1 Introduction

In the standard approach to classification of complex semisimple Lie groups the problem is reduced to an equivalent problem of classification of root data. In other words, the root data (roots/co-roots and lattices of characters/co-characters of a maximal torus \(H\subset G\)) define the corresponding semisimple Lie group up to an isomorphism. Curtis, Wiederhold and Williams [5] demonstrated that for classification of compact connected semisimple Lie groups G it is enough to classify the normalizers \(N_G(H)\) of maximal tori \(H\subset G\). The normalizer provides information about the action of the Weyl group \(W_G:=N_G(H)/H\) on H, but this is not enough for classification, since one also needs the precise structure of the group extension of \(W_G\) by H. Thus, for the classification problem one might replace the original object, the semisimple Lie group G, by the group extension of the finite group \(W_G\) by the abelian Lie group H. One perspective to grasp this equivalence is to look at \(N_G(H)\) as a kind of degeneration of G [5]. An apparently related but more conceptual approach is based on attempts to look at \(N_G(H)\) as the Lie group G defined over some non-standard base field (akin to the mysterious field \(\mathbb {F}_1\) “with one element” introduced by Tits [11], probably with regard to this subject). In this way an equivalence of the two classification problems for compact semisimple Lie groups and normalizers looks like a manifestation of a general principle (due to Claude Chevalley [4]), saying that a classification of semisimple algebraic groups should not essentially depend on the nature of the base algebraically closed field.

The reasoning above demands a more detailed study of the group extension structure on \(N_G(H)\). The key fact is that this extension does not split in general [1, 5, 6, 9, 10]. To get a universal description of \(N_G(H)\) one should look for a set-theoretic section of the projection \(N_G(H)\rightarrow W_G\) generating an extension of \(W_G\). Such construction was proposed by Demazure [6] and Tits [9, 10]. It may be naturally formulated in terms of the Tits extension \(W^{\mathrm{T}}_G\) of Weyl group \(W_G\) by the subgroup of order 2 elements in H. This construction allows an explicit presentation of \(N_G(H)\) by generators and relations.

Although the Tits construction is known for a long time, its forthright explanation involves scheme-theoretic arguments. Precisely, for a Chevalley group scheme \(\mathscr {G}\) over \(\mathbb {Z}\) the Tits extension is the group of \(\mathbb {Z}\)-points of the normalizer of the \(\mathbb {Z}\)-split torus of \(\mathscr {G}\). In this paper we use set-theoretic arguments to explain the Tits construction in the case of complex reductive Lie groups (for recent discussion on Tits groups see e.g. [1, 7, 8]). After reminding general results on normalizers of maximal tori in Sect. 2 we revisit the Tits construction in Sect. 3. We stress that the Tits group construction is defined for the split real form \(G(\mathbb {R})\subset G\) of a complex semisimple group G. This enables us to present in Proposition 3.5 a simple purely topological description of the Tits extension of the Weyl group \(W_G\) (our considerations appear to be pretty close to the final section of [2]).

As a variation of the Tits construction we consider its analog for the maximal compact group \(U\subset G\). Precisely, we define an extension \(W_G^U\) of \(W_G\) embedded into a semi-direct product \({\widetilde{U}}=U\,{\rtimes }\, \Gamma \) of the maximal compact subgroup \(U\subset G\) and the Galois group \(\Gamma ={\mathrm{Gal}}\,(\mathbb {C}/\mathbb {R})\) acting in the standard (non-twisted) way on U. In this case the natural generators of the corresponding extension \(W_G^U\) of the Weyl group are squared to identity (in contrast with the case of the Tits group), while the non-trivial extension of \(W_G\) arises from the action of \(\Gamma \) via on \(W_G^U\) by outer automorphisms. The main result of this paper (Theorem 4.5 in Sect. 4) describes the structure of maximal tori normalizers in the Galois group extension \({\widetilde{U}}\) of the compact connected semisimple Lie group U. Note that our construction is dealing with the extended group \({\widetilde{U}}\) and thus, it differs from the constructions given in [1, 7]. In Sect. 5 we calculate explicitly the adjoint action of the Tits group and of its unitary analog on the Lie algebra \({\mathfrak g}={\mathrm{Lie}}\,(G)\). This action, in contrast with the adjoint action on \(\mathfrak {h}\subset {\mathfrak g}\), depends on the lift of \(W_G\) into G. Finally in Sect. 6 we provide details of the proof of Theorem 4.5.

2 Normalizers of maximal tori and Weyl groups

We start with recalling standard facts on normalizers of maximal tori and the associated Weyl groups. Let G be a complex connected semisimple Lie group, \(H\subset G\) be a maximal torus and \(N_G(H)\) be its normalizer in G. Then there is the following exact sequence:

$$\begin{aligned} 1\longrightarrow H\longrightarrow N_G(H)\xrightarrow { \ p \ } W_G\longrightarrow 1, \end{aligned}$$
(2.1)

where p is the projection on the finite group \(W_G:=N_G(H)/H\), the Weyl group of G. The Weyl group \(W_G\) does not actually depend on the choice of \(H\subset G\), and thus appears to be an invariant of G. Let \({\mathfrak g}:=\mathrm{Lie}(G)\) and let I be the set of vertices of the Dynkin diagram associated to G, where \(|I|=\mathrm{Lie}({\mathfrak g})\). Let \((\Delta ,\Delta ^\vee )\) be the root-coroot system corresponding to G, \(\{\alpha _i,i\in I\}\) be a set of positive simple roots, and \(\{\alpha _i^\vee \!,i\in I\}\) be the corresponding set of positive simple co-roots. Let \(\Vert a_{ij}\Vert \), \(a_{ij}=\langle \alpha _i^\vee \!,\alpha _j\rangle \), be the Cartan matrix of \((\Delta ,\Delta ^\vee )\). The Weyl group \(W_G\) has a simple presentation in terms of generators and relations. Precisely, \(W_G\) is generated by simple root reflections \(\{s_i,i\in I\}\) subjected to

$$\begin{aligned} s_i^2&=1, \end{aligned}$$
(2.2)
$$\begin{aligned} \underbrace{s_is_js_i\cdots }_{m_{ij}}&=\underbrace{s_js_is_j\cdots }_{m_{ij}},\quad i\ne j,\;\; i,j\in I, \end{aligned}$$
(2.3)

where \(m_{ij}=2,3,4,6\) for \(a_{ij}a_{ji}=0,1,2,3\), respectively. Equivalently these relations may be written in the Coxeter form:

$$\begin{aligned} s_i^2=1, \quad (s_is_j)^{m_{ij}}=1, \;\; i\ne j,\qquad i,j\in I. \end{aligned}$$

The exact sequence (2.1) defines the canonical action of \(W_G\) on H. Let \(h_i\in \mathfrak {h}={\mathrm{Lie}}\,(H)\) be the generators corresponding to the simple co-roots \({\alpha }_i^{\vee }\), then the \(W_G\)-action on \(\mathfrak {h}\subset \mathfrak {g}\) and on its dual is as follows:

$$\begin{aligned} \begin{aligned} s_i(h_j)&=h_j-\langle {\alpha }_i,{\alpha }_j^\vee \rangle \, h_i=h_j-a_{ji}h_i,\\ s_i({\alpha }_j)&={\alpha }_j-\langle {\alpha }_j,{\alpha }_i^\vee \rangle \, {\alpha }_i={\alpha }_j-a_{ij}{\alpha }_i. \end{aligned} \end{aligned}$$
(2.4)

The exact sequence (2.1) does not split in general, i.e. \(N_G(H)\) is not necessarily isomorphic to a semi-direct product of \(W_G\) and H. A delicate situation in this regard is described by the following result due to [1, 5].

