Remark

The group $T(n, K)^*$ is a closed subgroup of $\mathbf{GL}(n, K)$, and $\Delta_{T(n, K)^*}((z_{ij})) = \prod_{i=1}^n (\operatorname{mod} z_{ii})^{2i-n-1}$. We saw in Example 1 that $\Delta_{\mathbf{GL}(n, K)} = 1$. If $n > 1$, the function

on $T(n, K)^*$ cannot be extended to a continuous representation of $\mathbf{GL}(n, K)$ in $\mathbf{C}^*$ (because such a representation would be equal to 1 on $\mathbf{SL}(n, K)$ by Lemma 5, whereas $\operatorname{mod}(z_{11})^{1-n} \neq 1$ for $z_{11}$ suitably chosen). It follows that the homogeneous space $\mathbf{GL}(n, K)/T(n, K)^*$ admits no relatively invariant measure if $n > 1$ (§ 2, No. 6, Cor. 1 of Th. 3).

This homogeneous space may be identified, for $n = 2$, with the projective line over $K$. For, let $(e_1, e_2)$ be the canonical basis of $K^2$. The group $\mathbf{GL}(2, K)$ operates transitively on the set of lines of $K^2$ with 0 omitted, and the stabilizer of $Ke_1 - \{0\}$ is $T(2, K)^*$.