Twisted Weyl Groups of Compact Lie Groups and Nonabelian Cohomology

Abstract

Given a compact connected Lie group G with an $S^1$-module structure and a maximal compact torus T of $G^{S^1}$, we define twisted Weyl group $W(G,S^1,T)$ of G associated to $S^1$-module and show that two elements of T are $\delta$-conjugate if and only if they are in one $W(G,S^1,T)$-orbit. Based on this, we prove that the natural map $W(G,S^1,T)\backslash H^1(S^1,T)\to H^1(S^1,G)$ is bijective, which reduces the calculation for the nonabelian cohomology $H^1(S^1,G)$.

1. Introduction

Let G be a compact connected Lie group and $\sigma$ be an automorphism on G [1]. The twisted conjugate action ${\tau }^{\sigma }$ [2] of G on itself associated to $\sigma$ is defined as

$${\tau }^{\sigma }:G\times G\to G,(g,h)\mapsto gh\sigma {(g)}^{-1}\triangleq {\tau }_g^{\sigma }(h).$$

Two elements $g_1$ and $g_2$ of G are called $\sigma$-conjugate if they are in one twisted orbit of G associated to $\sigma$, i.e., there exists an element $g_0\in G$ such that

$$g_2={\tau }_{g_0}^{\sigma }(g_1).$$

Based on this, given a compact connected Lie group G with an additional ${\mathbb{Z}}_n$-module structure and T a maximal compact torus of $G_0^{{\mathbb{Z}}_n}$, An J. [3] defined the twisted Weyl group $W(G,{\mathbb{Z}}_n,T)$ of G associated to ${\mathbb{Z}}_n$-module and reduced the calculation of $H^1({\mathbb{Z}}_n,G)$ to the action of $W(G,{\mathbb{Z}}_n,T)$ on T, where $H^1({\mathbb{Z}}_n,G)$ is the first nonabelian cohomology of ${\mathbb{Z}}_n$ with coefficients in G [2].

Motivated by the underlying work, in this paper we consider the case of G with an nonabelian $S^1$-module structure, where $S^1$ is the one-dimensional torus. Picking a topological generator $\delta$ of $S^1$, which can be regarded as an automorphism of G [4], we first define the twisted Weyl group $W(G,\delta ,T)$ of G associated to $\delta$ and then define the twisted Weyl group $W(G,S^1,T)$ associated to $S^1$-module, where T is a maximal compact torus in $G^{S^1}$. Closed by the definition, we study the action of $W(G,S^1,T)$ on T and show that two elements in T are $\delta$-conjugate if and only if they are in one $W(G,S^1,T)$-orbit. Furthermore, we prove that $W(G,S^1,T)$ is a finite group, which is the same as the case of classical Weyl groups.

Based on the underlying properties of $W(G,S^1,T)$, we study the action of $W(G,S^1,T)$ on the first cohomology $H^1(S^1,T)$ of the compact Lie group $S^1$ with coefficients in T [5], and prove that the natural map

$$W(G,S^1,T)\backslash H^1(S^1,T)\to H^1(S^1,G)$$

induced by the natural embedding $T\hookrightarrow G$ is a bijection. Using the result, one can reduce the calculation for $H^1(S^1,G)$ to the calculation for the orbit space $W(G,S^1,T)\backslash H^1(S^1,T)$. Indeed, using this formula, one can simplify some calculations in dynamical systems theory, especially in fractional dynamics and fractional-wavelet analysis of some positive definite distributions [6,7,8].

In Section 2, we exhibit the definition of the twisted Weyl group $W(G,S^1,T)$ of G associated to $S^1$-module together with its some properties. In Section 3, we construct a one-to-one correspondence between the orbit space of the action of $W(G,S^1,T)$ on $H^1(S^1,T)$ and $H^1(S^1,G)$. In Section 4, we discuss some new developments in the field as well as its relations with amenability of groups [9,10,11,12]. For basic knowledge on compact Lie groups and twisted conjugate actions, one can refer [2,13,14]; for the nonabelian cohomology of Lie groups, one can refer [5,15,16,17].

2. Twisted Weyl Groups of Compact Lie Groups

Let G be a compact connected Lie group with an $S^1$-module structure and $\delta$ be a topological generator of $S^1$. From the definition of $S^1$-module structure, $\delta$ can be regarded as an automorphism on G. Denote by ${\tau }^{\delta }$ the twisted conjugate action of G associated to $\delta$.

