I am trying to understand the Borel-Harish Chandra theorem about arithmetic subgroups being lattices.

Suppose $G$ is an algebraic group inside $GL_n(\mathbb{C})$ such that it is definable as a zero set of polynomials with coefficients in $\mathbb{Q}$. Let $G_\mathbb{R} = G \cap G_\mathbb{R}$ and $G_\mathbb{Z} = {G} \cap GL_n(\mathbb{Z})$.

The Borel-Harish Chandra theorem says that $X_\mathbb{Q}(G^0) = \{ e \} \Leftrightarrow G_\mathbb{R} / G_\mathbb{Z}$ has a finite natural measure. Here $X_\mathbb{Q}(G^0)$ is the multiplicative group of $\mathbb{Q}$-morphisms of the algebraic group $G^0$ to $GL_1(\mathbb{C})$.

I am wondering if the "$\Leftarrow$" part can be proven shown easily under the assumption that $G$ is closed. Here is my attempt:

Since $G$ is closed, any $\mathbb{Q}$-morphism $\chi:G^0 \rightarrow GL_1(\mathbb{C})$ can be written as a polynomal, instead of a regular function. However, on the points of $G_\mathbb{Z}^0$, this polynomial will actually take values in $N^{-1} \mathbb{Z}$ for some $N \in \mathbb{N}$ (the lcm of the denominators of all the rational coefficients) but since the image is a multiplicatively closed set of $\mathbb{Q}^*$, $\chi$ can only be $\{ \pm 1\}$ on $G_\mathbb{Z}^0$. But then, $\chi:G^0_\mathbb{R}/G^0_\mathbb{Z} \rightarrow \mathbb{R}^*$ is a continuous map that might induce a finite Haar measure on $\mathbb{R}^*$.

This approach might work, but saying that all $\mathbb{Q}$-characters on $G_\mathbb{Z}^0$ are trivial seems like a big deal and I could not find any such statements in Borel's "Intro. to Arithmetic groups". Do you have any counterexamples that show that the integer points of closed connected $\mathbb{Q}$ algebraic groups may admit a non-trivial $\mathbb{Q}$-character?

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