Discrete complex reflection groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 12 May 2014

Notes and References

This is an excerpt of the lecture notes Discrete complex reflection groups by V.L. Popov. Lectures delivered at the Mathematical Institute, Rijksuniversiteit Utrecht, October 1980.

Several auxiliary results and the classification of irreducible infinite noncrystallographic complex r-groups

Now we move on to proofs. The greater part of these proofs relies heavily on the fact that for an infinite r-group W the subgroup TranW is sufficiently "massive".

The subgroup of translations

In order to clarify the last statement we need the following principal fact:

Theorem. Let WA(E) be an infinite r-group. Then TranW0.


Assume that TranW=0. Then Lin:WU(V) is a monomorphism.

It follows from the discreteness of W that (LinW)0 is a torus (here bar denotes the closure and 0 denotes the connected component of unity). (See [Bou1972], Ch. III §4, ex 13a.)

Set S=LinW(LinW)0.

Then LinWS and [LinW:S]<. Let also V0= {vV|(LinW)0v=v} and let V1 be the subspace of V determined by V=V0V1, V0V1. The subspaces V0 and V1 are S-invariant. Definitely, V10 (indeed, if not, then S=1, and hence |LinW|< which is a contradiction with |LinW|=|W|).

The set { P(LinW)0 |HP=V0 } is dense in (LinW)0. Therefore {PS|HP=V0} is dense in (LinW)0. Hence {PS|HP=V0} 0 (because S is dense in (LinW)0).

We claim that there exists a point aE with γ(a)-aV0 for every γW.

Consider an arbitrary point bE and an operator P0S such that HP0=V0. Take γ0W with Linγ0=P0. We have γ0(b)-b=t0+t1 where t0V0, t1V1. But the restriction of 1-P0 to V1 is nondegenerate. Hence there exists a vector tV1 such that (1-P0)t=t1. So, we have γ0(b+t)-(b+t)=(γ0(b)+P0t)-(b+t)=(γ0(b)-b)+(P0-1)t=t0+t1-t1=t0V0. Put a=b+t. We have γ0(a)-aV0. We shall prove that a is a point as wanted.

Let us first check that γ(a)-aV0 if γW, LinγS. Write P=Linγ, v0=γ0(a)-aV0 and v=γ(a)-a. We need to prove that vV0. But S is commutative, so PP0=P0P. Hence γγ0=γ0γ. We have now: κa(γ0)= (P0,v0)= κa(γγ0γ-1) =(P,v)(P0,v0) (P-1,-P-1v) = ( PP0P-1,-PP0 P-1v+Pv0+v ) =(P0,-P0v+Pv0+v). So, -P0v+Pv0+v=v0, i.e. (1-P0)v=(1-P)v0. But PS, hence (1-P)v0=0. It follows from Ker(1-P0)=V0 that vV0. This establishes the claim.

Now we can prove γ(a)-aV0 for arbitrary γW. Indeed, write P=Linγ, v=γ(a)-a, as before. We have: κa(γγ0γ-1)= ( PP0P-1, -PP0P-1v +Pv0+v ) . It follows from SLinW that V0 is P-invariant. Therefore we have Pv0V0. Moreover, if γ=γγ0γ-1, then LinγS and, by the claim, γ(a)-a=-PP0P-1v+Pv0+vV0. Therefore, -PP0P-1v+vV0 and hence -P0P-1v+P-1vP-1V0=V0, i.e. (1-P0)P-1vV0. But the image of 1-P0 is V1, hence (1-P0)P-1v=0. Therefore P-1vV0, i.e. vPV0=V0 and we are done.

Now we consider two subgroups of W: W, resp. W is the subgroup generated by those reflections γ for which Hγa, resp. Hγa.

The subgroup W is finite. Indeed, identifying A(E) and GL(V)V by means of κa, we have κa(γ)=(R,0)U(V) for any reflection γW and hence for any γW. So κa(W) is a discrete subgroup of a compact group U(V), hence finite.

