I am struggling to understand the following passage on Weil's text "Sur la formule de Siegel dans la théorie des groupes classiques" on page 16.

If I understood correctly, in the second sentence Weil writes that a "convenable" (suitable) relatively invariant measures on $g(\omega)_\mathbb{A}/g(\omega)_k$ and $G_\mathbb{A}/g(\omega)_\mathbb{A}$ will allow us to unfold the integral at (10). For convenience, I reproduce the integral of Equation (10) here:

I am a bit confused here because there must be justification of why such relatively invariant measures must exist. In Bourbaki cited there, and also in the standard literature here, a relatively invariant measure exists on a homogeneous space $G/H$ only if the modular character on $H$ can be extended to be made a character of $G$. I don't see where Weil justifies that this will be true for $G_\mathbb{A}/g(\omega)_\mathbb{A}$.

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