First of all, we define the twisted Weyl group of G associated to $\delta$.

Definition 1.

Define

$$N_{\delta ,G}(T):=\{g\in G\mid {\tau }_g^{\delta }(T)=T\},$$

$$Z_{\delta ,G}(T):=\{g\in G\mid {\tau }_g^{\delta }(t)=t,\forall t\in T\},$$

where T is a maximal compact torus of $G^{\delta }:=\{g\in G\mid \delta (g)=g\}$.

It is easy to know that $N_{\delta ,G}(T)$ and $Z_{\delta ,G}(T)$ are both closed subgroups of G and that $Z_{\delta ,G}(T)$ is a normal subgroup of $N_{\delta ,G}(T)$.

Definition 2.

Define

$$W(G,\delta ,T):=N_{\delta ,G}(T)/Z_{\delta ,G}(T).$$

$W(G,\delta ,T)$ is called the twisted Weyl group of G associated to δ.

As an abstract group, the group operations on $W(G,\delta ,T)$ are defined as following. For all $w_1,w_2\in W(G,\delta ,T)$,

$$w_1·w_2:=\left[g_1{g}_2\right],$$

where $g_1\in w_1$, $g_2\in w_2$, $\left[g_1{g}_2\right]$ represents the equivalence class of $g_1{g}_2$ in $N_{\delta ,G}(T)/Z_{\delta ,G}(T)$; for all $w\in W(G,\delta ,T)$,

$$w^{-1}:=\left[g^{-1}\right],$$

where $g\in w$.

Next, we exhibit the definition of the twisted Weyl group of G associated to $S^1$-module.

Definition 3.

Define,

$$N_{S^1}(T):=\{g\in G\mid {\tau }_g^{\delta }(T)=T,\forall \delta \in S^1\},$$

$$Z_{S^1}(T):=\{g\in G\mid {\tau }_g^{\delta }(t)=t,\forall t\in T,\forall \delta \in S^1\},$$

where T is a maximal compact torus of $G^{S^1}:=\{g\in G\mid a(g)=g,\forall a\in S^1\}$.

It is clear that they are both closed subgroups of G and that $Z_{S^1}(T)$ is a normal subgroup of $N_{S^1}(T)$.

Proposition 1.

Let δ be a topological generator of $S^1$. Then,

$$G^{S^1}=G^{\delta }.$$

Proof. 

It suffices to prove $G^{S^1}\supset G^{\delta }$. Let $g\in G^{\delta }$ and set

$$S_g^1:=\{\delta \in S^1\mid \delta (g)=g\}.$$

For the underlying $g\in G^{\delta }$, it is obvious that $\delta (g)=g$. Thus, for all $n\in \mathbb{Z}$,

$${\delta }^n(g)={\delta }^{n-1}(\delta (g))={\delta }^{n-1}(g)=\cdots =g.$$

Since $\delta$ is a topological generator of $S^1$, $S_g^1$ is dense in $S^1$. Again $S_g^1$ is a closed subgroup of $S^1$,

$$S_g^1=S^1.$$

Then, $a(g)=g$,$\forall a\in S^1,g\in G^{S^1}$. So,

$$G^{S^1}\supset G^{\delta }.$$

 □

By Proposition 1, $G^{S^1}$ is connected. For the underlying two subgroups $N_{S^1}(T)$ and $Z_{S^1}(T)$ of G, we claim that:

Proposition 2.

For a given topological generator ${\delta }_0$ of $S^1$ and a maximal compact torus T of $G^{S^1}$,

(i) 

$N_{S^1}(T)=N_{{\delta }_0,G}(T)$;

(ii) 

$Z_{S^1}(T)=Z_{{\delta }_0,G}(T)$.

Proof. 

(i) It suffices to show that $N_{S^1}(T)\supset N_{{\delta }_0,G}(T)$.

