Example 2

Affine group. For every $X \in \mathbf{GL}(n, K)$ and every $x \in K^n$, let $(X, x)$ be the affine linear mapping $\xi \mapsto X\xi + x$ in $K^n$. The set of $(X, x)$ is the affine group $G$ of $K^n$ (A, II, §9, No. 4). The set $T$ of translations is a closed normal subgroup of $G$, canonically isomorphic to $K^n$; on the other hand, $\mathbf{GL}(n, K)$ is a closed subgroup of $G$, and $G$ is the semi-direct product of $\mathbf{GL}(n, K)$ and $T = K^n$. One equips $G$ with the (locally compact) topology for which $G$ is the topological semi-direct product of $\mathbf{GL}(n, K)$ and $T$ (GT, III, §2, No. 10). One has

On the other hand, if $X \in \mathbf{GL}(n, K)$ and $x \in T$ then, for every $\xi \in K^n$, therefore the automorphism $(1, x) \mapsto (X, 0)(1, x)(X, 0)^{-1}$ of $T$ has modulus $\mod(\det X)$ (§1, No. 10, Prop. 15). In view of Example 1 and §2, No. 9, Remark, the measure is a left Haar measure on $G$. On the other hand, by Prop. 14 of §2, No. 9, or

Thus, a right Haar measure on $G$ is given by