We claim now that W is infinite. Before proving this, we shall show how to finish the proof of the theorem if the statement holds. Thus, assume W is an infinite r-group.

Note that TranW=0. Using the above arguments and constructions we obtain a torus (LinW)0, a subgroup S=LinW(LinW)0 and a decomposition V=V0V1, V0={vV|(LinW)0v=v}, V0V1. Also, we have V10 and {PS|HP=V0}. But WW, therefore LinWLinW, whence (LinW)0(LinW)0. So SS and V0V0. It follows now that γ(a)-aV0 and γ(a)-a0 for every γW.

Let γW be a reflection. Then κa(γ)=(Linγ,γ(a)-a), and in view of γ(a)-a0, we have: (root line of Linγ)=(γ(a)-a)V0. Therefore HLinγV1 and Linγ acts on V1 trivially. But W is generated by reflections; therefore LinW acts trivially on V1 which contradicts the existence of PSLinW such that HP=V0 and V10.

It remains to show that |W|=. Let us assume that this is not the case. Then W has a fixed point bE. By construction, ba. We shall show that W and W commute. This then yields that W=WW is a finite group (for W and W clearly generate W), which is a contradiction.

Let γW and γW be reflections. We have aHγ, bHγ. γ(b) γ(Hγ) a Hγ γ-1(b) b Hγ

But γγγ-1 is a reflection with mirror γ(Hγ).

Hence γγγ-1 is either in W or in W. If this element is in W, then γ(Hγ)a, i.e., Hγγ-1(a)=a, which is a contradiction. Therefore, γ(Hγ)b and Hγγ-1(b). We also have bHγ and bγ-1(b) (for otherwise bHγ which is absurd).

Consequently, there is a unique line on b and γ-1(b). This line lies in Hγ and is orthogonal to Hγ-1=Hγ. Hence HγHγ, i.e. γγ=γγ.

It follows from this theorem that TranW is "big enough":

Theorem. Let W be an infinite irreducible r-group. Then T=TranW is a lattice of rank n if k= and of rank n or 2n if k= (here n=dimkE).


Let k=. Then T is an invariant subspace of the LinW-module W. This subspace is nontrivial according to the previous theorem. Therefore T=V, because of irreducibility (see the theorem in 1.4), and the assertion follows from the equality rkT=dimT.

Let k=. As above, we have 0T=T+iT=V. Hence 2n=dimVdimT+dimiT=2rkT. Therefore rkTn. But TiT is a LinW-invariant complex subspace of V and hence either TiT=0 or TiT=V, i.e. T=V. If rkT=dimT>n then TiT0, hence T=V, i.e. rkT=2n.


1) |LinW|<;
2) if k= then W is a crystallographic group iff rkT=2n.


1) follows from the fact that LinW is contained in a compact group and has an invariant lattice of a full rank.

Some auxiliary results

Let KGL(V) be a finite r-group and be the set of mirrors of all reflections from K. Let H.

Then it is easy to see that the subgroup of K generated by all reflections RK with HR=H, is a cyclic group. Let m(H) be the order of this group.

Theorem. Let {Rj}jJ be a generating system of reflections of K such that the order of Rj is equal to m(HRj) for every jJ. Then each reflection RK is conjugate to Rj for certain j and lj.


Let 𝒪 be the K-orbit of HR in (it follows from PHR=HPRP-1 that K acts on ). Let χ𝒪 be the product of linear equations of the mirrors in 𝒪, normalized by the condition χ𝒪(1)=1. Then χ𝒪 is a character of K, i.e. a homomorphism of K into the multiplicative group of k. The group χ𝒪(K) is generated by χ𝒪(Rj), jJ. We have also χ𝒪(R)1. Therefore there exists a jJ with χ𝒪(Rj)1. But χ𝒪(Rj)1 iff 𝒪 is the orbit of HRj, see [Spr1977]. Therefore gHR=HRj for a certain gK and the statement follows.

