P--ADIC LATTICES OF PSEUDO REFLECTION GROUPS
D. Notbohm
Abstract. Let U be a vector space over the p-adic rationals, and let W -!*
* Gl(U)
be faithful representation of a finite group such that W is generated by *
*pseudo re-
flections. For odd primes we study the p-adic W -sublattice of this repre*
*sentation
and achieve a complete classification. Examples of such situations are gi*
*ven by the
Weyl group acting on the 1-dimesional homology of the maximal torus of a *
*connected
compact Lie group, or of the so called p-compact groups, a homotopy theor*
*etic gen-
eralisation of compact Lie groups. The associated lattices are an importa*
*nt algebraic
invariant in the study of these geometric object.
Introduction.
Let U be a finite dimensional vector space over the p-adic rationals Q^p. Fo*
*r a
faithful representation ae : W!- Gl(U) of a group W , an element 1 6= oe 2 W *
*is
called a pseudo reflection if oe or ae(oe) has finite order and if the kernel o*
*f ae(oe) - idU
has codimension 1. The element oe is called a honest reflection or a reflectio*
*n if
oe has order 2. Because we are working in characteristic 0, the order of oe di*
*vides
p - 1 and the linear transformation ae(oe) is diagonalzable. i.e. U has a bas*
*is of
eigenvectors with respect to ae(oe).
The representation ae : W!- Gl(U) represents W as a pseudo reflection group,
if W is generated by pseudo reflections. If we say that W is a pseudo reflecti*
*on
group, then we always have a representation in mind.
In this work we are concerned with the classification of all p-adic W -subla*
*ttices
L U of a given finite pseudo reflection group W!- Gl(U). Our motivation to
study this question comes from homotopy theory. The Weyl group WG of a con-
nected compact Lie group G acting on the tangent space of the maximal torus T
of G or on the 1-dimensional homology H1(T ; Z) provides an example of a honest
reflection group and also of an integral sublattice. This action is an importa*
*nt
algebraic invariant in the study of connected compact Lie groups. In [4], Dwyer
and Wilkerson gave the notion of p-compact groups, which is the homotopy the-
oretic generalisation of the notion of compact Lie groups. In their work, pseu*
*do
reflection groups occured in the same manner as honest reflection groups for co*
*n-
nected compact Lie groups, namely as Weyl groups acting on a `maximal torus'.
These p-compact groups provide examples of pseudo reflection groups and associ-
ated p-adic sublattices. Besides these geometric and homotopy theoretic aspects
we believe that the study of p-adic W -lattices has interest from it's own.
We will use the following notation and definitions in this paper.
______________
1980 Mathematics Subject Classification (1985 Revision). 20C11, 06B15, 55R35.
Key words and phrases. reflection group, pseudo reflection group, lattice, p*
*-adic representa-
tions, p-compact groups.
Typeset by AM S-T*
*EX
1
2
1.1 Notation, Definitions and Remarks.
We always denote by U a finite dimensional vector space over Q^p, and by W
a finite pseudo reflection group represented by a faithful representation ae : *
*W!-
Gl(U). By a sublattice or a W -sublattice L U we always understand a sublattice
of maximal rank, i.e. LQ := L Z Q ~= U as W -modules. If we say that W is a
pseudo reflection group, then this always includes that W is finite.
1.1.1: The representation W!- Gl(U) is called fixed point free if the fixed*
* point
set UW = 0 is trivial. A sublattice L U is called fixed point free if LW = 0.
1.1.2 The pseudo reflection group W!- Gl(U) is called irreducible if the r*
*ep-
resentation is irreducible. By [3], this is equivalent to the fact that the ass*
*ociated
complex representation U Q^pC is irreducible.
1.1.3: Let Z=p1 S1 denote the subgroup of all elements of p-power order.
Then, for any n, the group T1 := (Z=p1 )n is called a p-discrete torus and the
inclusion T1 := (Z=p1 )n (S1)n =: T is a p-discrete approximation in the sens*
*e of
[4], i.e. the map BT1 !- BT between the classifying spaces induces an isomorph*
*ism
H2(BT1 ; Z^p) ~= H2(BT ; Z^p) and an equivalence after p-adic completion. The
completion Tp^ is called a p-adic torus.
For every W -lattice L, we have a short exact sequence
1!- L!- LQ!- LQ =L =: TL;1!- 1
of W -modules. Completing the classifying spaces of the p-discrete torus TL;1 *
* ~=
(Z=p1 )n and passing to 2-dimesional homology establishes an isomorphism
H2(BTL;1 ; Z^p) ~=L of W -modules.
