Note that N/T can be identified with the group of type-preserving
0
automorphisms of Σ , and hence with W.
0
Lemma 2.3. For a subgroup H of G, the following are equivalent:
(i) H acts weakly transitively on ∆.
4 PETERABRAMENKOANDMATTHEWC.B.ZAREMSKY
(ii) There exists an element g ∈G such that g(nT)g−1∩H 6=∅ for all n∈N.
Proof. Let Σ be any element of A. By assumption, there exists g ∈ G such that
Σ=gΣ . HenceStab (Σ)=gNg−1,andthepointwisefixerofΣinGisgTg−1. So
0 G
Stab (Σ)=gNg−1∩H,andthisgroupactstransitivelyonthesetC(Σ)ofchambers
H
of Σ if and only if Stab (Σ)(gTg−1)=gNg−1. (Here we use that gNg−1/gTg−1,
H
whichcanbeidentifiedwiththegroupoftype-preservingautomorphismsofΣ,acts
simply transitively on C(Σ).) But (gNg−1 ∩H)(gTg−1) = gNg−1 if and only if
each coset (gng−1)(gTg−1) (with n∈N) in gNg−1/gTg−1 has a representative in
H, i.e. if and only if gnTg−1∩H 6=∅ for all n∈N. (cid:3)
Corollary 2.4. If there exists an element n ∈N\T such that all elements of n T
0 0
have finite order in G, then no torsionfree subgroup of G acts weakly transitively
on ∆. (cid:3)
In the following sections we shall apply this set-up to the Chevalley group G=
g(Q ) and the associatedaffine building ∆, in whichcase ann as in Corollary2.4
p 0
can be found. On the other hand, we shall exhibit torsionfree subgroups of g(Q )
p
which still act Weyl transitively on ∆. In order to establish the latter, we apply
Proposition 3.4 of [AB1]. For the convenience of the reader, we restate this result
below.
Lemma 2.5. Suppose that G is a topological group and that the stabilizer B of
some chamber C of ∆ is an open subgroup of G. Then any dense subgroup H of G
acts Weyl transitively on ∆.
3. Chevalley Groups and VRGD systems
SincethecurrentgroupofinterestisanarbitraryChevalleygroup,inthissection
somebackgroundinformationaboutChevalleygroupsiscollected. Chevalleygroups
will also be established as examples of RGD systems, and p-adic Chevalley groups
as examples of VRGD systems. For a more comprehensive review of Chevalley
groups, one may consult [S], and for a quick overview (the notation of which we
will use here) see [AB2, Section 7.9.2]. For an overview of RGD (root group data)
systems one may consult [AB2, Chapter 7], and a reference for what we will call
VRGD (valuated root group data) systems can be found in [W, Chapter 3].
Given a complex semisimple Lie algebra g with root system Φ, there is a family
of groups g(K) = g(Φ,Λ,K) parameterized by K and Λ. Here, K is an arbitrary
field and Λ ⊇ Φ is a weight lattice arising from some faithful finite-dimensional
representation V of g. Let V = V be the weight space decomposition of
γ γ
V, and let E be the Euclidean space spanned by the roots, with Euclidean inner
L
product{,}. Alsolets denotethereflectioninE aboutthehyperplaneorthogonal
α
to α, and define hv,v′i:=2{v,v′}/{v′,v′}. Note that hv,vi=2 for all v ∈E. The
group g(K) has generators x (λ) for α ∈ Φ, λ ∈ K, and a series of relations that
α
TRANSITIVITY OF GROUP ACTIONS ON BUILDINGS FROM CHEVALLEY GROUPS 5
we will reference as they become necessary. For notational convenience, we define
some important elements of g(K). For α∈Φ, λ∈K∗, let
m (λ):=x (λ)x (−λ−1)x (λ) and h (λ):=m (λ)m (1)−1.
α α −α α α α α
Define for each root α a subgroup U := {x (λ)|λ ∈ K}. As proved in [S,
α α
Corollary 1 to Lemma 18], the map (K,+) → U given by λ 7→ x (λ) is an
α α
isomorphism. Also, for fixed α one has by [S, Lemma 28(a)] that h (λ)h (µ) =
α α
h (λµ). In this way, the algebraic structure of K is reflected in the structure of
α
the Chevalley group. Now we will show that if K has a discrete valuation, the
additional structure given by the valuation can also be encoded into the Chevalley
group.