Theorem 2.1

Assume G is a simple complex Lie group and let Z(G) be the center of G. Then the exact sequence (2.1) splits in the following and only the following cases:

  • Type \(A_{\ell }\), \(\ell \geqslant 1\), \(\ell \ne 3\), such that |Z(G)| is odd;

  • Type \(B_{\ell }\), \(\ell >1\), for the adjoint form;

  • Type \(D_{\ell }\), \(\ell >2\), for all forms except \({\mathrm{Spin}}\,(2\ell )\);

  • Type \(G_2\).

Thus to have an explicit description of the normalizer \(N_G(H)\) one should look for appropriate set-theoretic section of the projection map p in (2.1) generating some extension of \(W_G\). In the following section we provide the construction of the resulting extension of the Weyl group by a finite group. Let us note that for a normal finite subgroup \(Z\subset G\) one has: if (2.1) splits for G then it splits for G/Z as well. In the following, for simplicity, we consider only the case of simply-connected complex groups.

3 The Tits extension of Weyl group

To describe the extension (2.1) in terms of generators and relations Tits proposed the following extension \(W^{\mathrm{T}}_G\) of the Weyl group \(W_G\) by a discrete group [9, 10] (closely related results were obtained by Demazure [6]).

Definition 3.1

Let \(A=\Vert a_{ij}\Vert \) be the Cartan matrix corresponding to a semisimple Lie algebra \({\mathfrak g}=\mathrm{Lie}(G)\) and let \(m_{ij}=2,3,4,6\) for \(a_{ij}a_{ji}=0,1,2,3\), respectively. The Tits group \(W^{\mathrm{T}}_G\) is an extension of the Weyl group \(W_G\) by an abelian group \(\mathbb {Z}_2^{|I|}\) generated by \(\{\tau _i, \theta _i, i\in I\}\) subjected to the following relations:

$$\begin{aligned}&(\tau _i)^2=\theta _i,\quad \theta _i\theta _j=\theta _j\theta _i, \quad \theta _i^2=1, \end{aligned}$$
(3.1)
$$\begin{aligned}&\tau _i\theta _j=\theta _i^{-a_{ji}}\theta _j\tau _i, \end{aligned}$$
(3.2)
$$\begin{aligned}&\underbrace{\tau _i\tau _j\tau _i\cdots }_{m_{ij}}=\underbrace{\tau _j\tau _i\tau _j\cdots }_{m_{ij}},\quad i\ne j,\qquad i,j\in I, \end{aligned}$$
(3.3)

where the abelian subgroup is generated by \(\{\theta _i,i\in I\}\).

Let \(\{h_i,e_i,f_i\,{:}\, i\,{\in }\, I\}\) be the Chevalley–Serre generators of the Lie algebra \({\mathfrak g}=\mathrm{Lie}\,(G)\), satisfying the standard relations

$$\begin{aligned}&[h_i,e_j]=a_{ij}e_j,\quad [h_i,f_j]={}-a_{ij}f_j,\quad [e_i,f_j]=\delta _{ij}h_j, \end{aligned}$$
(3.4)
$$\begin{aligned}&\mathrm{ad}_{e_i}^{1-a_{ij}}(e_j)=0, \quad \mathrm{ad}_{f_i}^{1-a_{ij}}(f_j)=0, \end{aligned}$$
(3.5)

where \(A=\Vert a_{ij}\Vert \) is the Cartan matrix, i.e. \(a_{ij}=\langle {\alpha }_i^\vee \!,{\alpha }_j\rangle \).

According to [2] (see also [9]) there exists a subset \(\{\zeta _i,i\in I\}\subset H\) of canonical elements of order two satisfying the following relations:

$$\begin{aligned} s_i(\zeta _j)=\zeta _j\zeta _i^{-a_{ji}}, \end{aligned}$$

where \(s_i\), \(i\in I\), are the generators of the Weyl group \(W_G\) (2.2), (2.3).

Theorem 3.2

(Demazure–Tits) Let \(W^\mathrm{T}_G\) be the Tits group associated with the complex semisimple simply connected Lie group G, then the map

$$\begin{aligned} \tau _i\mapsto \dot{s}_i:=e^{f_i}\,e^{-e_i}\,e^{f_i}, \quad \theta _i\mapsto \zeta _i,\qquad i\in I, \end{aligned}$$
(3.6)

defines an embedding of the Tits group \(W^\mathrm{T}_G\) into \(N_G(H)\) such that \(p(W_G^\mathrm{T})=W_G\) for the projection p in (2.1). In particular, the normalizer group \(N_G(H)\) is generated by H and by the image of the Tits group, so that the following relations hold:

$$\begin{aligned} {\mathop {\mathrm{Ad}}}_{\dot{s}_i}(h)=s_i(h),\quad \text {for all}\;\; h\in \mathfrak {h}=\mathrm{Lie}(H), \;\; i\in I. \end{aligned}$$

Example 3.3

In the standard faithful two-dimensional representation \(\phi :\mathrm{SL}_2(\mathbb {C})\rightarrow \mathrm{End}\,(\mathbb {C}^2)\) given by (6.3) we have

$$\begin{aligned} \phi (\dot{s})= \begin{pmatrix} 0 &{} -1 \\ 1 &{} 0\end{pmatrix},\quad \phi (\dot{s})^2=\phi (\zeta )=\begin{pmatrix} -1 &{} 0 \\ 0 &{} -1\end{pmatrix}. \end{aligned}$$

The appearance of the Tits extension \(W_G^\mathrm{T}\) via a specific choice of a set-theoretic section looks a bit ad hoc. As it is mentioned in Introduction one may use a scheme-theoretic argument to support this particular choice of extension of \(W_G\). In the following we propose a set-theoretic argument based on consideration of the split real form \(G(\mathbb {R})\) of G to elucidate the construction of \(W_G^\mathrm{T}\). For the split real form \(G(\mathbb {R})\subset G\) there is an analog of (2.1):

$$\begin{aligned} 1\longrightarrow H(\mathbb {R})\longrightarrow N_{G(\mathbb {R})}(H(\mathbb {R})) \xrightarrow { \ p \ } W_G\longrightarrow 1, \end{aligned}$$
(3.7)

with the real split maximal torus given by the intersection

$$\begin{aligned} H(\mathbb {R})=H\cap G(\mathbb {R}), \end{aligned}$$

of the complex maximal torus with the split real subgroup. Note that the set-theoretic section of (3.7) defining \(W_G^\mathrm{T}\) provides a set-theoretic section of (2.1) thus embedding \(W_G^\mathrm{T}\) into \(N_{G(\mathbb {R})}(H(\mathbb {R}))\). The group \(H(\mathbb {R})\) allows the product decomposition

$$\begin{aligned} H(\mathbb {R})=MA, \quad M:=H(\mathbb {R})\cap K, \end{aligned}$$

where \(K\subset G(\mathbb {R})\) is a maximal compact subgroup of \(G(\mathbb {R})\), M is isomorphic to the group \(\mathbb {Z}_2^{|I|}\) and \(A\subset H(\mathbb {R})\) is the connected exponential group \(A=\exp \,(\mathfrak {h}(\mathbb {R}))\) without torsion. Therefore, \(H(\mathbb {R})\) is not connected and consists of \(2^{|I|}\) components, and the group M may be identified with the discrete group of connected components of \(H(\mathbb {R})\):

$$\begin{aligned} M=\pi _0(H(\mathbb {R})). \end{aligned}$$

Considering the groups of connected components of the topological groups entering (3.7) we obtain the induced exact sequence

$$\begin{aligned} 1\longrightarrow \pi _0(H(\mathbb {R}))\longrightarrow \pi _0(N_{G(\mathbb {R})}(H(\mathbb {R}))) \longrightarrow W_G\longrightarrow 1, \end{aligned}$$
(3.8)

so that \(|\pi _0(N_{G(\mathbb {R})}(H(\mathbb {R})))|=|M|\,{\cdot }\,|W_G|=|W_G^\mathrm{T}|\). This provides a canonical extension of \(W_G\) by the abelian group M of order \(2^{|I|}\).