1.1.4 For a W -sublattice L U, the fixed point set Z(L) := (TL;1 )W is cal*
*led
the center of L. The sublattice L is called centerfree if Z(L) = 0.
1.1.5 For a W -sublattice L U, the covariants LW are given by the quotient
L=SL, where SL L is the sublattice generated by all elements of the form l -w(*
*l)
with l 2 L and w 2 W . The lattice L is called simply connected if LW = 0.
1.1.6 A monomorphism L!- M of W -lattices is called a W -trivial restrictio*
*n or
W -trivial extension if W acts trivially on the quotient M=L and if M=L is fini*
*te.
Most of these notions are motivated by an analogy to connected compact Lie
groups. Let WG denote the Weyl group and TG the maximal torus of a connected
compact Lie group G. Then, the action of WG on UG := H2(BTG ; Z^p) Q rep-
resents W as a finite reflection group, and LG := H2(BTG ; Z^p) UG is a p-ad*
*ic
sublattice. For odd primes, the center Z(LG ) of LG is a p-discrete approximat*
*ion
of the center of G [7]. Because the fundamental group ss1(G) of G is isomorphic
to the quotient of ss1(T ) by the translations of the extended Weyl group, one *
*can
also show that, for odd primes, (LG )WG is a p-discrete approximation of ss1(G*
*), i.e.
(LG )WG Z^p~= (LG Z^p)WG ~=ss1(G) Z^p.
1.2 Theorem. Let p be an odd prime. Let W!- Gl(U) be a finite fixed-point
free pseudo reflection group. Then the following holds:
(1) There exists a centerfree sublattice P U, unique up to isomorphism.
(2) There exists a simply connected sublattice S U, unique up to isomor-
phism.
3
(3) For every W -sublattice L U there exist a composition S!- L!- P
of W -trivial restrictions such that P=L ~= Z(L) and L=S ~= LW as W -
modules.
(4) Let S; P U be sublattices and S!- P a W -trivial restriction. If every
sublattice L U fits between these both, i.e. there exist W -trivial re*
*stric-
tions S!- L and L!- P such that the composition gives S!- P , then S
is simply connected and P is centerfree.
It is well known, that, for a finite pseudo reflection group W!- Gl(U), th*
*ere
exists splitting U ~= UW U1 ::: Un of U as W -modules and a splitting W ~=
W1 x ::: x Wn of W such that Wi acts on Ui as an irreducible pseudo reflection
group and trivially on every Uj for j 6= i.
For the proof of the first two parts of Theorem 1.2, we first study irreduci*
*ble
pseudo reflection groups. After having done this, the general case is a corolla*
*ry of
the following statement;
1.3 Theorem. Let p be an odd prime. Let W!-Q Gl(U) be aLfinite fixed-point
free pseudo reflection group, and let W ~= iWi and U ~= iUi be the associat*
*ed
splittings into irreducible pseudo reflection groups. Then the following holds:
L
(1) Every centerfree sublattice P U splits into a direct sum P ~= iPi of
centerfree sublattices Pi Ui. The summands Pi are uniquely determined
up to order and isomorphisms. L
(2) Every simply connected sublattice S U splits into a direct sum S ~= *
*iSi
of simply connected sublattices Si Ui. The summands Si are uniquely
determined up to order and isomorphisms.
Finally we consider the general case of a pseudo reflection group W!- Gl(U*
*).
Let U ~=UW U0 be the splitting into the fixed-point set and the fixed point f*
*ree
factor U0. Let S U0 be the unique simply connected sublattice and P U0 the
unique centerfree sublattice given by Theorem 1.2. Because W acts trivially on
UW , all sublattices of UW are isomorphic as W -modules. We choose a sublatti*
*ce
of UW and denote it by Z.
1.4 Theorem. Let p be an odd prime. Let W!- Gl(U) be a pseudo reflection
group, and let L be a W -sublattice. Then the following holds:
(1) There exists a W -trivial restriction Z S!- L with quotient L=(Z S) *
*~=
(L=LW )W .
(2) There exists a W -trivial restriction L!- Z P .
The classification of lattices of pseudo reflection groups can be done analo*
*gously
as for connected compact Lie groups. Irreducible honest reflection groups corr*
*e-
spond to simple connected compact Lie groups. The following corollary is obvious
and classifies all lattices of finite pseudo reflection groups.
1.5 Corollary. Let p be an odd prime. Then, every W -lattice is a W -trivial
extension of a trivial W -lattice and a simply connected W -lattice, and every *
*simply
connected W -lattice is a direct sum of simply connected lattices of irreducible
pseudo reflection groups.