Suppose that $g_0\in N_{{\delta }_0,G}(T)$. Then

$$g_{0}T{\delta }_0{(g_0)}^{-1}=T.$$

Define

$$S_{g_0,T}^1:=\{\delta \in S^1\mid g_{0}T\delta {(g_0)}^{-1}=T\}.$$

Then, $S_{g_0,T}^1$ is a dense subset of $S^1$. In fact, for all $n\in \mathbb{Z}$,

$$\begin{array}{cc}\hfill & g_{0}T{\delta }_0^n{(g_0)}^{-1}\hfill \\ \hfill =& g_{0}T{\delta }_0^n(g_0^{-1})\hfill \\ \hfill =& (g_{0}T{\delta }_0(g_0^{-1}))({\delta }_0(g_0)T{\delta }_0^2(g_0^{-1}))\cdots ({\delta }_0^{n-1}(g_0)T{\delta }_0^n(g_0^{-1}))\hfill \\ \hfill =& (g_{0}T{\delta }_0(g_0^{-1}))\delta (g_{0}T{\delta }_0(g_0^{-1}))\cdots {\delta }^{n-1}(g_{0}T{\delta }_0(g_0^{-1}))\hfill \\ \hfill =& T^n\hfill \\ \hfill =& T.\hfill \end{array}$$

Since ${\delta }_0$ is a topological generator of $S^1$, the set which is generated by ${\delta }_0$ is a dense subset of $S^1$. Again for all $n\in \mathbb{Z}$,

$$g_{0}T{\delta }_0^n{(g_0)}^{-1}=T,$$

$S_{g_0,T}^1$ is dense in $S^1$. It is obvious that $S_{g_0,T}^1$ is also a closed subgroup of $S^1$. Hence,

$$S_{g_0,T}^1=S^1,$$

which shows that

$${\tau }_{g_0}^{\delta }(T)=T,$$

for all $\delta \in S^1$. So, $g_0\in N_{S^1}(T)$, i.e.,

$$N_{S^1}(T)\supset N_{{\delta }_0,G}(T).$$

(ii) Similarly, it suffices to show that $Z_{S^1}(T)\supset Z_{{\delta }_0,G}(T)$.

Suppose that $g_0\in Z_{{\delta }_0,G}(T)$. Then we have

$$g_{0}t{\delta }_0{(g_0)}^{-1}=t,$$

for all $t\in T$.

For $g_0\in Z_{{\delta }_0,G}(T)$, for any $t\in T$, define

$$S_{g_0,t}^1:=\{\delta \in S^1\mid g_{0}t\delta {(g_0)}^{-1}=t\}.$$

Analogously,

$$S_{g_0,t}^1=S^1.$$

So,

$$Z_{S^1}(T)\supset Z_{{\delta }_0,G}(T).$$

 □

From Proposition 2, one can get that the subgroups $N_{S^1}(T)$ and $Z_{S^1}(T)$ of G are independent with the choice of the topological generator $\delta$ of $S^1$.

Definition 4.

Let δ be a topological generator of $S^1$ and $W(G,\delta ,T)$ be defined as above. Define,

$$W(G,S^1,T):=W(G,\delta ,T).$$

$W(G,S^1,T)$ is called the twisted Weyl group of G associated to $S^1$-module.

Following, we present some properties of $W(G,S^1,T)$.

Proposition 3.

$W(G,S^1,T)$ is independent with the choice of T.

Proof. 

Let $T^{\prime }$ is another maximal compact torus of $G^{\delta }$. Then by ([3], Proposition 2.11), there exists an element $g_0\in G^{\delta }$ such that $T^{\prime }=g_{0}T{g}_0^{-1}$. Thus,

$$\begin{array}{ccc}{\tau }_{g_{0}g{g}_0^{-1}}^{\delta }(T^{\prime })\hfill & =\hfill & g_{0}g{g}_0^{-1}{T}^{\prime }\delta {(g_{0}g{g}_0^{-1})}^{-1}\hfill \\ \hfill & =\hfill & g_{0}g{g}_0^{-1}{T}^{\prime }{g}_0\delta {(g)}^{-1}{g}_0^{-1}\hfill \\ \hfill & =\hfill & g_{0}gT\delta {(g)}^{-1}{g}_0^{-1}\hfill \\ \hfill & =\hfill & g_{0}T{g}_0^{-1}\hfill \\ \hfill & =\hfill & T^{\prime },\hfill \end{array}$$