Explicitly, the group of connected components may be identified with the quotients by the connected normal subgroup A

$$\begin{aligned} \pi _0(N_{G(\mathbb {R})}(H(\mathbb {R}))) \simeq N_{G(\mathbb {R})}(H(\mathbb {R}))/A, \end{aligned}$$

and we have the exact sequence

$$\begin{aligned} 1\longrightarrow A\longrightarrow N_{G(\mathbb {R})}(H(\mathbb {R})) \longrightarrow \pi _0(N_{G(\mathbb {R})}(H(\mathbb {R})))\longrightarrow 1. \end{aligned}$$
(3.9)

Lemma 3.4

The exact sequence (3.9) splits and thus \(\pi _0(N_{G(\mathbb {R})}(H(\mathbb {R})))\) allows an embedding into \(N_{G(\mathbb {R})}(H(\mathbb {R}))\).

Proof

The extension (3.9) is an instance of extensions of \(\pi _0(N_{G(\mathbb {R})}(H(\mathbb {R})))\) by A. Such extensions are classified by the group \(H^2(\pi _0(N_{G(\mathbb {R})}(H(\mathbb {R}))),A)\). The triviality of this group follows from the fact that \(\pi _0(N_{G(\mathbb {R})}(H(\mathbb {R})))\) is a finite group and A is an instance of abelian torsion-free group. By the standard cohomology argument (see e.g. [3, Chapter VI]), the second cohomology of any finite group with coefficients in a torsion-free abelian group is trivial. Thus the extension (3.9) is necessarily trivial and therefore the required embedding exists. \(\square \)

Up to now we have constructed a canonical extension of \(W_G\) given by (3.8). It is easy to see that this extension is isomorphic to the Tits group.

Proposition 3.5

The following isomorphism holds:

$$\begin{aligned} \pi _0(N_{G(\mathbb {R})}(H(\mathbb {R})))\simeq W^\mathrm{T}_G. \end{aligned}$$
(3.10)

Proof

Recall that the images \(\dot{s}_i,\zeta _i\), \(i\in I\), of the Tits generators belong to the maximally split real subgroup \(G(\mathbb {R})\subset G\). Thus we have the embedding \(W_G^\mathrm{T}\) into \(N_{G(\mathbb {R})}(H(\mathbb {R}))\) providing section of the exact sequence

$$\begin{aligned} 1\longrightarrow A\longrightarrow N_{G(\mathbb {R})}(H(\mathbb {R})) \longrightarrow W^\mathrm{T}_G\longrightarrow 1. \end{aligned}$$

Note that splitting of this exact sequence may be independently verified by using an argument similar to the one used in our proof of Lemma 3.4. Therefore the group \(N_{G(\mathbb {R})}(H(\mathbb {R}))\) allows a representation as semidirect product \(N_{G(\mathbb {R})}(H(\mathbb {R}))=A\,{\rtimes }\, W^\mathrm{T}_G\). By considering the connected components we deduce the assertion (3.10).\(\square \)

Example 3.6

For maximal split form \(\mathrm{SL}_2(\mathbb {R})\subset \mathrm{SL}_2(\mathbb {C})\) we have

$$\begin{aligned} H(\mathbb {R})&=\left\{ \begin{pmatrix} \lambda &{} 0\\ 0&{} \lambda ^{-1}\end{pmatrix},\, \lambda \in \mathbb {R}^*\right\} , \quad H(\mathbb {R})=MA, \\ A&=\left\{ \begin{pmatrix} \lambda &{} 0\\ 0&{} \lambda ^{-1}\end{pmatrix},\, \lambda \in \mathbb {R}^*_+\right\} ,\quad M=\left\{ \pm \mathrm{Id}\right\} . \end{aligned}$$

Elements \(g\in N_{\mathrm{SL}_2(\mathbb {R})}(H(\mathbb {R}))\) are defined by the condition that for each \(\lambda \in \mathbb {R}^*\) there exists \({\tilde{\lambda }}\in \mathbb {R}^*\) such that

$$\begin{aligned} g\begin{pmatrix}\lambda &{} 0 \\ 0 &{} \lambda ^{-1}\end{pmatrix} =\begin{pmatrix}{\tilde{\lambda }} &{} 0 \\ 0 &{} {\tilde{\lambda }}^{-1}\end{pmatrix}g, \quad g=\begin{pmatrix}a &{} b \\ c &{} d\end{pmatrix},\;\; ad-bc=1. \end{aligned}$$

One might check that the normalizer group \(N_{\mathrm{SL}_2(\mathbb {R})}(H(\mathbb {R}))\) is a union of two components

$$\begin{aligned} N_{\mathrm{SL}_2}(H(\mathbb {R}))=N_1\sqcup N_s, \end{aligned}$$

where \(N_1\) is a set of diagonal elements with \(c=b=0\), \(ad=1\ne 0\), and \(N_s\) is the set of anti-diagonal elements with \(a=d=0\), \(cb=-1\). Each of the co-sets \(N_1,N_s\) splits further into two connected components

$$\begin{aligned} N_1=N^+_1\sqcup N^-_1, \quad N_s=N^+_s\sqcup N^-_s, \end{aligned}$$

depending on the sign of the entries c, d in the last row of g.

The group \(\pi _0(N_{\mathrm{SL}_2(\mathbb {R})}(H(\mathbb {R})))\) is isomorphic to the quotient of \(N_{\mathrm{SL}_2(\mathbb {R})}(H(\mathbb {R}))\) by \(A\simeq \mathbb {R}^*_+\) and consists of four elements corresponding to the connected components \(N_1^{\pm }\), \(N_s^{\pm }\) of the group \(N_{\mathrm{SL}_2}(H(\mathbb {R}))\), allowing the following parameterization:

$$\begin{aligned} N_1^+=A,\quad N_s^+=\dot{s}A, \quad N_1^-=\theta A, \quad N_s^-=\theta \dot{s} A, \end{aligned}$$

where

$$\begin{aligned} \dot{s}=\begin{pmatrix} 0 &{} -1 \\ 1&{} 0\end{pmatrix}, \quad \theta =(\dot{s})^2 =\begin{pmatrix} -1 &{} 0 \\ 0&{}- 1 \end{pmatrix}, \quad \theta \dot{s} =\begin{pmatrix} 0 &{} 1\\ -1&{} 0 \end{pmatrix}. \end{aligned}$$

Clearly, the group \(\pi _0(N_{\mathrm{SL}_2(\mathbb {R})}(H(\mathbb {R})))\) is generated by representatives of the connected components \(N_{1,s}^\pm \), so indeed it is isomorphic to the order four cyclic group generated by \(\dot{s}\), in concordance with (3.10).

4 Weyl group and Galois extension of the compact real form

As we have demonstrated in the previous section the Tits group extension \(W^\mathrm{T}_G\) appears quite naturally if we consider the split real subgroup \(G(\mathbb {R})\subset G\). This motivates to look for analogs of the Tits construction associated with other real forms of G.