4
The analogy is given by the fact that every connected compact Lie group is a
qoutient of a product of simply connected simple Lie groups and a torus.
Our main theorems are statements about odd primes. This comes simply from
the following lemma, which plays a litle but important role in several proofs.
1.4 Lemma. Let p be an odd prime, let W be a pseudo reflection group and let M
be a Z^p-module with trivial W -action. Then, we have H1(W ; M) = H1(W ; M) =
0
Proof. Without loss of gnerality we can assume that M is finitely generated. As
a pseudo reflection group for an odd prime, W is generated by elements of order
coprime to p. By the Hurewicz theorem, the first homology group H1(W; Z) is
isomorphic to the abelinization of W , which is a finite abelian group of order*
* coprime
to p. Universal coefiicient theorems imply the statement.
The paper is organized as follows: In Section 2, we discuss centerfree latti*
*ces and
prove part (1) of Theorem 1.2 and of Theorem 1.3. In Section 3 we study simply
connected lattices and prove part (2) of Theorem 1.2 and of Theorem 1.3. ` The
last section contains the proof of the rest of Theorem 1.2 and of Theorem 1.4.
Although the nature of this paper is mostly algebriac, sometimes we deal with
completed topological spaces. Completion is always meant in the sense of Bousfi*
*eld
and Kan [1].
Independently of us, Dwyer and Wilkerson got also proofs for the main results
of this work, which are not published yet.
Finally a warning: We are only dealing with odd primes, i.e. p always denotes
an odd prime.
2. Centerfree sublattices.
As mentioned in the introduction W!- Gl(U) is a finite pseudo reflection gr*
*oup.
2.1 Lemma.
(1) For a W -lattice L, the center Z(L) is a finite abelian p-group if and *
*only
if L is fixed point free.
(2) If L is fixed-point free, then we have Z(L) ~=H1(W ; L).
Proof. Taking fixed-points in the short exact sequence
0!- L!- LQ!- TL;1!- 0
gives rise to an exact sequence
0!- LW !- LWQ!- (TL;1 )W !- H1(W ; L)!- H1(W ; LQ ) = 0 :
The first two terms vanishes if L is fixed point free. Otherwise the qoutient L*
*WQ=LW
is a p-discrete torus. Inparticular, the qoutient is not finite. Because H1(W ;*
* L) is
always finite, both parts follow.
For a W -lattice L we denote by L=p := L Z^pFp the associated Fp[W ]-module.
5
2.2 Lemma. A W -lattice P is centerfree if and only if (P=p)W = 0.
Proof. The multiplication p : P!- P by p establishes a short exact sequence
p
0!- P -! P!- P=p!- 0 :
Passing to fixed points gives an exact sequence
p p
0!- P W -! P W!- (P=p)W !- H1(W ; P ) -! H1(W ; P ) :
p
If (P=p)W = 0 then p : P W !- P W and p : H1(W ; P ) -! H1(W ; P ) are
isomorphisms. From the first isomorphism follows that P is fixed-point free. Be-
cause H1(W ; P ) is a finite abelian p-group, the second isomorphism implies th*
*at
H1(W ; P ) = 0. The other direction follows from Lemma 2.1.
__
Let L be W -lattice. The qoutient T 1 := TL;1 =TLW;1 carries_a W -action and
is a p-discrete torus which gives rise to a p-adic torus T . Passing to class*
*ifying
spaces establishes a fibration
__
BZ(L)^p-! BTL!- BT :
Passing to 2-dimesional homology yields a W -equivariant map
__ ^
L ~=H2(BTL ; Z^p)!- H2(BT ; Zp ) =: P L :
The lattice P L is called the associated centerfree lattice of L.
2.3 Proposition. Let L be a fixed point free W lattice.
(1) There exists an exact sequence
0!- L!- P L!- Z(L)!- 0 ;
and hence, L is a W -trivial restriction of P L.
(2) The lattice P L is centerfree and H1(W ; P L) = 0.
Proof. Because L is fixed point free, the center_Z(L) is finite. The Serre spec*
*tral
sequence of the_fibration BZ(L)!- BTL!- BT has a differential
d : P L = H2(BT : Z^p)!- H1(BZ(L); Z^p) ~= Z(L) which is an epimorphism and
has kernel L ~= H2(BT ; Z^p). This establishes the desired exact sequence of p*
*art
(1).
