# Weil's "Sur la formule de Siegel dans la théorie des groupes classiques"

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 $$G_\mathbb{A}/g(\omega)_\mathbb{A}$$.

• Maybe they're both reductive, so unimodular? Mar 18 at 19:42
• The assumption he works with is that $G$ is a connected algebraic group acting on a variety $X$ (both over a field $k$) and $g(\omega)$ is the stabilizer of a point $\omega \in X$. Mar 18 at 19:52
• @paulgarrett, I have seen in the surrounding literature that the working assumption is that stabilizers in an algebraic group $G$ acting on a vector space are unimodular. Does this assumption fail for some pathological examples? Weil seems to somehow prove that when $G$ is semisimple, stabilizers of rational vectors are also unimodular. Mar 19 at 23:54
• Well, no, stabilizers of rational vectors can fail to be unimodular in interesting cases, for example, already for $SL(2,\mathbb Q)$. Many traditional overviews of such things pointedly ignore this possibility, indeed. And, yes, such subgroups do cause trouble in making precise "trace formula" assertions, etc. Mar 20 at 0:07
• @paulgarrett I'm still trying to understand your comment. If you mean the upper-triangular matrices in $SL_2$ then this is a stabilizer of a projective representation, but not a stabilizer of a usual representation as pointed out here: mathoverflow.net/questions/409499/… Mar 20 at 3:53