First we will see that (g(K),(U ) ) is an RGD system. A pair (G,(U ) ),
α α∈Φ α α∈Φ
where (U ) is a family of subgroups of G, is called an RGD system provided
α α∈Φ
the following axioms hold.
(RGD0): For each α∈Φ, U 6={1}.
α
(RGD1): For all α,β ∈ Φ with α 6= ±β, [U ,U ] ⊆ U , where (α,β) is
α β γ
γ∈Y(α,β)
defined as in [AB2, Section 7.7.2]
(RGD2): For eachα∈Φ there is a function m:U∗ →G suchthat for α∈Φ and
α
u∈U∗, m(u)∈U uU and m(u)U m(u)−1 =U .
α −α −α β sα(β)
(RGD3): For each fundamental root α, U 6≤U , where U =hU |α∈Φ+i.
−α + + α
(RGD4): G=ThU |α∈Φi, where T = N (U ).
α G α
α∈Φ
\
A proof that (g(K),(U ) ) satisfies the axioms is given in [AB2, Section 7.9.2],
α α∈Φ
though here the axioms are in a slightly different (equivalent) form.
We now assume that K has a (surjective) discrete valuation ν : K ։Z∪{∞}.
In later sections we will use K = Q and ν = ν . Let (V1) denote the property
p p
ν(λµ) = ν(λ)+ν(µ), let (V2) denote the property ν(λ+µ) ≥ max(ν(λ),ν(µ)),
let (V3) denote the property ν(c) ≥ 0 for c ∈Z, and let (V4) denote the property
ν(−λ)=ν(λ).
The family of maps (φ ) with φ : U∗ → Z is called a root group valuation
α α∈Φ α α
provided that the following axioms hold:
(VRGD0): Each φ is surjective.
α
(VRGD1): For each α ∈ Φ and each k ∈ Z, U := hu ∈ U |φ (u) ≥ ki is a
α,k α α
subgroup of U , where φ (1) is considered to be ∞.
α α
(VRGD2): For all α,β ∈ Φ with α 6= ±β, [U ,U ] ⊆ U ,
α,k β,ℓ γ,pγk+qγℓ
γ∈Y(α,β)
where p and q are as defined in [W, Chapter 3].
γ γ
(VRGD3): Forα,β ∈Φ,u∈U∗,x∈U∗,onehasthatφ (m(u)−1xm(u))−
α β sα(β)
φ (x) is independent of x, where m:U∗ →g(K) is as defined in (RGD2).
β α
(VRGD4): For α ∈ Φ, u ∈ U∗, x ∈ U∗, we have that φ (m(u)−1xm(u))−
α α −α
φ (x)=−2φ (u), independent of x.
α α
6 PETERABRAMENKOANDMATTHEWC.B.ZAREMSKY
Definition 3.1. Let G be a group with a family of subgroups (U ) . Let
α α∈Φ
(φ ) be a family of maps φ : U → Z. If (G,(U ) ) is an RGD system
α α∈Φ α α α α∈Φ
and (φ ) is a root group valuation then (G,(U ) ,(φ ) ) is called a VRGD
α α∈Φ α α α α
system.
Now, in the particular case of G = g(K) where K has (surjective) discrete
valuation ν, define for each α ∈ Φ a map φ : U → Z by x (λ) 7→ ν(λ). Since
α α α
Uα ∼= (K,+) via xα(λ) 7→ λ, φα is clearly well-defined. It is easy to check that
(G,(U ) ,(φ ) ) is a VRGD system.
α α∈Φ α α∈Φ
Proposition 3.2. (G,(U ) ,(φ ) ) is a VRGD system.
α α∈Φ α α∈Φ
Proof. (VRGD0)holdstrivially. (VRGD1)followsfrom(V2),(V4),andthecanon-
icalisomorphism(K,+)→U . (VRGD2) followsfromthe Chevalley relation(R2)
α
found after Lemma 20 of [S], and from (V1) and (V3).
Next we check (VRGD3). By [AB2, Equation 7.36], we have that
mα(λ)xβ(µ)mα(λ)−1 =xsα(β)(±λhsα(β),αiµ)
for any α,β ∈Φ, λ∈K∗, µ∈K. Denote this relationby (R). Let u=x (λ)∈U
α α
and set m(u)=m (−λ−1), as in the proof that (G,(U ) ) is an RGD system.