Taking fixed points in the exact sequence of (1) gives rise to the exact seq*
*uence
0 = P LW !- Z(L)W = Z(L)!- H1(W ; L)!- H1(W ; P L)!- H1(W ; Z(L)) :
By Lemma 2.1, the second arrow is an isomorphism. By Lemma 1.4, the last term
vanishes. Hence, we have H1(W ; P L) = 0. Again by Lemma 2.1, the lattice P L is
centerfree.
The following statement is part (1) of Theorem 1.2.
6
2.4 Theorem. Let W !- Gl(U) be a finite fixed-point free pseudo reflection
group. Then, up to isomorphisms, there exists a unique centerfree W -sublattice
P U.
The existence of a centerfree sublattice follows from Proposition 2.3. The *
*key
for the proof of the uniqueness is the following technical result.
2.5 Proposition. Let W!- Gl(U) be an irreducible pseudo reflection group, and
let P be a centerfree sublattice. If there exists an exact sequence
0!- V0!- P=p!- V1!- 0
of Fp[W ]-modules, such that V0W = 0 = V1W , then either V0 = 0 or V1 = 0.
Proof. Let assume that V0 and V1 are nontrivial vector spaces. We choose a basi*
*s for
V0 and extend it to a basis of P=p. Then every element w 2 W can be represented
by a matrix of the form
Aw Cw
0 Bw
where Aw decribes the action of w on V0, Bw the action on V1 and Cw : V1!- V0
the twisting, i.e. the failure to be a direct product. This description estab*
*lishes
a homomorphism OE : W !- Gl(V0) x Gl(V1) given by OE(w) := (Aw ; Bw ). Let
Wi be the image of W in the factor Gl(Vi). That is we have a homomorphism
OE : W!- W0 x W1. Because V0 and V1 have no non trivial fixed point, both grou*
*ps
W0 and W1 are non trivial.
The kernel K of OE consists of those elements which are desribed by a matrix*
* of
the form id0 Cid . Therefore, every element of the kernel has order p and the
kernel is an elementary abelian p-group and a normal subgroup of W .
Now let oe 2 W be a p-adic pseudo reflection. The matrix
oe - id = Aoe-0id B Coe
oe- id
has rank 1. That is that all columns and all rows are multiple of one column or*
* one
row. We have Aoe- id 6= 0 if and only if Boe= id. The equivalence follows from *
*the
fact that the order of oe is coprime to p. Therefore, W0 and W1 are generated by
p-adic reflections. Let (w0; w1) 2 W0 x W1. We can assume that w0 is the image
of a product of p-adic reflections which are mapped onto the identity in W1, and
similiar for w1. This shows that OE is an epimorphism.
The above considerations show that W allows a short exact sequence
(*) 1!- K!- W!- W0 x W1!- 1 :
where W0 and W1 are nontrivial groups, where both are generated by elements
coming from pseudo reflections in W , and where K W is an elementary abelian
normal subgroup. For abbreviation, we say that W has the property (*).
We want to show that either W0 or W1 is the trivial group. This would imply
that either V0 = 0 or V1 = 0. The proof of this conclusion splits into two part*
*, the
7
nonmodular case, i.e. (|W |; p) = 1, and the modular case. For the modular ca*
*se
we use the classification of the irreducible pseudo reflection groups by Clark *
*and
Ewing [3]. We also use there numbering of the different cases.
First let (|W |; p) = 1. Then K = 0 and W ~= W0 x W1 splits into a product of
pseudo reflection groups. Because the representation W!- Gl(U) is irreducibl*
*e,
this implies that either W0 or W1 is the trivial group.
Now let p divide |W |. If K W is a central subgroup, then every element of
K establishes a W -equivariant self map U!- U of the irreducible representati*
*on
U Q^pC, which therefore is given by a p-adic multiple of the identity. That is *
*to
say that there exists a homomorphism K!- Z^p*~= Z=p - 1 x Z^p. Because W is
finite and because W acts faithfully on U, this homomorphism is injective, and *
*the
kernel K is trivial. We can proceed as in the nonmodular case.
If K!- W is not central, then there exists a pseudo reflection oe 2 W acti*
*ng
nontrivially on K. Because the order of oe is coprime to p, the representation *
*K of
the group < oe >, generated by oe, splits into 1-dimesional irreducible summand*
*s.
Let K0 K be one of the summands with a nontrivial action of oe and let x 2 K0
be a generator. The subgroup D :=< oe; xoex-1 >=< oe; xoe >=< oe; x > of W ,
generated by two pseudo reflections, fits into a short exact sequence
1!- K0!- D!- < oe >-! 1 :
The order m = |oe| of oe is coprime to p. Therefore, the sequence splitsLand D*
* ~=
Z=p o Z=m acts on U as a pseudo relection group. As a Z=p-module, U ~= iUi
splits into a direct sum of irreducible Z=p-modules which are permuted by Z=m.