Let \(\{h_i, e_i, f_i ; i\in I\}\) be the Chevalley–Serre generators of the Lie algebra \({\mathfrak g}=\mathrm{Lie}(G)\), satisfying (3.4), (3.5). We fix the split real structure by assuming the generators \(\{h_i,-e_i,-f_i-; i\in I\}\) to be real. Let \(\top :{\mathfrak g}\rightarrow {\mathfrak g}\) be the Cartan anti-involution, associated with the split real structure:

$$\begin{aligned} \top :e_i\mapsto f_i,\quad f_i \mapsto e_i,\quad h_i \mapsto h_i,\quad \; i\in I. \end{aligned}$$
(4.1)

Let \(U\subset G\) be the connected compact real form of the Lie group G:

$$\begin{aligned} U=\{g\in G : g^\dagger g=1\}, \end{aligned}$$

where \(g\mapsto g^\dagger \) is the composition of the Cartan anti-involution (4.1) with the complex conjugation associated with the split real structure. The Galois group \(\Gamma :=\mathrm{Gal}(\mathbb {C}/\mathbb {R})\simeq \mathbb {Z}_2\) of the extension \(\mathbb {R}\subset \mathbb {C}\) is generated by \(\gamma \), \(\gamma ^2=1\). The group \(\Gamma \) acts both on G and on U by the complex conjugation, so let us introduce the following semidirect products:

$$\begin{aligned} {\widetilde{U}} :=U\,{\rtimes }\, \Gamma \subset {\widetilde{G}}:=G\,{\rtimes }\, \Gamma . \end{aligned}$$
(4.2)

Since the generators \(\{e_i,f_i,h_i ; i\in I\}\) are real, they are fixed by \(\gamma \in \Gamma \). Let us note that the \(\Gamma \)-fixed subgroup of U is a maximal compact subgroup \(K\subset G(\mathbb {R})\) of the split real form \(G(\mathbb {R})\).

Definition 4.1

Let \(\Vert a_{ij}\Vert \) be the Cartan matrix corresponding to a semisimple Lie algebra \({\mathfrak g}=\mathrm{Lie}(G)\). Let \(W^U_G\) be a group generated by \(\{\sigma _i,\bar{\sigma }_i;i\in I\}\) subjected to

$$\begin{aligned} \sigma _i^2&=\bar{\sigma }_i^2=1,\quad \sigma _i\bar{\sigma }_i=\bar{\sigma }_i\sigma _i, \end{aligned}$$
(4.3)
$$\begin{aligned} \sigma _j\sigma _i\sigma _j&=\bar{\sigma }_j\Pi ^{1-a_{ji}}(\bar{\sigma }_i)\,\bar{\sigma }_j,\quad i\ne j,\quad \; i,j\in I, \end{aligned}$$
(4.4)
$$\begin{aligned} \underbrace{\sigma _i\sigma _j\sigma _i\cdots }_{m_{ij}}&= \underbrace{\bar{\sigma }_j\bar{\sigma }_i\bar{\sigma }_j\cdots }_{m_{ij}}, \quad i\ne j,\quad \; i,j\in I, \end{aligned}$$
(4.5)

where in (4.5), \(m_{ij}=2,3,4,6\) for \(a_{ij}a_{ji}=0,1,2,3\), respectively. Here \(\Pi \) is the involutive map of the generators given by \(\Pi (\sigma _i)=\bar{\sigma }_i\), \(\Pi (\bar{\sigma }_i)=\sigma _i\), \(i\in I\).

Lemma 4.2

For all \(i,j\in I\) the following holds:

$$\begin{aligned} (\sigma _j\bar{\sigma }_j)(\sigma _i\bar{\sigma }_i)=(\sigma _i\bar{\sigma }_i)(\sigma _j\bar{\sigma }_j). \end{aligned}$$
(4.6)

Proof

For both \(a_{ji}\) and \(a_{ij}\) odd, (4.4) reads

$$\begin{aligned} (\sigma _j\bar{\sigma }_j)\,\sigma _i(\sigma _j\bar{\sigma }_j)^{-1}=\bar{\sigma }_i,\quad (\sigma _j\bar{\sigma }_j)\,\bar{\sigma }_i(\sigma _j\bar{\sigma }_j)^{-1}=\sigma _i, \end{aligned}$$

which yields \((\sigma _j\bar{\sigma }_j)(\sigma _i\bar{\sigma }_i)=(\sigma _i\bar{\sigma }_i)(\sigma _j\bar{\sigma }_j)\).

For \(a_{ji}\) even, we have two cases: \(a_{ji}=a_{ij}=0\) and \(a_{ji}=-2\), \(a_{ij}=-1\). In the former case (4.5) gives \(\sigma _i\sigma _j=\bar{\sigma }_j\bar{\sigma }_i\) and \(\sigma _j\sigma _i=\bar{\sigma }_i\bar{\sigma }_j\), which implies

$$\begin{aligned} (\sigma _j\bar{\sigma }_j)(\sigma _i\bar{\sigma }_i)=\sigma _j(\bar{\sigma }_j\bar{\sigma }_i)\,\sigma _i=(\sigma _j\sigma _i)(\sigma _j\sigma _i) =\bar{\sigma }_i(\bar{\sigma }_j\bar{\sigma }_i)\,\bar{\sigma }_j=(\bar{\sigma }_i\sigma _i)(\sigma _j\bar{\sigma }_j). \end{aligned}$$

In the case \(a_{ji}=-2\), \(a_{ij}=-1\) (4.4) gives \((\sigma _j\bar{\sigma }_j)\,\sigma _i=\sigma _i(\sigma _j\bar{\sigma }_j)\) and \((\sigma _i\bar{\sigma }_i)\,\sigma _j=\bar{\sigma }_j(\sigma _i\bar{\sigma }_i)\), so that the latter equality entails \(\bar{\sigma }_j(\sigma _i\bar{\sigma }_i)=(\sigma _i\bar{\sigma }_i)\,\sigma _j\). Thus we have

$$\begin{aligned} (\sigma _i\bar{\sigma }_i)(\sigma _j\bar{\sigma }_j)=\bar{\sigma }_j(\sigma _i\bar{\sigma }_i)\,\bar{\sigma }_j=(\sigma _j\bar{\sigma }_j)(\sigma _i\bar{\sigma }_i). \end{aligned}$$

This completes our proof. \(\square \)

Lemma 4.3

The map \(\Pi \) acting on the generators by

$$\begin{aligned} \Pi (\sigma _i)=\bar{\sigma }_i, \quad \Pi (\bar{\sigma }_i)=\sigma _i, \end{aligned}$$

extends to an involutive automorphism of the group \(W^U_G\).

Proof

Clearly, the relations (4.3), (4.5) are invariant under the action of \(\Pi \). The relation (4.4) transforms under the \(\Pi \)-action into

$$\begin{aligned} \bar{\sigma }_j\bar{\sigma }_i\bar{\sigma }_j=\sigma _j\Pi ^{1-a_{ji}}(\sigma _i)\,\sigma _j,\quad i\ne j, \;\; i,j\in I, \end{aligned}$$

which may be equivalently written as follows:

$$\begin{aligned} (\sigma _j\bar{\sigma }_j)\,\bar{\sigma }_i(\sigma _j\bar{\sigma }_j)^{-1}=\Pi ^{-a_{ji}}(\bar{\sigma }_i),\quad i\ne j, \;\; i,j\in I. \end{aligned}$$
(4.7)

For \(a_{ji}\) odd (4.7) reads

$$\begin{aligned} (\sigma _j\bar{\sigma }_j)\,\bar{\sigma }_i(\sigma _j\bar{\sigma }_j)^{-1}=\sigma _i, \end{aligned}$$

which follows from (4.4) in the form \((\sigma _j\bar{\sigma }_j)\,\sigma _i(\sigma _j\bar{\sigma }_j)^{-1}=\bar{\sigma }_i\).

For \(a_{ji}\) even we have to prove (4.7), which reads

$$\begin{aligned} (\sigma _j\bar{\sigma }_j)\,\bar{\sigma }_i(\sigma _j\bar{\sigma }_j)^{-1}=\bar{\sigma }_i. \end{aligned}$$

This follows from (4.4), \((\sigma _j\bar{\sigma }_j)\,\sigma _i(\sigma _j\bar{\sigma }_j)^{-1}=\sigma _i\), by multiplying both sides by \(\sigma _i\bar{\sigma }_i\) and using (4.6). \(\square \)

Proposition 4.4

The group \(W^U_G\) is given by the following extension:

$$\begin{aligned} 1\longrightarrow \mathbb {Z}_2^{|I|}\longrightarrow W^U_G\longrightarrow W_G \longrightarrow 1, \end{aligned}$$
(4.8)

of the Weyl group \(W_G\) by the abelian group \(\mathbb {Z}_2^{|I|}\subset W_G^U\).

