Proposition 6

a) The group $\mathbf{SL}(n, K)$ is unimodular.

b) Let $\mu_1$ and $\mu_2$ be left Haar measures on the upper strict triangular group $T_1(n, K)$ and the lower special triangular group $G'_1$, respectively. The image of $\mu_1 \otimes \mu_2$ under the homeomorphism $(M, N) \mapsto M \cdot N^{-1}$ of $T_1(n, K) \times G'_1$ onto $\Omega'$ is the restriction to $\Omega'$ of a Haar measure on $\mathbf{SL}(n, K)$.

c) The complement of $\Omega'$ in $\mathbf{SL}(n, K)$ is negligible for the Haar measure of $\mathbf{SL}(n, K)$.

The group $\mathbf{GL}(n, K)$ is unimodular (Example 1), and $\mathbf{SL}(n, K)$ is a normal subgroup of $\mathbf{GL}(n, K)$, hence is unimodular (§ 2, No. 7, Prop. 10 b)). The assertion b) follows from a), Lemma 8, and Prop. 13 of § 2, No. 9. Let us prove c). By Lemma 8 a), it suffices to prove the following: if $p((u_{ij})_{1 \leq i,j \leq n})$ is a polynomial, not identically zero on $\mathbf{SL}(n, K)$, then the set E of $U \in \mathbf{SL}(n, K)$ such that $p(U) = 0$ is negligible for the Haar measure. Taking into account § 1, No. 10, Cor. of Prop. 13, the topology of $\mathbf{SL}(n, K)$ has a countable base. It therefore suffices to prove that for every $U_0 \in E$, there exists a neighborhood of $U_0$ in $\mathbf{SL}(n, K)$ whose intersection with E is negligible; or again that there exists a neighborhood W of I in $\mathbf{SL}(n, K)$ such that $U_0^{-1}E \cap W$ is negligible. Let us take $W = \Omega'$. In view of b), it all comes down to showing that the set of pairs $(M, N) \in T_1(n, K) \times G'_1$ such that $p(U_0 MN) = 0$ is negligible for $\mu_1 \otimes \mu_2$. By the expressions for $\mu_1$ and $\mu_2$ (calculated in Examples 3 and 5), this will result from the following lemma: