## Discrete complex reflection groups

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 $r\text{-groups}$

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

### The subgroup of translations

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

Theorem. Let $W\subset A\left(E\right)$ be an infinite $r\text{-group.}$ Then $\text{Tran} W\ne 0\text{.}$

 Proof. Assume that $\text{Tran} W=0\text{.}$ Then $\text{Lin}:W\to U\left(V\right)$ is a monomorphism. It follows from the discreteness of $W$ that ${\left(\stackrel{‾}{\text{Lin} W}\right)}^{0}$ is a torus (here bar denotes the closure and $0$ denotes the connected component of unity). (See [Bou1972], Ch. III §4, ex 13a.) Set $S=\text{Lin} W\cap {\left(\stackrel{‾}{\text{Lin} W}\right)}^{0}\text{.}$ Then $\text{Lin} W⊳S$ and $\left[\text{Lin} W:S\right]<\infty \text{.}$ Let also $V0= {v∈V | (Lin W‾)0v=v}$ and let ${V}_{1}$ be the subspace of $V$ determined by $V=V0⊕V1, V0⊥V1.$ The subspaces ${V}_{0}$ and ${V}_{1}$ are $S\text{-invariant.}$ Definitely, ${V}_{1}\ne 0$ (indeed, if not, then $S=1,$ and hence $|\text{Lin} W|<\infty$ which is a contradiction with $|\text{Lin} W|=|W|\text{).}$ The set ${ P∈(Lin W‾)0 | HP=V0 }$ is dense in ${\left(\stackrel{‾}{\text{Lin} W}\right)}^{0}\text{.}$ Therefore $\left\{P\in S | {H}_{P}={V}_{0}\right\}$ is dense in ${\left(\stackrel{‾}{\text{Lin} W}\right)}^{0}\text{.}$ Hence ${P∈S | HP=V0} ≠0$ (because $S$ is dense in ${\left(\stackrel{‾}{\text{Lin} W}\right)}^{0}\text{).}$ We claim that there exists a point $a\in E$ with $\gamma \left(a\right)-a\in {V}_{0}$ for every $\gamma \in W\text{.}$ Consider an arbitrary point $b\in E$ and an operator ${P}_{0}\in S$ such that ${H}_{{P}_{0}}={V}_{0}\text{.}$ Take ${\gamma }_{0}\in W$ with $\text{Lin} {\gamma }_{0}={P}_{0}\text{.}$ We have ${\gamma }_{0}\left(b\right)-b={t}_{0}+{t}_{1}$ where ${t}_{0}\in {V}_{0},$ ${t}_{1}\in {V}_{1}\text{.}$ But the restriction of $1-{P}_{0}$ to ${V}_{1}$ is nondegenerate. Hence there exists a vector $t\in {V}_{1}$ such that $\left(1-{P}_{0}\right)t={t}_{1}\text{.}$ So, we have ${\gamma }_{0}\left(b+t\right)-\left(b+t\right)=\left({\gamma }_{0}\left(b\right)+{P}_{0}t\right)-\left(b+t\right)=\left({\gamma }_{0}\left(b\right)-b\right)+\left({P}_{0}-1\right)t={t}_{0}+{t}_{1}-{t}_{1}={t}_{0}\in {V}_{0}\text{.}$ Put $a=b+t\text{.}$ We have ${\gamma }_{0}\left(a\right)-a\in {V}_{0}\text{.}$ We shall prove that $a$ is a point as wanted. Let us first check that $\gamma \left(a\right)-a\in {V}_{0}$ if $\gamma \in W,$ $\text{Lin} \gamma \in S\text{.}$ Write $P=\text{Lin} \gamma ,$ ${v}_{0}={\gamma }_{0}\left(a\right)-a\in {V}_{0}$ and $v=\gamma \left(a\right)-a\text{.}$ We need to prove that $v\in {V}_{0}\text{.}$ But $S$ is commutative, so $P{P}_{0}={P}_{0}P\text{.}$ Hence $\gamma {\gamma }_{0}={\gamma }_{0}\gamma \text{.}$ 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, $-{P}_{0}v+P{v}_{0}+v={v}_{0},$ i.e. $\left(1-{P}_{0}\right)v=\left(1-P\right){v}_{0}\text{.}$ But $P\in S,$ hence $\left(1-P\right){v}_{0}=0\text{.}$ It follows from $\text{Ker}\left(1-{P}_{0}\right)={V}_{0}$ that $v\in {V}_{0}\text{.}$ This establishes the claim. Now we can prove $\gamma \left(a\right)-a\in {V}_{0}$ for arbitrary $\gamma \in W\text{.}$ Indeed, write $P=\text{Lin} \gamma ,$ $v=\gamma \left(a\right)-a,$ as before. We have: $κa(γγ0γ-1)= ( PP0P-1, -PP0P-1v +Pv0+v ) .$ It follows from $S⊲\text{Lin} W$ that ${V}_{0}$ is $P\text{-invariant.}$ Therefore we have $P{v}_{0}\in {V}_{0}\text{.}$ Moreover, if $\gamma \prime =\gamma {\gamma }_{0}{\gamma }^{-1},$ then $\text{Lin} \gamma \prime \in S$ and, by the claim, $\gamma \prime \left(a\right)-a=-P{P}_{0}{P}^{-1}v+P{v}_{0}+v\in {V}_{0}\text{.}$ Therefore, $-P{P}_{0}{P}^{-1}v+v\in {V}_{0}$ and hence $-{P}_{0}{P}^{-1}v+{P}^{-1}v\in {P}^{-1}{V}_{0}={V}_{0},$ i.e. $\left(1-{P}_{0}\right){P}^{-1}v\in {V}_{0}\text{.}$ But the image of $1-{P}_{0}$ is ${V}_{1},$ hence $\left(1-{P}_{0}\right){P}^{-1}v=0\text{.}$ Therefore ${P}^{-1}v\in {V}_{0},$ i.e. $v\in P{V}_{0}={V}_{0}$ and we are done. Now we consider two subgroups of $W\text{:}$ $W\prime ,$ resp. $W″$ is the subgroup generated by those reflections $\gamma$ for which ${H}_{\gamma }\ni a,$ resp. ${H}_{\gamma }\not\ni a\text{.}$ The subgroup $W\prime$ is finite. Indeed, identifying $A\left(E\right)$ and $GL\left(V\right)V$ by means of ${\kappa }_{a},$ we have ${\kappa }_{a}\left(\gamma \right)=\left(R,0\right)\in U\left(V\right)$ for any reflection $\gamma \in W\prime$ and hence for any $\gamma \in W\prime \text{.}$ So ${\kappa }_{a}\left(W\prime \right)$ is a discrete subgroup of a compact group $U\left(V\right),$ 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 $r\text{-group.}$ Note that $\text{Tran} W″=0\text{.}$ Using the above arguments and constructions we obtain a torus ${\left(\stackrel{‾}{\text{Lin} W″}\right)}^{0},$ a subgroup $S″=\text{Lin} W″\cap {\left(\stackrel{‾}{\text{Lin} W″}\right)}^{0}$ and a decomposition $V={V}_{0}^{″}\oplus {V}_{1}^{″},$ ${V}_{0}^{″}=\left\{v\in V | {\left(\stackrel{‾}{\text{Lin} W″}\right)}^{0}v=v\right\},$ ${V}_{0}^{″}\perp {V}_{1}^{″}\text{.}$ Also, we have ${V}_{1}^{″}\ne 0$ and $\left\{P\in S″ | {H}_{P}={V}_{0}^{″}\right\}\ne \varnothing \text{.}$ But $W″\subset W,$ therefore $\text{Lin} W″\subset \text{Lin} W,$ whence ${\left(\stackrel{‾}{\text{Lin} W″}\right)}^{0}\subset {\left(\stackrel{‾}{\text{Lin} W}\right)}^{0}\text{.}$ So $S″\subset S$ and ${V}_{0}^{″}\supset {V}_{0}\text{.}$ It follows now that $\gamma \left(a\right)-a\in {V}_{0}^{″}$ and $\gamma \left(a\right)-a\ne 0$ for every $\gamma \in W″\text{.}$ Let $\gamma \in W″$ be a reflection. Then ${\kappa }_{a}\left(\gamma \right)=\left(\text{Lin} \gamma ,\gamma \left(a\right)-a\right),$ and in view of $\gamma \left(a\right)-a\ne 0,$ we have: (root line of $\text{Lin} \gamma \text{)}=ℂ\left(\gamma \left(a\right)-a\right)\subset {V}_{0}^{″}\text{.}$ Therefore ${H}_{\text{Lin} \gamma }\supset {V}_{1}$ and $\text{Lin} \gamma$ acts on ${V}_{1}$ trivially. But $W″$ is generated by reflections; therefore $\text{Lin} W″$ acts trivially on ${V}_{1}$ which contradicts the existence of $P\in S″\subset \text{Lin} W″$ such that ${H}_{P}={V}_{0}^{″}$ and ${V}_{1}^{″}\ne 0\text{.}$ It remains to show that $|W″|=\infty \text{.}$ Let us assume that this is not the case. Then $W″$ has a fixed point $b\in E\text{.}$ By construction, $b\ne a\text{.}$ We shall show that $W\prime$ and $W″$ commute. This then yields that $W=W\prime W″$ is a finite group (for $W\prime$ and $W″$ clearly generate $W\text{),}$ which is a contradiction. Let $\gamma \prime \in W\prime$ and $\gamma ″\in W″$ be reflections. We have $a\in {H}_{\gamma \prime },$ $b\in {H}_{\gamma ″}\text{.}$ $\gamma \prime \left(b\right) \gamma \prime \left({H}_{\gamma ″}\right) a {H}_{\gamma \prime } {\gamma \prime }^{-1}\left(b\right) b {H}_{\gamma ″}$ But $\gamma \prime \gamma ″{\gamma \prime }^{-1}$ is a reflection with mirror $\gamma \prime \left({H}_{\gamma ″}\right)\text{.}$ Hence $\gamma \prime \gamma ″{\gamma \prime }^{-1}$ is either in $W\prime$ or in $W″\text{.}$ If this element is in $W\prime ,$ then $\gamma \prime \left({H}_{\gamma ″}\right)\ni a,$ i.e., ${H}_{\gamma ″}\ni {\gamma \prime }^{-1}\left(a\right)=a,$ which is a contradiction. Therefore, $\gamma \prime \left({H}_{\gamma ″}\right)\ni b$ and ${H}_{\gamma ″}\ni {\gamma \prime }^{-1}\left(b\right)\text{.}$ We also have $b\in {H}_{\gamma ″}$ and $b\ne {\gamma \prime }^{-1}\left(b\right)$ (for otherwise $b\in {H}_{\gamma \prime }$ which is absurd). Consequently, there is a unique line on $b$ and ${\gamma \prime }^{-1}\left(b\right)\text{.}$ This line lies in ${H}_{\gamma \prime }$ and is orthogonal to ${H}_{{\gamma \prime }^{-1}}={H}_{\gamma \prime }\text{.}$ Hence ${H}_{\gamma ″}\perp {H}_{\gamma \prime },$ i.e. $\gamma \prime \gamma ″=\gamma ″\gamma \prime \text{.}$ $\square$

It follows from this theorem that $\text{Tran} W$ is "big enough":

Theorem. Let $W$ be an infinite irreducible $r\text{-group.}$ Then $T=\text{Tran} W$ is a lattice of rank $n$ if $k=ℝ$ and of rank $n$ or $2n$ if $k=ℂ$ (here $n={\text{dim}}_{k} E\text{).}$

 Proof. Let $k=ℝ\text{.}$ Then $ℝT$ is an invariant subspace of the $\text{Lin} W\text{-module}$ $W\text{.}$ This subspace is nontrivial according to the previous theorem. Therefore $ℝT=V,$ because of irreducibility (see the theorem in 1.4), and the assertion follows from the equality $\text{rk} T=\text{dim} ℝT\text{.}$ Let $k=ℂ\text{.}$ As above, we have $0\ne ℂT=ℝT+iℝT=V\text{.}$ Hence $2n={\text{dim}}_{ℝ}V\le {\text{dim}}_{ℝ}ℝT+{\text{dim}}_{ℝ}iℝT=2\text{rk} T\text{.}$ Therefore $\text{rk} T\ge n\text{.}$ But $ℝT\cap iℝT$ is a $\text{Lin} W\text{-invariant}$ complex subspace of $V$ and hence either $ℝT\cap iℝT=0$ or $ℝT\cap iℝT=V,$ i.e. $ℝT=V\text{.}$ If $\text{rk} T=\text{dim} ℝT>n$ then $ℝT\cap iℝT\ne 0,$ hence $ℝT=V,$ i.e. $\text{rk} T=2n\text{.}$ $\square$

Corollary.