Proof

Introduce the elements \(\eta _i:=\sigma _i\bar{\sigma }_i\), \(i\in I\). They have order two, and they pairwise commute by (4.6):

$$\begin{aligned} \eta _i^2=1, \quad \eta _i\eta _j=\eta _j\eta _i,\quad \; i,j\in I, \end{aligned}$$

and are invariant under involution \(\Pi \). Consider a \(\Pi \)-stable subgroup \(H_\eta \subset W^U_G\) generated by \(\{\eta _i,i\in I\}\). The relation (4.4) may be equivalently written in the following form:

$$\begin{aligned} \sigma _i\eta _j=\eta _j\eta _i^{-a_{ji}}\sigma _i,\quad \bar{\sigma }_i\eta _j=\eta _j\eta _i^{-a_{ji}}\bar{\sigma }_i. \end{aligned}$$
(4.9)

Indeed, from (4.4) we have \(\sigma _i\eta _j=\eta _j\Pi ^{1-a_{ji}}(\bar{\sigma }_i)\), which for \(a_{ji}\) even reads \(\sigma _i\eta _j=\eta _j \sigma _i \) and for \(a_{ji}\) odd (4.4) is equivalent to \(\sigma _i\eta _j=\eta _j\bar{\sigma }_i=\eta _j\eta _i\sigma _i\). This implies the first equation in (4.9). The second relation in (4.9) is obtained by applying the automorphism \(\Pi \) to the first one. The identities (4.9) yield that the subgroup \(H_\eta \subset W^U_G\) generated by \(\eta _i\), \(i\in I\), is normal. A proof of the fact that \(|H_\eta |=2^{|I|}\) is given in Lemma 6.4 below.

Next, we consider the quotient group \(W_G^U/\mathbb {Z}_2^{|I|}\). It is generated by \(s_i:=\pi (\sigma _i)\), \(i\in I\), and satisfying the standard relations (2.2), (2.3) of the group \(W_G\). Indeed, \(\pi (\sigma _i)=\pi (\bar{\sigma }_i)\) implies that the relations (4.3) are mapped to the relations (2.2), relations (4.4) equivalent to (4.9) become identities, and the braid relations (4.5) of \(W_G^U\) are mapped to the braid relations (2.3) of \(W_G\). Thus, we have a surjective homomorphism \(\pi \)

$$\begin{aligned} \pi :W_G^U\rightarrow W_G,\quad \eta _i\mapsto 1,\;\; i\in I. \end{aligned}$$

This gives the exact sequence (4.8). \(\square \)

The following analog of Theorem 3.2 holds.

Theorem 4.5

Let \(U\subset G\) be a maximal compact subgroup of the complex semisimple simply connected Lie group G. Let \((\Delta ,W_G)\) be the root system of \({\mathfrak g}=\mathrm{Lie}(G)\) with the Cartan matrix \(\Vert a_{ij}\Vert \). Let \(\gamma \) be the generator of the Galois group \(\Gamma =\mathrm{Gal}(\mathbb {C}/\mathbb {R})\) of the field extension \(\mathbb {R}\subset \mathbb {C}\) and let \(\imath \in \mathbb {C}\) be the imaginary unit. Then the following map:

$$\begin{aligned} \sigma _i\mapsto \varsigma _i:=e^{\frac{\imath \pi }{2}(e_i+f_i)} \,\gamma ,\quad \bar{\sigma }_i\mapsto \bar{\varsigma }_i:=e^{-\frac{\imath \pi }{2}(e_i+f_i)} \,\gamma ,\qquad i\in I, \end{aligned}$$
(4.10)

defines an injective homomorphism \(W^U_G\rightarrow {\widetilde{U}}\), with \({\widetilde{U}}\) given by (4.2). The elements \(\varsigma _i\), \(i\in I\), and \(\gamma \) together with the maximal torus H generate the group \(N_G(H)\,{\rtimes }\, \Gamma \).

We give a proof of Theorem 4.5 in Sect. 6 below.

Let us stress a clear analogy between the constructions of \(W_G^\mathrm{T}\) and \(W_G^U\). On the one hand, in the Tits setting the finite group is embedded into the maximal compact subgroup \(K\subset G(\mathbb {R})\) of the maximally split real form \(G(\mathbb {R})\subset G\). On the other hand, the extension \(W^U_G\) constructed above is embedded into Galois group extension \({\widetilde{U}}\) of maximal compact subgroup \(U\subset G\) (note that the action of complex conjugation on U may be equivalently represented by the action of the Cartan anti-involution (4.1)). In this way \({\widetilde{U}}\) plays the role analogous to \(K\subset G(\mathbb {R})\) in the Tits construction, while \(W^U_G\) looks like a “complex” analog of the finite group \(W^\mathrm{T}_G\), where the relations \(\tau _i^2=\theta _i\) are replaced by \(\sigma _i\bar{\sigma }_i=\eta _i\). Let us note that for each \(i\in I\) we have the cyclic subgroup \(\langle \dot{s}_i\,{:}\, (\dot{s}_i)^4=1\rangle =\mathbb {Z}_4\subset W_G^\mathrm{T}\) in the Tits construction. Meanwhile in the case of \(W_G^U\) for each \(i\in I\) the group \(\langle \sigma _i,\bar{\sigma }_i\,{:}\,\sigma _i^2=\bar{\sigma }_i^2=(\sigma _i\bar{\sigma }_i)^2=1\rangle =(\mathbb {Z}_2)^2\subset W_G^U\) appears. Note that these two instances exhaust possible extensions of \(\mathbb {Z}_2\) by \(\mathbb {Z}_2\). It is natural to expect that with other real forms of the complex group G one can associate appropriate extensions of the Weyl group \(W_G\). These extensions presumably would be combinations of both constructions considered above.

5 Adjoint action of the extended Weyl groups

While the action of \(W_G\) on the maximal commutative subalgebra \(\mathfrak {h}=\mathrm{Lie}(H)\) is defined canonically (2.4) and does not depend on a lift of \(W_G\) into \(N_G(H)\) its action on the whole Lie algebra \({\mathfrak g}=\mathrm{Lie}(G)\) does depend on the lift. Above we have considered two extensions of the Weyl group \(W_G\) together with their homomorphisms into the corresponding Lie group. Here we describe their induced adjoint actions on \({\mathfrak g}\).

Proposition 5.1

The adjoint action of the Tits group \(W_G^\mathrm{T}\) on the Lie algebra \({\mathfrak g}=\mathrm{Lie}(G)\) via homomorphism (3.6) is given by

$$\begin{aligned}&{\mathop {\mathrm{Ad}}}_{\dot{s}_i}(e_i)={}-f_i, \quad {\mathop {\mathrm{Ad}}}_{\dot{s}_i}(f_i)={}-e_i , \end{aligned}$$
(5.1)
$$\begin{aligned}&{\mathop {\mathrm{Ad}}}_{\dot{s}_i}(e_j)=e_j, \qquad {\mathop {\mathrm{Ad}}}_{\dot{s}_i}(f_j)=f_j,\quad a_{ij}=0, \end{aligned}$$
(5.2)
$$\begin{aligned}&{\mathop {\mathrm{Ad}}}_{\dot{s}_i}(e_j)=\frac{(-1)^{|a_{ij}|}}{|a_{ij}|!} \underbrace{\bigl [e_i,[\ldots [e_i}_{|a_{ij}|},\,e_j]\ldots ]\bigr ],\nonumber \\ {\mathop {\mathrm{Ad}}}_{\dot{s}_i}(f_j)&=\frac{1}{|a_{ij}|!}\underbrace{\bigl [f_i,[\ldots [f_i}_{|a_{ij}|},\,f_j] \ldots ]\bigr ],\quad i\ne j. \end{aligned}$$
(5.3)