Theorem. Let W be an r-group. Then for any reflection RLinW there exists a reflection γW with Linγ=R.


Let {ρj}jJ be the set of all reflections of W. Then {Rj=Linρj}jJ is a generating system of reflections of LinW. Let 𝒪 be the LinW-orbit of HR in . Then there exists a number l with HRl𝒪 and PlLinW such that PlHRl=HPlRlPl-1=HR. Let πlW be such that Linπl=Pl. We have:
Linπlρlπl-1=PlRlPl-1 and πlρlπl-1 is a reflect~on, too. Therefore χ𝒪(LinW) is generated by χ𝒪(Rj) where jJ={jJ|HRj=HR}. We have χ𝒪(R)=χ𝒪(Rj1)χ𝒪(Rjs)=χ𝒪(Rj1Rjs) for certain j1,,jsJ. But Rj1,,Rjs are the reflections whose mirrors are HR. It follows now (see [Spr1977]) that R=Rj1Rjs. Let us consider the element ρ=ρj1ρjs. We have Linρ=R and it easily follows from the parallelity of the mirrors Hρj1,,Hρjs that ρ is a reflection.

Semidirect products

Let W be a subgroup of A(E). We have 0TranWWLinW 1. When is W a semidirect product of LinW and TranW? We have the following criterion:

Theorem. Let |LinW|<. Then:

a) W is a semidirect product iff there exists a point aE such that Lin induces an isomorphism of the stabilizer Wa of a with LinW.
b) For every finite group KU(V) and every k-invariant subgroup T of V there exists a unique group WA(E) (up to equivalence) such that LinW=K, TranW=T and W is a semidirect product of LinW and TranW.


Proof is left to the reader.

The point a from part a) of this theorem is called a special point of W, see [Bou1968].

Using this theorem we can clarify the structure of infinite r-groups in a number of important cases:

Theorem. Let WA(E) be a group generated by reflections. Assume that |LinW|< and that LinW is an essential group (i.e. {vV|(LinW)v=v}={0}) generated by n=dimkE reflections. Then W is a semidirect product of LinW and TranW.


Let R1,,Rn be a generating system of reflections of LinW. Then there exists a reflection γjW such that Linγj=Rj, j=1,,n, see Section 3.2. We have j=1nHRj=0 because LinW is essential. Therefore j=1nHγj; more precisely, this intersection is a single point aE.

We have now that Lin:WaLinW is a surjective map, hence an a isomorphism. Thus we are done in view of the theorem above.

The conditions of this theorem are always fulfilled if k= and W is an infinite irreducible r-group. In general this is not the case if k=, though "in most cases", it is.

Classification of the irreducible infinite complex noncrystallographic r-groups.

Let k= and let W be a group as in the title. Set T=TranW. Then rkT=n=dimE, by 3.1.

T is a LinW-invariant -submodule of V. Let us consider the restriction of | to T. We claim that this restriction has only real values. Indeed, Re| defines an euclidean structure on T such that LinW is orthogonal with respect to this structure. But (T)=V because of irreducibility. Hence, there exists a canonical extension of Re|, determined up to a hermitian LinW-invariant scalar product, say (|), on V. Thus we have two Lin-invariant hermitian structures, | and (|), on V. They are proportional because of irreducibility of LinW:|=λ(|), λ. Taking restrictions to T, we have: λ=1. In other words, T is a real form of V and the restriction of the action of LinW to T gives an irreducible real finite r-group (and LinW itself is the complexification of this group). It follows from the classification that this group is generated by n reflections, see 1.5. Therefore LinW is generated by n reflections, too. It follows from the theorem above (section 3.3) that W is a semidirect product. Let aE be a special point of W. Then a+T is a real form of E. It is clear that a+T is W-invariant. The restriction of W to a+T is a real form of W. Therefore, this restriction is an affine Weyl group and W is its complexification. This completes the proof of the theorem in Section 2.2.

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