−α α α∈Φ
Also let x = x (µ). We claim that the integer given by φ (m(u)−1xm(u))−
β sα(β)
φ (x) is independent of µ. We know that this quantity equals
β
φ (m (−λ−1)−1x (µ)m (−λ−1))−ν(µ)
sα(β) −α β −α
=φ (x (±λ−hsα(β),−αiµ))−ν(µ)
sα(β) sα(β)
by (R), since s =s . This equals
−α α
ν(±λhsα(β),αiµ)−ν(µ)=ν(±λhsα(β),αi)
by (V1) and (V4). This quantity is indeed independent of µ and so (VRGD3)
follows.
Note that when α=β we have s (β)=s (α)=−α. Thus
α α
ν(±λhsα(β),αi)=ν(±λ−hα,αi)=ν(±λ−2)=−2ν(λ)=−2φα(u)
by (V1) and (V4), so (VRGD4) holds. Thus, all the axioms are satisfied and
(G,(U ) ,(φ ) ) is a VRGD system. (cid:3)
α α∈Φ α α∈Φ
Since(G,(U ) ,(φ ) )isaVRGDsystemonecanuse[W,Theorem14.38]
α α∈Φ α α∈Φ
to conclude that G has an affine BN-pair (B ,N), with affine Weyl group W =
a a
N/B ∩N. By [AB2, Theorem6.56]one gets anaffine building ∆ onwhichG acts
a
strongly transitively with respect to G.Σ . If one makes the further assumption
0
that K is complete with respect to the metric induced by ν, then by [W, Theo-
rem 17.7 and 17.9] one knows that G in fact acts strongly transitively on ∆ with
TRANSITIVITY OF GROUP ACTIONS ON BUILDINGS FROM CHEVALLEY GROUPS 7
respect to the complete apartment system. Since Q is complete, one can apply
p
Corollary 2.4 to the setup of g(Q ) acting on the corresponding building ∆.
p
Of course, by virtue of (G,(U ) ) being an RGD system, there is also a
α α∈Φ
spherical BN-pair (B,N) with spherical Weyl group W = N/B∩N. Looking at
theconstructionoftheaffineandsphericalBN-pairs,oneseesthatthesubgroupN
is the same in both cases, namely N =hm (λ)|α∈Φ,λ∈K∗i; see [S, Lemma 22]
α
and [W, 14.3]. Also, T ⊆ T, where T = B ∩N = hh (λ)|α ∈ Φ,λ ∈ A×i and
a a a α
T =B∩N =hh (λ)|α∈Φ,λ∈K∗i, with A the valuation ring of K.
α
By [S, Lemma 22(b,c)] W is isomorphic to the Weyl groupof the root systemΦ
(thus the name), and so any w ∈ W acts as an orthogonal transformation on the
EuclideanvectorspaceE spannedbyΦ. We willsee inthe nextsectionthatitwill
beusefulto thinkofW interchangeablyasboththe quotientN/T andasthe Weyl
groupofthe rootsystemΦ, acting onE. We alsonote that W is in factthe affine
a
Weyl group associated to Φ, though we will not use this explicitly.
4. Strongly transitive subgroups of g(Q ) have torsion
p
The goal of this section is to establish Proposition 1.1. Let G = g(Q ), with
p
affine BN-pair (B ,N) and affine Weyl group W , and let ∆ be the canonical
a a
affine building associatedto G as describedin the previous section. We restate the
proposition:
Proposition 4.1. No torsionfree subgroup of G acts strongly transitively on ∆.
By Corollary 2.4 it suffices to exhibit a non-trivial element w in W = N/T
a a
such that all representatives of w in N have finite order in G. Recall that T ≤T,
a
so if there exists n in N\T suchthat all elements of the cosetnT havefinite order,
then also all elements of the non-trivial coset nT will have finite order. We may
a
thusshiftoursearchtothesphericalWeylgroupW =N/T. Infactitdoeshappen
that W can be realized as a subgroup of W , though we will not need to use this
a
explicitly.
First some additional setup is necessary.
Let N ≥ N :=hm (1)|α ∈Φi and let T := hh (−1)|α ∈Φi. Since m (−1)=
0 α 0 α α
m (1)−1, in fact h (−1) = m (1)−2, so T ≤ N . Also, by [S, Lemma 20(a)],
α α α 0 0
m (1)h (−1)m (1)−1 =h (−1), so T ⊳N .
α β α sα(β) 0 0
Lemma 4.2. W =N /T .