Each factor is either 1-dimensional with trivial Z=p-action (Q^pcontains no p-th
root of unity) or isomorphic to U0 ~=(Q^p)p-1 where we consider U0 as the kernel
of the map (Q^p)p!- Q^pgiven by summing up the coordinates and where Z=p acts
via cyclic permutation on (Q^p)n. The factors with trivial Z=p-action does not *
*lead
to a faithful representation of D. Every factor isomorphic to U0 is fixed under*
* the
action of Z=m, and Z=m acts on U0 via permutation associated to the action on
Z=p considered as a set. Therefore, U0 represents D as a pseudo reflection gro*
*up
if and only if m = 2. That is to say that D ~= D2p is a dieder group. By the
classification list of irreducible pseudo reflection groups [3] the only modula*
*r cases
are given by D6 and D12. Hence, we have p = 3 and D ~=D6.
By the above arguments it is only left to consider modular cases for p = 3. *
*We
will finish the proof by a case by case checking following the list of [3]. We*
* only
have to discuss the numbers 2a, 2b, 12, 28, 35, 36 and 37.
Case_number_2a__. In this case n W Z=l o n where l divides p - 1. Inpar-
ticular, the subgroup ss W is a normal subgroup of n as well as of An n.
Here, An denotes the group of permutations of positive sign. For n 5, the group
An is simple. For n = 4, we have A4 ~= (Z=2 x Z=2) o Z=3. Therefore, in these
cases there exists no normal elementary abelian 3-subgroup, and we can proceed
as in the nonmodular case.
For n = 3, the representation U is 2-dimensional and is irreducible even con-
sidered as a 3 ~= D6-module. Let oe; o 2 3 be two tanspositions generating
3. Let L U be the standard sublattice with the action given by oe = 01 10
8
and o = 10 -1-1. A straight forward calculation shows that L Fp is an ir-
reducible Fp[3]-module. Therefore, every other sublattice of U also produces a
mod-p irreducible module. This shows that either V0 or V1 is trivial.
For later purpose we note the following observation: For n 5, the above arg*
*u-
ment shows that, if W has the property (*), every pseudo reflection representat*
*ion
of W splits into two summands, where both factors carry a nontrivial W -action.
Case_number_2b__. In this case, we have W = D6 or W = D12 and U is 2-
dimensional. In particular, U is an irreducible D6-module. We can argue as in t*
*he
above case.
Case_number_12__. In this case we have dimQ^pU = 2. By the above arguments,
we have a subgroup D6 W . Moreover, U is irreducible as D6-module. Again we
can procedd as above to show that either V0 or V1 is trivial.
Case_number_28__. In this case, we have W = WF4 ~= ((Z=2)3 o 4) o 3. The
last isomorphism may be found in [5, p. 45]. A straight forward calculations sh*
*ows
that K = 0. We can proceed as in the nonmodular case.
Case_number_35,_36,_37_. In this case we have W = WE6 , W = WE7 or W = WE8 .
We describe two maximal subgroups of maximal rank for each of these connected
compact Lie groups.
__G__ _H0___ __H00__
E6 S1 xZ=2 Spin(10) SU(2) xZ=2 SU(6)
E7 S1 xZ=2 Spin(12) S1 xZ=3 E6
E8 SSpin(16) SU(2) xZ=2 E7
A list of all maiximal subgroups of maximal rank may be found in [6]. This esta*
*b-
lishes subgroups of W as follows:
__W___ _W_0__ __W_00_
WE6 WH0 ~=(Z=2)5 o 5 WSU(6) ~=6
WE7 WH0 ~=(Z=2)5 o 6 WE6
WE8 WH0 ~=(Z=2)7 o 8 WE7
In all cases, the two groups W 0and W 00generate W . This follows because H0 G
is maximal of maximal rank. Moreover, the intersection W 0\ W 00is nonempty. We
want to show that there exists no epimorphism W!- W0xW1 as in (*) with kernel
given by an elementary abelian p-group.