for all $g\in N_{\delta ,G}(T)$. Hence,

$$g_0{N}_{\delta ,G}(T)g_0^{-1}\subset N_{\delta ,G}(T^{\prime }).$$

Similarly, we have

$$g_0{N}_{\delta ,G}(T)g_0^{-1}\supset N_{\delta ,G}(T^{\prime }).$$

Therefore,

$$N_{\delta ,G}(T^{\prime })=g_0{N}_{\delta ,G}(T)g_0^{-1}.$$

Analogously,

$$Z_{\delta ,G}(T^{\prime })=g_0{Z}_{\delta ,G}(T)g_0^{-1}.$$

Then,

$$\begin{array}{ccc}{W}_{\delta }(T^{\prime })\hfill & =\hfill & N_{\delta ,G}(T^{\prime })/Z_{\delta ,G}(T^{\prime })\hfill \\ \hfill & =\hfill & g_0{N}_{\delta ,G}(T)g_0^{-1}/g_0{Z}_{\delta ,G}(T)g_0^{-1}\hfill \\ \hfill & \cong \hfill & W_{\delta }(T),\hfill \end{array}$$

which shows that $W(G,S^1,T)$ is independent with the choice of T. □

For the reason, we write $W(G,S^1,T)$ as $W(S^1)$ and write $W(G,\delta ,T)$ as $W(\delta )$ in simplified forms, respectively.

Lemma 1.

Denote by $\mathfrak{L}$ Lie functor. Then,

$$\mathfrak{L}(Z_{\delta ,G}(T))=\mathfrak{L}(N_{\delta ,G}(T))=\mathfrak{L}(T).$$

Proof. 

Above all, we show the first equality. It suffices to show the reverse inclusion for the clear fact that $\mathfrak{L}(Z_{\delta ,G}(T))\subset \mathfrak{L}(N_{\delta ,G}(T))$. For all $X\in \mathfrak{L}(N_{\delta ,G}(T)),s\in \mathbb{R},t\in T$, we have

$$\begin{array}{cc}\hfill & {\tau }_{e^{sX}}(t)\hfill \\ \hfill =& e^{sX}t\delta {(e^{sX})}^{-1}\hfill \\ \hfill =& e^{sX}t{e}^{-sd\delta (X)}\hfill \\ \hfill =& e^{sX}{e}^{-sAd(t)d\delta (X)}t\in T.\hfill \end{array}$$

Then, $(1-Ad(t)d\delta )(X)\in \mathfrak{L}(T)$. Now, we show

$$(1-Ad(t)d\delta )(X)=0,$$

for all $t\in T$. In fact, $\mathfrak{L}(T)\subset ker(1-Ad(t)d\delta )$ implies that ${(1-Ad(t)d\delta )}^2(X)=0$. Since $d\delta$ and $Ad(t)$ are both $1-$semisimple (see in [3]), $Ad(t)\circ d\delta$ is $1-$semisimple. Hence, $ker(1-Ad(t)d\delta )=ker{(1-Ad(t)d\delta )}^2$. Then, we have $(1-Ad(t)d\delta )(X)=0$ for all $t\in T$.

Next, we show the equality $\mathfrak{L}(Z_{\delta ,G}(T))=\mathfrak{L}(T)$. For all $g\in Z_{\delta ,G}(T),t\in T$, ${\tau }_g^{\delta }(t)=t$. Take e for the unit of T. Then ${\tau }_g(e)=e$, i.e., $g\delta {(g)}^{-1}=e$. Thus, $\delta (g)=g$ and hence $g\in G^{\delta }$. So, $Z_{\delta ,G}(T)\subset G^{\delta }.$ Moreover,

$$Z_{\delta ,G}(T)=Z_{\delta ,G}(T)\cap G^{\delta }=Z_{\delta ,G^{\delta }}(T)=T,$$

which shows $Z_{\delta ,G}(T)=T$. Then,

$$\mathfrak{L}(Z_{\delta ,G}(T))=\mathfrak{L}(T).$$

Therefore, $\mathfrak{L}(Z_{\delta ,G}(T))=\mathfrak{L}(N_{\delta ,G}(T))=\mathfrak{L}(T).$ □

Proposition 4.

As an abstract group, $W(G,S^1,T)$ is a finite group.

Proof. 

By Lemma 1, it is clear. □

Remark 1.