0 0
Proof. The proof is similar to [S, Lemma 22(b,c)]. Define a homomorphism φ :
0
W → N /T by φ (s ) = T m (1). To check this is well-defined one checks that
0 0 0 α 0 α
the relations in W are satisfied in N /T . Note that
0 0
T m (1)=T h (−1)m (1)=T m (−1)=T m (1)−1
0 α 0 α α 0 α 0 α
8 PETERABRAMENKOANDMATTHEWC.B.ZAREMSKY
so for any α, s2 7→T m (1)m (1)−1 =T . Also,
α 0 α α 0
s s s−1s−1 7→T m (1)m (1)m (1)−1m (1)−1.
α β α sα(β) 0 α β α sα(β)
But this is just T since m (1)m (1)m (1)−1 = m (c) by [S, Lemma 20(b)],
0 α β α sα(β)
where c = ±1. If c = −1 one must also again use the fact that T m (−1) =
0 α
T m (1). These relations define W, so φ is well-defined, and is clearly surjective.
0 α 0
Now suppose w = s ...s 7→ T , so m (1)...m (1) ∈ T . In particular,
α1 αk 0 α1 αk 0
m (1)...m (1) ∈ T and so by the proof of [S, Lemma 22(c)], w = 1. Thus, φ
α1 αk 0
is an isomorphism. (cid:3)
Now one can establish a criterion on w whereby all representatives in N will
have finite order.
Theorem 4.3. Let W = N/T be the spherical Weyl group corresponding to the
Chevalley group g(K). Let w ∈W. Then the following are equivalent:
(i) As an orthogonal transformation of E, w does not have eigenvalue 1.
(ii) For any field K, every representative of w in N has finite order in N.
Proof of the forward implication. Let w ∈ W have order m, and suppose 1 is not
an eigenvalue of w. Since for any v ∈E we have
w(v+w(v)+···+wm−1(v))=v+w(v)+···+wm−1(v),
the hypothesis forces1+w+···+wm−1 to be zero. Now,since W =N /T , there
0 0
exists a representative n ∈ N of w. Since wm = 1, nm ∈ T . But T is abelian
0 0 0 0 0
and h (−1)2 = 1, so everything in T has order 1 or 2. Since m must divide the
α 0
order of n , we know that n has order m or 2m. Now let h be any element of T.
0 0
Say n = m (ǫ )...m (ǫ ) and h = h (λ )...h (λ ). Here each ǫ is either 1
0 α1 1 αk k β1 1 βℓ ℓ i
or -1, since n ∈ N and m (1)−1 = m (−1). Note that since n represents w in
0 0 α α 0
W, we have that w =s ...s .
α1 αk
Using the Chevalley relation
m (1)h (λ)m (1)−1 =m (1)−1h (λ)m (1)=h (λ),
α β α α β α sα(β)
one gets that n h=h (λ )...h (λ )n . Repeating this, one gets that
0 w(β1) 1 w(βℓ) ℓ 0
m ℓ ℓ m
(n h)m = h (λ ) nm = h (λ ) nm.
0  wi(βj) j  0  wi(βj) j  0
i=1j=1 j=1i=1
YY YY
This last step follows since T is abelian. Now, for any j and for anyweight γ,
m
λhγ,wi(βj)i =λPmi=1hγ,wi(βj)i.
j j
i=1
Y
Since w is an orthogonaltransformation and {,} is bilinear,
TRANSITIVITY OF GROUP ACTIONS ON BUILDINGS FROM CHEVALLEY GROUPS 9
m m
hγ,wi(β )i = 2{γ,wi(β )}/{wi(β ),wi(β )}
j j j j
i=1 i=1
X X
m
= 2{γ,wi(β )}/{β ,β }
j j j
i=1
X
m
2
= γ, wi(β ) =0
j
{β ,β }
j j ( )
i=1
X
m
foreachj. Thus, λhγ,wi(βj)i =1,regardlessofthefieldK. By[S,Lemma19(c)],
j
i=1
Y
elements h (λ) of T act on the weight space V via multiplication by λhγ,αi, and
α γ
m
so in fact h (λ )=1 for each j. One concludes that (n h)m =nm for any
wi(βj) j 0 0
i=1
Y
h ∈ T. Since w has order m, this implies that all representatives of w must have
the same order, and the result follows. (cid:3)
Note that this proves something stronger. Every representativehas finite order,
and in fact they all have the same order, either m or 2m. It is a quick exercise to
check that such a w exists, in fact any Coxeter element of W will work, as seen in
[H, Section 3.16 Lemma]. For completeness we will prove the reverse implication
of Theorem 4.3, though it is not needed to prove Proposition 4.1.