Let us look at the case W = WE6 . By the observation at the end of case numb*
*er
2a, the W 0-module U splits into a direct sum of nontrivial W 0-modules, if W 0*
*has
the property (*). The same is true for W 00. But by the choice of the groups, b*
*oth
belong to case 2a with n 5, we only can split of a trivial summand of U consid*
*ered
as a W 0or W 00-module. Therefore, an epimorphism WE6!- W0 x W1 maps W 0
and W 00only into one factor. Because W 0\ W 00is nonempty, both are mapped
into the same factor, let us say into W0. Because WE6 is generated by W 0and W *
*00,
the group W is only mapped into W0, too. Hence, W1 is trivial, This proves the
statement in this case. In particular, this argument also shows that there exis*
*ts no
epimorphism of the form (*) with kernel given by an elementary abelian p-group.
For WE7 and WE8 , we can argue analogously using the result for WE6 or WE7 .
This finishes the discusion of all possible cases and the proof of the statemen*
*t.
9
Remark. The last proposition as well as the proof originates in a discussion w*
*ith
C.Broto and J.Aguade on a similar question.
2.6 Lemma. Let P!- L be a monomorphism between W -sublattices of U. If P
is centerfree, then we have (L=P )W = 0.
Proof. Because P is centerfree, every sublattice of U ~=P Z Q is fixed-point fr*
*ee.
The short exact sequence P!- L!- L=P gives rise to an exact sequence
LW = 0!- (L=P )W !- H1(W ; P ) = 0. Thus, the quotient L=P has no fixed
points.
Proof of Theorem 1.7 for irreducible pseudo reflection groups.
Let P and Q be two centerfree sublattices of an irreducible pseudo reflecti*
*on
group W!- Gl(U). Then, there exists a W -equivariant monomorphism ff : P!-
Q such that rk(Q=P ) < rk(Q) = rk(P ). Here, rk(M) denotes the rank of a
module, which we define to be the dimension of M=p over Fp. Otherwise we have
P pQ := {px : x 2 Q} and we can replace Q by pQ. Because P is centerfree
we know that (Q=P )W = 0 (Lemma 2.6). Because U is irreducible we know that
ff
P!- Q is rationally an isomorphism and that Q=P is finite.
Applying the functor Fp yields an exact sequence
__ff
0!- T or(Q=P; Fp)!- P=p!- Q=p!- Q=P Fp!- 0
of W -modules. Let V0 := T or(Q=P ; Fp) and let V1 := Im(__ff) be the image of
ff which is isomorphic to the kernel of Q=p!- Q=P Fp. Because P and Q
are centerfree we have V0W = 0 = V1W (Lemma 2.2). Applying Proposition 2.5
shows that either V0 or V1 are trivial vector spaces. If V1 = 0 then rk(Q=P ) =
rk(T or(Q=P ; Fp) = rk(P ), which is a contradiction. Thus, V0 = 0 and Q=P = 0.
That is to say that ff : P!- Q is an isomorphism. This proves the statement f*
*or
irreducible pseudo reflection groups.
Next we consider the case of a reducible fixed-point free pseudo reflection *
*group
W , i.e. W ~= W1 x W2 splits into a nontrivial product of pseudo reflection gro*
*ups.
Moreover, U ~=U1xU2 also splits into a direct sum where U1 = UW2 and U2 = UW1 .
2.7 Lemma. Let W!- Gl(U) be a reducible pseudo reflection group, and let P
be a centerfree W -sublattice of U = U1 U2. Then, the fixed-point set P W1 is
centerfree with respect to the W2-action and P ~= P W1 P W2 as W -modules.
Proof. The qoutient P=P W1 is torsionfree. Hence, the sequence of W -modules
0!- P W1=p!- P=p!- (P=p)=(P W1=p)!- 0
is short exact . Taking fixed-points yields an exact sequence
0!- (P W1=p)W ~=(P W1=p)W2 !- (P=p)W = 0 :
The last fixed point set vanishes because P is centerfree and because of Lemma
2.2. Again by Lemma 2.2, the fixed-point set P W1 is centerfree with respect to*
* the
W2-action.
10
Applying the functor Q establishes an exact sequence
0!- P W1 Q!- P Q!- (P=P W1) Q!- 0 :
Because P W1 Q ~=(P Q)W1 , this sequence splits and shows that (P=P W1) Q
as well as P=P W1 are trivial W2-module. Taking W2-fixed points establishes the
exact sequence
0 = P W = (P W1)W2 !- P W2!- (P=P W1)W2 = P=P W1!- H1(W2; P W1) = 0 :
The last identity follows from Lemma 2.1 since P W1 is W2-centerfree. This im-
plies that the middle arrow is an isomorphism, and that P W1 P W2!- P is an
isomorphism of W -modules.