Proof

Relations (5.1) are actually relations for \(\mathfrak {sl}_2\) Lie subalgebras generated by \(\{e_i,h_i,f_i,i\in I\}\) and may easily be checked using for example the standard faithful representation (6.3). Relations (5.2) trivially follow from the Lie algebra relations (3.4). Thus we need to prove (5.3). Let us introduce the following notation: \(\dot{s}_i(a):={\mathop {\mathrm{Ad}}}_{\dot{s}_i}(a)\). Then for the conjugated generators we have

$$\begin{aligned}{}[h_k,\dot{s}_i(e_j)]&=\dot{s}_i([h_{s_i(k)},e_j])=\langle s_i(\alpha _k^{\vee }),\alpha _j\rangle \, \dot{s}_i(e_j)=(a_{kj}-a_{ki}a_{ij})\,\dot{s}_i(e_j),\\ [h_k,\dot{s}_i(f_j)]&=\dot{s}_i([h_{s_i(k)},f_j])={}-\langle s_i(\alpha _k^{\vee }),\alpha _j\rangle \, \dot{s}_i(f_j)={}-(a_{kj}-a_{ki}a_{ij})\,\dot{s}_i(f_j). \end{aligned}$$

These relations fix the r.h.s. of (5.3) up to coefficients. Let us calculate the coefficients by taking into account only the terms of the right weights. We have

$$\begin{aligned} \dot{s}_i(e_j)=e^{f_i}e^{-e_i}e^{f_i}\,e_j\,e^{-f_i}e^{e_i}e^{-f_i}= \frac{(-1)^{|a_{ij}|}}{|a_{ij}|!}\, \bigl (\mathrm{Ad}_{e^{f_i}e_ie^{-f_i}}^{|a_{ij}|}(e_j)\bigr )+\cdots , \end{aligned}$$

where we have used the Serre relations (3.5) and denote by \(\cdots \) the terms of the “wrong” weight. Taking into account

$$\begin{aligned} {\mathop {\mathrm{Ad}}}_{e^{f_i}}(e_i)=e_i+\cdots , \end{aligned}$$

we obtain the first relation in (5.3). The second relation is obtained quite similarly using the following equality (for a proof see Lemma 6.1):

$$\begin{aligned} e^{f_i}e^{-e_i}e^{f_i}=e^{-e_i}e^{f_i}e^{-e_i}. \end{aligned}$$

In this case we have

$$\begin{aligned} \dot{s}_i(f_j)=e^{-e_i}e^{f_i}e^{-e_i}f_j\,e^{e_i}e^{-f_i}e^{e_i}= \frac{1}{|a_{ij}|!}\,\bigl (\mathrm{Ad}_{e^{-e_i}f_ie^{e_i}}^{|a_{ij}|}(f_j)\bigr )+ \cdots . \end{aligned}$$

Taking into account

$$\begin{aligned} {\mathop {\mathrm{Ad}}}_{e^{-e_i}}(f_i)=f_i+\cdots , \end{aligned}$$

we obtain the second relation in (5.3). \(\square \)

Let us stress that there is a simple way to get rid of sign factors in (5.1) and (5.3). Define a new set of generators \({\tilde{e}}_i=-e_i\), \({\tilde{f}}_i=f_i\). Then we have

$$\begin{aligned} {\mathop {\mathrm{Ad}}}_{\dot{s}_i}({\tilde{e}}_i)&={\tilde{f}}_i, \quad {\mathop {\mathrm{Ad}}}_{\dot{s}_i}({\tilde{f}}_i)={\tilde{e}}_i, \\ {\mathop {\mathrm{Ad}}}_{\dot{s}_i}({\tilde{e}}_j)&={\tilde{e}}_j, \quad {\mathop {\mathrm{Ad}}}_{\dot{s}_i}({\tilde{f}}_j)={\tilde{f}}_j,\quad a_{ij}=0, \\ {\mathop {\mathrm{Ad}}}_{\dot{s}_i}({\tilde{e}}_j)&=\frac{1}{|a_{ij}|!} \underbrace{\bigl [{\tilde{e}}_i,[\ldots [{\tilde{e}}_i}_{|a_{ij}|},\,{\tilde{e}}_j] \ldots ]\bigr ],\\ {\mathop {\mathrm{Ad}}}_{\dot{s}_i}({\tilde{f}}_j)&=\frac{1}{|a_{ij}|!}\underbrace{\bigl [{\tilde{f}}_i,[ \ldots [{\tilde{f}}_i}_{|a_{ij}|},\,{\tilde{f}}_j] \ldots ]\bigr ],\quad i\ne j. \end{aligned}$$

Now we describe the action on \({\mathfrak g}\) of the Weyl group extension \(W_G^U\) introduced in Sect. 4. It is convenient to express it in terms of purely imaginary generators \(\imath e_i\), \(\imath f_i\), \(i\in I\).

Proposition 5.2

The elements of the group \(W^U_G\) act on the Lie algebra \({\mathfrak g}=\mathrm{Lie}(G)\) via homomorphism (4.10) as follows:

$$\begin{aligned} {\mathop {\mathrm{Ad}}}_{\varsigma _i}(\imath e_i)={}-\imath f_i,\quad {\mathop {\mathrm{Ad}}}_{\varsigma _i}(\imath f_i)={}-\imath e_i, \end{aligned}$$
(5.4)

and

$$\begin{aligned} {\mathop {\mathrm{Ad}}}_{\varsigma _i}(\imath e_j)&={}-\imath e_j, \quad {\mathop {\mathrm{Ad}}}_{\varsigma _i}(\imath f_j)={}-\imath f_j,\quad a_{ij}=0, \end{aligned}$$
(5.5)
$$\begin{aligned} {\mathop {\mathrm{Ad}}}_{\varsigma _i}(\imath e_j)&={}- \frac{1}{|a_{ij}|!}\, \underbrace{\bigl [\imath e_i,[\ldots [\imath e_i}_{|a_{ij}|},\, \imath e_j]\ldots ]\bigr ], \nonumber \\ {\mathop {\mathrm{Ad}}}_{\varsigma _i}(\imath f_j)&={}-\frac{1}{|a_{ij}|!}\, \underbrace{\bigl [\imath f_i,[\ldots [\imath f_i}_{|a_{ij}|},\, \imath f_j]\ldots ]\bigr ],\quad i\ne j. \end{aligned}$$
(5.6)

Proof

Taking into account (3.4) we have

$$\begin{aligned} e^{\imath \pi t\, \mathrm{ad}_{h_i}}(e_j)=e_j e^{\imath \pi ta_{ij}} , \quad e^{\imath \pi t\, \mathrm{ad}_{h_i}} (f_j) =f_j e^{-\imath \pi ta_{ij}} . \end{aligned}$$

Next, we use the following representation for \(\varsigma _i\) (see Lemma 6.1 for details):

$$\begin{aligned} \varsigma _i=\dot{s}_i e^{\frac{\imath \pi }{2}h_i}\gamma ,\quad i\in I, \end{aligned}$$

and Proposition 5.1 to obtain (5.6) and (5.4). \(\square \)

6 Proof of Theorem 4.5

We start with establishing an explicit relation between the generators \(\varsigma _i,\bar{\varsigma }_i\) (4.10) and the Tits generators \(\dot{s}_i\).