 1) $|\text{Lin} W|<\infty \text{;}$ 2) if $k=ℂ$ then $W$ is a crystallographic group iff $\text{rk} T=2n\text{.}$

 Proof. 1) follows from the fact that $\text{Lin} W$ is contained in a compact group and has an invariant lattice of a full rank. $\square$

### Some auxiliary results

Let $K\subset GL\left(V\right)$ be a finite $r\text{-group}$ and $ℋ$ be the set of mirrors of all reflections from $K\text{.}$ Let $H\in ℋ\text{.}$

Then it is easy to see that the subgroup of $K$ generated by all reflections $R\in K$ with ${H}_{R}=H,$ is a cyclic group. Let $m\left(H\right)$ be the order of this group.

Theorem. Let ${\left\{{R}_{j}\right\}}_{j\in J}$ be a generating system of reflections of $K$ such that the order of ${R}_{j}$ is equal to $m\left({H}_{{R}_{j}}\right)$ for every $j\in J\text{.}$ Then each reflection $R\in K$ is conjugate to ${R}_{j}$ for certain $j$ and ${l}_{j}\text{.}$

 Proof. Let $𝒪$ be the $K\text{-orbit}$ of ${H}_{R}$ in $ℋ$ (it follows from $P{H}_{R}={H}_{PR{P}^{-1}}$ that $K$ acts on $ℋ\text{).}$ Let ${\chi }_{𝒪}$ be the product of linear equations of the mirrors in $𝒪,$ normalized by the condition ${\chi }_{𝒪}\left(1\right)=1\text{.}$ Then ${\chi }_{𝒪}$ is a character of $K,$ i.e. a homomorphism of $K$ into the multiplicative group of $k\text{.}$ The group ${\chi }_{𝒪}\left(K\right)$ is generated by ${\chi }_{𝒪}\left({R}_{j}\right),$ $j\in J\text{.}$ We have also ${\chi }_{𝒪}\left(R\right)\ne 1\text{.}$ Therefore there exists a $j\in J$ with ${\chi }_{𝒪}\left({R}_{j}\right)\ne 1\text{.}$ But ${\chi }_{𝒪}\left({R}_{j}\right)\ne 1$ iff $𝒪$ is the orbit of ${H}_{{R}_{j}},$ see [Spr1977]. Therefore $g{H}_{R}={H}_{{R}_{j}}$ for a certain $g\in K$ and the statement follows. $\square$

Theorem. Let $W$ be an $r\text{-group.}$ Then for any reflection $R\in \text{Lin} W$ there exists a reflection $\gamma \in W$ with $\text{Lin} \gamma =R\text{.}$