In the case of classical Weyl groups, let G be a compact connected Lie group, T be a maximal torus of G, and ${\mathfrak{g}}_{\mathfrak{0}}$ and ${\mathfrak{t}}_{\mathfrak{0}}$ be the Lie algebras of G and T respectively. Denote by $\mathfrak{g}$ and $\mathfrak{t}$ the complexifications of ${\mathfrak{g}}_{\mathfrak{0}}$ and ${\mathfrak{t}}_{\mathfrak{0}}$, and denote $\Delta (\mathfrak{g},\mathfrak{t})$ be the set of roots of $\mathfrak{g}$ with respect to $\mathfrak{t}$. In analytical level, Weyl group $W(G,T)$ is defined as the quotient of normalizer by centralizer $W(G,T)=N_G(T)/Z_G(T)$. In algebraical level, Weyl group $W(\Delta (\mathfrak{g},\mathfrak{t}))$ is defined as the subgroup of the orthogonal group on $\mathfrak{g}$ generated by the root reflections $s_{\alpha }$ for $\alpha \in \Delta (\mathfrak{g},\mathfrak{t})$. When $W(G,T)$ is considered as acting on ${(i{\mathfrak{t}}_{\mathfrak{0}})}^{*}$, $W(G,T)$ coincides with $W(\Delta (\mathfrak{g},\mathfrak{t}))$ [14].

In the case of twisted Weyl groups, let G be a compact connected Lie group with an ${\mathbb{Z}}_n-$ module structure, T be a maximal torus of $G^{{\mathbb{Z}}_n}$, and $\mathfrak{g}$, $\mathfrak{t}$, ${\mathfrak{g}}_{\mathfrak{0}}$ and ${\mathfrak{t}}_{\mathfrak{0}}$ described as above. In [3], An J. defined the twisted Weyl group

$$W(G,{\mathbb{Z}}_n,T):=W(G,\delta ,T)=N_{\delta }(T)/Z_{\delta }(T),$$

where δ is a generator of ${\mathbb{Z}}_n$. Motivated by the underlying equality between $W(G,T)$ and $W(\Delta (\mathfrak{g},\mathfrak{t}))$, one can also consider the algebraic twisted Weyl group. Since δ can be regarded as an automorphism on G, its differential $d\delta$ can be thought as an automorphism on $\mathfrak{g}$. Define

$${\mathfrak{g}}_{\alpha }^{\delta }:=\{X\in \mathfrak{g}\mid ad(d\delta )(H)X=\alpha (H)X,\forall H\in \mathfrak{t}\}.$$

If ${\mathfrak{g}}_{\alpha }^{\delta }\ne 0$, α is called a twisted root of $\mathfrak{g}$ with respect to $\mathfrak{t}$. Denote by $\Delta (\mathfrak{g},\delta ,\mathfrak{t})$ the set of twisted roots of $\mathfrak{g}$ with respect to $\mathfrak{t}$. For $\alpha \in \Delta (\mathfrak{g},\delta ,\mathfrak{t})$, the twisted root reflection is defined as

$$s_{\alpha }^{\delta }(\gamma ):=\gamma -\frac{2<d\delta (\gamma ),\alpha >}{\mid \alpha {\mid }^2}\alpha .$$

Define $W(\Delta (\mathfrak{g},\delta ,\mathfrak{t}))$ as the group generated by the twisted root reflections $s_{\alpha }^{\delta }$ for $\alpha \in \Delta (\mathfrak{g},\delta ,\mathfrak{t})$. Similar as the proof in ([14], Theorem 4.54), one can obtain that

$$W(G,{\mathbb{Z}}_n,T)=W(\Delta (\mathfrak{g},\delta ,\mathfrak{t})),$$

when $W(G,{\mathbb{Z}}_n,T)$ is considered as acting on ${(i{\mathfrak{t}}_{\mathfrak{0}})}^{*}$. For analytical twisted Weyl group $W(G,S^1,T)$, we have an analogous algebraical counterpart described as above.

The twisted conjugate action ${\tau }^{\delta }$ of G associated to $\delta$ naturally induces the action of twisted Weyl group $W(S^1)$ on $T:$

$$W(S^1)\times T\to T,w·t:={\tau }_g^{\delta }(t),$$

where $\delta$ is a topological generator of $S^1$, $w\in W(S^1),t\in T,g\in w$. Now, we show the following property of the action of $W(S^1)$ on T.

Proposition 5.

Let G be a compact connected Lie group associated to an $S^1$-module, δ be a topological generator of $S^1$, T be a maximal compact torus of $G^{S^1}$ and $W(S^1)$ be the twisted Weyl group of G associated to $S^1$-module. Then two elements of T are δ-conjugate if and only if they are in one $W(S^1)$-orbit.

Proof. 

