Lemma 5

The continuous representations of $\mathbf{GL}(n, K)$ in $\mathbf{C}^*$ are the mappings of the form $X \mapsto \chi(\det X)$, where $\chi$ is a continuous representation of $K^*$ in $\mathbf{C}^*$.

Such a mapping is obviously a continuous representation of $\mathbf{GL}(n, K)$ in $\mathbf{C}^*$. Conversely, suppose that $\psi$ is a continuous representation of $\mathbf{GL}(n, K)$ in $\mathbf{C}^*$. For $x \in K^*$, set

and $\chi(x) = \psi(\widetilde{x})$. Then, for every matrix $X \in \mathbf{GL}(n, K)$, we have $(\det X^{-1}) \sim \cdot X \in \mathbf{SL}(n, K)$. Since $\mathbf{SL}(n, K)$ is the commutator subgroup of $\mathbf{GL}(n, K)$ (A, III, §8, No. 9, Cor. of Prop. 17), $\psi((\det X^{-1}) \sim \cdot X) = 1$, whence

This established, Cor. 1 of Prop. 10 of §1, No. 8 proves that the relatively invariant measures on $\mathbf{GL}(n, K)$ are, up to a constant factor, the measures of the form where $\chi$ is a continuous representation of $K^*$ in $\mathbf{C}^*$.