 Proof. Let ${\left\{{\rho }_{j}\right\}}_{j\in J}$ be the set of all reflections of $W\text{.}$ Then ${\left\{{R}_{j}=\text{Lin} {\rho }_{j}\right\}}_{j\in J}$ is a generating system of reflections of $\text{Lin} W\text{.}$ Let $𝒪$ be the $\text{Lin} W\text{-orbit}$ of ${H}_{R}$ in $ℋ\text{.}$ Then there exists a number $l$ with ${H}_{{R}_{l}}\in 𝒪$ and ${P}_{l}\in \text{Lin} W$ such that ${P}_{l}{H}_{{R}_{l}}={H}_{{P}_{l}{R}_{l}{P}_{l}^{-1}}={H}_{R}\text{.}$ Let ${\pi }_{l}\in W$ be such that $\text{Lin} {\pi }_{l}={P}_{l}\text{.}$ We have: $\text{Lin} {\pi }_{l}{\rho }_{l}{\pi }_{l}^{-1}={P}_{l}{R}_{l}{P}_{l}^{-1}$ and ${\pi }_{l}{\rho }_{l}{\pi }_{l}^{-1}$ is a reflect~on, too. Therefore ${\chi }_{𝒪}\left(\text{Lin} W\right)$ is generated by ${\chi }_{𝒪}\left({R}_{j}\right)$ where $j\in J\prime =\left\{j\in J | {H}_{{R}_{j}}={H}_{R}\right\}\text{.}$ We have ${\chi }_{𝒪}\left(R\right)={\chi }_{𝒪}\left({R}_{{j}_{1}}\right)\dots {\chi }_{𝒪}\left({R}_{{j}_{s}}\right)={\chi }_{𝒪}\left({R}_{{j}_{1}}\dots {R}_{{j}_{s}}\right)$ for certain ${j}_{1},\dots ,{j}_{s}\in J\prime \text{.}$ But ${R}_{{j}_{1}},\dots ,{R}_{{j}_{s}}$ are the reflections whose mirrors are ${H}_{R}\text{.}$ It follows now (see [Spr1977]) that $R={R}_{{j}_{1}}\dots {R}_{{j}_{s}}\text{.}$ Let us consider the element $\rho ={\rho }_{{j}_{1}}\dots {\rho }_{{j}_{s}}\text{.}$ We have $\text{Lin} \rho =R$ and it easily follows from the parallelity of the mirrors ${H}_{{\rho }_{{j}_{1}}},\dots ,{H}_{{\rho }_{{j}_{s}}}$ that $\rho$ is a reflection. $\square$

### Semidirect products

Let $W$ be a subgroup of $A\left(E\right)\text{.}$ We have $0→Tran W↪W→Lin W →1.$ When is $W$ a semidirect product of $\text{Lin} W$ and $\text{Tran} W\text{?}$ We have the following criterion:

Theorem. Let $|\text{Lin} W|<\infty \text{.}$ Then:

 a) $W$ is a semidirect product iff there exists a point $a\in E$ such that $\text{Lin}$ induces an isomorphism of the stabilizer ${W}_{a}$ of $a$ with $\text{Lin} W\text{.}$ b) For every finite group $K\subset U\left(V\right)$ and every $k\text{-invariant}$ subgroup $T$ of $V$ there exists a unique group $W\subset A\left(E\right)$ (up to equivalence) such that $\text{Lin} W=K,$ $\text{Tran} W=T$ and $W$ is a semidirect product of $\text{Lin} W$ and $\text{Tran} W\text{.}$

 Proof. Proof is left to the reader. $\square$

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

Using this theorem we can clarify the structure of infinite $r\text{-groups}$ in a number of important cases:

Theorem. Let $W\subset A\left(E\right)$ be a group generated by reflections. Assume that $|\text{Lin} W|<\infty$ and that $\text{Lin} W$ is an essential group (i.e. $\left\{v\in V | \left(\text{Lin} W\right)v=v\right\}=\left\{0\right\}\text{)}$ generated by $n={\text{dim}}_{k} E$ reflections. Then $W$ is a semidirect product of $\text{Lin} W$ and $\text{Tran} W\text{.}$

 Proof. Let ${R}_{1},\dots ,{R}_{n}$ be a generating system of reflections of $\text{Lin} W\text{.}$ Then there exists a reflection ${\gamma }_{j}\in W$ such that $\text{Lin} {\gamma }_{j}={R}_{j},$ $j=1,\dots ,n,$ see Section 3.2. We have $\underset{j=1}{\overset{n}{\cap }}{H}_{{R}_{j}}=0$ because $\text{Lin} W$ is essential. Therefore $\underset{j=1}{\overset{n}{\cap }}{H}_{{\gamma }_{j}}\ne \varnothing \text{;}$ more precisely, this intersection is a single point $a\in E\text{.}$ We have now that $\text{Lin}:{W}_{a}\to \text{Lin} W$ is a surjective map, hence an a isomorphism. Thus we are done in view of the theorem above. $\square$

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

### Classification of the irreducible infinite complex noncrystallographic $r\text{-groups.}$

Let $k=ℂ$ and let $W$ be a group as in the title. Set $T=\text{Tran} W\text{.}$ Then $\text{rk} T=n={\text{dim}}_{ℂ} E,$ by 3.1.

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