Exercise 3, § 5

Let $f: \widetilde{G}\rightarrow G$ be a universal covering of G and $\pi$ the kernel of $f$ (isomorphic to $\pi_1(G)$); put C = C(G) and $\widetilde{C} = C( \widetilde{G})$. Let $\sigma$ be an automorphism of $G, \widetilde{\sigma}$ the automorphism of $\widetilde{G}$ induced by $\sigma$. Denote by $G_{\sigma},C_{\sigma},\widetilde{G}_{\sigma},\widetilde{C}_{\sigma}$ the set of points of $G,C,\widetilde{G},\widetilde{C}$ fixed by $\sigma , \sigma ,\widetilde{\sigma ,}\widetilde{\sigma}$, respectively.

a) Show that the identity component $(G_{\sigma})_0$ of $G_{\sigma}$ is $f( \widetilde{G}_{\sigma})$.

b) Denote by $s$ the endomorphism of the $\mathbf{Z}$-module $\pi$ induced by $\widetilde{\sigma}$. Show that the quotient group $G_{\sigma}/(G_{\sigma})_0$ is isomorphic to a subgroup of Coker(1 $-s$), and in particular is commutative. If $1-s$ is surjective, $G_{\sigma}$ is connected.

c) Let $n\in \mathbf{N}$ be such that $s^n=$ Id$_{\pi}$. Show that the group $G_{\sigma}/(G_{\sigma})_0$ can be identified with a subgroup of the quotient Ker(1 $+s+\cdots +s^{n-1}$)$/$Im(1 $-s$); deduce that it is annihilated by $n$. If $n$ is prime to the order of the torsion subgroup of $\pi$, then $G_{\sigma}$ is connected.

d) Recover the results of b) and c) by using Exerc. 23 of Algèbre, Chap. X, p. 194.

e) Show that $G_{\sigma}$ is connected in each of the following cases:

(i) G is semi-simple of type $A_{2n}(n\geq 1)$ and $\sigma$ is not inner;

(ii) G is semi-simple of type $E_6$ and $\sigma$ is not inner;

(iii) G is semi-simple of type $D_4$ and $\sigma$ is a triality (that is, of order 3 modulo Int(G)).

f) Define an isomorphism from $C_{\sigma}/(C_{\sigma}\cap (G_{\sigma})_0)$ to $((1-\widetilde{\sigma}) \widetilde{C}\cap \pi )/(1-\widetilde{\sigma})\pi$. Deduce that if $1-s$ is not surjective and $\pi \subset (1-\widetilde{\sigma})C$, then $C_{\sigma}$ is not contained in $(G_{\sigma})_0$. For $G =\mathbf{S}\mathbf{O}(2n,\mathbf{R})$ and $\sigma$ not inner, we have $-I_{2n}\notin(G_{\sigma})_0$ and $G_{\sigma}$ is not connected.