Proof of Theorem 1.7 in the general case. Let P and Q be centerfree W =
W1 x W2 lattice. Let P Wi =: Pi and QWi =: Qi. By Lemma 2.7, we know that
P ~= P1P2 and that Q ~=Q1Q2. Because P1 and Q1 are both W1-centerfree, they
are isomorphic as W1-modules by induction over the order of W . Analogously, we
have P2 ~=Q2 as W2-modules. Putting this together gives the desired W -module
isomorphism P ~= Q.
Proof of Theorem 1.3 (1). Let W!- Gl(U) be a reducible pseudo reflection
group. Using an induction over the number of irredicble summands of U, the
statement follows from Lemma 2.7 and Theorem 1.7 for irreducible pseudo reflect*
*ion
groups.
3. Simply connected sublattices.
Again, W!- Gl(U) denotes a finite pseudo reflection group. The situation f*
*or
simply connected lattices is somehow dual to the case of centerfree lattices (s*
*ee
Proposition 4.1).
3.1 Lemma.
(1) For a W -lattice L, the group LW of covariants is finite if and only i*
*f L is
fixed point free.
(2) If L is fixed point free, then we have LW ~=H1(W; TL;1 ) .
Proof. Passing to covariants and using the fact that LW ~= H0(W; L), the short
exact sequence
0!- L!- L Q := LQ!- TL;1!- 0
gives rise to the exact sequence
0 = H1(W ; LQ )!- H1(W ; TL;1 )!- LW !- (LQ )W !- (TL;1 )W !- 0 :
Because every exact sequence of W -modules over Q^psplits, we have (LQ )W ~=LW*
*Q.
The cohomology group H1(W ; TL;1 ) is finite. Thus, LW is finite if and only i*
*f L
is fixed-point free. The second part is obvious.
In the introduction, for a W -lattice L, we defined SL to be the kernel of L*
*!-
LW .
11
3.2 Propostion. Let L be a fixed-point free W -lattice.
(1) There exists an exact sequence
0!- SL!- L!- LW !- 0 ;
and L is a W -trivial extension of SL.
(2) The lattice SL is simply connected.
Proof. The first part follows from the definition of LW and SL. Passing to co*
*in-
variants, the short exact sequence of (1) establishes the exact sequence
H1(W; LW )!- SLW !- LW !- LW !- 0 :
The first term vanishes (Lemma 3.1 ) and the second last arrow is an isomor-
phism.
The next results connects simply connected and centerfree lattices.
3.3 Proposition. Let S be a simply connected W -lattice. Let P := P S be the
associated centerfree lattice. Then, we have SP ~= S and Z(S) ~=P=W .
Proof. By construction there exists an short exact sequence
qS
0!- S!- P -! Z(S)!- 0 :
Because Z(S) is a trivial W -module, the map qS factors over the covariants PW .
This establishes a commutative diagram of short exact sequences
0 --! SP --! P --! PW ~=Z(SP ) --! 0
?? fl ?
y flfl ?y
0 --! S --! P --! Z(S) --! 0
where the cokernel S=SP of the monomorphism SP!- S is a W -equivariant quo-
tient of PW . Therefore, the quotient S=SP is a module with trivial W -action, *
*and
the epimorphism S!- S=SP factors over SW = 0. This shows that all vertical
arrows are isomorphisms.
We finish this section with proofs of Part (2) of Theorem 1.2 and Part (2) of
Theorem 1.3.
Proof of Theorem 1.2 (2). The existence of a simply connected sublattice follo*
*ws
from Proposition 3.2. Let S and S0 be two simply connected sublattices. Let P
and P 0be the associated centerfree lattices. By Theorem 1.7 we know that P ~= *
*P 0,
and by Proposition 3.3 follows that S ~=SP ~= SP 0~=S0.
Proof of Theorem 1.3 (2). Let S U be a simply connected sublattices and let
P := P S U be theLassociated centerfree sublattice. By Theorem 1.3 (1), we have
a splitting P ~= iPi of P into centerfree sublattices Pi Ui of the irreduci*
*ble
pseudo reflection groups Wi!- Gl(Ui) and the summands are uniquely determined
up to order and isomorphisms.LThe sublatticesLSi := SPi Ui are simply connected
and the sequence S ~= SP ~= iSPi = iSi proves the statement. The first
isomorphism follows from Proposition 3.3, and the second from Lemma 2.7 and
Lemma 3.2.
12
4. Proof of the main theorems.
In this section we want to prove the last two parts of Theorem 1.2 and Theor*
*em
1.4.
Proof of Theorem 1.2.
