Exercise 3, § 4

Let $(H_\alpha)$ be a real filtration on a group $H$ and let $v_H$ be the corresponding order function. Suppose that $G$ operates on $H$ on the left and that, for all $g \in G$, the mapping $h \mapsto g(h)$ is an automorphism of the filtered group $H$.

(a) If $\alpha \in \mathbf{R}$, we denote by $G_\alpha$ the set of $g \in G$ such that

Show that $(G_\alpha)$ is a real filtration of $G$ and that $G_\alpha = G$ if $\alpha \leq 0$. Let $v$ denote the corresponding order function.

(b) Suppose that the filtration $(H_\alpha)$ is central and that $G_0^+ = G$. Show that $(G_\alpha)$ is a central filtration of $G$. If $\xi \in \operatorname{gr}_\alpha(G)$ and $\eta \in \operatorname{gr}_\beta(H)$, let $g$ (resp. $h$) be a representative of $\xi$ (resp. $\eta$) in $G_\alpha$ (resp. $H_\beta$); show that the image of $h^{-1}g(h)$ in $\operatorname{gr}_{\alpha+\beta}(H)$ is independent of the choice of $g$ and $h$; if it is denoted by $D_\xi(\eta)$, show that $D_\xi$ can be extended to a derivation of the Lie algebra $\operatorname{gr}(H)$ of degree $\alpha$ and that $\xi \mapsto D_\xi$ is a homomorphism of the Lie algebra $\operatorname{gr}(G)$ into the Lie algebra of derivations of $\operatorname{gr}(H)$. If $v_H(H) \cap \mathbf{R}$ is contained in a discrete subgroup $\Gamma$ of $\mathbf{R}$, so is $v(G) \cap \mathbf{R}$ and, for all $\alpha \in \mathbf{R}$, the mapping $\xi \mapsto D_\xi$ defined above is injective.