Assume that $\text{Tran}\hspace{0.17em}W=0\text{.}$ Then $\text{Lin}:W\to U\left(V\right)$ is a monomorphism.

It follows from the discreteness of $W$ that ${\left(\overline{\text{Lin}\hspace{0.17em}W}\right)}^0$ is a torus (here bar denotes the closure and $0$ denotes the connected component of unity). (See [Bou1972], Ch. III §4, ex 13a.)

Set $S=\text{Lin}\hspace{0.17em}W\cap {\left(\overline{\text{Lin}\hspace{0.17em}W}\right)}^0\text{.}$

Then $\text{Lin}\hspace{0.17em}W⊳S$ and $[\text{Lin}\hspace{0.17em}W:S]<\infty \text{.}$ Let also $$V_0=\{v\in V\hspace{0.17em}|\hspace{0.17em}{\left(\overline{\text{Lin}\hspace{0.17em}W}\right)}^{0}v=v\}$$ and let $V_1$ be the subspace of $V$ determined by $$V=V_0\oplus V_1,\hspace{0.17em}{V}_0\perp V_1\text{.}$$ The subspaces $V_0$ and $V_1$ are $S\text{-invariant.}$ Definitely, $V_1\ne 0$ (indeed, if not, then $S=1,$ and hence $\left|\text{Lin}\hspace{0.17em}W\right|<\infty$ which is a contradiction with $\left|\text{Lin}\hspace{0.17em}W\right|=\left|W\right|\text{).}$

The set $$\{P\in {\left(\overline{\text{Lin}\hspace{0.17em}W}\right)}^0\hspace{0.17em}|\hspace{0.17em}{H}_P=V_0\}$$ is dense in ${\left(\overline{\text{Lin}\hspace{0.17em}W}\right)}^0\text{.}$ Therefore $\{P\in S\hspace{0.17em}|\hspace{0.17em}{H}_P=V_0\}$ is dense in ${\left(\overline{\text{Lin}\hspace{0.17em}W}\right)}^0\text{.}$ Hence $$\{P\in S\hspace{0.17em}|\hspace{0.17em}{H}_P=V_0\}\ne 0$$ (because $S$ is dense in ${\left(\overline{\text{Lin}\hspace{0.17em}W}\right)}^0\text{).}$

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 $\text{Lin}\hspace{0.17em}{\gamma }_0=P_0\text{.}$ 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 $(1-P_0)t=t_1\text{.}$ So, we have ${\gamma }_0(b+t)-(b+t)=({\gamma }_0\left(b\right)+P_{0}t)-(b+t)=({\gamma }_0\left(b\right)-b)+(P_0-1)t=t_0+t_1-t_1=t_0\in V_0\text{.}$ 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 $\gamma \in W,$ $\text{Lin}\hspace{0.17em}\gamma \in S\text{.}$ Write $P=\text{Lin}\hspace{0.17em}\gamma ,$ $v_0={\gamma }_0\left(a\right)-a\in V_0$ 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: $$\begin{array}{c}{\kappa }_a\left({\gamma }_0\right)=(P_0,v_0)={\kappa }_a\left(\gamma {\gamma }_0{\gamma }^{-1}\right)=(P,v)(P_0,v_0)(P^{-1},-P^{-1}v)\\ =(P{P}_0{P}^{-1},-P{P}_0{P}^{-1}v+P{v}_0+v)=(P_0,-P_{0}v+P{v}_0+v)\text{.}\end{array}$$ So, $-P_{0}v+P{v}_0+v=v_0,$ i.e. $(1-P_0)v=(1-P)v_0\text{.}$ But $P\in S,$ hence $(1-P)v_0=0\text{.}$ It follows from $\text{Ker}(1-P_0)=V_0$ 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 $P=\text{Lin}\hspace{0.17em}\gamma ,$ $v=\gamma \left(a\right)-a,$ as before. We have: $${\kappa }_a\left(\gamma {\gamma }_0{\gamma }^{-1}\right)=(P{P}_0{P}^{-1},-P{P}_0{P}^{-1}v+P{v}_0+v)\text{.}$$ It follows from $S⊲\text{Lin}\hspace{0.17em}W$ 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 $\text{Lin}\hspace{0.17em}\gamma \prime \in S$ 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. $(1-P_0)P^{-1}v\in V_0\text{.}$ But the image of $1-P_0$ is $V_1,$ hence $(1-P_0)P^{-1}v=0\text{.}$ 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 ${\kappa }_a\left(\gamma \right)=(R,0)\in U\left(V\right)$ 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 $\text{Tran}\hspace{0.17em}W″=0\text{.}$ Using the above arguments and constructions we obtain a torus ${\left(\overline{\text{Lin}\hspace{0.17em}W″}\right)}^0,$ a subgroup $S″=\text{Lin}\hspace{0.17em}W″\cap {\left(\overline{\text{Lin}\hspace{0.17em}W″}\right)}^0$ and a decomposition $V=V_0^{″}\oplus V_1^{″},$ $V_0^{″}=\{v\in V\hspace{0.17em}|\hspace{0.17em}{\left(\overline{\text{Lin}\hspace{0.17em}W″}\right)}^{0}v=v\},$ $V_0^{″}\perp V_1^{″}\text{.}$ Also, we have $V_1^{″}\ne 0$ and $\{P\in S″\hspace{0.17em}|\hspace{0.17em}{H}_P=V_0^{″}\}\ne \varnothing \text{.}$ But $W″\subset W,$ therefore $\text{Lin}\hspace{0.17em}W″\subset \text{Lin}\hspace{0.17em}W,$ whence ${\left(\overline{\text{Lin}\hspace{0.17em}W″}\right)}^0\subset {\left(\overline{\text{Lin}\hspace{0.17em}W}\right)}^0\text{.}$ 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 ${\kappa }_a\left(\gamma \right)=(\text{Lin}\hspace{0.17em}\gamma ,\gamma \left(a\right)-a),$ and in view of $\gamma \left(a\right)-a\ne 0,$ we have: (root line of $\text{Lin}\hspace{0.17em}\gamma \text{)}=ℂ(\gamma \left(a\right)-a)\subset V_0^{″}\text{.}$ Therefore $H_{\text{Lin}\hspace{0.17em}\gamma }\supset V_1$ and $\text{Lin}\hspace{0.17em}\gamma$ acts on $V_1$ trivially. But $W″$ is generated by reflections; therefore $\text{Lin}\hspace{0.17em}W″$ acts trivially on $V_1$ which contradicts the existence of $P\in S″\subset \text{Lin}\hspace{0.17em}W″$ such that $H_P=V_0^{″}$ and $V_1^{″}\ne 0\text{.}$