Proof of reverse implication. Let K = Q. Suppose 0 6= v ∈ E is a 1-eigenvector.
Then v +w(v)+···+wm−1(v) = mv 6= 0, and so 1+w +···+wm−1 6= 0 as
a linear transformation. Since the roots span E, there exists a root β such that
β +w(β) +···+wm−1(β) 6= 0. Choose a representative n ∈ N as before, so
0 0
nm ∈T , say nm =h (−1)...h (−1). Then for any r ∈N,
0 0 0 α1 αk
rm
(n h (2))rm = h (2) nrm.
0 β wi(β) 0
!
i=1
Y
(We chose K = Q but in fact, any K that is not an algebraic extension of a
finite field will work;we just need anelement with infinite multiplicative order;for
K = Q we have used the number 2.) Suppose this equals 1 for some r. Then by
[S, Lemma 19(c)], for any weight γ we have
rm k rm
1= 2hγ,wi(β)i (−1)rhγ,αki =± 2hγ,wi(β)i.
i=1 j=1 i=1
Y Y Y
By the same argument as before, this equals
±2{β2,β}{γ,Pri=m1wi(β)} =±2{β2,β}{γ,rPmi=1wi(β)}.
10 PETERABRAMENKOANDMATTHEWC.B.ZAREMSKY
m m
Theonlywaythiscanequal1isif γ,r wi(β) =0. Butsincer wi(β)6=0,
( )
i=1 i=1
X X
this is impossible, since one can always choose a weight γ to be not orthogonal to
m
r wi(β). Since the rm are the only candidates for a finite order of n h (2),
0 β
i=1
X
in fact it has infinite order. Since n h (2) is a representative of w, the theorem
0 β
follows. (cid:3)
Proposition 4.1 now follows from Corollary 2.4 and Theorem 4.3, and the fact
that Coxeter elements do not have eigenvalue 1.
Remark 4.4. There is also a very nice, shorter proof due to A. Rapinchuk that all
the representatives of w have finite order if 1 is not an eigenvalue of w [R]. In fact
his proof shows that each representative has order dividing m2, where m = |w|.
Coupling the two proofs, we conclude that if m is odd, then since representatives
cannot have order 2m they must all have order m.
5. Torsionfree Weyl transitive subgroups of g(Q )
p
One now has the tools to produce examples of Weyl transitive group actions on
buildings that are not strongly transitive with respect to any apartment system,
proving Proposition 1.2. Let (B ,N) be the affine BN-pair of G = g(Q ) as
a p
described in Section 3. Think of G as a subgroup of SL (Q ) for some d, and let
d p
(B,N) be the usual affine BN-pair of SL (Q ) described in [AB2, Section 6.9].
d p
The construction of B shows that it is contained in B∩G. Also, B∩G contains
a
e e
no nontrivial representatives of the affine Weyl group of G. Thus by looking at
e e
the affine Bruhat decomposition given by (G,B ) one sees that B =G∩B. This
a a
proves that B is open in G, and so by Lemma 2.5 any dense subgroup of G acts
a
e
Weyl transitively on ∆. Thus by Proposition 4.1, any dense, torsionfree subgroup
of G will act Weyl transitively but not strongly transitively on ∆. We now exhibit
a number of such subgroups, establishing Proposition 1.2.
Let Γ = g(Z[1]). While we technically have only been considering Chevalley
p
groups over fields, this is allowed; see [S, Section 3]. Let q be any nonzero integer
prime to p, so it makes sense to reduce the entries of matrices in Γ mod q. Define
the congruence subgroup Γ to be Γ := {A∈Γ : A ≡ I mod q}, where matrices
q q d
aretakenmod q entry-wise. This is the kernelofthe restrictionto Γ ofthe natural
grouphomomorphismSL (Z[1])→SL (Z[1]/qZ[1]),soitreallyisasubgroup. We
d p d p p
will show that for any q >2 prime to p, Γ is both torsionfree and dense in G.
q
Lemma 5.1. For any nonzero q in Z, Γ is dense in G.
q
Proof. Since the topological closure of Z[1] contains Z and 1/p, Z[1] is dense
p p p
in Q . Also Q = qQ , so qZ[1] is dense in Q . Thus for any α ∈ Φ, the set
p p p p p
{x (λ) | λ ∈ qZ[1]} is dense in {x (λ) | λ ∈ Q }. Since the latter set generates
α p α p