Denote by $O_t^W$ the $W(S^1)$-orbit over $t\in T$, i.e.,

$$O_t^W:=\{{\tau }_g^{\delta }(t)\mid w\in W(S^1),g\in w\}.$$

⇒) If $t_1,t_2\in O_t^W$, then there exist $g_1\in w_1\in W(S^1)$, $g_2\in w_2\in W(S^1)$ such that $t_1={\tau }_{g_1}^{\delta }(t)$, $t_2={\tau }_{g_2}^{\delta }(t)$. Hence, $t=g_1^{-1}{t}_1\delta (g_1)$. So,

$$\begin{array}{ccc}{t}_2\hfill & =\hfill & g_{2}t\delta {(g_2)}^{-1}\hfill \\ \hfill & =\hfill & g_2{g}_1^{-1}{t}_1\delta (g_1)\delta {(g_2)}^{-1}\hfill \\ \hfill & =\hfill & (g_2{g}_1^{-1})t_1\delta {(g_2{g}_1^{-1})}^{-1},\hfill \end{array}$$

which shows that $t_2={\tau }_{g_2{g}_1^{-1}}^{\delta }(t_1)$, $g_2{g}_1^{-1}\in G$. So, $t_1,t_2$ are $\delta$-conjugate.

⇐) If $t_1,t_2$ are $\delta$-conjugate. i.e., there exists an element $g\in G$ such that $t_2={\tau }_g^{\delta }(t_1)$. It needs to show that $t_1,t_2$ are in one $W(S^1)$-orbit, i.e., to find an element $w_0\in W(S^1)$ such that $t_2=w_0·t_1$ or to find an element $g_0\in N_{\delta ,G}(T)$ such that

$$t_2={\tau }_{g_0}^{\delta }(t_1).$$

Define

$$G_{t_1}:=\{h\in G\mid {\tau }_h^{\delta }(t_1)=t_1\}.$$

Then $T\subset G_{t_1}$, which holds for the fact T is connected and ${\tau }_t^{\delta }(t_1)=t_1$ for all $t\in T$. Again,

$$\begin{array}{ccc}h\in G_{t_1}\hfill & \iff \hfill & {\tau }_h^{\delta }(t_1)=h{t}_1\delta {(h)}^{-1}=t_1\hfill \\ \hfill & \iff \hfill & h=t_1\delta (h)t_1^{-1}\hfill \\ \hfill & \iff \hfill & h\in G^{{ı}_{t_1}\circ \delta },\hfill \end{array}$$

where ${ı}_{t_1}$ represents the automorphism induced by $t_1$. Thus,

$$G_{t_1}=G^{{ı}_{t_1}\circ \delta }.$$

Then by ([2], Theorem 2.1),

$$rank(G^{\delta })=rank(G^{{ı}_{t_1}\circ \delta })=rank(G_{t_1}),$$

where $rank(G^{\delta })$ represents the dimension of the maximal compact torus of $G^{\delta }$. Hence, T is maximal compact torus of $G_{t_1}$. In fact, for the underlying $g\in G$, $g^{-1}Tg$ is also a maximal compact torus of $G_{t_1}:$ for any $t\in T$, we have

$$\begin{array}{ccc}{\tau }_{g^{-1}tg}^{\delta }(t_1)\hfill & =\hfill & g^{-1}tg{t}_1\delta {(g^{-1}tg)}^{-1}\hfill \\ \hfill & =\hfill & g^{-1}tg{t}_1\delta {(g)}^{-1}\delta {(t)}^{-1}\delta {(g^{-1})}^{-1}\hfill \\ \hfill & =\hfill & g^{-1}t{t}_2{t}^{-1}\delta {(g^{-1})}^{-1}\hfill \\ \hfill & =\hfill & g^{-1}{t}_2\delta {(g^{-1})}^{-1}\hfill \\ \hfill & =\hfill & t_1,\hfill \end{array}$$

which shows that $g^{-1}Tg\subset G_{t_1}$. Thus, T and $g^{-1}Tg$ are both torus of $G_{t_1}$. Then there exists an element $x\in G_{t_1}$ such that $xT{x}^{-1}=g^{-1}Tg$. Pick $g_0=gx$, then $g_{0}T{g}_0^{-1}=T$.

