canonical Haar Measure on a subset of $K_\mathbb{R}$

Question

I'm currently working on the understanding of a proof in Neukirch's Algebraic Number Theory (VII, §5, 5.6) and in this proof the concept of Haar measure is introduced.

It says " we defined $d^*x$ as the unique Haar measure on S such that the canonical Haar measure $dy/y$ on $R_+^*$ becomes the product measure $dy/y = d^*x \times dt/t$ (on $S \times \mathbb{R}_+^*$)."

In this context $R_+^*= \{ x \in K_\mathbb{R} |(x_\tau)=(x_{\bar{\tau}}) \}$ with $x=(x_\tau)$ and $S= \{ x \in R_+^* | N(x)=1 \}$ is the norm-one hypersurface. I've read about Haar measure, but I don't understand what exactly is meant by the canonical Haar measure and how, in general, I should understand the statement above (in particular the uniqueness).

Thank you for your help and explanations!

Answer 1

First, I hope that $K$ is your number field, and $K_{\mathbb R}=K\otimes_{\mathbb Q}\mathbb R$. Or maybe that's what you mean by $\mathbb R^*_+$? Your condition that $x_{\overline{\tau}}=x_\tau$ makes me think that you are indexing real-or-complex numbers not by all (real or) complex imbeddings, but by (real or) complex imbeddings mod complex conjugation, which is what the tensor product would give you.

Whichever way, I think the underlying issue is not so much about those details, as the idea that, for $f\in C^o_c(G)$, $$ \int_G f(g)\,dg \;=\; \int_{G/H}\int_H f(\dot{g}\,h)\;dh\;d\dot{g} $$ for (e.g.) abelian topological group $G$ and closed subgroup $H$. (More generally, this holds whenever the restriction of the modular function on $G$ to $H$ is equal to the modular function on $H$.) The point is that, specification of any two of the implied measures/integrals uniquely specifies the third. Thus, given any two, the third might be said to be "canonical", though this is slightly misleading, since the more genuine assertion is about uniqueness of the third given any two.

(After all, there is no canonical Haar measure in most situations. Of course, there are exceptions: on a finite group, counting measure is canonical. On compact groups, total-mass $1$ is canonical (and in conflict with the finite case. On adele groups, there is Tamagawa measure...)

Answer 2

Haar measure is not usually unique, because if you have one Haar measure $\mu$ (on a given topological group) then you can multiply it by a positive scalar $r$ to get another Haar measure $r\mu$. The basic uniqueness theorem is that these are the only Haar measures.

Sometimes one wants to pin down one particular choice of Haar measure $r\mu$ and declare that one to be the "canonical" one. Depending on your situation there might be various ways to do this. In the compact case one of the popular choices is the one whose total measure equals $1$, but I don't know if that applies in your situation. Since I'm not sure what $S$ is and what the notation $dy/y$ means, it's hard to say why that notation represents the "canonical" one in this case, or what the author intended to be "canonical" about it.