Lemma 6.1

For each \(i\in I\) the following identities hold:

$$\begin{aligned} \dot{s}_i:=e^{f_i}e^{-e_i}e^{f_i}=e^{-e_i}e^{f_i}e^{-e_i} = e^{\frac{\imath \pi }{4}h_i} e^{\frac{\imath \pi }{2}(e_i+f_i)} e^{-\frac{\imath \pi }{4}h_i},\;\; \dot{s}_i^2=e^{\imath \pi h_i}. \end{aligned}$$
(6.1)

Thus the generators \(\{\varsigma _i,\bar{\varsigma }_i,i\in I\}\) defined by (4.10) may be represented as follows:

$$\begin{aligned} \begin{aligned} \varsigma _i&=e^{-\frac{\imath \pi }{2}h_i}\dot{s}_i\gamma =e^{-\frac{\imath \pi }{4}h_i} \dot{s}_i\,e^{\frac{\imath \pi }{4}h_i}\gamma ,\\ \bar{\varsigma }_i&=e^{\frac{\imath \pi }{2}h_i}\dot{s}_i \gamma =e^{\frac{\imath \pi }{4}h_i} \dot{s}_i e^{-\frac{\imath \pi }{4}h_i}\gamma ,\\ \varsigma _i\bar{\varsigma }_i&=\dot{s}_i^2=e^{\imath \pi h_i}. \end{aligned} \end{aligned}$$
(6.2)

Proof

The identities (6.1) follow from the corresponding relations in \(\mathrm{SL}_2\subset G\), using the standard faithful two-dimensional representation \(\phi :\mathrm{SL}_2\rightarrow \mathrm{End}(\mathbb {C}^2)\),

$$\begin{aligned} \phi (e)=\begin{pmatrix} 0 &{}1 \\ 0&{} 0 \end{pmatrix}, \quad \phi (f)=\begin{pmatrix} 0 &{}0 \\ 1&{}0 \end{pmatrix}, \quad \phi (h)=\begin{pmatrix} 1 &{}0 \\ 0&{}-1 \end{pmatrix}. \end{aligned}$$
(6.3)

Direct calculations show that

$$\begin{aligned} \phi (\dot{s})&=\phi (e^{f}e^{-e}e^{f})=\phi ( e^{-e}e^{f}e^{-e})= \begin{pmatrix} 0&{} -1 \\ 1 &{} 0 \end{pmatrix}, \\ \phi (\dot{s}^2)&=\phi ( e^{-e}e^{f}e^{-e}e^{-e}e^{f}e^{-e})= \begin{pmatrix} -1&{} 0 \\ 0 &{} -1 \end{pmatrix} =\phi (e^{\imath \pi h}), \\ \phi \bigl (e^{-\frac{\imath \pi }{4}h}\dot{s}e^{\frac{\imath \pi }{4}h}\bigr )&= \phi \bigl (e^{\frac{\imath \pi }{2}(e+f)}\bigr )= \begin{pmatrix} 0&{} \imath \\ \imath &{} 0 \end{pmatrix},\quad \phi \bigl (e^{\frac{-\imath \pi }{2}(e+f)}\bigr )= \begin{pmatrix} 0&{} -\imath \\ -\imath &{} 0 \end{pmatrix}. \end{aligned}$$

Then (6.1), (6.2) follow from the faithfulness of \(\phi \). \(\square \)

Lemma 6.2

The following relations hold:

$$\begin{aligned} \varsigma _i^2=\bar{\varsigma }_i^2=1, \quad i\in I. \end{aligned}$$
(6.4)

Proof

Direct calculation gives

$$\begin{aligned} \varsigma _i^2&= e^{-\frac{\imath \pi }{4}h_i} \dot{s}_i e^{\frac{\imath \pi }{4}h_i}\gamma e^{-\frac{\imath \pi }{4}h_i} \dot{s}_i e^{\frac{\imath \pi }{4}h_i}\gamma \\ {}&= e^{-\frac{\imath \pi }{4}h_i} \dot{s}_i e^{\frac{\imath \pi }{4}h_i} e^{\frac{\imath \pi }{4}h_i} \dot{s}_i e^{-\frac{\imath \pi }{4}h_i}\\ {}&= e^{-\frac{\imath \pi }{4}h_i} \dot{s}_i e^{\frac{\imath \pi }{2}h_i} \dot{s}_i e^{-\frac{\imath \pi }{4}h_i} \\&= e^{-\frac{\imath \pi }{4}h_i} e^{-\frac{\imath \pi }{2}h_i}e^{-\frac{\imath \pi }{4}h_i} (\dot{s}_i)^2=e^{-\imath \pi h_i}\,{\cdot }\, e^{\imath \pi h_i}=1. \end{aligned}$$

The identity \(\bar{\varsigma }_i^2=1\) follows from \(\bar{\varsigma }_i={\gamma }\varsigma _i{\gamma }\), \(i\in I\). \(\square \)

Now let us verify that the generators \(\varsigma _i,\bar{\varsigma }_i\) and \(\xi _i=\varsigma _i\bar{\varsigma }_i\), \(i\in I\), satisfy the remaining defining relations (4.9), (4.5) for the group \(W_G^U\):

$$\begin{aligned} \varsigma _i\xi _j=\xi _j\xi _i^{-a_{ji}}\varsigma _i,\quad \bar{\varsigma }_i\xi _j=\xi _j\xi _i^{-a_{ji}}\bar{\varsigma }_i, \end{aligned}$$
(6.5)

and

$$\begin{aligned} \underbrace{\varsigma _i\varsigma _j\varsigma _i\cdots }_{m_{ij}}= \underbrace{\bar{\varsigma }_j\bar{\varsigma }_i\bar{\varsigma }_j\cdots }_{m_{ij}}, \quad i\ne j,\;\; i,j\in I, \end{aligned}$$
(6.6)

where \(m_{ij}=2,3,4,6\) for \(a_{ij}a_{ji}=0,1,2,3\), respectively.

The first identity in (6.5) follows from (6.2) and (3.2):

$$\begin{aligned} \varsigma _i\xi _j\varsigma _i^{-1} =e^{-\frac{\pi \imath }{2}h_i}\dot{s}_i{\gamma }e^{\pi \imath h_j}{\gamma }\dot{s}_i^{-1}e^{\frac{\pi \imath }{2}h_i} =e^{-\frac{\pi \imath }{2}h_i}e^{\pi \imath (h_j-a_{ji}h_i)} e^{\frac{\pi \imath }{2}h_i}=\xi _j\xi _i^{-a_{ji}}. \end{aligned}$$

The other identity in (6.5) follows from \(\bar{\varsigma }_i={\gamma }\varsigma _i{\gamma }\), \(i\in I\).

For the relation (6.6), on the left-hand side we have

$$\begin{aligned} \underbrace{\varsigma _i\varsigma _j\varsigma _i\cdots }_{m_{ij}} =\bigl (\underbrace{e^{-\frac{\pi \imath }{2}h_i}\dot{s}_i e^{\frac{\pi \imath }{2}h_j}\dot{s}_j\cdots }_{m_{ij}}\bigr )\,{\gamma }^{m_{ij}}\! =\exp \frac{\pi \imath }{2} (\underbrace{-h_i+s_ih_j-s_is_jh_i+\ldots }_{m_{ij}}) (\underbrace{\dot{s}_i\dot{s}_j\cdots }_{m_{ij}})\,{\gamma }^{m_{ij}}, \end{aligned}$$

and on the right-hand side:

$$\begin{aligned} \underbrace{\bar{\varsigma }_j\bar{\varsigma }_i\bar{\varsigma }_j\cdots }_{m_{ij}} =\bigl (\underbrace{e^{\frac{\pi \imath }{2}h_j}\dot{s}_j e^{-\frac{\pi \imath }{2}h_i}\dot{s}_i\cdots }_{m_{ij}}\bigr )\,{\gamma }^{m_{ij}}\! =\exp \frac{\pi \imath }{2}(\underbrace{h_j-s_jh_i+s_js_ih_j-\ldots }_{m_{ij}}) (\underbrace{\dot{s}_j\dot{s}_i\cdots }_{m_{ij}})\,{\gamma }^{m_{ij}}. \end{aligned}$$

Since \(\underbrace{\dot{s}_i\dot{s}_j\cdots }_{m_{ij}}=\underbrace{\dot{s}_j\dot{s}_i\cdots }_{m_{ij}}\) holds due to (3.3), the identity (6.6) reduces to the following:

$$\begin{aligned} \exp \frac{\pi \imath }{2}\bigl (\underbrace{-h_i+s_ih_j-s_is_jh_i+\ldots }_{m_{ij}}\bigr )= \exp \frac{\pi \imath }{2}\bigl (\underbrace{h_j-s_jh_i+s_js_ih_j-\ldots }_{m_{ij}}\bigr ). \end{aligned}$$
(6.7)

In turn the identity (6.7) may be proved by invoking the following fact.