Now we show $g_0\in N_{\delta ,G}(T)$ and ${\tau }_{g_0}^{\delta }(t_1)=t_2$. By

$$\begin{array}{ccc}{\tau }_{g_0}^{\delta }(T)\hfill & =\hfill & g_{0}T\delta {(g_0)}^{-1}\hfill \\ \hfill & =\hfill & g_{0}T{g}_0^{-1}{g}_0{t}_1\delta {(g_0)}^{-1}\hfill \\ \hfill & =\hfill & Tgx{t}_1\delta {(x)}^{-1}\delta {(g)}^{-1}\hfill \\ \hfill & =\hfill & Tg{t}_1\delta {(g)}^{-1}\hfill \\ \hfill & =\hfill & T{t}_2\hfill \\ \hfill & =\hfill & T,\hfill \end{array}$$

we get $g_0\in N_{\delta ,G}(T)$; by

$$\begin{array}{ccc}{\tau }_{g_0}^{\delta }(t_1)\hfill & =\hfill & g_0{t}_1\delta {(g_0)}^{-1}\hfill \\ \hfill & =\hfill & gx{t}_1\delta {(gx)}^{-1}\hfill \\ \hfill & =\hfill & gx{t}_1\delta {(x)}^{-1}\delta {(g)}^{-1}\hfill \\ \hfill & =\hfill & g{t}_1\delta {(g)}^{-1}\hfill \\ \hfill & =\hfill & t_2,\hfill \end{array}$$

we get ${\tau }_{g_0}^{\delta }(t_1)=t_2$. □

3. Twisted Weyl Groups of Compact Lie Groups and Nonabelian Cohomology

In this section, we discuss the relationship between twisted Weyl group $W(S^1)$ and the first nonabelian cohomology $H^1(S^1,G)$ of $S^1$ with coefficients in G.

Let T be a maximal compact torus of $G^{S^1}$. As T is abelian and $S^1$ acts trivially on T, $H^1(S^1,T)$ coincides with $Z^1(S^1,T)$ and $\gamma \in Z^1(S^1,T)$ is a group homomorphism $\gamma :S^1\to T$. Thus, $W(S^1)$ naturally acts on $H^1(S^1,T)=Z^1(S^1,T)$, i.e.,

$$W(S^1)\times H^1(S^1,T)\to H^1(S^1,T),$$

$$(w·\gamma )(a):=g\gamma (a)a{(g)}^{-1},$$

where $w\in W(S^1),\gamma \in H^1(S^1,T),g\in w,a\in S^1$. Now, we show that the definition is well-defined. It suffices to show that $w·\gamma \in Z^1(S^1,T)$.

In fact, for any $a_1,a_2\in S^1$, we have

$$\begin{array}{ccc}(w·\gamma )(a_1)a_1((w·\gamma )(a_2))\hfill & =\hfill & g\gamma (a_1)a_1{(g)}^{-1}{a}_1(g\gamma (a_2)a_2{(g)}^{-1})\hfill \\ \hfill & =\hfill & g\gamma (a_1)a_1{(g)}^{-1}{a}_1(g)a_1(\gamma (a_2))a_1(a_2{(g)}^{-1})\hfill \\ \hfill & =\hfill & g\gamma (a_1)\gamma (a_2)(a_1{a}_2){(g)}^{-1}\hfill \\ \hfill & =\hfill & g\gamma (a_1{a}_2)(a_1{a}_2){(g)}^{-1}\hfill \\ \hfill & =\hfill & (w·\gamma )(a_1{a}_2),\hfill \end{array}$$

which shows that the underlying definition is well-defined. Thus, we have

$$(w·\gamma )(a_1{a}_2)=(w·\gamma )(a_1)(w·\gamma )(a_2).$$

Hence, $w·\gamma$ is a group homomorphism.

The natural embedding

$$i:T\hookrightarrow G$$

induces the natural map

$$i_1:H^1(S^1,T)\to H^1(S^1,G),\left[\gamma \right]\mapsto [i\circ \gamma ],$$

and $i_1$ can be reduced to

$$i_1^{W(S^1)}:W(S^1)\backslash H^1(S^1,T)\to H^1(S^1,G),O_{\gamma }^W\mapsto [i\circ \gamma ],$$

where $O_{\gamma }^W:=\{w·\gamma \mid w\in W\}$ is the $W(S^1)$-orbit over $\gamma$ in $H^1(S^1,T)$. Thus, we construct a correspondence between $W(S^1)\backslash H^1(S^1,T)$ and $H^1(S^1,G)$ by $i_1^{W(S^1)}$.

Theorem 1.