It remains to show that $\left|W″\right|=\infty \text{.}$ 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 $a\in H_{\gamma \prime },$ $b\in H_{\gamma ″}\text{.}$ $$\begin{array}{c} γ′(b) γ′(Hγ″) a Hγ′ γ′-1(b) b Hγ″ \end{array}$$

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. $\gamma \prime \gamma ″=\gamma ″\gamma \prime \text{.}$

$\square$

Let $k=ℝ\text{.}$ Then $ℝT$ is an invariant subspace of the $\text{Lin}\hspace{0.17em}W\text{-module}$ $W\text{.}$ 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 $\text{rk}\hspace{0.17em}T=\text{dim}\hspace{0.17em}ℝT\text{.}$

Let $k=ℂ\text{.}$ As above, we have $0\ne ℂT=ℝT+iℝT=V\text{.}$ Hence $2n={\text{dim}}_{ℝ}V\le {\text{dim}}_{ℝ}ℝT+{\text{dim}}_{ℝ}iℝT=2\text{rk}\hspace{0.17em}T\text{.}$ Therefore $\text{rk}\hspace{0.17em}T\ge n\text{.}$ But $ℝT\cap iℝT$ is a $\text{Lin}\hspace{0.17em}W\text{-invariant}$ complex subspace of $V$ and hence either $ℝT\cap iℝT=0$ or $ℝT\cap iℝT=V,$ i.e. $ℝT=V\text{.}$ If $\text{rk}\hspace{0.17em}T=\text{dim}\hspace{0.17em}ℝT>n$ then $ℝT\cap iℝT\ne 0,$ hence $ℝT=V,$ i.e. $\text{rk}\hspace{0.17em}T=2n\text{.}$

$\square$

1) follows from the fact that $\text{Lin}\hspace{0.17em}W$ is contained in a compact group and has an invariant lattice of a full rank.

$\square$

Let $𝒪$ be the $K\text{-orbit}$ of $H_R$ in $\mathscr{H}$ (it follows from $P{H}_R=H_{PR{P}^{-1}}$ that $K$ acts on $\mathscr{H}\text{).}$ 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.

$\square$

Let ${\left\{{\rho }_j\right\}}_{j\in J}$ be the set of all reflections of $W\text{.}$ Then ${\{R_j=\text{Lin}\hspace{0.17em}{\rho }_j\}}_{j\in J}$ is a generating system of reflections of $\text{Lin}\hspace{0.17em}W\text{.}$ Let $𝒪$ be the $\text{Lin}\hspace{0.17em}W\text{-orbit}$ of $H_R$ in $\mathscr{H}\text{.}$ Then there exists a number $l$ with $H_{R_l}\in 𝒪$ and $P_l\in \text{Lin}\hspace{0.17em}W$ 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 $\text{Lin}\hspace{0.17em}{\pi }_l=P_l\text{.}$ We have:
$\text{Lin}\hspace{0.17em}{\pi }_l{\rho }_l{\pi }_l^{-1}=P_l{R}_l{P}_l^{-1}$ and ${\pi }_l{\rho }_l{\pi }_l^{-1}$ is a reflect~on, too. Therefore ${\chi }_{𝒪}\left(\text{Lin}\hspace{0.17em}W\right)$ is generated by ${\chi }_{𝒪}\left(R_j\right)$ where $j\in J\prime =\{j\in J\hspace{0.17em}|\hspace{0.17em}{H}_{R_j}=H_R\}\text{.}$ 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 $\text{Lin}\hspace{0.17em}\rho =R$ and it easily follows from the parallelity of the mirrors $H_{{\rho }_{j_1}},\dots ,H_{{\rho }_{j_s}}$ that $\rho$ is a reflection.

$\square$

Proof is left to the reader.

$\square$

Let $R_1,\dots ,R_n$ be a generating system of reflections of $\text{Lin}\hspace{0.17em}W\text{.}$ Then there exists a reflection ${\gamma }_j\in W$ such that $\text{Lin}\hspace{0.17em}{\gamma }_j=R_j,$ $j=1,\dots ,n,$ see Section 3.2. We have $\underset{j=1}{\overset{n}{\cap }}{H}_{R_j}=0$ because $\text{Lin}\hspace{0.17em}W$ 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 $\text{Lin}:W_a\to \text{Lin}\hspace{0.17em}W$ is a surjective map, hence an a isomorphism. Thus we are done in view of the theorem above.

$\square$