Lemma 6.3

For each pair \(i,j\in I\), \(i\ne j\), the following holds:

$$\begin{aligned} (\underbrace{1-s_j+s_is_j-\ldots }_{m_{ij}})\,h_i=0. \end{aligned}$$
(6.8)

Proof

Consider the order \(2m_{ij}\) Coxeter subgroup of \(W_G\) generated by a pair of the simple root reflections \(s_i,s_j\), \(i\ne j\):

$$\begin{aligned} \langle s_i,s_j\,{:}\,(s_is_j)^{m_{ij}}\!=s_i^2=s_j^2=1\rangle \subset W_G. \end{aligned}$$

This group is isomorphic to the dihedral group \(D_{m_{ij}}\!\subset O_2(\mathbb {R})\) of symmetries of \(m_{ij}\)-gone in the real plane \(V_{ij}=\mathbb {R}h_i\,{\oplus }\mathbb {R}h_j\). The dihedral group may be equivalently written in the following form:

$$\begin{aligned} D_{m_{ij}}=\bigl \langle t,r:t^{m_{ij}}\!=r^2=1, rtr^{-1}=t^{-1}\bigr \rangle ,\quad t=s_is_j,\;\; r=s_i. \end{aligned}$$

We have two projectors in the plane \(V_{ij}=\mathbb {R}h_i\,{\oplus }\mathbb {R}h_j\):

$$\begin{aligned} P_{\pm }=\frac{1\pm s_i}{2}:V_{ij}\rightarrow V_{ij},\quad P_{\pm }^2=P_{\pm }, \;\; P_{\pm } P_{\mp }=0, \end{aligned}$$

such that

$$\begin{aligned} P_+h_i=0, \quad P_-h_i=h_i. \end{aligned}$$

Therefore, the identity (6.8) is equivalent to the following:

$$\begin{aligned} (\underbrace{1-s_j+s_is_j-\cdots }_{m_{ij}})(1-s_i)\,{\cdot }\, h_i \,=\!\!\sum _{g\in D_{m_{ij}}}\!\!\!(-1)^gg{\cdot }h_i=0, \end{aligned}$$
(6.9)

where \((-1)^g:=\mathrm{det}(g)\) is the sign character of \(D_{m_{ij}}\!\subset O_2(\mathbb {R})\). The kernel of the sign character is a normal subgroup,

$$\begin{aligned} \mathbb {Z}_{m_{ij}}\!=\langle t\,{:}\,t^{m_{ij}}\!=1\rangle \subset D_{m_{ij}}, \end{aligned}$$

which consists of the rotations by \(\frac{2\pi k}{m_{ij}}\), \(0\leqslant k<m_{ij}\) of the plane \(V_{ij}\). The non-trivial co-set in \(D_{m_{ij}}/\langle t\rangle \) consists of the reflections \(\{r_k=t^kr:0\leqslant k<m_{ij}\}\) with \(r_0=r\) being a reflection sending \(h_i\) to \(-h_i\). Thus we have \((-1)^{r_k}=\mathrm{det}(r_k)=-1\) and

$$\begin{aligned} D_{m_{ij}}\!=\bigl \{t^k\,{:}\,0\leqslant k<m_{ij}\bigr \}\sqcup \bigl \{r_k=rt^k\,{:}\,0\leqslant k<m_{ij}\bigr \}, \end{aligned}$$

hence the identity (6.9) reads

$$\begin{aligned} \sum _{g\in D_{m_{ij}}}\!\!(-1)^gg\cdot h_i =(1-r)\!\!\sum _{k=0}^{m_{ij}-1}\! t^k\cdot h_i. \end{aligned}$$

Now in the group algebra \(\mathbb {C}[D_{m_{ij}}]\) the following identity holds:

$$\begin{aligned} t^{m_{ij}}\!-1=(t-1)\!\sum _{k=0}^{m_{ij}-1} \! t^k=0. \end{aligned}$$

Since t acts in the faithful representation \(V_{ij}\) without fixed vectors, we infer that \(\sum _{k=0}^{m_{ij}-1}t^k\cdot h_i=0\) and thus prove (6.9).\(\square \)

Lemma 6.4

The elements \(\eta _i=\sigma _i\bar{\sigma }_i\), \(i\in I\), generate a subgroup \(H_\eta \subset W_G^U\) of order \(|H_\eta |=2^{|I|}\).

Proof

By (6.4), (6.5) and (6.6), the elements \(\{\varsigma _i,\bar{\varsigma }_i,i\in I\}\subset {\widetilde{U}}=U\,{\rtimes }\, \Gamma \) satisfy the defining relations of the group \(W_G^U\) from Definition 4.1. Moreover, the images \(\varsigma _i\bar{\varsigma }_i=e^{\imath \pi h_i}\in U\) of the elements \(\eta _i=\sigma _i\bar{\sigma }_i\), \(i\in I\), generate the subgroup \(H_{\eta }\simeq H^{(2)}\) of order two points in the maximal torus \(H\subset G\), so that the order of \(H_\eta \) should be not less then \(2^{|I|}\) and thus \(|H_\eta |=2^{|I|}\) holds.\(\square \)

We complete our proof of Theorem 4.5 by verifying injectivity of the homomorphism \(\psi :W^{U}_G\rightarrow {\widetilde{U}}\). By Proposition (4.4), \(W^{U}_G\) has a structure of the group extension:

$$\begin{aligned} 1\longrightarrow \mathbb {Z}_2^{|I|}\longrightarrow W_G^U\longrightarrow W_G\longrightarrow 1. \end{aligned}$$
(6.10)

Let \(\mathscr {W}_G^U=\psi (W_G^U)\subset {\widetilde{U}}\), then the \(\mathscr {W}_G^U\)-action on \(\mathfrak {h}=\mathrm{Lie}\,(H)\) implies the existence of the surjective homomorphism \(\pi :\mathscr {W}_G^U\rightarrow W_G\). By Lemma 6.4, \(\mathscr {W}_G^U\) contains a normal abelian subgroup generated by \(\{\psi (\eta _i)=\varsigma _i\bar{\varsigma }_i=e^{\imath \pi h_i}\!,i\in I\}\subset U\), which is isomorphic to \(H^{(2)}\simeq \mathbb {Z}_2^{|I|}\). Clearly, the normal abelian subgroup \(H^{(2)}\) acts trivially on \(\mathfrak {h}\), hence it is in the kernel of the surjective homomorphism \(\pi \), which entails \(|W_G^U|\leqslant |\mathscr {W}_G^U|\). On the other hand, the existence of homomorphism \(\psi :W_G^U \rightarrow \mathscr {W}_G^U\) implies that \(|W_G^U|\geqslant |\mathscr {W}_G^U|\) and hence \(|W_G^U|=|\mathscr {W}_G^U|\). Thus for \(\mathscr {W}_G^U\) we have the following exact sequence:

$$\begin{aligned} 1\longrightarrow \mathbb {Z}_2^{|I|}\longrightarrow \mathscr {W}_G^U\longrightarrow W_G\longrightarrow 1. \end{aligned}$$

Taking into account (6.10) this provides a proof of injectivity of \(\psi \), and therefore, of Theorem 4.5.