Let G be a compact connected Lie group associated to an $S^1$-module, δ be a topological generator of $S^1$, T be a maximal compact torus of $G^{S^1}$ and $W(S^1)$ be the twisted Weyl group of G associated to $S^1$-module. Then the map

$$i_1^{W(S^1)}:W(S^1)\backslash H^1(S^1,T)\to H^1(S^1,G)$$

is a bijection.

Proof. 

By ([17], Theorem 2.5), the natural map $i_1$ is a surjection and hence $i_1^{W(S^1)}$ is also a surjection.

Now we show $i_1^{W(S^1)}$ is an injection. Suppose that ${\gamma }_1,{\gamma }_2\in H^1(S^1,T)=Z^1(S^1,T)$ have the same image under $i_1$. Then, there exists an element $g\in G$ such that

$${\gamma }_2(a)=g{\gamma }_{(}{a)a(g)}^{-1},$$

for all $a\in A$. Picking a topological generator ${\delta }_0$ of $S^1$. Thus, we have

$${\gamma }_2({\delta }_0)=g{\gamma }_1({\delta }_0){\delta }_0{(g)}^{-1},$$

which shows that ${\gamma }_1({\delta }_0)$ and ${\gamma }_2({\delta }_0)$ are ${\delta }_0-$cojugate. By Proposition 5, ${\gamma }_1({\delta }_0)$ and ${\gamma }_2({\delta }_0)$ are in one $W(S^1)=W({\delta }_0)$-orbit. In other words, there exists an element $g_0\in N_{{\delta }_0,G}(T)$ such that

$${\gamma }_2({\delta }_0)=g_0{\gamma }_1({\delta }_0){\delta }_0{(g_0)}^{-1},$$

which also means that there exists $w_0\in W$ such that $g_0\in w_0$ and

$${\gamma }_2({\delta }_0)=(w_0·{\gamma }_1)({\delta }_0).$$

Since ${\delta }_0$ is a topological generator of $S^1$, one can obtain that

$${\gamma }_2(a)=(w_0·{\gamma }_1)(a),$$

for all $a\in S^1$. Thus,

$${\gamma }_2=w_0·{\gamma }_1,$$

which shows that ${\gamma }_1$ and ${\gamma }_2$ are in one $W(S^1)$-orbit. So,

$$i_1^{W(S^1)}(O_{{\gamma }_1}^W)=i_1^{W(S^1)}(O_{{\gamma }_2}^W),$$

and thus,

$$O_{{\gamma }_1}^{S^1}=O_{{\gamma }_2}^{S^1},$$

which shows that $i_1^{W(S^1)}$ is an injection. □

Remark 2.

Let A be a general compact group [4] and G be a compact connected Lie group with an A-module structure. Our motivation for this paper is to define the twisted Weyl group of G associated A-module. For this aim, we have to deal with the existence of the maximal compact torus in $G_0^A$, where $G_0^A$ is the the identity connected component of

$$G^A:=\{g\in G\mid a(g)=g,\forall a\in A\}.$$

However, for a general compact group A, maximal compact torus in $G_0^A$ may not exist [1]. Even for the existence of $A-$invariant maximal compact torus in G, it is not certain too [17]. From the discussions for the existence of maximal compact torus in $G_0^A$ and $G^A$, we find the following two open problems.

Problem 1.

Under what conditions there exists a nontrivial maximal compact torus in $G_0^A$?

Problem 2.

If T is a nontrivial maximal compact torus of $G_0^A$. Can the maximal compact torus T generalize to an $A-$invariant maximal compact torus $T^{\prime }$ of G?

4. Discussions

In [9], Bartholdi studied the amenability of $\mathsf{\Gamma }$-set, here $\mathsf{\Gamma }$ is a group, which was induced by John Von Neumann in 1929. Fundamentally, the notion exhibited the following property of a group acting on a $\mathsf{\Gamma }$-set X: The $\mathsf{\Gamma }$-set X right is called amenable if there exists a $\mathsf{\Gamma }$-invariant mean m on the power set $2^X$ of X, namely a function

$$m:2^X\to [0,1],$$

satisfying $m(A\bigsqcup B)=m(A)+m(B),m(X)=1$ and

$$m(Ag)=m(A),$$

for all $A\subset X$ and $g\in \mathsf{\Gamma }$.

In [9], Bartholdi presented some criterions to show a $\mathsf{\Gamma }$-set X amenable. By ([9], Proposition 2.3), one can get that the underlying twisted Weyl group $W(S^1)$ is indeed amenable. For the amenability of twisted Weyl groups, we will study it in a sole paper.