Part (1) and part (2) are already proved in the previous sections. Let L U *
*be a
W -sublattice, and let P and S the centerfree and the simply connected sublatti*
*ces
of U. The composition SL!- L!- P L consist of W -trivial restrictions with
L=SL ~=LW and P L=L ~=Z(L) (Propositions 2.3 and 3.2). By (1) and (2) follows
that SL ~=S and that P L ~=P . This proves the third part.
Now let S!- P be a W -trivial restriction of sublattices of U, satisfying*
* the
assumptions of (4). Let L be a W -sublattice of U. Then, by assumption, there
exists a W -trivial restriction P L!- P . Inparticular, the qoutient P=P L is*
* finite
with trivial W -action. Passing to cohomology gives an exact sequence
H1(W ; P L) = 0!- H1(W ; P )!- H1(W ; P=P L) = 0 :
The first term vanishes because P L is centerfree and because of Lemma 2.1, the
last term vanishes by Lemma 1.4. Thus H1(W ; P ) = 0 and P is centerfree (Lemma
2.1). On the other hand there exists a W -trivial restriction S!- SL. Taking
covariants gives an exact sequence H1(W ; SL=S) = 0!- SW !- SLW = 0. The
first term vanishes because of Lemma 1.4 and the last term because SL is simply
connected and because of Proposition 3.2 . Thus, we have SW = 0 and S is simply
connected.
Before we discuss the case of general finite pseudo reflection groups, i.e. *
*before we
prove Theorem 1.4, we need some informations about the dual representations. Let
W!- Gl(U) be a pseudo reflection group. We consider U as a left Q^p[W ]-module.
Then the set HomQ^p(U; Q^p) becomes a left Q^p[W ]-module by defininig w(x*) :=
x*w-1 for x* 2 U* and w 2 W . The vector space U* again represents W as a pseudo
reflection group. For a W -sublattice L U we define L* := HomZ^p(L; Z^p) which
becomes analogously as above a left Z^p[W ]-module. Because L* Q ~=(L Q)*
as Q^p[W ]-modules, the lattice L* is a sublattice of U* .
4.1 Proposition. Let W!- Gl(U) be a finite fixed point free pseudo reflection
group.
(1) A sublattice P U is centerfree if and only if P * U* is simply connect*
*ed.
(2) A sublattice S U is simply connected if and only if S* U* is centerfr*
*ee.
Proof. Let S!- P be the W -trivial restriction between the simply connected su*
*b-
lattice S and the centerfree sublattice P . For any W -lattice L, there exists*
* a
natural W -equivariant isomorphism L** ~= L. Let M be a W -sublattice of U* .
By Theorem 1.2 (3) the dual M* fits between S and P . Therefore, the lattice M
sits between P *and S*, which shows that P *is simply connected and that S* is
centerfree (Theorem 1.2 (4)). This proves one direction of both parts. The ot*
*her
follows by dualizing again.
13
__
Proof of Theorem 1.4. Let L U be a W -sublattice. The quotient L=LW =: L
is a fixed point free W -sublattice of U0, where U ~=UW U0 splits into the di*
*rect
sum of the fixed points UW and a fixed point free part U0. Let S; P U0 be
the simply connected and centerfree_sublattices. By Theorem 1.2, there exists a
W -trivial restriction S!- L. Using pullbacks establishes a commutaive diagr*
*am
of short exact sequences
0 --! LW ~=Z --! L0 --! S --! 0
flfl ? ?
fl ?y ?y
__
0 --! LW ~=Z --! L --! L --! 0 :
The top row describes an element of the group ExtZ^p[W](S; Z) of extensions. We
have the following sequence of isomorphisms:
ExtZ^p[W](S; Z)~= H1(W ; HomZ^p(S; Z))
~= H1(W; S* Z)
~= H1(W ; S*) Z
= 0 :
The first identity follows, because S and Z are free modules over Z^p[2, III; 2*
*.2],
the second from the isomorphism between the coefficients, the third because W
acts trivially on Z and because of Lemma 1.4, and the last because S* is center*
*free
(Proposition 4.1 and Lemma 2.1)._That is_to say that L0 ~=Z S. Moreover, we
have an isomorphism L=(Z S) ~=L =S ~= LW which shows that Z S!- L is a
W -trivial restriction. This proves part (1).
For the second statement we dualize the above argument. There exists a W -
trivial restriction Z* P *-! L* (Proposition 4.1 and part (1)). Dualizing ag*
*ain
gives a short exact sequence
0!- L!- Z P!- Ext(L*=(Z* P *); Z^p)!- 0 ;
which shows that the first arrow is a W -trivial restriction.
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[5]J.E. Humphries, Reflection groups and Coxeter groups, Cambridge University
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